diff --git a/.github/workflows/build.yaml b/.github/workflows/build.yaml index 3f784ff32f..01644f8b66 100644 --- a/.github/workflows/build.yaml +++ b/.github/workflows/build.yaml @@ -75,6 +75,13 @@ jobs: configuration: ${{ matrix.configuration }} directory: Boost + - name: Download core-math artifact + uses: mockingbirdnest/actions/windows/download_artifact@main + with: + name: core-math + configuration: ${{ matrix.configuration }} + directory: Inria + - name: Download gipfeli artifact uses: mockingbirdnest/actions/windows/download_artifact@main with: diff --git a/functions/core_math_accuracy_test.cpp b/functions/core_math_accuracy_test.cpp index 82d12e1786..66b22bb578 100644 --- a/functions/core_math_accuracy_test.cpp +++ b/functions/core_math_accuracy_test.cpp @@ -2,9 +2,9 @@ #include #include -#include "functions/cos.hpp" +#include "core-math/cos.h" +#include "core-math/sin.h" #include "functions/multiprecision.hpp" -#include "functions/sin.hpp" #include "glog/logging.h" #include "gmock/gmock.h" #include "gtest/gtest.h" @@ -17,8 +17,6 @@ namespace functions { namespace _multiprecision { using namespace boost::multiprecision; -using namespace principia::functions::_cos; -using namespace principia::functions::_sin; using namespace principia::testing_utilities::_approximate_quantity; using namespace principia::testing_utilities::_is_near; diff --git a/functions/cos.cc b/functions/cos.cc deleted file mode 100644 index e749893639..0000000000 --- a/functions/cos.cc +++ /dev/null @@ -1,2067 +0,0 @@ -/* Correctly-rounded cosine function for binary64 value. - -Copyright (c) 2022-2023 Paul Zimmermann and Tom Hubrecht - -This file is part of the CORE-MATH project -(https://core-math.gitlabpages.inria.fr/). - -Permission is hereby granted, free of charge, to any person obtaining a copy -of this software and associated documentation files (the "Software"), to deal -in the Software without restriction, including without limitation the rights -to use, copy, modify, merge, publish, distribute, sublicense, and/or sell -copies of the Software, and to permit persons to whom the Software is -furnished to do so, subject to the following conditions: - -The above copyright notice and this permission notice shall be included in all -copies or substantial portions of the Software. - -THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR -IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, -FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE -AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER -LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, -OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE -SOFTWARE. -*/ - -// This code has been adapted to C++ and MSVC. - -/* stdio.h and stdlib.h are needed in case the rounding test of the accurate - step fails, to print the corresponding input and exit. */ -#include "functions/cos.hpp" - -#include -#include -#include -#include - -#include "absl/numeric/int128.h" -#include "base/macros.hpp" // 🧙 For PRINCIPIA_COMPILER_MSVC. -#include "numerics/fma.hpp" - -// Warning: clang also defines __GNUC__ -#if defined(__GNUC__) && !defined(__clang__) -#pragma GCC diagnostic ignored "-Wunknown-pragmas" -#endif - -#if PRINCIPIA_COMPILER_MSVC -#define __builtin_clzl(x) __lzcnt64(x) -#define __builtin_fma(x, y, z) \ - principia::numerics::_fma::FusedMultiplyAdd(x, y, z) -#endif -#define __builtin_expect(x, y) x -#define __builtin_fabs(x) std::abs(x) -#define __builtin_floor(x) std::floor(x) - -/******************** code copied from dint.h and pow.[ch] *******************/ - -namespace principia { -namespace functions { -namespace _cos { -namespace internal { - -typedef absl::uint128 u128; - -typedef union { - struct { - u128 r; - int64_t _ex; - uint64_t _sgn; - }; - struct { - uint64_t lo; - uint64_t hi; - int64_t ex; - uint64_t sgn; - }; -} dint64_t; - -typedef union { - u128 r; - struct { - uint64_t l; - uint64_t h; - }; -} uint128_t; - -typedef union { - double f; - uint64_t u; -} f64_u; - -// Extract both the mantissa and exponent of a double -static inline void fast_extract (int64_t *e, uint64_t *m, double x) { - f64_u _x = {.f = x}; - - *e = (_x.u >> 52) & 0x7ff; - *m = (_x.u & (~0ull >> 12)) + (*e ? (1ull << 52) : 0); - *e = *e - 0x3fe; -} - -// Return non-zero if a = 0 -static inline int -dint_zero_p (const dint64_t *a) -{ - return a->hi == 0; -} - -static inline int cmp(int64_t a, int64_t b) { return (a > b) - (a < b); } - -static inline int cmpu128 (u128 a, u128 b) { return (a > b) - (a < b); } - -/* ZERO is a dint64_t representation of 0, which ensures that - dint_tod(ZERO) = 0 */ -static const dint64_t ZERO = {.lo = 0x0, .hi = 0x0, .ex = -1076, .sgn = 0x0}; -// MAGIC is a dint64_t representation of 1/2^11 -static const dint64_t MAGIC = {.lo = 0x0, .hi = 0x8000000000000000, .ex = -10, .sgn = 0x0}; - -// Compare the absolute values of a and b -// Return -1 if |a| < |b| -// Return 0 if |a| = |b| -// Return +1 if |a| > |b| -static inline signed char -cmp_dint_abs (const dint64_t *a, const dint64_t *b) { - if (dint_zero_p (a)) - return dint_zero_p (b) ? 0 : -1; - if (dint_zero_p (b)) - return +1; - char c1 = cmp (a->ex, b->ex); - return c1 ? c1 : cmpu128 (a->r, b->r); -} - -// Copy a dint64_t value -static inline void cp_dint(dint64_t *r, const dint64_t *a) { - r->ex = a->ex; - r->r = a->r; - r->sgn = a->sgn; -} - -// Add two dint64_t values, with error bounded by 2 ulps (ulp_128) -// (more precisely 1 ulp when a and b have same sign, 2 ulps otherwise) -// Moreover, when Sterbenz theorem applies, i.e., |b| <= |a| <= 2|b| -// and a,b are of different signs, there is no error, i.e., r = a-b. -static inline void -add_dint (dint64_t *r, const dint64_t *a, const dint64_t *b) { - if (!(a->hi | a->lo)) { - cp_dint (r, b); - return; - } - - switch (cmp_dint_abs (a, b)) { - case 0: - if (a->sgn ^ b->sgn) { - cp_dint (r, &ZERO); - return; - } - - cp_dint (r, a); - r->ex++; - return; - - case -1: // |A| < |B| - { - // swap operands - const dint64_t *tmp = a; a = b; b = tmp; - break; // fall through the case |A| > |B| - } - } - - // From now on, |A| > |B| thus a->ex >= b->ex - - u128 A = a->r, B = b->r; - uint64_t k = a->ex - b->ex; - - if (k > 0) { - /* Warning: the right shift x >> k is only defined for 0 <= k < n - where n is the bit-width of x. See for example - https://developer.arm.com/documentation/den0024/a/The-A64-instruction-set/Data-processing-instructions/Shift-operations - where it is said that k is interpreted modulo n. */ - B = (k < 128) ? B >> k : 0; - } - - u128 C; - unsigned char sgn = a->sgn; - - r->ex = a->ex; /* tentative exponent for the result */ - - if (a->sgn ^ b->sgn) { - /* a and b have different signs C = A + (-B) - Sterbenz case |a|/2 <= |b| <= |a| can occur only when: - * k=0: then B is not truncated, and C is exact below - * k=1 and ex>0 below: then we ensure C is exact - */ - C = A - B; - uint64_t ch = absl::Uint128High64(C); - /* We can't have C=0 here since we excluded the case |A| = |B|, - thus __builtin_clzl(C) is well-defined below. */ - uint64_t ex = ch ? __builtin_clzl(ch) : 64 + __builtin_clzl(absl::Uint128Low64(C)); - /* The error from the truncated part of B (1 ulp) is multiplied by 2^ex, - thus by 2 ulps when ex <= 1. */ - if (ex > 0) - { - if (k == 1) /* Sterbenz case */ - C = (A << ex) - (b->r << (ex - 1)); - else - C = (A << ex) - (B << ex); - /* If C0 is the previous value of C, we have: - (C0-1)*2^ex < A*2^ex-B*2^ex <= C0*2^ex - since some neglected bits from B might appear which contribute - a value less than ulp(C0)=1. - As a consequence since 2^(127-ex) <= C0 < 2^(128-ex), because C0 had - ex leading zero bits, we have 2^127-2^ex <= A*2^ex-B*2^ex < 2^128. - Thus the value of C, which is truncated to 128 bits, is the right - one (as if no truncation); moreover in some rare cases we need to - shift by 1 bit to the left. */ - r->ex -= ex; - ex = __builtin_clzl (absl::Uint128High64(C)); - /* Fall through with the code for ex = 0. */ - } - C = C << ex; - r->ex -= ex; - /* The neglected part of B is bounded by 2 ulp(C) when ex=0, 1 ulp - when ex > 0 but ex=0 at the end, and by 2*ulp(C) when ex > 0 and there - is an extra shift at the end (in that case necessarily ex=1). */ - } else { - C = A + B; - if (C < A) - { - C = ((u128) 1 << 127) | (C >> 1); - r->ex ++; - } - } - - /* In the addition case, we loose the truncated part of B, which - contributes to at most 1 ulp. If there is an exponent shift, we - might also loose the least significant bit of C, which counts as - 1/2 ulp, but the truncated part of B is now less than 1/2 ulp too, - thus in all cases the error is less than 1 ulp(r). */ - - r->sgn = sgn; - r->r = C; -} - -// Multiply two dint64_t numbers, with error bounded by 6 ulps -// on the 128-bit floating-point numbers. -// Overlap between r and a is allowed -static inline void -mul_dint (dint64_t *r, const dint64_t *a, const dint64_t *b) { - u128 bh = b->hi, bl = b->lo; - - /* compute the two middle terms */ - u128 m1 = (u128)(a->hi) * bl; - u128 m2 = (u128)(a->lo) * bh; - - /* put the 128-bit product of the high terms in r */ - r->r = (u128)(a->hi) * bh; - - /* there can be no overflow in the following addition since r <= (B-1)^2 - with B=2^64, (m1>>64) <= B-1 and (m2>>64) <= B-1, thus the sum is - bounded by (B-1)^2+2*(B-1) = B^2-1 */ - r->r += (m1 >> 64) + (m2 >> 64); - - // Ensure that r->hi starts with a 1 - uint64_t ex = r->hi >> 63; - r->r = r->r << (1 - ex); - - // Exponent and sign - // if ex=1, then ex(r) = ex(a) + ex(b) - // if ex=0, then ex(r) = ex(a) + ex(b) - 1 - r->ex = a->ex + b->ex + ex - 1; - r->sgn = a->sgn ^ b->sgn; - - /* The ignored part can be as large as 3 ulps before the shift (one - for the low part of a->hi * bl, one for the low part of a->lo * bh, - and one for the neglected a->lo * bl term). After the shift this can - be as large as 6 ulps. */ -} - -// Multiply two dint64_t numbers, assuming the low part of b is zero -// with error bounded by 2 ulps -static inline void -mul_dint_21 (dint64_t *r, const dint64_t *a, const dint64_t *b) { - u128 bh = b->hi; - u128 hi = (u128) (a->hi) * bh; - u128 lo = (u128) (a->lo) * bh; - - /* put the 128-bit product of the high terms in r */ - r->r = hi; - - /* add the middle term */ - r->r += lo >> 64; - - // Ensure that r->hi starts with a 1 - uint64_t ex = r->hi >> 63; - r->r = r->r << (1 - ex); - - // Exponent and sign - r->ex = a->ex + b->ex + ex - 1; - r->sgn = a->sgn ^ b->sgn; - - /* The ignored part can be as large as 1 ulp before the shift (truncated - part of lo). After the shift this can be as large as 2 ulps. */ -} - -// Convert a non-zero double to the corresponding dint64_t value -static inline void dint_fromd (dint64_t *a, double b) { - fast_extract (&a->ex, &a->hi, b); - - /* |b| = 2^(ex-52)*hi */ - - uint32_t t = __builtin_clzl (a->hi); - - a->sgn = b < 0.0; - a->hi = a->hi << t; - a->ex = a->ex - (t > 11 ? t - 12 : 0); - /* b = 2^ex*hi/2^64 where 1/2 <= hi/2^64 < 1 */ - a->lo = 0; -} - -static inline void subnormalize_dint(dint64_t *a) { - if (a->ex > -1023) - return; - - uint64_t ex = -(1011 + a->ex); - - uint64_t hi = a->hi >> ex; - uint64_t md = (a->hi >> (ex - 1)) & 0x1; - uint64_t lo = (a->hi & (~0ull >> ex)) || a->lo; - - switch (fegetround()) { - case FE_TONEAREST: - hi += lo ? md : hi & md; - break; - case FE_DOWNWARD: - hi += a->sgn & (md | lo); - break; - case FE_UPWARD: - hi += (!a->sgn) & (md | lo); - break; - } - - a->hi = hi << ex; - a->lo = 0; - - if (!a->hi) { - a->ex++; - a->hi = (1ll << 63); - } -} - -// Convert a dint64_t value to a double -static inline double dint_tod(dint64_t *a) { - subnormalize_dint (a); - - f64_u r = {.u = (a->hi >> 11) | (0x3ffll << 52)}; - - double rd = 0.0; - if ((a->hi >> 10) & 0x1) - rd += 0x1p-53; - - if (a->hi & 0x3ff || a->lo) - rd += 0x1p-54; - - if (a->sgn) - rd = -rd; - - r.u = r.u | a->sgn << 63; - r.f += rd; - - f64_u e; - - if (a->ex > -1022) { // The result is a normal double - if (a->ex > 1024) - if (a->ex == 1025) { - r.f = r.f * 0x1p+1; - e.f = 0x1p+1023; - } else { - r.f = 0x1.fffffffffffffp+1023; - e.f = 0x1.fffffffffffffp+1023; - } - else - e.u = ((a->ex + 1022) & 0x7ff) << 52; - } else { - if (a->ex < -1073) { - if (a->ex == -1074) { - r.f = r.f * 0x1p-1; - e.f = 0x1p-1074; - } else { - r.f = 0x0.0000000000001p-1022; - e.f = 0x0.0000000000001p-1022; - } - } else { - e.u = 1ll << (a->ex + 1073); - } - } - - return r.f * e.f; -} - -/**************** end of code copied from dint.h and pow.[ch] ****************/ - -/**************** the following is copied from sin.c *************************/ - -typedef union {double f; uint64_t u;} b64u64_u; - -/* This table approximates 1/(2pi) downwards with precision 1280: - 1/(2*pi) ~ T[0]/2^64 + T[1]/2^128 + ... + T[i]/2^((i+1)*64) + ... - Computed with computeT() from sin.sage. */ -static const uint64_t T[20] = { - 0x28be60db9391054a, // i=0 - 0x7f09d5f47d4d3770, - 0x36d8a5664f10e410, - 0x7f9458eaf7aef158, - 0x6dc91b8e909374b8, - 0x1924bba82746487, // i=5 - 0x3f877ac72c4a69cf, - 0xba208d7d4baed121, - 0x3a671c09ad17df90, - 0x4e64758e60d4ce7d, - 0x272117e2ef7e4a0e, // i=10 - 0xc7fe25fff7816603, - 0xfbcbc462d6829b47, - 0xdb4d9fb3c9f2c26d, - 0xd3d18fd9a797fa8b, - 0x5d49eeb1faf97c5e, // i=15 - 0xcf41ce7de294a4ba, - 0x9afed7ec47e35742, - 0x1580cc11bf1edaea, - 0xfc33ef0826bd0d87, // i=19 -}; - -/* Table containing 128-bit approximations of sin2pi(i/2^11) for 0 <= i < 256 - (to nearest). - Each entry is to be interpreted as (hi/2^64+lo/2^128)*2^ex*(-1)*sgn. - Generated with computeS() from sin.sage. */ -static const dint64_t S[256] = { - {.lo = 0x0, .hi = 0x0, .ex = 128, .sgn=0}, - {.lo = 0x480f7956b6470765, .hi = 0xc90fc5f66525d257, .ex = -8, .sgn=0}, - {.lo = 0xcb3ff35bd4d81baa, .hi = 0xc90f87f3380388d5, .ex = -7, .sgn=0}, - {.lo = 0xb767005691b9d9d1, .hi = 0x96cb587284b81770, .ex = -6, .sgn=0}, - {.lo = 0xf1d7d06db39ea9fc, .hi = 0xc90e8fe6f63c2330, .ex = -6, .sgn=0}, - {.lo = 0xd784e031f9af76d6, .hi = 0xfb514b55ccbe541a, .ex = -6, .sgn=0}, - {.lo = 0xf91ee371d6467dca, .hi = 0x96c9b5df1877e9b5, .ex = -5, .sgn=0}, - {.lo = 0xf56e3c87ae3c56df, .hi = 0xafea690fd5912ef3, .ex = -5, .sgn=0}, - {.lo = 0xc539edcbfda0cf2c, .hi = 0xc90aafbd1b33efc9, .ex = -5, .sgn=0}, - {.lo = 0x850021e392744a4f, .hi = 0xe22a7a6729d8e453, .ex = -5, .sgn=0}, - {.lo = 0xb21ccebc9caac3, .hi = 0xfb49b98e8e7807f6, .ex = -5, .sgn=0}, - {.lo = 0xde5b1068d174be9c, .hi = 0x8a342eda160bf5ae, .ex = -4, .sgn=0}, - {.lo = 0x37b2dd49d5fca3c0, .hi = 0x96c32baca2ae68b4, .ex = -4, .sgn=0}, - {.lo = 0xb56007d16d4ad5a3, .hi = 0xa351cb7fc30bc889, .ex = -4, .sgn=0}, - {.lo = 0xcd34d2751c2e1da7, .hi = 0xafe00694866a1b44, .ex = -4, .sgn=0}, - {.lo = 0xf10bfca3d6464012, .hi = 0xbc6dd52c3a342eb5, .ex = -4, .sgn=0}, - {.lo = 0x6a17954b2b7c5171, .hi = 0xc8fb2f886ec09f37, .ex = -4, .sgn=0}, - {.lo = 0x73d1472472f4a390, .hi = 0xd5880deafc18b534, .ex = -4, .sgn=0}, - {.lo = 0x438b4a73aecd2541, .hi = 0xe214689606bf1676, .ex = -4, .sgn=0}, - {.lo = 0xc4e92d01a2f42935, .hi = 0xeea037cc04764844, .ex = -4, .sgn=0}, - {.lo = 0xf0a0e36a000c7350, .hi = 0xfb2b73cfc106ff68, .ex = -4, .sgn=0}, - {.lo = 0x60e782313f6161af, .hi = 0x83db0a7231831d8f, .ex = -3, .sgn=0}, - {.lo = 0x77724a2b2a669bc4, .hi = 0x8a2009a6b84d9402, .ex = -3, .sgn=0}, - {.lo = 0x56e0a8b0d177b55d, .hi = 0x9064b3a76a22640c, .ex = -3, .sgn=0}, - {.lo = 0xf77574094d3c35c4, .hi = 0x96a9049670cfae65, .ex = -3, .sgn=0}, - {.lo = 0x50ffe4f5caa7f1fa, .hi = 0x9cecf8962d14c822, .ex = -3, .sgn=0}, - {.lo = 0xdec1b7f2768bdafa, .hi = 0xa3308bc93904ad69, .ex = -3, .sgn=0}, - {.lo = 0x76f8c63986598c79, .hi = 0xa973ba526a6850d9, .ex = -3, .sgn=0}, - {.lo = 0xfdd2fc0936594c2d, .hi = 0xafb68054d520c60b, .ex = -3, .sgn=0}, - {.lo = 0x924bef13600f9852, .hi = 0xb5f8d9f3cd8945d6, .ex = -3, .sgn=0}, - {.lo = 0xeb13e106732687f1, .hi = 0xbc3ac352ead90abe, .ex = -3, .sgn=0}, - {.lo = 0xb228a03916371f6f, .hi = 0xc27c389609850433, .ex = -3, .sgn=0}, - {.lo = 0xc7396c894bbf7389, .hi = 0xc8bd35e14da15f0e, .ex = -3, .sgn=0}, - {.lo = 0x6b47b8c44e5b037e, .hi = 0xcefdb7592542e1e9, .ex = -3, .sgn=0}, - {.lo = 0x7337412cf70716cb, .hi = 0xd53db9224ae01bca, .ex = -3, .sgn=0}, - {.lo = 0xbb286d23e11c8337, .hi = 0xdb7d3761c7b263b6, .ex = -3, .sgn=0}, - {.lo = 0x31883b30137c6e62, .hi = 0xe1bc2e3cf616a7ac, .ex = -3, .sgn=0}, - {.lo = 0xeeb8f9c33340a2f2, .hi = 0xe7fa99d983ee098f, .ex = -3, .sgn=0}, - {.lo = 0xed16b994af6c18ae, .hi = 0xee38765d74fe4897, .ex = -3, .sgn=0}, - {.lo = 0x14e1a5488eaeab96, .hi = 0xf475bfef2551f5b9, .ex = -3, .sgn=0}, - {.lo = 0x704729ae56d78a37, .hi = 0xfab272b54b9871a2, .ex = -3, .sgn=0}, - {.lo = 0x3eac8308f1113e5e, .hi = 0x8077456b7dc2d967, .ex = -2, .sgn=0}, - {.lo = 0xdb1f70118c9c2198, .hi = 0x8395023dd418e919, .ex = -2, .sgn=0}, - {.lo = 0xc5a9decdfaad4db5, .hi = 0x86b26de5933c2e8e, .ex = -2, .sgn=0}, - {.lo = 0x97965c9860c34e44, .hi = 0x89cf8676d7abb55b, .ex = -2, .sgn=0}, - {.lo = 0xdcdca90cc73b116a, .hi = 0x8cec4a05f12739e8, .ex = -2, .sgn=0}, - {.lo = 0xa6e3df5975cca9da, .hi = 0x9008b6a763de75b7, .ex = -2, .sgn=0}, - {.lo = 0x899c4de737feec22, .hi = 0x9324ca6fe9a04b4e, .ex = -2, .sgn=0}, - {.lo = 0xa89a11e07c1fe, .hi = 0x964083747309d113, .ex = -2, .sgn=0}, - {.lo = 0x49c4863de522b217, .hi = 0x995bdfca28b53a54, .ex = -2, .sgn=0}, - {.lo = 0xe7bc08111d0bfca4, .hi = 0x9c76dd866c689dcc, .ex = -2, .sgn=0}, - {.lo = 0xf3ff913a4aadb85e, .hi = 0x9f917abeda4498df, .ex = -2, .sgn=0}, - {.lo = 0xa5dbee6084ee1260, .hi = 0xa2abb58949f2ced7, .ex = -2, .sgn=0}, - {.lo = 0x69fcb11e19f58619, .hi = 0xa5c58bfbcfd4436a, .ex = -2, .sgn=0}, - {.lo = 0xcd12a1f6ab6b095, .hi = 0xa8defc2cbe2f8fcc, .ex = -2, .sgn=0}, - {.lo = 0x8c95c4c91179176b, .hi = 0xabf80432a65ef190, .ex = -2, .sgn=0}, - {.lo = 0x3feef3bb58b1f10d, .hi = 0xaf10a22459fe32a6, .ex = -2, .sgn=0}, - {.lo = 0x16031a34d4fc855d, .hi = 0xb228d418ec1869ad, .ex = -2, .sgn=0}, - {.lo = 0xcd73fb5d8d45d302, .hi = 0xb5409827b25591f0, .ex = -2, .sgn=0}, - {.lo = 0x187e26d290714d70, .hi = 0xb857ec684627fa4c, .ex = -2, .sgn=0}, - {.lo = 0xbddd8a0365d6b1d3, .hi = 0xbb6ecef285f98a3a, .ex = -2, .sgn=0}, - {.lo = 0xdfe1b074e22fc666, .hi = 0xbe853dde9658dc60, .ex = -2, .sgn=0}, - {.lo = 0xad5a41de48f6b26f, .hi = 0xc19b3744e3262dcd, .ex = -2, .sgn=0}, - {.lo = 0xdab4e426409b23a0, .hi = 0xc4b0b93e20c0213f, .ex = -2, .sgn=0}, - {.lo = 0x5cc8c00e4fccd850, .hi = 0xc7c5c1e34d3055b2, .ex = -2, .sgn=0}, - {.lo = 0xfa6171200ab2efc3, .hi = 0xcada4f4db157cf77, .ex = -2, .sgn=0}, - {.lo = 0x65a3132adfb7dfd5, .hi = 0xcdee5f96e21b332c, .ex = -2, .sgn=0}, - {.lo = 0xaadb580a1eba209f, .hi = 0xd101f0d8c18ed1c1, .ex = -2, .sgn=0}, - {.lo = 0xdf4005ef6a64aa02, .hi = 0xd415012d802284f0, .ex = -2, .sgn=0}, - {.lo = 0x1779df36d1cc8912, .hi = 0xd7278eaf9dcd5b55, .ex = -2, .sgn=0}, - {.lo = 0xcbabaeb97af8e8aa, .hi = 0xda399779eb391377, .ex = -2, .sgn=0}, - {.lo = 0xece7f445cecf1e28, .hi = 0xdd4b19a78aed6515, .ex = -2, .sgn=0}, - {.lo = 0xebc61ade6ca83cd, .hi = 0xe05c1353f27b17e5, .ex = -2, .sgn=0}, - {.lo = 0x26a0eecdb4f16266, .hi = 0xe36c829aeba6e720, .ex = -2, .sgn=0}, - {.lo = 0x82b0aecadf808123, .hi = 0xe67c659895943123, .ex = -2, .sgn=0}, - {.lo = 0xb91caf23416e7e80, .hi = 0xe98bba6965ef725f, .ex = -2, .sgn=0}, - {.lo = 0x7244ee20f591983b, .hi = 0xec9a7f2a2a188aeb, .ex = -2, .sgn=0}, - {.lo = 0x1050cdf22f34182f, .hi = 0xefa8b1f8084ccdfc, .ex = -2, .sgn=0}, - {.lo = 0x587f3fa044e2d27d, .hi = 0xf2b650f080d0da8d, .ex = -2, .sgn=0}, - {.lo = 0x643720de93ba81bd, .hi = 0xf5c35a316f1a3c80, .ex = -2, .sgn=0}, - {.lo = 0x4221dc4ba772598d, .hi = 0xf8cfcbd90af8d57a, .ex = -2, .sgn=0}, - {.lo = 0xd24d3023da491920, .hi = 0xfbdba405e9c00cca, .ex = -2, .sgn=0}, - {.lo = 0x8b74fe2508ab8fc2, .hi = 0xfee6e0d6ff6fc5a4, .ex = -2, .sgn=0}, - {.lo = 0xfd958d68e8b49e6b, .hi = 0x80f8c035cfee8d76, .ex = -1, .sgn=0}, - {.lo = 0xfb4c92369f0cf008, .hi = 0x827dc071bfed6ffa, .ex = -1, .sgn=0}, - {.lo = 0xcb07b25a7b0372a7, .hi = 0x8402702f5b30f2a9, .ex = -1, .sgn=0}, - {.lo = 0x9d3dc689006896f4, .hi = 0x8586ce7ededc809d, .ex = -1, .sgn=0}, - {.lo = 0x9d52755ece3f70, .hi = 0x870ada70ba4e6d49, .ex = -1, .sgn=0}, - {.lo = 0x984156f553344306, .hi = 0x888e93158fb3bb04, .ex = -1, .sgn=0}, - {.lo = 0xa66d1d936c38c329, .hi = 0x8a11f77e349bc245, .ex = -1, .sgn=0}, - {.lo = 0x575f33366be0afef, .hi = 0x8b9506bbb28bb922, .ex = -1, .sgn=0}, - {.lo = 0xcb590d74f64e77c9, .hi = 0x8d17bfdf47921ac8, .ex = -1, .sgn=0}, - {.lo = 0xf2be3ecae62789d4, .hi = 0x8e9a21fa66d9ee8d, .ex = -1, .sgn=0}, - {.lo = 0x632b9cff5cfee724, .hi = 0x901c2c1eb93dee39, .ex = -1, .sgn=0}, - {.lo = 0x609c464b3dd676ec, .hi = 0x919ddd5e1ddb8b33, .ex = -1, .sgn=0}, - {.lo = 0x6a1ff8bfe6396e28, .hi = 0x931f34caaaa5d23a, .ex = -1, .sgn=0}, - {.lo = 0xae4ba773da6bf754, .hi = 0x94a03176acf82d45, .ex = -1, .sgn=0}, - {.lo = 0xe06a955a5b8e301d, .hi = 0x9620d274aa290339, .ex = -1, .sgn=0}, - {.lo = 0xfc8b7184b21f2d50, .hi = 0x97a116d7601c3515, .ex = -1, .sgn=0}, - {.lo = 0x9dd1eedf18a2e4df, .hi = 0x9920fdb1c5d5783d, .ex = -1, .sgn=0}, - {.lo = 0x9ffa0d23f3c26c62, .hi = 0x9aa086170c0a8d86, .ex = -1, .sgn=0}, - {.lo = 0xdab6b478577e7be5, .hi = 0x9c1faf1a9db554af, .ex = -1, .sgn=0}, - {.lo = 0xdb895384528d0d60, .hi = 0x9d9e77d020a5bbe6, .ex = -1, .sgn=0}, - {.lo = 0x98dbd3555ebcdefe, .hi = 0x9f1cdf4b76138b02, .ex = -1, .sgn=0}, - {.lo = 0x2f895f44a303cc0b, .hi = 0xa09ae4a0bb300a19, .ex = -1, .sgn=0}, - {.lo = 0xd29d23a624acd00c, .hi = 0xa21886e449b78316, .ex = -1, .sgn=0}, - {.lo = 0x2be036401ba87cc2, .hi = 0xa395c52ab8829dfc, .ex = -1, .sgn=0}, - {.lo = 0x82d9495ead5be348, .hi = 0xa5129e88dc17976a, .ex = -1, .sgn=0}, - {.lo = 0x17218792857f4c5a, .hi = 0xa68f1213c73b5124, .ex = -1, .sgn=0}, - {.lo = 0x3269f4702b88324a, .hi = 0xa80b1ee0cb823c27, .ex = -1, .sgn=0}, - {.lo = 0x8e3bdf8085321556, .hi = 0xa986c40579e11c0a, .ex = -1, .sgn=0}, - {.lo = 0xc1654b64a0081b46, .hi = 0xab020097a33da341, .ex = -1, .sgn=0}, - {.lo = 0x811f953984eff83e, .hi = 0xac7cd3ad58fee7f0, .ex = -1, .sgn=0}, - {.lo = 0x9a5318ac6fe94e4d, .hi = 0xadf73c5ced9db0f3, .ex = -1, .sgn=0}, - {.lo = 0x9fe5f4ea48965e2c, .hi = 0xaf7139bcf5349ac6, .ex = -1, .sgn=0}, - {.lo = 0x63c66682bae74898, .hi = 0xb0eacae4461013ed, .ex = -1, .sgn=0}, - {.lo = 0x695a5332090bb09b, .hi = 0xb263eee9f93e3088, .ex = -1, .sgn=0}, - {.lo = 0x992d96e5021e3c37, .hi = 0xb3dca4e56b1e54bb, .ex = -1, .sgn=0}, - {.lo = 0x971f4da709ad4378, .hi = 0xb554ebee3bf0b58e, .ex = -1, .sgn=0}, - {.lo = 0x35ebacd79f209137, .hi = 0xb6ccc31c5065afee, .ex = -1, .sgn=0}, - {.lo = 0x9cc3ef36746de3b8, .hi = 0xb8442987d22cf576, .ex = -1, .sgn=0}, - {.lo = 0xcdb0531c4e58484b, .hi = 0xb9bb1e4930848ead, .ex = -1, .sgn=0}, - {.lo = 0x55b92083658bb897, .hi = 0xbb31a07920c7b256, .ex = -1, .sgn=0}, - {.lo = 0xa4b0d21fc5036a5, .hi = 0xbca7af309efd7182, .ex = -1, .sgn=0}, - {.lo = 0xd1f90f79f46c7e01, .hi = 0xbe1d4988ee67380c, .ex = -1, .sgn=0}, - {.lo = 0x91a1b5eb79658c67, .hi = 0xbf926e9b9a0f2127, .ex = -1, .sgn=0}, - {.lo = 0x721853f8e528a934, .hi = 0xc1071d8275561f9b, .ex = -1, .sgn=0}, - {.lo = 0xcdc2bd470675104d, .hi = 0xc27b55579c81f96d, .ex = -1, .sgn=0}, - {.lo = 0x3122c2a59efddc37, .hi = 0xc3ef1535754b168d, .ex = -1, .sgn=0}, - {.lo = 0xf4ff2895ab6ebe89, .hi = 0xc5625c36af6a222f, .ex = -1, .sgn=0}, - {.lo = 0x14d24739de27e2e9, .hi = 0xc6d5297645257e8d, .ex = -1, .sgn=0}, - {.lo = 0x4ce0246ad4fa74, .hi = 0xc8477c0f7bde8a98, .ex = -1, .sgn=0}, - {.lo = 0x4319e5ad5b0dcb84, .hi = 0xc9b9531de49eb968, .ex = -1, .sgn=0}, - {.lo = 0xfaa3dfe675a65ee2, .hi = 0xcb2aadbd5ca47af5, .ex = -1, .sgn=0}, - {.lo = 0x2e663b3c7555a6c3, .hi = 0xcc9b8b0a0deff5d4, .ex = -1, .sgn=0}, - {.lo = 0x3c540a9eec47af38, .hi = 0xce0bea206fcf9192, .ex = -1, .sgn=0}, - {.lo = 0xa81290bdbaad62e4, .hi = 0xcf7bca1d476c516d, .ex = -1, .sgn=0}, - {.lo = 0xb9302788604e88f1, .hi = 0xd0eb2a1da855fefd, .ex = -1, .sgn=0}, - {.lo = 0x721fc87ba1d42456, .hi = 0xd25a093ef50f2482, .ex = -1, .sgn=0}, - {.lo = 0x87967926fdcecec4, .hi = 0xd3c8669edf98d680, .ex = -1, .sgn=0}, - {.lo = 0x1df22346611c6b4b, .hi = 0xd536415b69fe4c54, .ex = -1, .sgn=0}, - {.lo = 0x3090d44db12c418c, .hi = 0xd6a39892e6e04764, .ex = -1, .sgn=0}, - {.lo = 0xa573f2aa90434ba5, .hi = 0xd8106b63fa0048a0, .ex = -1, .sgn=0}, - {.lo = 0x2e349483e3fb2a6a, .hi = 0xd97cb8ed98cb93f5, .ex = -1, .sgn=0}, - {.lo = 0x362cb974182e3030, .hi = 0xdae8804f0ae6015b, .ex = -1, .sgn=0}, - {.lo = 0x3ccca3982328ed8b, .hi = 0xdc53c0a7eab49b35, .ex = -1, .sgn=0}, - {.lo = 0x1a5bd9269d408d7e, .hi = 0xddbe791825e8099e, .ex = -1, .sgn=0}, - {.lo = 0xcce2634be2bf54df, .hi = 0xdf28a8bffe06ca56, .ex = -1, .sgn=0}, - {.lo = 0x8aa895d5bf3e84ea, .hi = 0xe0924ec008f734fd, .ex = -1, .sgn=0}, - {.lo = 0xf7a1f9bd9ba13b6b, .hi = 0xe1fb6a3931894b38, .ex = -1, .sgn=0}, - {.lo = 0x7b32c72e31824e51, .hi = 0xe363fa4cb8005482, .ex = -1, .sgn=0}, - {.lo = 0xd40e9e6b989f89e5, .hi = 0xe4cbfe1c329c453a, .ex = -1, .sgn=0}, - {.lo = 0x2872ce1bfc7ad1cd, .hi = 0xe63374c98e22f0b4, .ex = -1, .sgn=0}, - {.lo = 0xf1b65cc5fd780262, .hi = 0xe79a5d770e6905dc, .ex = -1, .sgn=0}, - {.lo = 0x431626c10485bdda, .hi = 0xe900b7474edad637, .ex = -1, .sgn=0}, - {.lo = 0xcc39cfcc29960b1, .hi = 0xea66815d4304e6c8, .ex = -1, .sgn=0}, - {.lo = 0x1d90f780ae951140, .hi = 0xebcbbadc371c4aaa, .ex = -1, .sgn=0}, - {.lo = 0xc71debc372b6f9d4, .hi = 0xed3062e7d086c6f0, .ex = -1, .sgn=0}, - {.lo = 0x2a24164daec85ccb, .hi = 0xee9478a40e62bf86, .ex = -1, .sgn=0}, - {.lo = 0x527233b40d3432bb, .hi = 0xeff7fb354a0eecb1, .ex = -1, .sgn=0}, - {.lo = 0x6c48e9e3420b0f1e, .hi = 0xf15ae9c037b1d8f0, .ex = -1, .sgn=0}, - {.lo = 0x7f232aee178c6323, .hi = 0xf2bd4369e6c126d3, .ex = -1, .sgn=0}, - {.lo = 0x3c7f10db458c337c, .hi = 0xf41f0757c2889e84, .ex = -1, .sgn=0}, - {.lo = 0x93fa6107c4327527, .hi = 0xf58034af92b102a7, .ex = -1, .sgn=0}, - {.lo = 0xe1079824233fef46, .hi = 0xf6e0ca977bc6ac45, .ex = -1, .sgn=0}, - {.lo = 0xa9a56012067c570c, .hi = 0xf840c835ffbfed66, .ex = -1, .sgn=0}, - {.lo = 0x8da894471de1a18, .hi = 0xf9a02cb1fe833a0d, .ex = -1, .sgn=0}, - {.lo = 0x343fbf4a7d42af3, .hi = 0xfafef732b66d1742, .ex = -1, .sgn=0}, - {.lo = 0x27c07c911290b8d1, .hi = 0xfc5d26dfc4d5cfda, .ex = -1, .sgn=0}, - {.lo = 0x2377c3799c052fa, .hi = 0xfdbabae12696eea4, .ex = -1, .sgn=0}, - {.lo = 0xa9c6ba50490539f, .hi = 0xff17b25f38907dad, .ex = -1, .sgn=0}, - {.lo = 0x6f53873e2f1477ff, .hi = 0x803a06415c170525, .ex = 0, .sgn=0}, - {.lo = 0x5ca183dc973abc22, .hi = 0x80e7e43a61f5b6cb, .ex = 0, .sgn=0}, - {.lo = 0x9fba97fdf0c4d24c, .hi = 0x819572af6decac84, .ex = 0, .sgn=0}, - {.lo = 0x6fb2123fedfa6e22, .hi = 0x8242b1357110d372, .ex = 0, .sgn=0}, - {.lo = 0x91a965931f1a200a, .hi = 0x82ef9f618dc5b70e, .ex = 0, .sgn=0}, - {.lo = 0xbfd79717f2880abf, .hi = 0x839c3cc917ff6cb4, .ex = 0, .sgn=0}, - {.lo = 0x246efcff30cb064a, .hi = 0x8448890195846099, .ex = 0, .sgn=0}, - {.lo = 0x51917cac857fd5f5, .hi = 0x84f483a0be2f0403, .ex = 0, .sgn=0}, - {.lo = 0x327888fe4b62687b, .hi = 0x85a02c3c7c2f5ca5, .ex = 0, .sgn=0}, - {.lo = 0x85043222c9bdd18d, .hi = 0x864b826aec4c74e5, .ex = 0, .sgn=0}, - {.lo = 0x7e0b9b07548471a2, .hi = 0x86f685c25e25acf5, .ex = 0, .sgn=0}, - {.lo = 0x4e091160e2430712, .hi = 0x87a135d95473ec89, .ex = 0, .sgn=0}, - {.lo = 0x4f14c8afe4560291, .hi = 0x884b9246854ab50b, .ex = 0, .sgn=0}, - {.lo = 0xb892ca8361d8c84c, .hi = 0x88f59aa0da591421, .ex = 0, .sgn=0}, - {.lo = 0xc88302a31afce54a, .hi = 0x899f4e7f712a765e, .ex = 0, .sgn=0}, - {.lo = 0x660558a02136130a, .hi = 0x8a48ad799b6759f3, .ex = 0, .sgn=0}, - {.lo = 0x545f7d79ead8fa19, .hi = 0x8af1b726df15e13c, .ex = 0, .sgn=0}, - {.lo = 0x21a6675f51580bc4, .hi = 0x8b9a6b1ef6da4502, .ex = 0, .sgn=0}, - {.lo = 0x101a5adbcb9ffb43, .hi = 0x8c42c8f9d2372644, .ex = 0, .sgn=0}, - {.lo = 0x4d49cbaf15aecd80, .hi = 0x8cead04f95cdbf66, .ex = 0, .sgn=0}, - {.lo = 0xde2d43c6b67a7cbe, .hi = 0x8d9280b89b9df49b, .ex = 0, .sgn=0}, - {.lo = 0xbba4cfecbff54867, .hi = 0x8e39d9cd73464364, .ex = 0, .sgn=0}, - {.lo = 0xaf0e2345f3bd24b4, .hi = 0x8ee0db26e24390f8, .ex = 0, .sgn=0}, - {.lo = 0x9311a82459aa0f72, .hi = 0x8f87845de430d777, .ex = 0, .sgn=0}, - {.lo = 0xb144016c7a30b39a, .hi = 0x902dd50bab06b1b7, .ex = 0, .sgn=0}, - {.lo = 0x9d1072e09b72292, .hi = 0x90d3ccc99f5ac58b, .ex = 0, .sgn=0}, - {.lo = 0x6714fe6925b78cc4, .hi = 0x91796b31609f0c54, .ex = 0, .sgn=0}, - {.lo = 0x33d0a284a8c954ad, .hi = 0x921eafdcc560f9c5, .ex = 0, .sgn=0}, - {.lo = 0x1f8481e704e4a767, .hi = 0x92c39a65db88809d, .ex = 0, .sgn=0}, - {.lo = 0xb17821911e71c16e, .hi = 0x93682a66e896f544, .ex = 0, .sgn=0}, - {.lo = 0x1489a97671a42, .hi = 0x940c5f7a69e5ce1c, .ex = 0, .sgn=0}, - {.lo = 0xd6c7af02d5c16fd9, .hi = 0x94b0393b14e54156, .ex = 0, .sgn=0}, - {.lo = 0xac0106650f4ef023, .hi = 0x9553b743d75ac03f, .ex = 0, .sgn=0}, - {.lo = 0xd9f8e1a446e973b9, .hi = 0x95f6d92fd79f4fba, .ex = 0, .sgn=0}, - {.lo = 0xa7a7556c3b33abc1, .hi = 0x96999e9a74ddbde3, .ex = 0, .sgn=0}, - {.lo = 0xc0a03934f0cce19b, .hi = 0x973c071f4750b49c, .ex = 0, .sgn=0}, - {.lo = 0xd243aa0843a2c144, .hi = 0x97de125a2080a8ed, .ex = 0, .sgn=0}, - {.lo = 0x19cec845ac87a5c6, .hi = 0x987fbfe70b81a708, .ex = 0, .sgn=0}, - {.lo = 0xc4b992a37fb9b9bd, .hi = 0x99210f624d30facb, .ex = 0, .sgn=0}, - {.lo = 0x1ab42d43235757b6, .hi = 0x99c200686472b4a8, .ex = 0, .sgn=0}, - {.lo = 0x7e92c655656e6b85, .hi = 0x9a6292960a6f0ab0, .ex = 0, .sgn=0}, - {.lo = 0x698b94f50326a043, .hi = 0x9b02c58832cf95c0, .ex = 0, .sgn=0}, - {.lo = 0x9a5614e8ffbeac6f, .hi = 0x9ba298dc0bfc6a88, .ex = 0, .sgn=0}, - {.lo = 0xc7fd954194e6d8aa, .hi = 0x9c420c2eff590e5f, .ex = 0, .sgn=0}, - {.lo = 0x3e93627de8fd5779, .hi = 0x9ce11f1eb18147b1, .ex = 0, .sgn=0}, - {.lo = 0xe25e39549638ae68, .hi = 0x9d7fd1490285c9e3, .ex = 0, .sgn=0}, - {.lo = 0x2cad377d5c9c35d8, .hi = 0x9e1e224c0e28bc94, .ex = 0, .sgn=0}, - {.lo = 0xcc141e10c6460c8b, .hi = 0x9ebc11c62c1a1dfb, .ex = 0, .sgn=0}, - {.lo = 0xa88d5f46834bbf8d, .hi = 0x9f599f55f0340061, .ex = 0, .sgn=0}, - {.lo = 0x22cc118a0c118aa0, .hi = 0x9ff6ca9a2ab6a26d, .ex = 0, .sgn=0}, - {.lo = 0x7cec6df5bea167cf, .hi = 0xa0939331e8846237, .ex = 0, .sgn=0}, - {.lo = 0x71acea2819360c35, .hi = 0xa12ff8bc735d8af6, .ex = 0, .sgn=0}, - {.lo = 0x166c36e7bb3c402f, .hi = 0xa1cbfad9521bfd1b, .ex = 0, .sgn=0}, - {.lo = 0x3b5167ee359a234e, .hi = 0xa267992848eeb0c0, .ex = 0, .sgn=0}, - {.lo = 0x9443372e20d4377c, .hi = 0xa302d34959951243, .ex = 0, .sgn=0}, - {.lo = 0xca9a8a720d4c69c, .hi = 0xa39da8dcc39a38e5, .ex = 0, .sgn=0}, - {.lo = 0xbf623cf5301a2dde, .hi = 0xa4381983048ff747, .ex = 0, .sgn=0}, - {.lo = 0x23d251cc8d7975cc, .hi = 0xa4d224dcd849c5b0, .ex = 0, .sgn=0}, - {.lo = 0x189d39ffe11aaa2b, .hi = 0xa56bca8b391785db, .ex = 0, .sgn=0}, - {.lo = 0x8c33ebf3aa8501fb, .hi = 0xa6050a2f60002049, .ex = 0, .sgn=0}, - {.lo = 0x9b3ad6e4022183d9, .hi = 0xa69de36ac4fbfadc, .ex = 0, .sgn=0}, - {.lo = 0x149f6e75993468a3, .hi = 0xa73655df1f2f489e, .ex = 0, .sgn=0}, - {.lo = 0x6b2a39f856a69781, .hi = 0xa7ce612e65243291, .ex = 0, .sgn=0}, - {.lo = 0x3463a2c2e6e9cc55, .hi = 0xa86604facd04d969, .ex = 0, .sgn=0}, - {.lo = 0x6cc14c4f53e2e82d, .hi = 0xa8fd40e6ccd52ffd, .ex = 0, .sgn=0}, - {.lo = 0xd147625fda929af8, .hi = 0xa99414951aacae5e, .ex = 0, .sgn=0}, - {.lo = 0xb714ee81b53b4b9d, .hi = 0xaa2a7fa8acefdd63, .ex = 0, .sgn=0}, - {.lo = 0xe1b3dfc4dbda9bfd, .hi = 0xaac081c4ba89ba8a, .ex = 0, .sgn=0}, - {.lo = 0xf17cee69b0d2ecde, .hi = 0xab561a8cbb24f410, .ex = 0, .sgn=0}, - {.lo = 0x1becda8089c1a94c, .hi = 0xabeb49a46764fd15, .ex = 0, .sgn=0}, - {.lo = 0xf86ba0dde982fb59, .hi = 0xac800eafb91ef9a9, .ex = 0, .sgn=0}, - {.lo = 0x44bf16268608db96, .hi = 0xad146952eb9282af, .ex = 0, .sgn=0}, - {.lo = 0x9d30d4cfeb04f1fb, .hi = 0xada859327ba24151, .ex = 0, .sgn=0}, - {.lo = 0x3d53817865422565, .hi = 0xae3bddf3280c620d, .ex = 0, .sgn=0}, - {.lo = 0xf74d099042e8f326, .hi = 0xaecef739f1a2df10, .ex = 0, .sgn=0}, - {.lo = 0xa89a9b8f726b95bf, .hi = 0xaf61a4ac1b83a1de, .ex = 0, .sgn=0}, - {.lo = 0x8c679e67fc462d51, .hi = 0xaff3e5ef2b507c06, .ex = 0, .sgn=0}, - {.lo = 0xe4cad00d5c94bcd2, .hi = 0xb085baa8e966f6da, .ex = 0, .sgn=0}, - {.lo = 0x8d8be132d576e614, .hi = 0xb117227f6117f9f9, .ex = 0, .sgn=0}, - {.lo = 0x24784f32c3e3e5bd, .hi = 0xb1a81d18e0df4889, .ex = 0, .sgn=0}, - {.lo = 0x8cc7d4bd05ffd5ae, .hi = 0xb238aa1bfa9ad507, .ex = 0, .sgn=0}, - {.lo = 0xac9f7ebbc469ef59, .hi = 0xb2c8c92f83c1eb87, .ex = 0, .sgn=0}, - {.lo = 0x5d6635109164f740, .hi = 0xb35879fa959c323c, .ex = 0, .sgn=0}, - {.lo = 0xa156468ef6c18c60, .hi = 0xb3e7bc248d78802e, .ex = 0, .sgn=0}, - {.lo = 0x4a85350f69018c55, .hi = 0xb4768f550ce389fd, .ex = 0, .sgn=0}, -}; - -/* Table containing 128-bit approximations of cos2pi(i/2^11) for 0 <= i < 256 - (to nearest). - Each entry is to be interpreted as (hi/2^64+lo/2^128)*2^ex*(-1)*sgn. - Generated with computeC() from sin.sage. */ -static const dint64_t C[256] = { - {.lo = 0x0, .hi = 0x8000000000000000, .ex = 1, .sgn=0}, - {.lo = 0x3031437d7eccb9df, .hi = 0xffffb10b10e80e95, .ex = 0, .sgn=0}, - {.lo = 0x38e310779edfec68, .hi = 0xfffec42c7454926b, .ex = 0, .sgn=0}, - {.lo = 0x69fff9ae0dedb047, .hi = 0xfffd3964bc6275ba, .ex = 0, .sgn=0}, - {.lo = 0xb47903f7a19f8ee2, .hi = 0xfffb10b4dc96dabb, .ex = 0, .sgn=0}, - {.lo = 0x8cc193c5d508e13f, .hi = 0xfff84a1e29de8571, .ex = 0, .sgn=0}, - {.lo = 0x43366df666fd54ff, .hi = 0xfff4e5a25a8d095b, .ex = 0, .sgn=0}, - {.lo = 0x5428ed0647c9e5d1, .hi = 0xfff0e343865bbb13, .ex = 0, .sgn=0}, - {.lo = 0x5657552366961732, .hi = 0xffec4304266865d9, .ex = 0, .sgn=0}, - {.lo = 0x53aa9423bb0adc21, .hi = 0xffe704e71533c508, .ex = 0, .sgn=0}, - {.lo = 0x7d209f32d42d864e, .hi = 0xffe128ef8e9fc17a, .ex = 0, .sgn=0}, - {.lo = 0x4fd8f038449ec436, .hi = 0xffdaaf212fed72db, .ex = 0, .sgn=0}, - {.lo = 0x664649b4d541b9c5, .hi = 0xffd3977ff7bae4e9, .ex = 0, .sgn=0}, - {.lo = 0x5595ca3f421ae09c, .hi = 0xffcbe2104600a0a9, .ex = 0, .sgn=0}, - {.lo = 0x1c676208aa3be545, .hi = 0xffc38ed6dc0ef98b, .ex = 0, .sgn=0}, - {.lo = 0xccfed60a91097c48, .hi = 0xffba9dd8dc8b1e83, .ex = 0, .sgn=0}, - {.lo = 0x421e8edaaf59453e, .hi = 0xffb10f1bcb6bef1d, .ex = 0, .sgn=0}, - {.lo = 0xd2c665c2da3e7844, .hi = 0xffa6e2a58df6947d, .ex = 0, .sgn=0}, - {.lo = 0x1e1862cca089938b, .hi = 0xff9c187c6abade6a, .ex = 0, .sgn=0}, - {.lo = 0x2dabd3195a05710f, .hi = 0xff90b0a7098f6443, .ex = 0, .sgn=0}, - {.lo = 0x519c314973ccae6b, .hi = 0xff84ab2c738d6a03, .ex = 0, .sgn=0}, - {.lo = 0x3ea4f30adda3016f, .hi = 0xff780814130c893c, .ex = 0, .sgn=0}, - {.lo = 0x1b9d5851979f28fb, .hi = 0xff6ac765b39e1e19, .ex = 0, .sgn=0}, - {.lo = 0x50a7bb6a6ee3b0f1, .hi = 0xff5ce92982087867, .ex = 0, .sgn=0}, - {.lo = 0xf668633f1ab858a, .hi = 0xff4e6d680c41d0a9, .ex = 0, .sgn=0}, - {.lo = 0xb085c1828f69296a, .hi = 0xff3f542a416b0134, .ex = 0, .sgn=0}, - {.lo = 0x27e31939e2eec09c, .hi = 0xff2f9d7971ca0364, .ex = 0, .sgn=0}, - {.lo = 0xf5971326a3540ea9, .hi = 0xff1f495f4ec430d7, .ex = 0, .sgn=0}, - {.lo = 0x1f1901544271c3f8, .hi = 0xff0e57e5ead848d1, .ex = 0, .sgn=0}, - {.lo = 0xe0abd3a9b64df725, .hi = 0xfefcc917b99839a5, .ex = 0, .sgn=0}, - {.lo = 0xec34413e87ef2740, .hi = 0xfeea9cff8fa2ae54, .ex = 0, .sgn=0}, - {.lo = 0x2f88b949a72ff96c, .hi = 0xfed7d3a8a29c603b, .ex = 0, .sgn=0}, - {.lo = 0x41390efdc726e9ef, .hi = 0xfec46d1e89292cf0, .ex = 0, .sgn=0}, - {.lo = 0xb7b6cc53c3abc817, .hi = 0xfeb0696d3ae4f04d, .ex = 0, .sgn=0}, - {.lo = 0xd3af6ee4f2101c20, .hi = 0xfe9bc8a1105c22a5, .ex = 0, .sgn=0}, - {.lo = 0xb4f70c910505e10, .hi = 0xfe868ac6c3043b2e, .ex = 0, .sgn=0}, - {.lo = 0x2907cf2b3f6feac2, .hi = 0xfe70afeb6d33d6a2, .ex = 0, .sgn=0}, - {.lo = 0xd54faa364b7da8f6, .hi = 0xfe5a381c8a1aa224, .ex = 0, .sgn=0}, - {.lo = 0x87b8875373a818a4, .hi = 0xfe432367f5b90a62, .ex = 0, .sgn=0}, - {.lo = 0x8598c2c429caf7, .hi = 0xfe2b71dbecd7aefc, .ex = 0, .sgn=0}, - {.lo = 0x90cd1d959db674ef, .hi = 0xfe1323870cfe9a3d, .ex = 0, .sgn=0}, - {.lo = 0x9bfe5c51e91cbdcd, .hi = 0xfdfa3878546c3d28, .ex = 0, .sgn=0}, - {.lo = 0xe276d247626a23fd, .hi = 0xfde0b0bf220c2fd4, .ex = 0, .sgn=0}, - {.lo = 0x499ddb331d19539d, .hi = 0xfdc68c6b356db62f, .ex = 0, .sgn=0}, - {.lo = 0xfac7397cc07a6470, .hi = 0xfdabcb8caeba091b, .ex = 0, .sgn=0}, - {.lo = 0xd6e270740a186977, .hi = 0xfd906e340eaa6401, .ex = 0, .sgn=0}, - {.lo = 0x61beb8cd2696fc78, .hi = 0xfd747472367dd6c5, .ex = 0, .sgn=0}, - {.lo = 0x6c696582f346fd91, .hi = 0xfd57de5867eedc39, .ex = 0, .sgn=0}, - {.lo = 0xeae6bd951c1dabbe, .hi = 0xfd3aabf84528b50b, .ex = 0, .sgn=0}, - {.lo = 0x863b87258f11ad7e, .hi = 0xfd1cdd63d0bc8735, .ex = 0, .sgn=0}, - {.lo = 0xa06fab9f9d106709, .hi = 0xfcfe72ad6d9641f2, .ex = 0, .sgn=0}, - {.lo = 0xa4e064308f4999f4, .hi = 0xfcdf6be7def1464c, .ex = 0, .sgn=0}, - {.lo = 0xa3e22b4d38917e73, .hi = 0xfcbfc926484cd43a, .ex = 0, .sgn=0}, - {.lo = 0x5d582cac7cb4391c, .hi = 0xfc9f8a7c2d603c60, .ex = 0, .sgn=0}, - {.lo = 0x2880268f2e62955, .hi = 0xfc7eaffd720ed673, .ex = 0, .sgn=0}, - {.lo = 0x1c0d254b6c8da4bd, .hi = 0xfc5d39be5a5bbc4b, .ex = 0, .sgn=0}, - {.lo = 0x256778ffcb5c1769, .hi = 0xfc3b27d38a5d49ab, .ex = 0, .sgn=0}, - {.lo = 0x9433b49289417ea2, .hi = 0xfc187a52063060c2, .ex = 0, .sgn=0}, - {.lo = 0x25aafd7fdba12c5f, .hi = 0xfbf5314f31eb7375, .ex = 0, .sgn=0}, - {.lo = 0x7190c94899dff1b8, .hi = 0xfbd14ce0d191516e, .ex = 0, .sgn=0}, - {.lo = 0xe63ae8632b84473c, .hi = 0xfbaccd1d0903bb09, .ex = 0, .sgn=0}, - {.lo = 0x75df66f0ec3dd459, .hi = 0xfb87b21a5bf5b917, .ex = 0, .sgn=0}, - {.lo = 0x61ce9d5ef5a81487, .hi = 0xfb61fbefadddb985, .ex = 0, .sgn=0}, - {.lo = 0xb4b54683879c9c17, .hi = 0xfb3baab441e770f7, .ex = 0, .sgn=0}, - {.lo = 0x2172a361fd2a722f, .hi = 0xfb14be7fbae58156, .ex = 0, .sgn=0}, - {.lo = 0x2079880c450348ac, .hi = 0xfaed376a1b42e559, .ex = 0, .sgn=0}, - {.lo = 0x4a188aa367f90ab1, .hi = 0xfac5158bc4f4211f, .ex = 0, .sgn=0}, - {.lo = 0x10655ecd5cc771d8, .hi = 0xfa9c58fd796837d4, .ex = 0, .sgn=0}, - {.lo = 0x1fe196a53fb5b237, .hi = 0xfa7301d859796671, .ex = 0, .sgn=0}, - {.lo = 0xd24377c77a591e24, .hi = 0xfa491035e55da3a3, .ex = 0, .sgn=0}, - {.lo = 0x431c393c7f62da65, .hi = 0xfa1e842ffc96e4e0, .ex = 0, .sgn=0}, - {.lo = 0xba5dbf4510eddc8f, .hi = 0xf9f35de0dde328ab, .ex = 0, .sgn=0}, - {.lo = 0x4504ae08d19b2980, .hi = 0xf9c79d63272c4628, .ex = 0, .sgn=0}, - {.lo = 0x78685d850f80ecdc, .hi = 0xf99b42d1d57781eb, .ex = 0, .sgn=0}, - {.lo = 0x80e8c17bf80e8f02, .hi = 0xf96e4e4844d4e82a, .ex = 0, .sgn=0}, - {.lo = 0xc0e2a1352ed7f292, .hi = 0xf940bfe2304e6c45, .ex = 0, .sgn=0}, - {.lo = 0x68fc6e4d6a920bd2, .hi = 0xf91297bbb1d6cdbe, .ex = 0, .sgn=0}, - {.lo = 0x9701914c7f8fbcd7, .hi = 0xf8e3d5f1423842a0, .ex = 0, .sgn=0}, - {.lo = 0xac9f07f54ff5bc14, .hi = 0xf8b47a9fb902e76c, .ex = 0, .sgn=0}, - {.lo = 0xb36a9dfaadafc1e1, .hi = 0xf88485e44c7af48a, .ex = 0, .sgn=0}, - {.lo = 0xc7adc6b4988891bb, .hi = 0xf853f7dc9186b952, .ex = 0, .sgn=0}, - {.lo = 0xa776175bd284fe05, .hi = 0xf822d0a67b9c5cb5, .ex = 0, .sgn=0}, - {.lo = 0xa76f7efc19aed41c, .hi = 0xf7f110605caf6390, .ex = 0, .sgn=0}, - {.lo = 0x730785813f78aa1e, .hi = 0xf7beb728e51dfcb8, .ex = 0, .sgn=0}, - {.lo = 0x214cffcee9dd33ca, .hi = 0xf78bc51f239e12c6, .ex = 0, .sgn=0}, - {.lo = 0x4becad887680c197, .hi = 0xf7583a62852a23b2, .ex = 0, .sgn=0}, - {.lo = 0xf99107e50d631330, .hi = 0xf7241712d4edde49, .ex = 0, .sgn=0}, - {.lo = 0x50ca117eb18beed7, .hi = 0xf6ef5b503c328589, .ex = 0, .sgn=0}, - {.lo = 0x2c791f59cc1ffc23, .hi = 0xf6ba073b424b19e8, .ex = 0, .sgn=0}, - {.lo = 0xce8c455197cdf8a7, .hi = 0xf6841af4cc8048a4, .ex = 0, .sgn=0}, - {.lo = 0x119d358de0493956, .hi = 0xf64d969e1dfc2119, .ex = 0, .sgn=0}, - {.lo = 0x9dc7e5954c5a8f24, .hi = 0xf6167a58d7b59026, .ex = 0, .sgn=0}, - {.lo = 0xc8c615e72768d6b5, .hi = 0xf5dec646f85ba1c6, .ex = 0, .sgn=0}, - {.lo = 0xed0dd4bf62edd13f, .hi = 0xf5a67a8adc4088ca, .ex = 0, .sgn=0}, - {.lo = 0x275a2bbb2bab6c8a, .hi = 0xf56d97473d446cda, .ex = 0, .sgn=0}, - {.lo = 0x8da64484aaa0febc, .hi = 0xf5341c9f32bffeb9, .ex = 0, .sgn=0}, - {.lo = 0x163c5c7f03b718c5, .hi = 0xf4fa0ab6316ed2ec, .ex = 0, .sgn=0}, - {.lo = 0x890ac4aafa6a37bf, .hi = 0xf4bf61b00b5982b7, .ex = 0, .sgn=0}, - {.lo = 0xf8f9d3b87d11fd52, .hi = 0xf48421b0efbf939b, .ex = 0, .sgn=0}, - {.lo = 0x667e06866c07c369, .hi = 0xf4484add6b01254b, .ex = 0, .sgn=0}, - {.lo = 0x5019794a1f5896e5, .hi = 0xf40bdd5a6688662f, .ex = 0, .sgn=0}, - {.lo = 0x18ef535a7ffa7a3d, .hi = 0xf3ced94d28b2ce8a, .ex = 0, .sgn=0}, - {.lo = 0x50f29b4b49f31c37, .hi = 0xf3913edb54ba2242, .ex = 0, .sgn=0}, - {.lo = 0xd981acdcf6bc3e4, .hi = 0xf3530e2aea9d3966, .ex = 0, .sgn=0}, - {.lo = 0xa5486bdc455d56a2, .hi = 0xf314476247088f74, .ex = 0, .sgn=0}, - {.lo = 0x431be53f92ece9e6, .hi = 0xf2d4eaa8233e997d, .ex = 0, .sgn=0}, - {.lo = 0xebadcdbf915e8f6c, .hi = 0xf294f82394ffe320, .ex = 0, .sgn=0}, - {.lo = 0xaf0eed81e8c51e55, .hi = 0xf2546ffc0e72f286, .ex = 0, .sgn=0}, - {.lo = 0xe7112e89103cc0c7, .hi = 0xf21352595e0bf350, .ex = 0, .sgn=0}, - {.lo = 0x844e6a35ddc2b713, .hi = 0xf1d19f63ae7428a2, .ex = 0, .sgn=0}, - {.lo = 0x8f6bac72988088b0, .hi = 0xf18f574386712643, .ex = 0, .sgn=0}, - {.lo = 0x2730081c758fb42b, .hi = 0xf14c7a21c8cbd0f4, .ex = 0, .sgn=0}, - {.lo = 0x67127db35b287316, .hi = 0xf1090827b43725fd, .ex = 0, .sgn=0}, - {.lo = 0xc4e557b119ef3185, .hi = 0xf0c5017ee336ca0f, .ex = 0, .sgn=0}, - {.lo = 0x973ea9903ed5125f, .hi = 0xf08066514c055f7e, .ex = 0, .sgn=0}, - {.lo = 0x992d39ec5c561d28, .hi = 0xf03b36c9407aa3e8, .ex = 0, .sgn=0}, - {.lo = 0x62aef7b55319d1d4, .hi = 0xeff573116df1555d, .ex = 0, .sgn=0}, - {.lo = 0xf03a18a5e16ab641, .hi = 0xefaf1b54dd2cdf0f, .ex = 0, .sgn=0}, - {.lo = 0x767c0e8ad33bc085, .hi = 0xef682fbef23ecda6, .ex = 0, .sgn=0}, - {.lo = 0xe2398bf0eeb28cde, .hi = 0xef20b07b6c6c0b37, .ex = 0, .sgn=0}, - {.lo = 0x86f8c20fb664b01b, .hi = 0xeed89db66611e307, .ex = 0, .sgn=0}, - {.lo = 0xa1d2c3d018a9279f, .hi = 0xee8ff79c548acd0f, .ex = 0, .sgn=0}, - {.lo = 0x7872773830d368be, .hi = 0xee46be5a0813016b, .ex = 0, .sgn=0}, - {.lo = 0xfee6a1eebfa13b4a, .hi = 0xedfcf21cabacd3b1, .ex = 0, .sgn=0}, - {.lo = 0x11815196b9fbf5df, .hi = 0xedb29311c504d652, .ex = 0, .sgn=0}, - {.lo = 0x7289102076a125e5, .hi = 0xed67a1673455c601, .ex = 0, .sgn=0}, - {.lo = 0xddffe98c4f8aa031, .hi = 0xed1c1d4b344c3d4f, .ex = 0, .sgn=0}, - {.lo = 0xa8392eb238578ab0, .hi = 0xecd006ec59ea306f, .ex = 0, .sgn=0}, - {.lo = 0x7e610231ac1d6181, .hi = 0xec835e79946a3145, .ex = 0, .sgn=0}, - {.lo = 0x278047ae3dd0889, .hi = 0xec3624222d227bd1, .ex = 0, .sgn=0}, - {.lo = 0x1e99ccb9adc62ca6, .hi = 0xebe85815c767cb00, .ex = 0, .sgn=0}, - {.lo = 0xdae311e656e0661, .hi = 0xeb99fa84606ff5ff, .ex = 0, .sgn=0}, - {.lo = 0x39e39c6c2ab3655d, .hi = 0xeb4b0b9e4f345617, .ex = 0, .sgn=0}, - {.lo = 0x3383bbb5156bf1d7, .hi = 0xeafb8b944453f52f, .ex = 0, .sgn=0}, - {.lo = 0x24db98ad3a0647a1, .hi = 0xeaab7a9749f584fe, .ex = 0, .sgn=0}, - {.lo = 0x4a0ca5ea449b1c83, .hi = 0xea5ad8d8c3a91f05, .ex = 0, .sgn=0}, - {.lo = 0x15ad45b4a1b5e823, .hi = 0xea09a68a6e49cd62, .ex = 0, .sgn=0}, - {.lo = 0xcd24d4bd1056c826, .hi = 0xe9b7e3de5fdedc8b, .ex = 0, .sgn=0}, - {.lo = 0x89a92b199adfbafa, .hi = 0xe9659107077cf60f, .ex = 0, .sgn=0}, - {.lo = 0xacb1c26a06e5ae02, .hi = 0xe912ae372d27045d, .ex = 0, .sgn=0}, - {.lo = 0xf8972affb3d98e1f, .hi = 0xe8bf3ba1f1aedfbb, .ex = 0, .sgn=0}, - {.lo = 0x9fec1e78c4376186, .hi = 0xe86b397ace95c46f, .ex = 0, .sgn=0}, - {.lo = 0xbfe8378abfb87b6f, .hi = 0xe816a7f595ec9232, .ex = 0, .sgn=0}, - {.lo = 0xdbfb0fe56c6f80fe, .hi = 0xe7c187467233d508, .ex = 0, .sgn=0}, - {.lo = 0x125129529d48a92f, .hi = 0xe76bd7a1e63b9786, .ex = 0, .sgn=0}, - {.lo = 0xe2ba81b9ce96e02e, .hi = 0xe715993ccd02fe9c, .ex = 0, .sgn=0}, - {.lo = 0x82fcedb4c6434d76, .hi = 0xe6becc4c5997af06, .ex = 0, .sgn=0}, - {.lo = 0xdd2a3e32c3859960, .hi = 0xe667710616f4fc59, .ex = 0, .sgn=0}, - {.lo = 0x7613b68f6ab03130, .hi = 0xe60f879fe7e2e1e5, .ex = 0, .sgn=0}, - {.lo = 0x9b695cd67c93bd79, .hi = 0xe5b7105006d4c560, .ex = 0, .sgn=0}, - {.lo = 0x5a7c210a3a15e7ea, .hi = 0xe55e0b4d05c80388, .ex = 0, .sgn=0}, - {.lo = 0xe1f5a58c80292554, .hi = 0xe50478cdce2246bc, .ex = 0, .sgn=0}, - {.lo = 0x122785ae67f5515d, .hi = 0xe4aa5909a08fa7b4, .ex = 0, .sgn=0}, - {.lo = 0x20d63b5b9e3cd6ac, .hi = 0xe44fac3814e09856, .ex = 0, .sgn=0}, - {.lo = 0x56992551ae074e99, .hi = 0xe3f4729119e798d9, .ex = 0, .sgn=0}, - {.lo = 0xd1197dc12c63176, .hi = 0xe398ac4cf556b732, .ex = 0, .sgn=0}, - {.lo = 0x36563e2ffad8351a, .hi = 0xe33c59a4439cd8ec, .ex = 0, .sgn=0}, - {.lo = 0xd6fe4dd22e60a4a2, .hi = 0xe2df7acff7c2cf83, .ex = 0, .sgn=0}, - {.lo = 0xfd39138aa2d508ed, .hi = 0xe28210095b483751, .ex = 0, .sgn=0}, - {.lo = 0xe0521df01a1be6f5, .hi = 0xe224198a0e002123, .ex = 0, .sgn=0}, - {.lo = 0xf4e8a8372f8c5810, .hi = 0xe1c5978c05ed8691, .ex = 0, .sgn=0}, - {.lo = 0xe2f9d4600f4d0325, .hi = 0xe1668a498f1f892c, .ex = 0, .sgn=0}, - {.lo = 0x6ba8a9d9ba877899, .hi = 0xe106f1fd4b8d7c96, .ex = 0, .sgn=0}, - {.lo = 0x6d6c98fe79817946, .hi = 0xe0a6cee232f2bb9c, .ex = 0, .sgn=0}, - {.lo = 0x55ff6038a5197367, .hi = 0xe046213392aa486c, .ex = 0, .sgn=0}, - {.lo = 0x720588ff6547d884, .hi = 0xdfe4e92d0d8a37f5, .ex = 0, .sgn=0}, - {.lo = 0xab01350f013d78dd, .hi = 0xdf83270a9bbee890, .ex = 0, .sgn=0}, - {.lo = 0x64a58b2f103485dd, .hi = 0xdf20db088aa60404, .ex = 0, .sgn=0}, - {.lo = 0x4b19aa71fec3ae6d, .hi = 0xdebe05637ca94cfb, .ex = 0, .sgn=0}, - {.lo = 0x4248f15548f69ca, .hi = 0xde5aa65869193805, .ex = 0, .sgn=0}, - {.lo = 0xd597b10a01676659, .hi = 0xddf6be249c075037, .ex = 0, .sgn=0}, - {.lo = 0x739c45b982193b5e, .hi = 0xdd924d05b620678a, .ex = 0, .sgn=0}, - {.lo = 0x49c6e0ea76cbcaac, .hi = 0xdd2d5339ac8692fd, .ex = 0, .sgn=0}, - {.lo = 0xb2069fd0b482b4e8, .hi = 0xdcc7d0fec8aaf2aa, .ex = 0, .sgn=0}, - {.lo = 0xaca8017e375b64e5, .hi = 0xdc61c693a82745d5, .ex = 0, .sgn=0}, - {.lo = 0xccb7fd40d543f4a1, .hi = 0xdbfb34373c974b0e, .ex = 0, .sgn=0}, - {.lo = 0x2c19b63253da43fc, .hi = 0xdb941a28cb71ec87, .ex = 0, .sgn=0}, - {.lo = 0x5a98479cbef2ecbc, .hi = 0xdb2c78a7ede238a9, .ex = 0, .sgn=0}, - {.lo = 0x5b267c1bcff0ab62, .hi = 0xdac44ff490a02710, .ex = 0, .sgn=0}, - {.lo = 0xe257bde73d83dc1a, .hi = 0xda5ba04ef3c929f4, .ex = 0, .sgn=0}, - {.lo = 0x28e81dcb6dab91ac, .hi = 0xd9f269f7aab88c29, .ex = 0, .sgn=0}, - {.lo = 0xc4e4dc69fc2fff6f, .hi = 0xd988ad2f9bdf9bbb, .ex = 0, .sgn=0}, - {.lo = 0x1bb35ad6d2e74b67, .hi = 0xd91e6a38009da15a, .ex = 0, .sgn=0}, - {.lo = 0x1ed1a8ff78f1b632, .hi = 0xd8b3a1526517a48b, .ex = 0, .sgn=0}, - {.lo = 0x24b9fe00663574a4, .hi = 0xd84852c0a80ffcdb, .ex = 0, .sgn=0}, - {.lo = 0xced12d2899b803db, .hi = 0xd7dc7ec4fabdb011, .ex = 0, .sgn=0}, - {.lo = 0xcb78e80e67ba1b8, .hi = 0xd77025a1e0a39d8b, .ex = 0, .sgn=0}, - {.lo = 0x6cb3bfd65b38562b, .hi = 0xd703479a2f6776cc, .ex = 0, .sgn=0}, - {.lo = 0x83f082b570611d7, .hi = 0xd695e4f10ea88570, .ex = 0, .sgn=0}, - {.lo = 0x7afbefc05e9f7d99, .hi = 0xd627fde9f7d63e7e, .ex = 0, .sgn=0}, - {.lo = 0x7190b755535d4f18, .hi = 0xd5b992c8b606a351, .ex = 0, .sgn=0}, - {.lo = 0x7d00ae97abaa4096, .hi = 0xd54aa3d165cc7018, .ex = 0, .sgn=0}, - {.lo = 0xf630e8b6dac83e69, .hi = 0xd4db3148750d1819, .ex = 0, .sgn=0}, - {.lo = 0xdc4663a3168698d2, .hi = 0xd46b3b72a2d68fc9, .ex = 0, .sgn=0}, - {.lo = 0xb77d4f6bd0ee8591, .hi = 0xd3fac294ff34e4d0, .ex = 0, .sgn=0}, - {.lo = 0xa8faac741a6394dc, .hi = 0xd389c6f4eb07a41c, .ex = 0, .sgn=0}, - {.lo = 0xeeeaddb72f00e0dd, .hi = 0xd31848d817d70e16, .ex = 0, .sgn=0}, - {.lo = 0x4300fd1c1ce507e5, .hi = 0xd2a6488487a91918, .ex = 0, .sgn=0}, - {.lo = 0x981ba7e42537275f, .hi = 0xd233c6408cd64236, .ex = 0, .sgn=0}, - {.lo = 0xda7485a5aeffeb4c, .hi = 0xd1c0c252c9de2c86, .ex = 0, .sgn=0}, - {.lo = 0x744fea20e8abef92, .hi = 0xd14d3d02313c0eed, .ex = 0, .sgn=0}, - {.lo = 0x77a18eb13d2ecde5, .hi = 0xd0d93696053af098, .ex = 0, .sgn=0}, - {.lo = 0x6b8a685f6cb61c21, .hi = 0xd064af55d7c9b43e, .ex = 0, .sgn=0}, - {.lo = 0xdaf200dd81212d10, .hi = 0xcfefa7898a4ef23c, .ex = 0, .sgn=0}, - {.lo = 0xdfcb60445c1bf973, .hi = 0xcf7a1f794d7ca1b1, .ex = 0, .sgn=0}, - {.lo = 0x4d27090f10c454e, .hi = 0xcf04176da12390ac, .ex = 0, .sgn=0}, - {.lo = 0xf5babff66def7892, .hi = 0xce8d8faf5406ab8b, .ex = 0, .sgn=0}, - {.lo = 0x93e391861a034684, .hi = 0xce16888783ae13b3, .ex = 0, .sgn=0}, - {.lo = 0x23af31db7179a4aa, .hi = 0xcd9f023f9c3a059e, .ex = 0, .sgn=0}, - {.lo = 0x649474e36b8db9d3, .hi = 0xcd26fd2158358e7d, .ex = 0, .sgn=0}, - {.lo = 0x83e907fbd7aaf0b0, .hi = 0xccae7976c0691177, .ex = 0, .sgn=0}, - {.lo = 0xf839ce18e08bfb50, .hi = 0xcc35778a2bac9ca1, .ex = 0, .sgn=0}, - {.lo = 0x70cbb7f3343451be, .hi = 0xcbbbf7a63eba0dd5, .ex = 0, .sgn=0}, - {.lo = 0x2293661be51140ab, .hi = 0xcb41fa15ebff0777, .ex = 0, .sgn=0}, - {.lo = 0xd9944be1631846d8, .hi = 0xcac77f24736eb553, .ex = 0, .sgn=0}, - {.lo = 0x5328edeb3e6784de, .hi = 0xca4c871d625361a9, .ex = 0, .sgn=0}, - {.lo = 0x8335241be1693225, .hi = 0xc9d1124c931fda7a, .ex = 0, .sgn=0}, - {.lo = 0x83b0e96e1249c2b0, .hi = 0xc95520fe2d40a74b, .ex = 0, .sgn=0}, - {.lo = 0xb562c00b34ee771, .hi = 0xc8d8b37ea4ed0f62, .ex = 0, .sgn=0}, - {.lo = 0x65862939b83382e0, .hi = 0xc85bca1abaf7f0a7, .ex = 0, .sgn=0}, - {.lo = 0x2b31bc86877fd2c, .hi = 0xc7de651f7ca06749, .ex = 0, .sgn=0}, - {.lo = 0xd5c149509e9059f1, .hi = 0xc76084da43624634, .ex = 0, .sgn=0}, - {.lo = 0xcfe6c1b1a6b4e2a4, .hi = 0xc6e22998b4c6608e, .ex = 0, .sgn=0}, - {.lo = 0xe993503baf5afb41, .hi = 0xc66353a8c232a43c, .ex = 0, .sgn=0}, - {.lo = 0x43da25d99267326b, .hi = 0xc5e40358a8ba05a7, .ex = 0, .sgn=0}, - {.lo = 0xab4906075507e74, .hi = 0xc56438f6f0ec3cca, .ex = 0, .sgn=0}, - {.lo = 0xdd40950cf1ed92fa, .hi = 0xc4e3f4d26ea553b6, .ex = 0, .sgn=0}, - {.lo = 0x9dd768f30ca8e85c, .hi = 0xc463373a40dd06a3, .ex = 0, .sgn=0}, - {.lo = 0xa87e78136665cdb2, .hi = 0xc3e2007dd175f5a4, .ex = 0, .sgn=0}, - {.lo = 0x8ac9e1386e4cbabb, .hi = 0xc36050ecd50ca830, .ex = 0, .sgn=0}, - {.lo = 0x74c8f010d986a9e0, .hi = 0xc2de28d74ac6628b, .ex = 0, .sgn=0}, - {.lo = 0xb7041e9bc8c18b0d, .hi = 0xc25b888d7c1fcd38, .ex = 0, .sgn=0}, - {.lo = 0xbdf0715cb8b20bd7, .hi = 0xc1d8705ffcbb6e90, .ex = 0, .sgn=0}, - {.lo = 0x17858573216e0a22, .hi = 0xc154e09faa2ff69a, .ex = 0, .sgn=0}, - {.lo = 0x2bda5328933c854a, .hi = 0xc0d0d99dabd65d44, .ex = 0, .sgn=0}, - {.lo = 0x6dd06968e0ed1957, .hi = 0xc04c5bab7297d322, .ex = 0, .sgn=0}, - {.lo = 0xe4e62d86dd136e78, .hi = 0xbfc7671ab8bb84c6, .ex = 0, .sgn=0}, - {.lo = 0xd46655d6b012455, .hi = 0xbf41fc3d81b430db, .ex = 0, .sgn=0}, - {.lo = 0x2715ef03f8543355, .hi = 0xbebc1b6619ed9116, .ex = 0, .sgn=0}, - {.lo = 0x29d7f7b67d43b177, .hi = 0xbe35c4e716999630, .ex = 0, .sgn=0}, - {.lo = 0xac85320f528d6d5d, .hi = 0xbdaef913557d76f0, .ex = 0, .sgn=0}, - {.lo = 0x2ea36923d5d8e213, .hi = 0xbd27b83dfcbe9279, .ex = 0, .sgn=0}, - {.lo = 0x4a48496734be336d, .hi = 0xbca002ba7aaf25ea, .ex = 0, .sgn=0}, - {.lo = 0x727c405ffc73af56, .hi = 0xbc17d8dc859ad583, .ex = 0, .sgn=0}, - {.lo = 0xfce8d84068e825b6, .hi = 0xbb8f3af81b93095c, .ex = 0, .sgn=0}, - {.lo = 0x5120e35e1c1a250c, .hi = 0xbb062961823b1ddc, .ex = 0, .sgn=0}, - {.lo = 0x33201477347447d8, .hi = 0xba7ca46d46946802, .ex = 0, .sgn=0}, - {.lo = 0x39db32d014440024, .hi = 0xb9f2ac703cca0db3, .ex = 0, .sgn=0}, - {.lo = 0x9de1e3b22b8bf4db, .hi = 0xb96841bf7ffcb21a, .ex = 0, .sgn=0}, - {.lo = 0xa726f4f0828585c9, .hi = 0xb8dd64b0720df647, .ex = 0, .sgn=0}, - {.lo = 0x1c041d1ea5fb3fdb, .hi = 0xb8521598bb6bce26, .ex = 0, .sgn=0}, - {.lo = 0x2e7a35723f3ed035, .hi = 0xb7c654ce4adba9f2, .ex = 0, .sgn=0}, - {.lo = 0x7f86f63bb23f496a, .hi = 0xb73a22a755457448, .ex = 0, .sgn=0}, - {.lo = 0xeb2d28ef943dc88c, .hi = 0xb6ad7f7a557e64f2, .ex = 0, .sgn=0}, - {.lo = 0xea7c015f12b987f7, .hi = 0xb6206b9e0c13a892, .ex = 0, .sgn=0}, - {.lo = 0x737dd2824b608d13, .hi = 0xb592e7697f14dd4a, .ex = 0, .sgn=0}, -}; - -/* The following is a degree-7 polynomial with odd coefficients - approximating sin2pi(x) for -2^-24 < x < 2^-11+2^-24 - with relative error 2^-77.306. - Generated with sin_fast.sollya. */ -static const double PSfast[] = { - 0x1.921fb54442d18p+2, 0x1.1a62645446203p-52, // degree 1 (h+l) - -0x1.4abbce625be53p5, // degree 3 - 0x1.466bc678d8d63p6, // degree 5 - -0x1.331554ca19669p6, // degree 7 -}; - -/* The following is a degree-6 polynomial with even coefficients - approximating cos2pi(x) for -2^-24 < x < 2^-11+2^-24 - with relative error 2^-75.188. - Generated with cos_fast.sollya. */ -static const double PCfast[] = { - 0x1p+0, -0x1.923015cp-77, // degree 0 - -0x1.3bd3cc9be45dep4, // degree 2 - 0x1.03c1f080ad892p6, // degree 4 - -0x1.55a5c590f9e6ap6, // degree 6 -}; - -/* The following is a degree-11 polynomial with odd coefficients - approximating sin2pi(x) for 0 <= x < 2^-11 with relative error 2^-127.75. - Generated with sin_accurate.sollya. */ -static const dint64_t PS[] = { - {.lo = 0xc4c6628b80dc1cd1, .hi = 0xc90fdaa22168c234, .ex = 3, .sgn=0}, // 1 - {.lo = 0x5dc72f712aa57db4, .hi = 0xa55de7312df295f5, .ex = 6, .sgn=1}, // 3 - {.lo = 0x3f33be0021aa54d2, .hi = 0xa335e33bad570e92, .ex = 7, .sgn=0}, // 5 - {.lo = 0xe59d6ab8509a2025, .hi = 0x9969667315ec2d9d, .ex = 7, .sgn=1}, // 7 - {.lo = 0x7d5f8f76fa7d74ed, .hi = 0xa83c1a43bf1c6485, .ex = 6, .sgn=0}, // 9 - {.lo = 0xa7f0339113b8b3c5, .hi = 0xf16ab2898eae62f9, .ex = 4, .sgn=1}, // 11 -}; - -/* The following is a degree-10 polynomial with even coefficients - approximating cos2pi(x) for 0 <= x < 2^-11 with relative error 2^-137.246. - Generated with cos_accurate.sollya. */ -static const dint64_t PC[] = { - {.lo = 0x0, .hi = 0x8000000000000000, .ex = 1, .sgn=0}, // degree 0 - {.lo = 0x56e26cd9808c1949, .hi = 0x9de9e64df22ef2d2, .ex = 5, .sgn=1}, // 2 - {.lo = 0x9980f00630cb655e, .hi = 0x81e0f840dad61d9a, .ex = 7, .sgn=0}, // 4 - {.lo = 0xa508509534006249, .hi = 0xaae9e3f1e5ffcfe2, .ex = 7, .sgn=1}, // 6 - {.lo = 0xe0603ce7044eeba, .hi = 0xf0fa83448dd1e094, .ex = 6, .sgn=0}, // 8 - {.lo = 0xec63157807ebffa, .hi = 0xd368f6f4207cfe49, .ex = 5, .sgn=1}, // 10 -}; - -/* Table generated with ./buildSC 15 using accompanying buildSC.c. - For each i, 0 <= i < 256, xi=i/2^11+SC[i][0], with - SC[i][1] and SC[i][2] approximating sin2pi(xi) and cos2pi(xi) - respectively, both with 53+15 bits of accuracy. */ -static const double SC[256][3] = { - {0x0p+0, 0x0p+0, 0x1p+0}, /* 0 */ - {-0x1.c0f6cp-35, 0x1.921f892b900fep-9, 0x1.ffff621623fap-1}, /* 1 */ - {-0x1.9c7935ep-35, 0x1.921f0ea27ce01p-8, 0x1.fffd8858eca2ep-1}, /* 2 */ - {-0x1.d14d1acp-34, 0x1.2d96af779b0bbp-7, 0x1.fffa72c986392p-1}, /* 3 */ - {-0x1.dba8f6a8p-33, 0x1.921d1ce2d0a1cp-7, 0x1.fff62169dddaap-1}, /* 4 */ - {0x1.a6b7cdfp-32, 0x1.f6a29bdb7377p-7, 0x1.fff0943c02419p-1}, /* 5 */ - {0x1.b49618dp-33, 0x1.2d936d1506f3dp-6, 0x1.ffe9cb44829cp-1}, /* 6 */ - {-0x1.398d6fcp-35, 0x1.5fd4d1e21de6dp-6, 0x1.ffe1c687174b1p-1}, /* 7 */ - {-0x1.e9e9a8c8p-31, 0x1.9215597791e0ap-6, 0x1.ffd886097afcfp-1}, /* 8 */ - {-0x1.34e844cp-32, 0x1.c454f2e9480c7p-6, 0x1.ffce09ce95933p-1}, /* 9 */ - {-0x1.989a8a4p-32, 0x1.f693709b94f92p-6, 0x1.ffc251dfbac0cp-1}, /* 10 */ - {0x1.04a9b99p-30, 0x1.146860e69a571p-5, 0x1.ffb55e40a5c43p-1}, /* 11 */ - {-0x1.56947cp-36, 0x1.2d865748774adp-5, 0x1.ffa72efff95d1p-1}, /* 12 */ - {-0x1.c348768p-35, 0x1.46a396d34121ap-5, 0x1.ff97c420a8451p-1}, /* 13 */ - {0x1.9e80552p-32, 0x1.5fc00e6e4c65cp-5, 0x1.ff871dacd8761p-1}, /* 14 */ - {0x1.3f11d74p-34, 0x1.78dbaa97099ebp-5, 0x1.ff753bb18af95p-1}, /* 15 */ - {0x1.c039af4p-33, 0x1.91f65fc0abc0ap-5, 0x1.ff621e370ca7ap-1}, /* 16 */ - {0x1.53e1f8p-35, 0x1.ab101bf74ac2ep-5, 0x1.ff4dc54b00181p-1}, /* 17 */ - {0x1.114a649p-29, 0x1.c428d7de920e9p-5, 0x1.ff3830f2e9043p-1}, /* 18 */ - {0x1.adf0ef4p-31, 0x1.dd40723a3cdfbp-5, 0x1.ff21614b9d9adp-1}, /* 19 */ - {-0x1.d21f5918p-30, 0x1.f656e1e9e59cdp-5, 0x1.ff09565e83d77p-1}, /* 20 */ - {-0x1.4f54d708p-30, 0x1.07b612d6be078p-4, 0x1.fef0102c634e3p-1}, /* 21 */ - {-0x1.1efec9ap-30, 0x1.1440118ba7bdp-4, 0x1.fed58ecf342dap-1}, /* 22 */ - {0x1.cc17ba88p-29, 0x1.20c96cf0a7eedp-4, 0x1.feb9d24646fa6p-1}, /* 23 */ - {0x1.121dbe4p-33, 0x1.2d5209628edfp-4, 0x1.fe9cdacf99cffp-1}, /* 24 */ - {-0x1.9ecf61p-34, 0x1.39d9f103bf7f7p-4, 0x1.fe7ea854e6b08p-1}, /* 25 */ - {-0x1.04ede8ep-31, 0x1.466116c629e5cp-4, 0x1.fe5f3af4ee201p-1}, /* 26 */ - {-0x1.1821cecp-31, 0x1.52e773c9920c7p-4, 0x1.fe3e92c0e4108p-1}, /* 27 */ - {0x1.cdec726p-31, 0x1.5f6d02131f0b2p-4, 0x1.fe1cafc7f1a24p-1}, /* 28 */ - {-0x1.edece4dp-31, 0x1.6bf1b2653648cp-4, 0x1.fdf99233c230cp-1}, /* 29 */ - {-0x1.2aa4d1cp-31, 0x1.787585bc45f0fp-4, 0x1.fdd53a01d11d9p-1}, /* 30 */ - {0x1.d461592p-32, 0x1.84f871e32cf68p-4, 0x1.fdafa74f16482p-1}, /* 31 */ - {0x1.f0cbd728p-29, 0x1.917a71d3d2956p-4, 0x1.fd88da29f302ep-1}, /* 32 */ - {-0x1.583247p-30, 0x1.9dfb6c9865b06p-4, 0x1.fd60d2e14a6b1p-1}, /* 33 */ - {-0x1.2e81bf4p-30, 0x1.aa7b706bfdbbap-4, 0x1.fd3791484ff5p-1}, /* 34 */ - {-0x1.13941418p-28, 0x1.b6fa680a05c27p-4, 0x1.fd0d15a4b8471p-1}, /* 35 */ - {0x1.71098ffp-30, 0x1.c3785eba12b42p-4, 0x1.fce15fceddccfp-1}, /* 36 */ - {-0x1.c3519e8p-32, 0x1.cff53302f059p-4, 0x1.fcb4703b969e1p-1}, /* 37 */ - {0x1.2f522a5p-27, 0x1.dc70fb84af16ep-4, 0x1.fc8646987fc1dp-1}, /* 38 */ - {-0x1.ae9bed8p-33, 0x1.e8eb7f8a589e2p-4, 0x1.fc56e3b91ca3ap-1}, /* 39 */ - {0x1.f8868b2p-30, 0x1.f564e87d2330fp-4, 0x1.fc264701f9a09p-1}, /* 40 */ - {-0x1.b07985f8p-29, 0x1.00ee8835051f4p-3, 0x1.fbf47105f7439p-1}, /* 41 */ - {0x1.cbdaa94p-30, 0x1.072a05e1d4d8ep-3, 0x1.fbc16172a9e36p-1}, /* 42 */ - {0x1.37c5b908p-28, 0x1.0d64df9619f0dp-3, 0x1.fb8d18b635327p-1}, /* 43 */ - {-0x1.068b5fc8p-28, 0x1.139f09bc617f5p-3, 0x1.fb5797351da85p-1}, /* 44 */ - {-0x1.8ea66818p-29, 0x1.19d8919fa4ec8p-3, 0x1.fb20dc7da8affp-1}, /* 45 */ - {0x1.6278ceb8p-28, 0x1.2011719d50b87p-3, 0x1.fae8e8bd4427fp-1}, /* 46 */ - {-0x1.096df84p-29, 0x1.264993433763ap-3, 0x1.faafbcbfca356p-1}, /* 47 */ - {0x1.9b2534fp-29, 0x1.2c810967bbf7p-3, 0x1.fa7557d8d987ep-1}, /* 48 */ - {0x1.215b4ep-34, 0x1.32b7bfa25c91bp-3, 0x1.fa39bac71954bp-1}, /* 49 */ - {-0x1.94db891p-30, 0x1.38edb9d29b39dp-3, 0x1.f9fce56700a6dp-1}, /* 50 */ - {0x1.7727f7b8p-29, 0x1.3f22f7c3cce3ap-3, 0x1.f9bed7b8c8d8cp-1}, /* 51 */ - {-0x1.0cb33038p-29, 0x1.45576971dd53p-3, 0x1.f97f925d53c83p-1}, /* 52 */ - {-0x1.9071106p-31, 0x1.4b8b175c71e22p-3, 0x1.f93f14feb8022p-1}, /* 53 */ - {0x1.62741e78p-29, 0x1.51bdfa7ea30d5p-3, 0x1.f8fd5fe3efac8p-1}, /* 54 */ - {0x1.f8e16d0cp-28, 0x1.57f00e80e6e12p-3, 0x1.f8ba733a1ceb1p-1}, /* 55 */ - {-0x1.76acbcap-31, 0x1.5e2143b7bc1c2p-3, 0x1.f8764fad5e9bfp-1}, /* 56 */ - {-0x1.0a0f73ap-30, 0x1.6451a76411746p-3, 0x1.f830f4ad232d8p-1}, /* 57 */ - {0x1.ca11d1bcp-28, 0x1.6a8135d7bd143p-3, 0x1.f7ea625eb5af7p-1}, /* 58 */ - {-0x1.02f23628p-29, 0x1.70afd74071191p-3, 0x1.f7a299d3f182ap-1}, /* 59 */ - {0x1.b34dcb8p-29, 0x1.76dda08544b5cp-3, 0x1.f7599a1ac7ecdp-1}, /* 60 */ - {0x1.161ff4p-32, 0x1.7d0a7bf2d4abap-3, 0x1.f70f64322da74p-1}, /* 61 */ - {-0x1.c49b8b4p-31, 0x1.83366ddb3de23p-3, 0x1.f6c3f7e7c2707p-1}, /* 62 */ - {0x1.21da851p-29, 0x1.8961743b1429p-3, 0x1.f6775552a6ba2p-1}, /* 63 */ - {0x1.ac63edap-30, 0x1.8f8b851098588p-3, 0x1.f6297cef0cdd6p-1}, /* 64 */ - {0x1.27ef489cp-27, 0x1.95b4a5b9f2cebp-3, 0x1.f5da6e7820551p-1}, /* 65 */ - {0x1.ae8937p-30, 0x1.9bdcc07900146p-3, 0x1.f58a2b0689c82p-1}, /* 66 */ - {0x1.eb48c7ep-29, 0x1.a203e4a4f950ep-3, 0x1.f538b1d392049p-1}, /* 67 */ - {-0x1.bfd282fp-29, 0x1.a829ffaad0d79p-3, 0x1.f4e603d51f1aap-1}, /* 68 */ - {0x1.7ccf638p-29, 0x1.ae4f1fa80e1b5p-3, 0x1.f492204c5ef9ep-1}, /* 69 */ - {-0x1.2435c578p-28, 0x1.b4732b72ebc86p-3, 0x1.f43d0890e1e72p-1}, /* 70 */ - {0x1.0293fecp-30, 0x1.ba9634155f866p-3, 0x1.f3e6bbb6c2ea4p-1}, /* 71 */ - {-0x1.7bb1f92p-29, 0x1.c0b82461f65ep-3, 0x1.f38f3ae6f9afcp-1}, /* 72 */ - {0x1.27aaebcp-29, 0x1.c6d906faacf65p-3, 0x1.f3368589e17a2p-1}, /* 73 */ - {-0x1.2e2bcd5p-27, 0x1.ccf8c3f74a6c9p-3, 0x1.f2dc9cfb5fa74p-1}, /* 74 */ - {-0x1.6f070acp-30, 0x1.d31773ba218a8p-3, 0x1.f2817fd4d045bp-1}, /* 75 */ - {0x1.469adfcp-29, 0x1.d935004779e57p-3, 0x1.f2252f59c122dp-1}, /* 76 */ - {0x1.4f51c18p-32, 0x1.df5164301377ap-3, 0x1.f1c7abdeaa3efp-1}, /* 77 */ - {0x1.78e44dap-29, 0x1.e56ca4202807cp-3, 0x1.f168f51c5d5d5p-1}, /* 78 */ - {0x1.49bb5f8p-32, 0x1.eb86b4a1b7e9bp-3, 0x1.f1090bc4b68p-1}, /* 79 */ - {-0x1.67ba541p-28, 0x1.f19f9369d5e93p-3, 0x1.f0a7effdc937fp-1}, /* 80 */ - {0x1.c0cab95p-29, 0x1.f7b74ab7219d2p-3, 0x1.f045a1219e594p-1}, /* 81 */ - {-0x1.2b77e32p-30, 0x1.fdcdc0ca3288dp-3, 0x1.efe220cf5c751p-1}, /* 82 */ - {-0x1.e0d8cbp-33, 0x1.01f18054c8362p-2, 0x1.ef7d6e54c347dp-1}, /* 83 */ - {-0x1.ecd5b9cp-29, 0x1.04fb7f6d35d68p-2, 0x1.ef178a6f9a987p-1}, /* 84 */ - {0x1.eb24de5p-29, 0x1.0804e1d369ff2p-2, 0x1.eeb074934fdfp-1}, /* 85 */ - {0x1.4a897c4p-30, 0x1.0b0d9d7b0d042p-2, 0x1.ee482e14bcdep-1}, /* 86 */ - {0x1.336c376p-30, 0x1.0e15b555e7becp-2, 0x1.eddeb6908ca8cp-1}, /* 87 */ - {-0x1.3952d9p-31, 0x1.111d25efd48b8p-2, 0x1.ed740e7eb8dd6p-1}, /* 88 */ - {0x1.fc2a5d4p-31, 0x1.1423ef5c7e1bdp-2, 0x1.ed0835dc24e89p-1}, /* 89 */ - {0x1.a88ed37p-29, 0x1.172a0eb8361dap-2, 0x1.ec9b2d0ec8288p-1}, /* 90 */ - {-0x1.8ca4cb94p-27, 0x1.1a2f7b10b6d7p-2, 0x1.ec2cf55d6117cp-1}, /* 91 */ - {0x1.0144524p-27, 0x1.1d3446fd0cd3fp-2, 0x1.ebbd8c1d62f96p-1}, /* 92 */ - {-0x1.abf810cp-28, 0x1.203855b85f89ap-2, 0x1.eb4cf57454132p-1}, /* 93 */ - {0x1.5d4c5d58p-28, 0x1.233bbcca40561p-2, 0x1.eadb2e40746cap-1}, /* 94 */ - {-0x1.a1b0c58p-29, 0x1.263e685b1d714p-2, 0x1.ea68396d87754p-1}, /* 95 */ - {-0x1.77c8dacp-29, 0x1.294061d2eb611p-2, 0x1.e9f41597393c8p-1}, /* 96 */ - {0x1.915540ep-30, 0x1.2c41a580014cfp-2, 0x1.e97ec348fb87fp-1}, /* 97 */ - {-0x1.abb6d9bp-28, 0x1.2f422b2d0990cp-2, 0x1.e90843c55b996p-1}, /* 98 */ - {-0x1.b8ee5d58p-28, 0x1.3241f8cea2836p-2, 0x1.e890962268c49p-1}, /* 99 */ - {-0x1.1cd29828p-28, 0x1.35410a8396266p-2, 0x1.e817baf85c094p-1}, /* 100 */ - {-0x1.e216afp-32, 0x1.383f5e08283e2p-2, 0x1.e79db2a188b0ap-1}, /* 101 */ - {-0x1.24afc3p-31, 0x1.3b3cef6993c0bp-2, 0x1.e7227dbf82004p-1}, /* 102 */ - {-0x1.aa1657cp-31, 0x1.3e39be4767224p-2, 0x1.e6a61c62d5274p-1}, /* 103 */ - {-0x1.c5b65fap-30, 0x1.4135c898485bbp-2, 0x1.e6288ee07fea5p-1}, /* 104 */ - {0x1.23e8978p-32, 0x1.44310de3c284bp-2, 0x1.e5a9d54bbd26cp-1}, /* 105 */ - {-0x1.2b1d77ap-29, 0x1.472b8976d498dp-2, 0x1.e529f06cb187dp-1}, /* 106 */ - {-0x1.daaa348p-31, 0x1.4a253cb97efd1p-2, 0x1.e4a8e007231a2p-1}, /* 107 */ - {-0x1.322f5708p-28, 0x1.4d1e2260c3422p-2, 0x1.e426a500f6e33p-1}, /* 108 */ - {0x1.64758e8p-29, 0x1.50163eca0b337p-2, 0x1.e3a33e996b722p-1}, /* 109 */ - {0x1.12486278p-28, 0x1.530d89a17e007p-2, 0x1.e31eae3fb917bp-1}, /* 110 */ - {-0x1.6c3416ccp-27, 0x1.5603fcf8cd8a3p-2, 0x1.e298f502a579bp-1}, /* 111 */ - {0x1.ab481ffp-29, 0x1.58f9a896aa209p-2, 0x1.e2121016e14fcp-1}, /* 112 */ - {-0x1.6eb838bp-29, 0x1.5bee77aaf890bp-2, 0x1.e18a032eb4df5p-1}, /* 113 */ - {-0x1.d159b8p-32, 0x1.5ee2734efeef5p-2, 0x1.e100ccaa6bd78p-1}, /* 114 */ - {-0x1.a42e4ap-34, 0x1.61d595bedeabcp-2, 0x1.e0766d944915ep-1}, /* 115 */ - {-0x1.43d0dcp-30, 0x1.64c7dd5cc0cd1p-2, 0x1.dfeae63903034p-1}, /* 116 */ - {-0x1.8c7bdb7p-27, 0x1.67b9453ca2122p-2, 0x1.df5e378482eaep-1}, /* 117 */ - {0x1.1c0ead6p-30, 0x1.6aa9d844c980ap-2, 0x1.ded05f6a23a52p-1}, /* 118 */ - {0x1.7d526p-31, 0x1.6d99867e90d92p-2, 0x1.de4160e97b2e2p-1}, /* 119 */ - {0x1.924e0368p-28, 0x1.7088555d3c816p-2, 0x1.ddb13afb14e37p-1}, /* 120 */ - {-0x1.74b7c3ep-30, 0x1.73763c09fba09p-2, 0x1.dd1fef5335416p-1}, /* 121 */ - {-0x1.7943adp-30, 0x1.766340685c982p-2, 0x1.dc8d7ccf2567ap-1}, /* 122 */ - {0x1.79dd614p-29, 0x1.794f5f7522b88p-2, 0x1.dbf9e402aa5c3p-1}, /* 123 */ - {0x1.7b64f32p-30, 0x1.7c3a939c32d81p-2, 0x1.db652607e0db1p-1}, /* 124 */ - {-0x1.2bea5ce8p-28, 0x1.7f24db825141cp-2, 0x1.dacf43268b5bp-1}, /* 125 */ - {0x1.733c024p-30, 0x1.820e3b8bf15ap-2, 0x1.da383a7aed887p-1}, /* 126 */ - {-0x1.eac0fc94p-27, 0x1.84f6a51d077b3p-2, 0x1.d9a00efd84537p-1}, /* 127 */ - {0x1.aca37338p-27, 0x1.87de2f4704f98p-2, 0x1.d906bbf17f4dap-1}, /* 128 */ - {-0x1.910c4fp-30, 0x1.8ac4b7dc0d986p-2, 0x1.d86c4862b5d6ep-1}, /* 129 */ - {-0x1.33bb86p-31, 0x1.8daa52b4dc041p-2, 0x1.d7d0b0374a559p-1}, /* 130 */ - {-0x1.69e1507p-27, 0x1.908ef408ad22p-2, 0x1.d733f5e71c3bcp-1}, /* 131 */ - {0x1.cffacf08p-27, 0x1.9372ab7784d36p-2, 0x1.d696161d786c9p-1}, /* 132 */ - {-0x1.8629d9fp-26, 0x1.965552b0849abp-2, 0x1.d5f7190eeae23p-1}, /* 133 */ - {0x1.415p-30, 0x1.99371687c64f3p-2, 0x1.d556f5155d9ddp-1}, /* 134 */ - {-0x1.bd37aad8p-27, 0x1.9c17cf40715cbp-2, 0x1.d4b5b2caf8386p-1}, /* 135 */ - {0x1.d02cde7p-26, 0x1.9ef79ea4d995dp-2, 0x1.d4134ac5eb246p-1}, /* 136 */ - {-0x1.10547acp-30, 0x1.a1d653d9adf5ep-2, 0x1.d36fc7d291602p-1}, /* 137 */ - {-0x1.01a1a228p-27, 0x1.a4b40f9c0120bp-2, 0x1.d2cb22b45236bp-1}, /* 138 */ - {0x1.3ce2bacp-29, 0x1.a790ce2056b9ap-2, 0x1.d2255c3ae11a5p-1}, /* 139 */ - {-0x1.ccb4a6p-32, 0x1.aa6c828db4ea8p-2, 0x1.d17e774d4e3e2p-1}, /* 140 */ - {0x1.5db4bp-29, 0x1.ad47321f29847p-2, 0x1.d0d672bc0b122p-1}, /* 141 */ - {0x1.32f6a6ep-29, 0x1.b020d7a285e23p-2, 0x1.d02d4fb84d334p-1}, /* 142 */ - {0x1.cf8e39bcp-26, 0x1.b2f97c27f7494p-2, 0x1.cf830c2248c5ep-1}, /* 143 */ - {0x1.8927bbp-30, 0x1.b5d10129a750ap-2, 0x1.ced7af22cb105p-1}, /* 144 */ - {-0x1.3dec3c1p-28, 0x1.b8a77f8d0bbc5p-2, 0x1.ce2b32e50d6cdp-1}, /* 145 */ - {-0x1.26ba536p-28, 0x1.bb7cf08f0290dp-2, 0x1.cd7d98fcf3b1ep-1}, /* 146 */ - {0x1.23c568ep-29, 0x1.be51524e3aa53p-2, 0x1.cccee1da3d56ep-1}, /* 147 */ - {-0x1.f3b3afp-29, 0x1.c1249c1f5f2f6p-2, 0x1.cc1f0f95e1e24p-1}, /* 148 */ - {-0x1.1286a47p-28, 0x1.c3f6d2ef7054bp-2, 0x1.cb6e20ff37e81p-1}, /* 149 */ - {0x1.641214ep-29, 0x1.c6c7f594003d9p-2, 0x1.cabc165bf1b6p-1}, /* 150 */ - {0x1.0cda7c9p-27, 0x1.c997ff2bffccbp-2, 0x1.ca08f0dee434cp-1}, /* 151 */ - {-0x1.5557ac9p-28, 0x1.cc66e7b42e8f1p-2, 0x1.c954b28bca62ep-1}, /* 152 */ - {0x1.555eb62p-28, 0x1.cf34bccc567a1p-2, 0x1.c89f57f6e20f3p-1}, /* 153 */ - {-0x1.4e0e361p-28, 0x1.d2016cbb5e39ap-2, 0x1.c7e8e59999e1fp-1}, /* 154 */ - {0x1.446da1ep-29, 0x1.d4cd039d0ed05p-2, 0x1.c731585f970ebp-1}, /* 155 */ - {0x1.103d328p-29, 0x1.d797767638decp-2, 0x1.c678b3174afe1p-1}, /* 156 */ - {0x1.5814d6p-28, 0x1.da60c7ae9dc22p-2, 0x1.c5bef522be6fbp-1}, /* 157 */ - {-0x1.5e2321ep-29, 0x1.dd28f054cbb3fp-2, 0x1.c5042052c8c42p-1}, /* 158 */ - {-0x1.a259ffep-29, 0x1.dfeff54854631p-2, 0x1.c44833611bc7dp-1}, /* 159 */ - {-0x1.4f28d8p-31, 0x1.e2b5d34665b35p-2, 0x1.c38b2f278ea7ep-1}, /* 160 */ - {-0x1.de571p-36, 0x1.e57a86d137f2p-2, 0x1.c2cd1493d05c2p-1}, /* 161 */ - {0x1.e0d8d14p-29, 0x1.e83e0ffb7bfb4p-2, 0x1.c20de3a08ea07p-1}, /* 162 */ - {-0x1.12a858ep-28, 0x1.eb0067e48baf4p-2, 0x1.c14d9e2bd511ep-1}, /* 163 */ - {0x1.9a17403p-27, 0x1.edc19997a4431p-2, 0x1.c08c413089b2ep-1}, /* 164 */ - {0x1.68c8636p-29, 0x1.f0819163d1bcp-2, 0x1.bfc9d21568f32p-1}, /* 165 */ - {0x1.4cc5eb8p-29, 0x1.f3405a482e11dp-2, 0x1.bf064dd580fc9p-1}, /* 166 */ - {-0x1.fce7cd8p-27, 0x1.f5fde8f3f11d4p-2, 0x1.be41b798f6b97p-1}, /* 167 */ - {-0x1.af8169p-29, 0x1.f8ba4c98a9816p-2, 0x1.bd7c0b1a7f14bp-1}, /* 168 */ - {0x1.6e39e2p-33, 0x1.fb7575d1ea75p-2, 0x1.bcb54cac5dde5p-1}, /* 169 */ - {0x1.30f9256p-28, 0x1.fe2f665dcd168p-2, 0x1.bbed7bd1e17bp-1}, /* 170 */ - {0x1.626de2p-31, 0x1.00740ca0d5fbbp-1, 0x1.bb2499f9fe7a3p-1}, /* 171 */ - {0x1.5cc703p-30, 0x1.01cfc8afeea0ep-1, 0x1.ba5aa650dd495p-1}, /* 172 */ - {-0x1.6191e6p-32, 0x1.032ae54fe4057p-1, 0x1.b98fa2065a5e6p-1}, /* 173 */ - {-0x1.6b1485p-31, 0x1.0485624c328c8p-1, 0x1.b8c38d39737bcp-1}, /* 174 */ - {-0x1.11fbc3ap-29, 0x1.05df3e66a716dp-1, 0x1.b7f668a580fdp-1}, /* 175 */ - {-0x1.0eca7fp-27, 0x1.07387825589ecp-1, 0x1.b728352c44517p-1}, /* 176 */ - {-0x1.8073bc9ep-25, 0x1.089109ef1284dp-1, 0x1.b658f630112edp-1}, /* 177 */ - {-0x1.9dcf0adp-27, 0x1.09e9051603e29p-1, 0x1.b588a13ab750fp-1}, /* 178 */ - {-0x1.06ea9fp-29, 0x1.0b405820e78e7p-1, 0x1.b4b740d3cc07bp-1}, /* 179 */ - {-0x1.36a8d0cp-30, 0x1.0c9704a1ea4e5p-1, 0x1.b3e4d40f5524dp-1}, /* 180 */ - {0x1.63d1f3p-30, 0x1.0ded0bc01a533p-1, 0x1.b3115a3a628afp-1}, /* 181 */ - {0x1.f3181f14p-26, 0x1.0f4270e4787bfp-1, 0x1.b23cd1314c779p-1}, /* 182 */ - {-0x1.f269b78p-29, 0x1.109723e75c5cfp-1, 0x1.b167430cfebdbp-1}, /* 183 */ - {0x1.1d84dc08p-27, 0x1.11eb36bc9db52p-1, 0x1.b090a4915ee88p-1}, /* 184 */ - {-0x1.08e60068p-27, 0x1.133e9ba0061d8p-1, 0x1.afb8fe69a6527p-1}, /* 185 */ - {0x1.cda72abp-27, 0x1.14915d557a7c9p-1, 0x1.aee049bc0aeep-1}, /* 186 */ - {-0x1.f32f95p-30, 0x1.15e36dfb6bb55p-1, 0x1.ae068f6991699p-1}, /* 187 */ - {0x1.138092dp-28, 0x1.1734d6f34d7fp-1, 0x1.ad2bc96c1e1f5p-1}, /* 188 */ - {0x1.6b382dd4p-26, 0x1.188595ae376a5p-1, 0x1.ac4ff962bdb6dp-1}, /* 189 */ - {-0x1.f12fafap-28, 0x1.19d59f592a587p-1, 0x1.ab7326685eb57p-1}, /* 190 */ - {-0x1.2909e5ap-28, 0x1.1b2500aed7ac6p-1, 0x1.aa954823cf815p-1}, /* 191 */ - {-0x1.d66a8978p-25, 0x1.1c73aa0150cf9p-1, 0x1.a9b668fb0503fp-1}, /* 192 */ - {0x1.311ea86p-27, 0x1.1dc1b7db74db1p-1, 0x1.a8d675d9c6cc8p-1}, /* 193 */ - {-0x1.41c02b8p-31, 0x1.1f0f08a1a06a4p-1, 0x1.a7f5853bb4309p-1}, /* 194 */ - {-0x1.ca1f4edp-26, 0x1.205ba57211271p-1, 0x1.a71391146958fp-1}, /* 195 */ - {-0x1.910ce77p-28, 0x1.21a7988f8326bp-1, 0x1.a63092626202fp-1}, /* 196 */ - {0x1.2bfadbeep-25, 0x1.22f2dc71afab6p-1, 0x1.a54c8cd9fd0d9p-1}, /* 197 */ - {-0x1.5f1c02a8p-27, 0x1.243d5df4afb93p-1, 0x1.a4678dbbe5e73p-1}, /* 198 */ - {-0x1.db12b9p-30, 0x1.2587347f493a4p-1, 0x1.a38184db0df23p-1}, /* 199 */ - {-0x1.7b29ep-30, 0x1.26d05490f2f61p-1, 0x1.a29a7a2f40b49p-1}, /* 200 */ - {-0x1.b3ddca4p-29, 0x1.2818be6930629p-1, 0x1.a1b26d8f070d7p-1}, /* 201 */ - {0x1.e112744p-29, 0x1.2960730ff2bcdp-1, 0x1.a0c95e3df5e0ep-1}, /* 202 */ - {-0x1.5269766p-28, 0x1.2aa76dafcbbf4p-1, 0x1.9fdf4fae1df6fp-1}, /* 203 */ - {-0x1.09777e1p-28, 0x1.2bedb1b6b4e15p-1, 0x1.9ef43f6cbe162p-1}, /* 204 */ - {0x1.ae2051fp-28, 0x1.2d333e4617f25p-1, 0x1.9e082e148680ep-1}, /* 205 */ - {-0x1.36f6ced8p-27, 0x1.2e780cb47180ep-1, 0x1.9d1b207f383c3p-1}, /* 206 */ - {-0x1.23fdc6bp-28, 0x1.2fbc23fba2f44p-1, 0x1.9c2d1197130a7p-1}, /* 207 */ - {0x1.bc540ep-33, 0x1.30ff7fd6d967dp-1, 0x1.9b3e0478b961bp-1}, /* 208 */ - {-0x1.cfb4ed7p-28, 0x1.32421da0bf0e9p-1, 0x1.9a4dfb1c89326p-1}, /* 209 */ - {0x1.55802aecp-26, 0x1.3384042a92b1dp-1, 0x1.995cf06920d11p-1}, /* 210 */ - {0x1.60719e4p-28, 0x1.34c52608e3a92p-1, 0x1.986aee6d6837ep-1}, /* 211 */ - {-0x1.cbf2e48p-30, 0x1.36058ac8863b6p-1, 0x1.9777ef832c986p-1}, /* 212 */ - {0x1.9061c32p-27, 0x1.374533ab707dp-1, 0x1.9683f2ad7e2ecp-1}, /* 213 */ - {-0x1.da84dfep-27, 0x1.3884160f9488fp-1, 0x1.958f000fdd50ap-1}, /* 214 */ - {0x1.92e8a74p-29, 0x1.39c23eba6b22ap-1, 0x1.94990dd9cee51p-1}, /* 215 */ - {-0x1.bff5d9ap-29, 0x1.3affa20756bddp-1, 0x1.93a225056084ap-1}, /* 216 */ - {0x1.4c462p-36, 0x1.3c3c4498e98ebp-1, 0x1.92aa41fbb951cp-1}, /* 217 */ - {-0x1.e4613e9p-28, 0x1.3d782261dff62p-1, 0x1.91b167e92d706p-1}, /* 218 */ - {0x1.0eb2964p-30, 0x1.3eb33ed579bbep-1, 0x1.90b794146043cp-1}, /* 219 */ - {-0x1.60abec2p-29, 0x1.3fed94c834d8ap-1, 0x1.8fbcca9583479p-1}, /* 220 */ - {0x1.6954977p-27, 0x1.4127281ddac03p-1, 0x1.8ec1085083553p-1}, /* 221 */ - {0x1.a16fec2p-29, 0x1.425ff1f841235p-1, 0x1.8dc452ca328d3p-1}, /* 222 */ - {-0x1.27bcdd3p-27, 0x1.4397f44aa44f2p-1, 0x1.8cc6a8771e165p-1}, /* 223 */ - {-0x1.60dded4p-28, 0x1.44cf317a563dbp-1, 0x1.8bc8076122736p-1}, /* 224 */ - {-0x1.9a8f405cp-26, 0x1.4605a2b02d705p-1, 0x1.8ac875232f3efp-1}, /* 225 */ - {0x1.32777dcp-27, 0x1.473b532bc5a67p-1, 0x1.89c7e8713120cp-1}, /* 226 */ - {-0x1.1418a7bp-26, 0x1.4870306ca20e2p-1, 0x1.88c670a0ea774p-1}, /* 227 */ - {-0x1.fed182ep-28, 0x1.49a44886b534p-1, 0x1.87c401fdf05e5p-1}, /* 228 */ - {0x1.86144d8p-27, 0x1.4ad796ea1410cp-1, 0x1.86c0a04dbacc5p-1}, /* 229 */ - {0x1.1bc2e6p-33, 0x1.4c0a14640d2afp-1, 0x1.85bc51aa114c2p-1}, /* 230 */ - {-0x1.f53d2fep-28, 0x1.4d3bc5aaa8cd5p-1, 0x1.84b7121b30a13p-1}, /* 231 */ - {-0x1.2e100ap-30, 0x1.4e6cab91556bep-1, 0x1.83b0e0e6b6cccp-1}, /* 232 */ - {-0x1.fa58c62p-29, 0x1.4f9cc1c69fddep-1, 0x1.82a9c1c1ab463p-1}, /* 233 */ - {0x1.bb491ep-33, 0x1.50cc09fdcbd92p-1, 0x1.81a1b3342f858p-1}, /* 234 */ - {0x1.a11541p-28, 0x1.51fa82c3aa029p-1, 0x1.8098b67ea8509p-1}, /* 235 */ - {0x1.ab0a5d3p-27, 0x1.53282b20b96b6p-1, 0x1.7f8ecc791953p-1}, /* 236 */ - {-0x1.cba0438p-28, 0x1.5454fe43a7d7cp-1, 0x1.7e83f96af78ap-1}, /* 237 */ - {-0x1.0dd83a4p-29, 0x1.5581033a81573p-1, 0x1.7d783712e20ecp-1}, /* 238 */ - {-0x1.e9a8299p-28, 0x1.56ac33fbb8253p-1, 0x1.7c6b8acf90fa6p-1}, /* 239 */ - {0x1.225c4aap-29, 0x1.57d6939d4b513p-1, 0x1.7b5df1da18065p-1}, /* 240 */ - {-0x1.82e66ep-27, 0x1.59001b9e64d79p-1, 0x1.7a4f72157cfdfp-1}, /* 241 */ - {0x1.51a6a354p-26, 0x1.5a28d5b36d597p-1, 0x1.794002a7c9023p-1}, /* 242 */ - {0x1.13917f4p-26, 0x1.5b50b4e10bec1p-1, 0x1.782faf6dc7ba2p-1}, /* 243 */ - {0x1.49310ccp-30, 0x1.5c77bc15ab4efp-1, 0x1.771e75c43942ep-1}, /* 244 */ - {0x1.24d493cp-30, 0x1.5d9dee9de49dbp-1, 0x1.760c529bc17bp-1}, /* 245 */ - {-0x1.04638f7p-26, 0x1.5ec347044e0f4p-1, 0x1.74f94b0af972p-1}, /* 246 */ - {-0x1.3f41b28p-29, 0x1.5fe7cb834600cp-1, 0x1.73e55936a516p-1}, /* 247 */ - {-0x1.a5f6f5cp-30, 0x1.610b7515d1562p-1, 0x1.72d083b8214ebp-1}, /* 248 */ - {0x1.19fb2ep-28, 0x1.622e459eafbc1p-1, 0x1.71bac8c7b0592p-1}, /* 249 */ - {-0x1.56d2c2bp-28, 0x1.6350396fe4e62p-1, 0x1.70a42bec51665p-1}, /* 250 */ - {-0x1.3c156c2p-28, 0x1.64715385bed93p-1, 0x1.6f8caa4969708p-1}, /* 251 */ - {-0x1.f23e576p-29, 0x1.659191d2fd57fp-1, 0x1.6e7445d74f711p-1}, /* 252 */ - {0x1.1e4be38p-30, 0x1.66b0f41d484c4p-1, 0x1.6d5afecd4938dp-1}, /* 253 */ - {-0x1.397cc8d8p-27, 0x1.67cf76eac73dfp-1, 0x1.6c40d89625f63p-1}, /* 254 */ - {-0x1.202f686p-28, 0x1.68ed1e0990551p-1, 0x1.6b25cf728c35p-1}, /* 255 */ -}; - -// Multiply exactly a and b, such that *hi + *lo = a * b. -static inline void a_mul(double *hi, double *lo, double a, double b) { - *hi = a * b; - *lo = __builtin_fma (a, b, -*hi); -} - -/* Multiply a double with a double double : a * (bh + bl) - with error bounded by ulp(lo) */ -static inline void s_mul (double *hi, double *lo, double a, double bh, - double bl) { - a_mul (hi, lo, a, bh); /* exact */ - *lo = __builtin_fma (a, bl, *lo); - /* the error is bounded by ulp(lo), where |lo| < |a*bl| + ulp(hi) */ -} - -// Returns (ah + al) * (bh + bl) - (al * bl) -// We can ignore al * bl when assuming al <= ulp(ah) and bl <= ulp(bh) -static inline void d_mul(double *hi, double *lo, double ah, double al, - double bh, double bl) { - double s, t; - - a_mul(hi, &s, ah, bh); - t = __builtin_fma(al, bh, s); - *lo = __builtin_fma(ah, bl, t); -} - -static inline void -fast_two_sum(double *hi, double *lo, double a, double b) -{ - double e; - - *hi = a + b; - e = *hi - a; /* exact */ - *lo = b - e; /* exact */ -} - -/* Put in h+l an approximation of sin2pi(xh+xl), - for 2^-24 <= xh+xl < 2^-11 + 2^-24, - and |xl| < 2^-52.36, with absolute error < 2^-77.09 - (see evalPSfast() in sin.sage). - Assume uh + ul approximates (xh+xl)^2. */ -static void -evalPSfast (double *h, double *l, double xh, double xl, double uh, double ul) -{ - double t; - *h = PSfast[4]; // degree 7 - *h = __builtin_fma (*h, uh, PSfast[3]); // degree 5 - *h = __builtin_fma (*h, uh, PSfast[2]); // degree 3 - s_mul (h, l, *h, uh, ul); - fast_two_sum (h, &t, PSfast[0], *h); - *l += PSfast[1] + t; - // multiply by xh+xl - d_mul (h, l, *h, *l, xh, xl); -} - -/* Put in h+l an approximation of cos2pi(xh+xl), - for 2^-24 <= xh+xl < 2^-11 + 2^-24, - and |xl| < 2^-52.36, with relative error < 2^-69.96 - (see evalPCfast() in sin.sage). - Assume uh + ul approximates (xh+xl)^2. */ -static void -evalPCfast (double *h, double *l, double uh, double ul) -{ - double t; - *h = PCfast[4]; // degree 6 - *h = __builtin_fma (*h, uh, PCfast[3]); // degree 4 - *h = __builtin_fma (*h, uh, PCfast[2]); // degree 2 - s_mul (h, l, *h, uh, ul); - fast_two_sum (h, &t, PCfast[0], *h); - *l += PCfast[1] + t; -} - -/* Put in Y an approximation of sin2pi(X), for 0 <= X < 2^-11, - where X2 approximates X^2. - Absolute error bounded by 2^-132.999 with 0 <= Y < 0.003068 - (see evalPS() in sin.sage), and relative error bounded by - 2^-124.648 (see evalPSrel(K=8) in sin.sage). */ -static void -evalPS (dint64_t *Y, dint64_t *X, dint64_t *X2) -{ - mul_dint_21 (Y, X2, PS+5); // degree 11 - add_dint (Y, Y, PS+4); // degree 9 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PS+3); // degree 7 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PS+2); // degree 5 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PS+1); // degree 3 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PS+0); // degree 1 - mul_dint (Y, Y, X); // multiply by X -} - -/* Put in Y an approximation of cos2pi(X), for 0 <= X < 2^-11, - where X2 approximates X^2. - Absolute/relative error bounded by 2^-125.999 with 0.999995 < Y <= 1 - (see evalPC() in sin.sage). */ -static void -evalPC (dint64_t *Y, dint64_t *X2) -{ - mul_dint_21 (Y, X2, PC+5); // degree 10 - add_dint (Y, Y, PC+4); // degree 8 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PC+3); // degree 6 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PC+2); // degree 4 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PC+1); // degree 2 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PC+0); // degree 0 -} - -// normalize X such that X->hi has its most significant bit set (if X <> 0) -static void -normalize (dint64_t *X) -{ - int cnt; - if (X->hi != 0) - { - cnt = __builtin_clzl (X->hi); - if (cnt) - { - X->hi = (X->hi << cnt) | (X->lo >> (64 - cnt)); - X->lo = X->lo << cnt; - } - X->ex -= cnt; - } - else if (X->lo != 0) - { - cnt = __builtin_clzl (X->lo); - X->hi = X->lo << cnt; - X->lo = 0; - X->ex -= 64 + cnt; - } -} - -/* Approximate X/(2pi) mod 1. If Xin is the input value, and Xout the - output value, we have: - |Xout - (Xin/(2pi) mod 1)| < 2^-126.67*|Xout| - Assert X is normalized at input, and normalize X at output. -*/ -static void -reduce (dint64_t *X) -{ - int e = X->ex; - u128 u; - - if (e <= 1) // |X| < 2 - { - /* multiply by T[0]/2^64 + T[1]/2^128, where - |T[0]/2^64 + T[1]/2^128 - 1/(2pi)| < 2^-130.22 */ - u = (u128) X->hi * (u128) T[1]; - uint64_t tiny = absl::Uint128Low64(u); - X->lo = absl::Uint128High64(u); - u = (u128) X->hi * (u128) T[0]; - X->lo += absl::Uint128Low64(u); - X->hi = absl::Uint128High64(u) + (X->lo < (uint64_t) u); - /* hi + lo/2^64 + tiny/2^128 = hi_in * (T[0]/2^64 + T[1]/2^128) thus - |hi + lo/2^64 + tiny/2^128 - hi_in/(2*pi)| < hi_in * 2^-130.22 - Since X is normalized at input, hi_in >= 2^63, and since T[0] >= 2^61, - we have hi >= 2^(63+61-64) = 2^60, thus the normalize() below - perform a left shift by at most 3 bits */ - int e = X->ex; - normalize (X); - e = e - X->ex; - // put the upper e bits of tiny into X->lo - if (e) - X->lo |= tiny >> (64 - e); - /* The error is bounded by 2^-130.22 (relative) + ulp(lo) (absolute). - Since now X->hi >= 2^63, the absolute error of ulp(lo) converts into - a relative error of less than 2^-127. - This yields a maximal relative error of: - (1 + 2^-130.22) * (1 + 2^-127) - 1 < 2^-126.852. - */ - return; - } - - // now 2 <= e <= 1024 - - /* The upper 64-bit word X->hi corresponds to hi/2^64*2^e, if multiplied by - T[i]/2^((i+1)*64) it yields hi*T[i]/2^128 * 2^(e-i*64). - If e-64i <= -128, it contributes to less than 2^-128; - if e-64i >= 128, it yields an integer, which is 0 modulo 1. - We thus only consider the values of i such that -127 <= e-64i <= 127, - i.e., (-127+e)/64 <= i <= (127+e)/64. - Up to 4 consecutive values of T[i] can contribute (only 3 when e is a - multiple of 64). */ - int i = (e < 127) ? 0 : (e - 127 + 64 - 1) / 64; // ceil((e-127)/64) - // 0 <= i <= 15 - uint64_t c[5]; - u = (u128) X->hi * (u128) T[i+3]; // i+3 <= 18 - c[0] = absl::Uint128Low64(u); - c[1] = absl::Uint128High64(u); - u = (u128) X->hi * (u128) T[i+2]; - c[1] += absl::Uint128Low64(u); - c[2] = absl::Uint128High64(u) + (c[1] < (uint64_t) u); - u = (u128) X->hi * (u128) T[i+1]; - c[2] += absl::Uint128Low64(u); - c[3] = absl::Uint128High64(u) + (c[2] < (uint64_t) u); - u = (u128) X->hi * (u128) T[i]; - c[3] += absl::Uint128Low64(u); - c[4] = absl::Uint128High64(u) + (c[3] < (uint64_t) u); - - /* up to here, the ignored part hi*(T[i+4]+T[i+5]+...) can contribute by - less than 2^64 in c[0], thus less than 1 in c[1] */ - - int f = e - 64 * i; // hi*T[i]/2^128 is multiplied by 2^f - /* {c, 5} = hi*(T[i]+T[i+1]/2^64+T[i+2]/2^128+T[i+3]/2^192) */ - /* now shift c[0..4] by f bits to the left */ - uint64_t tiny; - if (f < 64) - { - X->hi = (c[4] << f) | (c[3] >> (64 - f)); - X->lo = (c[3] << f) | (c[2] >> (64 - f)); - tiny = (c[2] << f) | (c[1] >> (64 - f)); - /* the ignored part was less than 1 in c[1], - thus less than 2^(f-64) <= 1/2 in tiny */ - } - else if (f == 64) - { - X->hi = c[3]; - X->lo = c[2]; - tiny = c[1]; - /* the ignored part was less than 1 in c[1], - thus less than 1 in tiny */ - } - else /* 65 <= f <= 127: this case can only occur when e >= 65 */ - { - int g = f - 64; /* 1 <= g <= 63 */ - /* we compute an extra term */ - u = (u128) X->hi * (u128) T[i+4]; // i+4 <= 19 - u = u >> 64; - c[0] += absl::Uint128Low64(u); - c[1] += (c[0] < u); - c[2] += (c[0] < u) && c[1] == 0; - c[3] += (c[0] < u) && c[1] == 0 && c[2] == 0; - c[4] += (c[0] < u) && c[1] == 0 && c[2] == 0 && c[3] == 0; - X->hi = (c[3] << g) | (c[2] >> (64 - g)); - X->lo = (c[2] << g) | (c[1] >> (64 - g)); - tiny = (c[1] << g) | (c[0] >> (64 - g)); - /* the ignored part was less than 1 in c[0], - thus less than 1/2 in tiny */ - } - /* The approximation error between X/in(2pi) mod 1 and - X->hi/2^64 + X->lo/2^128 + tiny/2^192 is: - (a) the ignored part in tiny, which is less than ulp(tiny), - thus less than 1/2^192; - (b) the ignored terms hi*T[i+4] + ... or hi*T[i+5] + ..., - which accumulate to less than ulp(tiny) too, thus - less than 1/2^192. - Thus the approximation error is less than 2^-191 (absolute). - */ - X->ex = 0; - normalize (X); - /* the worst case (for 2^25 <= x < 2^1024) is X->ex = -61, attained - for |x| = 0x1.6ac5b262ca1ffp+851 */ - if (X->ex < 0) // put the upper -ex bits of tiny into low bits of lo - X->lo |= tiny >> (64 + X->ex); - /* Since X->ex >= -61, it means X >= 2^-62 before the normalization, - thus the maximal absolute error of 2^-191 yields a relative error - bounded by 2^-191/2^-62 = 2^-129. - There is an additional truncation error (for tiny) of at most 1 ulp - of X->lo, thus at most 2^-127. - The relative error is thus bounded by 2^-126.67. */ -} - -/* Given Xin:=X with 0 <= Xin < 1, return i and modify X such that - Xin = i/2^11 + Xout, with 0 <= Xout < 2^-11. - This operation is exact. */ -static int -reduce2 (dint64_t *X) -{ - if (X->ex <= -11) - return 0; - int sh = 64 - 11 - X->ex; - int i = X->hi >> sh; - X->hi = X->hi & ((1ull << sh) - 1); - normalize (X); - return i; -} - -/* h+l <- c1/2^64 + c0/2^128 */ -static void -set_dd (double *h, double *l, uint64_t c1, uint64_t c0) -{ - uint64_t e, f, g; - b64u64_u t; - if (c1) - { - e = __builtin_clzl (c1); - if (e) - { - c1 = (c1 << e) | (c0 >> (64 - e)); - c0 = c0 << e; - } - f = 0x3fe - e; - t.u = (f << 52) | ((c1 << 1) >> 12); - *h = t.f; - c0 = (c1 << 53) | (c0 >> 11); - if (c0) - { - g = __builtin_clzl (c0); - if (g) - c0 = c0 << g; - t.u = ((f - 53 - g) << 52) | ((c0 << 1) >> 12); - *l = t.f; - } - else - *l = 0; - } - else if (c0) - { - e = __builtin_clzl (c0); - f = 0x3fe - 64 - e; - c0 = c0 << (e+1); // most significant bit shifted out - /* put the upper 52 bits of c0 into h */ - t.u = (f << 52) | (c0 >> 12); - *h = t.f; - /* put the lower 12 bits of c0 into l */ - c0 = c0 << 52; - if (c0) - { - int g = __builtin_clzl (c0); - c0 = c0 << (g+1); - t.u = ((f - 64 - g) << 52) | (c0 >> 12); - *l = t.f; - } - else - *l = 0; - } - else - *h = *l = 0; - /* Since we truncate from two 64-bit words to a double-double, - we have another truncation error of less than 2^-106, thus - the absolute error is bounded as follows: - | h + l - frac(x/(2pi)) | < 2^-75.999 + 2^-106 < 2^-75.998 */ -} - -/* Assuming 0x1.6a09e667f3bccp-27 < x < +Inf, - return i and set h,l such that i/2^11+h+l approximates frac(x/(2pi)). - If x <= 0x1.921fb54442d18p+2: - | i/2^11 + h + l - frac(x/(2pi)) | < 2^-104.116 * |i/2^11 + h + l| - with |h| < 2^-11 and |l| < 2^-52.36. - - Otherwise only the absolute error is bounded: - | i/2^11 + h + l - frac(x/(2pi)) | < 2^-75.998 - with 0 <= h < 2^-11 and |l| < 2^-53. - - In both cases we have |l| < 2^-51.64*|i/2^11 + h|. - - Put in err1 a bound for the absolute error: - | i/2^11 + h + l - frac(x/(2pi)) |. -*/ -static int -reduce_fast (double *h, double *l, double x, double *err1) -{ - if (__builtin_expect(x <= 0x1.921fb54442d17p+2, 1)) [[likely]] // x < 2*pi - { - /* | CH+CL - 1/(2pi) | < 2^-110.523 */ -#define CH 0x1.45f306dc9c883p-3 -#define CL -0x1.6b01ec5417056p-57 - a_mul (h, l, CH, x); // exact - *l = __builtin_fma (CL, x, *l); - /* The error in the above fma() is at most ulp(l), - where |l| <= CL*|x|+|l_in|. - Assume 2^(e-1) <= x < 2^e. - Then |h| < 2^(e-2) and |l_in| <= 1/2 ulp(2^(e-2)) = 2^(e-55), - where l_in is the value of l after a_mul. - Then |l| <= CL*x + 2^(e-55) <= 2^e*(CL+2-55) < 2^e * 2^-55.6. - The rounding error of the fma() is bounded by - ulp(l) <= 2^e * ulp(2^-55.6) = 2^(e-108). - The error due to the approximation of 1/(2pi) - is bounded by 2^-110.523*x <= 2^(e-110.523). - Adding both errors yields: - |h + l - x/(2pi)| < 2^e * (2^-108 + 2^-110.523) < 2^e * 2^-107.768. - Since |x/(2pi)| > 2^(e-1)/(2pi), the relative error is bounded by: - 2^e * 2^-107.768 / (2^(e-1)/(2pi)) = 4pi * 2^-107.768 < 2^-104.116. - - Bound on l: since |h| < 1, we have after |l| <= ulp(h) <= 2^-53 - after a_mul(), and then |l| <= |CL|*0x1.921fb54442d17p+2 + 2^-53 - < 2^-52.36. - - Bound on l relative to h: after a_mul() we have |l| <= ulp(h) - <= 2^-52*h. After fma() we have |l| <= CL*x + 2^-52*h - <= 2^-53.84*CH*x + 2^-52*h <= (2^-53.84+2^-52)*h < 2^-51.64*h. - */ - *err1 = 0x1.d9p-105 * *h; // error < 2^-104.116 * h - } - else // x > 0x1.921fb54442d17p+2 - { - b64u64_u t = {.f = x}; - int e = (t.u >> 52) & 0x7ff; /* 1025 <= e <= 2046 */ - /* We have 2^(e-1023) <= x < 2^(e-1022), thus - ulp(x) is a multiple of 2^(e-1075), for example - if x is just above 2*pi, e=1025, 2^2 <= x < 2^e, - and ulp(x) is a multiple of 2^-50. - On the other side 1/(2pi) ~ T[0]/2^64 + T[1]/2^128 + T[2]/2^192 + ... - Let i be the smallest integer such that 2^(e-1075)/2^(64*(i+1)) - is not an integer, i.e., e - 1139 - 64i < 0, i.e., - i >= (e-1138)/64. */ - uint64_t m = (1ull << 52) | (t.u & 0xffffffffffffful); - uint64_t c[3]; - u128 u; - // x = m/2^53 * 2^(e-1022) - if (e <= 1074) // 1025 <= e <= 1074: 2^2 <= x < 2^52 - { - /* In that case the contribution of x*T[2]/2^192 is less than - 2^(52+64-192) <= 2^-76. */ - u = (u128) m * (u128) T[1]; - c[0] = absl::Uint128Low64(u); - c[1] = absl::Uint128High64(u); - u = (u128) m * (u128) T[0]; - c[1] += absl::Uint128Low64(u); - c[2] = absl::Uint128High64(u) + (c[1] < (uint64_t) u); - /* | c[2]*2^128+c[1]*2^64+c[0] - m/(2pi)*2^128 | < m*T[2]/2^64 < 2^53 - thus: - | (c[2]*2^128+c[1]*2^64+c[0])*2^(e-1203) - x/(2pi) | < 2^(e-1150) - The low 1075-e bits of c[2] contribute to frac(x/(2pi)). - */ - e = 1075 - e; // 1 <= e <= 50 - // e is the number of low bits of C[2] contributing to frac(x/(2pi)) - } - else // 1075 <= e <= 2046, 2^52 <= x < 2^1024 - { - int i = (e - 1138 + 63) / 64; // i = ceil((e-1138)/64), 0 <= i <= 15 - /* m*T[i] contributes to f = 1139 + 64*i - e bits to frac(x/(2pi)) - with 1 <= f <= 64 - m*T[i+1] contributes a multiple of 2^(-f-64), - and at most to 2^(53-f) - m*T[i+2] contributes a multiple of 2^(-f-128), - and at most to 2^(-11-f) - m*T[i+3] contributes a multiple of 2^(-f-192), - and at most to 2^(-75-f) <= 2^-76 - */ - u = (u128) m * (u128) T[i+2]; - c[0] = absl::Uint128Low64(u); - c[1] = absl::Uint128High64(u); - u = (u128) m * (u128) T[i+1]; - c[1] += absl::Uint128Low64(u); - c[2] = absl::Uint128High64(u) + (c[1] < (uint64_t) u); - u = (u128) m * (u128) T[i]; - c[2] += absl::Uint128Low64(u); - e = 1139 + (i<<6) - e; // 1 <= e <= 64 - // e is the number of low bits of C[2] contributing to frac(x/(2pi)) - } - if (e == 64) - { - c[0] = c[1]; - c[1] = c[2]; - } - else - { - c[0] = (c[1] << (64 - e)) | c[0] >> e; - c[1] = (c[2] << (64 - e)) | c[1] >> e; - } - /* In all cases the ignored contribution from x*T[2] or x*T[i+3] - is less than 2^-76, - and the truncated part from the above shift is less than 2^-128 thus: - | c[1]/2^64 + c[0]/2^128 - frac(x/(2pi)) | < 2^-76+2^-128 < 2^-75.999 - */ - set_dd (h, l, c[1], c[0]); - /* set_dd() ensures |h| < 1 and |l| < ulp(h) <= 2^-53 */ - *err1 = 0x1.01p-76; - } - - double i = __builtin_floor (*h * 0x1p11); - *h = __builtin_fma (i, -0x1p-11, *h); - return i; -} - -/* Assume x is a regular number and x > 0x1.6a09e667f3bccp-27, - return a bound on the maximal absolute error err: - | h + l - cos(x) | < err */ -static double -cos_fast (double *h, double *l, double x) -{ - int neg = 0, is_cos = 1; - - double err1; - int i = reduce_fast (h, l, x, &err1); - /* err1 is an absolute bound for | i/2^11 + h + l - frac(x/(2pi)) |: - | i/2^11 + h + l - frac(x/(2pi)) | < err1 */ - - // if i >= 2^10: 1/2 <= frac(x/(2pi)) < 1 thus pi <= x <= 2pi - // we use cos(pi+x) = -cos(x) - neg = neg ^ (i >> 10); - i = i & 0x3ff; - // | i/2^11 + h + l - frac(x/(2pi)) | mod 1/2 < err1 - - // now i < 2^10 - // if i >= 2^9: 1/4 <= frac(x/(2pi)) < 1/2 thus pi/2 <= x <= pi - // we use cos(pi/2+x) = -sin(x) - is_cos = is_cos ^ (i >> 9); - neg = neg ^ (i >> 9); - i = i & 0x1ff; - // | i/2^11 + h + l - frac(x/(2pi)) | mod 1/4 < err1 - - // now 0 <= i < 2^9 - // if i >= 2^8: 1/8 <= frac(x/(2pi)) < 1/4 - // we use cos(pi/2-x) = sin(x) - if (i & 0x100) // case pi/4 <= x_red <= pi/2 - { - is_cos = !is_cos; - i = 0x1ff - i; - /* 0x1p-11 - h is exact below: indeed, reduce_fast first computes - a first value of h (say h0, with 0 <= h0 < 1), then i = floor(h0*2^11) - and h1 = h0 - 2^11*i with 0 <= h1 < 2^-11. - If i >= 2^8 here, this implies h0 >= 1/2^3, thus ulp(h0) >= 2^-55: - h0 and h1 are integer multiples of 2^-55. - Thus h1 = k*2^-55 with 0 <= k < 2^44 (since 0 <= h1 < 2^-11). - Then 0x1p-11 - h = (2^44-k)*2^-55 is exactly representable. - We can have a huge cancellation in 0x1p-11 - h, for example for - x = 0x1.61a3db8c8d129p+1023 where we have before this operation - h = 0x1.ffffffffff8p-12, and h = 0x1p-53 afterwards. But this - does not hurt since we bound the absolute error and not the - relative error at the end. */ - *h = 0x1p-11 - *h; - *l = -*l; - } - - /* Now 0 <= i < 256 and 0 <= h+l < 2^-11 - with | i/2^11 + h + l - frac(x/(2pi)) | cmod 1/4 < err1 - If is_cos=1, cos(x) = cos2pi(R + err1); - if is_cos=0, cos(x) = sin2pi (R + err1). - In both cases R = i/2^11 + h + l, 0 <= R < 1/4. - */ - double sh, sl, ch, cl; - /* since the SC[] table evaluates at i/2^11 + SC[i][0] and not at i/2^11, - we must subtract SC[i][0] from h+l */ - /* Here h = k*2^-55 with 0 <= k < 2^44, and SC[i][0] is an integer - multiple of 2^-62, with |SC[i][0]| < 2^-24, thus SC[i][0] = m*2^-62 - with |m| < 2^38. It follows h-SC[i][0] = (k*2^7 + m)*2^-62 with - 2^51 - 2^38 < k*2^7 + m < 2^51 + 2^38, thus h-SC[i][0] is exact. - Now |h| < 2^-11 + 2^-24. */ - *h -= SC[i][0]; - // now -2^-24 < h < 2^-11+2^-24 - // from reduce_fast() we have |l| < 2^-52.36 - double uh, ul; - a_mul (&uh, &ul, *h, *h); - ul = __builtin_fma (*h + *h, *l, ul); - // uh+ul approximates (h+l)^2 - evalPSfast (&sh, &sl, *h, *l, uh, ul); - /* the absolute error of evalPSfast() is less than 2^-77.09 from - routine evalPSfast() in sin.sage: - | sh + sh - sin(h+l) | < 2^-77.09 */ - evalPCfast (&ch, &cl, uh, ul); - /* the relative error of evalPCfast() is less than 2^-69.96 from - routine evalPCfast(rel=true) in sin.sage: - | ch + cl - cos(h+l) | < 2^-69.96 * |ch + cl| */ - double err; - if (!is_cos) - { - s_mul (&sh, &sl, SC[i][2], sh, sl); - s_mul (&ch, &cl, SC[i][1], ch, cl); - fast_two_sum (h, l, ch, sh); - *l += sl + cl; - /* absolute error bounded by 2^-68.588 - from global_error(is_sin=true,rel=false) in sin.sage: - | h + l - sin2pi (R) | < 2^-68.588 - thus: - | h + l - cos(x) | < 2^-68.588 + | sin2pi (R) - sin |x| | - < 2^-68.588 + err1 */ - err = 0x1.55p-69; // 2^-66.588 < 0x1.55p-69 - } - else - { - s_mul (&ch, &cl, SC[i][2], ch, cl); - s_mul (&sh, &sl, SC[i][1], sh, sl); - fast_two_sum (h, l, ch, -sh); - *l += cl - sl; - /* absolute error bounded by 2^-68.414 - from global_error(is_sin=false,rel=false) in sin.sage: - | h + l - cos2pi (R) | < 2^-68.414 - thus: - | h + l - cos(x) | < 2^-68.414 + | cos2pi (R) - sin |x| | - < 2^-68.414 * |h + l| + err1 */ - err = 0x1.81p-69; // 2^-68.414 < 0x1.81p-69 - } - static const double sgn[2] = {1.0, -1.0}; - *h *= sgn[neg]; - *l *= sgn[neg]; - return err + err1; -} - -/* Assume x is a regular number and x > 0x1.6a09e667f3bccp-27. */ -static double -cos_accurate (double x) -{ - dint64_t X[1]; - dint_fromd (X, x); - - /* reduce argument */ - reduce (X); - - // now |X - x/(2pi) mod 1| < 2^-126.67*X, with 0 <= X < 1. - - int neg = 0, is_cos = 1; - - // Write X = i/2^11 + r with 0 <= r < 2^11. - int i = reduce2 (X); // exact - - if (i & 0x400) // pi <= x < 2*pi: cos(x) = -cos(x-pi) - { - neg = 1; - i = i & 0x3ff; - } - - // now i < 2^10 - - if (i & 0x200) // pi/2 <= x < pi: cos(x) = -sin(x-pi/2) - { - neg = !neg; - is_cos = 0; - i = i & 0x1ff; - } - - // now 0 <= i < 2^9 - - if (i & 0x100) - // pi/4 <= x < pi/2: cos(x) = sin(pi/2-x), sin(x) = cos(pi/2-x) - { - is_cos = !is_cos; - X->sgn = 1; // negate X - add_dint (X, &MAGIC, X); // X -> 2^-11 - X - // here: 256 <= i <= 511 - i = 0x1ff - i; - // now 0 <= i < 256 - } - - // now 0 <= i < 256 and 0 <= X < 2^-11 - - /* If is_cos=1, cos |x| = cos2pi (R * (1 + eps)) - (cases 0 <= x < pi/4 and 3pi/4 <= x < pi) - if is_cos=0, cos |x| = sin2pi (R * (1 + eps)) - (case pi/4 <= x < 3pi/4) - In both cases R = i/2^11 + X, 0 <= R < 1/4, and |eps| < 2^-126.67. - */ - - dint64_t U[1], V[1], X2[1]; - mul_dint (X2, X, X); // X2 approximates X^2 - evalPC (U, X2); // cos2pi(X) - /* since 0 <= X < 2^-11, we have 0.999 < U <= 1 */ - evalPS (V, X, X2); // sin2pi(X) - /* since 0 <= X < 2^-11, we have 0 <= V < 0.0005 */ - if (!is_cos) - { - // sin2pi(R) ~ sin2pi(i/2^11)*cos2pi(X)+cos2pi(i/2^11)*sin2pi(X) - mul_dint (U, S+i, U); - /* since 0 <= S[i] < 0.705 and 0.999 < Uin <= 1, we have - 0 <= U < 0.705 */ - mul_dint (V, C+i, V); - /* For the error analysis, we distinguish the case i=0. - For i=0, we have S[i]=0 and C[1]=1, thus V is the value computed - by evalPS() above, with relative error < 2^-124.648. - - For 1 <= i < 256, analyze_sin_case1(rel=true) from sin.sage gives a - relative error bound of -122.797 (obtained for i=1). - In all cases, the relative error for the computation of - sin2pi(i/2^11)*cos2pi(X)+cos2pi(i/2^11)*sin2pi(X) is bounded by -122.797 - not taking into account the approximation error in R: - |U - sin2pi(R)| < |U| * 2^-122.797, with U the value computed - after add_dint (U, U, V) below. - - For the approximation error in R, we have: - cos(x) = sin2pi (R * (1 + eps)) - R = i/2^11 + X, 0 <= R < 1/4, and |eps| < 2^-126.67. - Thus cos(x) = sin2pi(R+R*eps) - = sin2pi(R)+R*eps*2*pi*cos2pi(theta), theta in [R,R+R*eps] - Since 2*pi*R/sin(2*pi*R) < pi/2 for R < 1/4, it follows: - | cos(x) - sin2pi(R) | < pi/2*R*|sin(2*pi*R)| - | cos(x) - sin2pi(R) | < 2^-126.018 * |sin2pi(R)|. - - Adding both errors we get: - | cos(x) - U | < |U| * 2^-122.797 + 2^-126.018 * |sin2pi(R)| - < |U| * 2^-122.797 + 2^-126.018 * |U| * (1 + 2^-122.797) - < |U| * 2^-122.650. - */ - } - else - { - // cos2pi(R) ~ cos2pi(i/2^11)*cos2pi(X)-sin2pi(i/2^11)*sin2pi(X) - mul_dint (U, C+i, U); - mul_dint (V, S+i, V); - V->sgn = 1 - V->sgn; // negate V - /* For 0 <= i < 256, analyze_sin_case2(rel=true) from sin.sage gives a - relative error bound of -123.540 (obtained for i=0): - |U - cos2pi(R)| < |U| * 2^-123.540, with U the value computed - after add_dint (U, U, V) below. - - For the approximation error in R, we have: - cos(x) = cos2pi (R * (1 + eps)) - R = i/2^11 + X, 0 <= R < 1/4, and |eps| < 2^-126.67. - Thus cos(x) = cos2pi(R+R*eps) - = cos2pi(R)-R*eps*2*pi*sin2pi(theta), theta in [R,R+R*eps] - Since we have R < 1/4, we have cos2pi(R) >= sqrt(2)/2, - and it follows: - | cos(x)/cos2pi(R) - 1 | < 2*pi*R*eps/(sqrt(2)/2) - < pi/2*eps/sqrt(2) [since R < 1/4] - < 2^-126.518. - Adding both errors we get: - | cos(x) - U | < |U| * 2^-123.540 + 2^-126.518 * |cos2pi(R)| - < |U| * 2^-123.540 + 2^-126.518 * |U| * (1 + 2^-123.540) - < |U| * 2^-123.367. - */ - } - add_dint (U, U, V); - /* If is_cos=0: - | cos(x) - U | < |U| * 2^-122.650 - If is_cos=1: - | cos(x) - U | < |U| * 2^-123.367. - In all cases the total error is bounded by |U| * 2^-122.650. - The term |U| * 2^-122.650 contributes to at most 2^(128-122.650) < 41 ulps - relatively to U->lo. - */ - uint64_t err = 41; - uint64_t hi0, hi1, lo0, lo1; - lo0 = U->lo - err; - hi0 = U->hi - (lo0 > U->lo); - lo1 = U->lo + err; - hi1 = U->hi + (lo1 < U->lo); - /* check the upper 54 bits are equal */ - if ((hi0 >> 10) != (hi1 >> 10)) - { - static const double exceptions[][3] = { - {0x1.8000000000009p-23, 0x1.fffffffffff7p-1, 0x1.b56666666666cp-143}, - {0x1.8000000000024p-22, 0x1.ffffffffffdcp-1, 0x1.b56666666667ep-137}, - {0x1.800000000009p-21, 0x1.ffffffffff7p-1, 0x1.b5666666666c4p-131}, - {0x1.20000000000f3p-20, 0x1.fffffffffebcp-1, 0x1.37642666666fdp-127}, - {0x1.800000000024p-20, 0x1.fffffffffdcp-1, 0x1.b5666666667ddp-125}, - }; - for (int i = 0; i < 5; i++) - { - if (__builtin_fabs (x) == exceptions[i][0]) - return exceptions[i][1] + exceptions[i][2]; - } - printf ("Rounding test of accurate path failed for cos(%la)\n", x); - printf ("Please report the above to core-math@inria.fr\n"); - exit (1); - } - - if (neg) - U->sgn = 1 - U->sgn; - - double y = dint_tod (U); - - return y; -} - -double __cdecl -cr_cos (double x) -{ - b64u64_u t = {.f = x}; - int e = (t.u >> 52) & 0x7ff; - - if (__builtin_expect (e == 0x7ff, 0)) [[unlikely]] /* NaN, +Inf and -Inf. */ - { - t.u = ~0ull; - return t.f; - } - - /* now x is a regular number */ - - /* For |x| <= 0x1.6a09e667f3bccp-27, cos(x) rounds to x (to nearest): - we can assume x >= 0 without loss of generality since cos(-x) = cos(x), - we have 1 - x^2/2 < cos(x) < 1 for say 0 < x <= 1 thus - |cos(x) - 1| < x^2/2. - Assume 0 < x < 1, and write x = c*2^e with 1/2 <= c < 1. - For 0 < x < 1, 1/2 < cos(x) < 1, thus ulp(cos(x)) = 2^-53, - and x^2/2 = c^2/2*2^(2e), thus - x^2/2 < ulp(cos(x))/2 rewrites as c^2/2*2^(2e) < 2^-54, - or c^2*2^(2e+53) < 1 (1). - For e <= -27, since c^2 < 1, we have c^2*2^(2e+53) < 1/2 < 1. - For e=-26, (1) rewrites c^2*2 < 1 which yields c <= 0x1.6a09e667f3bccp-1. - */ - t.u &= 0x7fffffffffffffff; - if (__builtin_expect (t.u <= 0x3e46a09e667f3bcc, 0)) - // |x| <= 0x1.6a09e667f3bccp-27 - return __builtin_fma (t.f, -0x1p-28, 1.0); - - double h, l, err; - err = cos_fast (&h, &l, t.f); - double left = h + (l - err), right = h + (l + err); - /* With SC[] from ./buildSC 15 we get 1100 failures out of 50000000 - random tests, i.e., about 0.002%. */ - if (__builtin_expect (left == right, 1)) - return left; - - return cos_accurate (t.f); -} - -} // namespace internal -} // namespace _cos -} // namespace functions -} // namespace principia - -#if PRINCIPIA_COMPILER_MSVC -#undef __builtin_clzl -#undef __builtin_fma -#endif -#undef __builtin_expect -#undef __builtin_fabs -#undef __builtin_floor diff --git a/functions/cos.hpp b/functions/cos.hpp deleted file mode 100644 index f84fa7e5c4..0000000000 --- a/functions/cos.hpp +++ /dev/null @@ -1,16 +0,0 @@ -#pragma once - -namespace principia { -namespace functions { -namespace _cos { -namespace internal { - -double __cdecl cr_cos(double x); - -} // namespace internal - -using internal::cr_cos; - -} // namespace _cos -} // namespace functions -} // namespace principia diff --git a/functions/functions.vcxproj b/functions/functions.vcxproj index f378eb733e..8da1fd7b48 100644 --- a/functions/functions.vcxproj +++ b/functions/functions.vcxproj @@ -7,16 +7,12 @@ - - - - diff --git a/functions/functions.vcxproj.filters b/functions/functions.vcxproj.filters index c736b4f6e0..4f6137aea6 100644 --- a/functions/functions.vcxproj.filters +++ b/functions/functions.vcxproj.filters @@ -30,12 +30,6 @@ Test Files - - Source Files - - - Source Files - Test Files @@ -50,11 +44,5 @@ Source Files - - Header Files - - - Header Files - \ No newline at end of file diff --git a/functions/sin.cc b/functions/sin.cc deleted file mode 100644 index dfceffbe34..0000000000 --- a/functions/sin.cc +++ /dev/null @@ -1,2063 +0,0 @@ -/* Correctly-rounded sine function for binary64 value. - -Copyright (c) 2022-2023 Paul Zimmermann and Tom Hubrecht - -This file is part of the CORE-MATH project -(https://core-math.gitlabpages.inria.fr/). - -Permission is hereby granted, free of charge, to any person obtaining a copy -of this software and associated documentation files (the "Software"), to deal -in the Software without restriction, including without limitation the rights -to use, copy, modify, merge, publish, distribute, sublicense, and/or sell -copies of the Software, and to permit persons to whom the Software is -furnished to do so, subject to the following conditions: - -The above copyright notice and this permission notice shall be included in all -copies or substantial portions of the Software. - -THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR -IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, -FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE -AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER -LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, -OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE -SOFTWARE. -*/ - -// This code has been adapted to C++ and MSVC. - -/* stdio.h and stdlib.h are needed in case the rounding test of the accurate - step fails, to print the corresponding input and exit. */ -#include "functions/sin.hpp" - -#include -#include -#include -#include - -#include "absl/numeric/int128.h" -#include "base/macros.hpp" // 🧙 For PRINCIPIA_COMPILER_MSVC. -#include "numerics/fma.hpp" - -// Warning: clang also defines __GNUC__ -#if defined(__GNUC__) && !defined(__clang__) -#pragma GCC diagnostic ignored "-Wunknown-pragmas" -#endif - -#if PRINCIPIA_COMPILER_MSVC -#define __builtin_clzl(x) __lzcnt64(x) -#define __builtin_fma(x, y, z) \ - principia::numerics::_fma::FusedMultiplyAdd(x, y, z) -#endif -#define __builtin_expect(x, y) x -#define __builtin_fabs(x) std::abs(x) -#define __builtin_floor(x) std::floor(x) - -/******************** code copied from dint.h and pow.[ch] *******************/ - -namespace principia { -namespace functions { -namespace _sin { -namespace internal { - -typedef absl::uint128 u128; - -typedef union { - struct { - u128 r; - int64_t _ex; - uint64_t _sgn; - }; - struct { - uint64_t lo; - uint64_t hi; - int64_t ex; - uint64_t sgn; - }; -} dint64_t; - -typedef union { - u128 r; - struct { - uint64_t l; - uint64_t h; - }; -} uint128_t; - -typedef union { - double f; - uint64_t u; -} f64_u; - -// Extract both the mantissa and exponent of a double -static inline void fast_extract (int64_t *e, uint64_t *m, double x) { - f64_u _x = {.f = x}; - - *e = (_x.u >> 52) & 0x7ff; - *m = (_x.u & (~0ull >> 12)) + (*e ? (1ull << 52) : 0); - *e = *e - 0x3fe; -} - -// Return non-zero if a = 0 -static inline int -dint_zero_p (const dint64_t *a) -{ - return a->hi == 0; -} - -static inline int cmp(int64_t a, int64_t b) { return (a > b) - (a < b); } - -static inline int cmpu128 (u128 a, u128 b) { return (a > b) - (a < b); } - -/* ZERO is a dint64_t representation of 0, which ensures that - dint_tod(ZERO) = 0 */ -static const dint64_t ZERO = {.lo = 0x0, .hi = 0x0, .ex = -1076, .sgn = 0x0}; -// MAGIC is a dint64_t representation of 1/2^11 -static const dint64_t MAGIC = {.lo = 0x0, .hi = 0x8000000000000000, .ex = -10, .sgn = 0x0}; - -// Compare the absolute values of a and b -// Return -1 if |a| < |b| -// Return 0 if |a| = |b| -// Return +1 if |a| > |b| -static inline signed char -cmp_dint_abs (const dint64_t *a, const dint64_t *b) { - if (dint_zero_p (a)) - return dint_zero_p (b) ? 0 : -1; - if (dint_zero_p (b)) - return +1; - char c1 = cmp (a->ex, b->ex); - return c1 ? c1 : cmpu128 (a->r, b->r); -} - -// Copy a dint64_t value -static inline void cp_dint(dint64_t *r, const dint64_t *a) { - r->ex = a->ex; - r->r = a->r; - r->sgn = a->sgn; -} - -// Add two dint64_t values, with error bounded by 2 ulps (ulp_128) -// (more precisely 1 ulp when a and b have same sign, 2 ulps otherwise) -// Moreover, when Sterbenz theorem applies, i.e., |b| <= |a| <= 2|b| -// and a,b are of different signs, there is no error, i.e., r = a-b. -static inline void -add_dint (dint64_t *r, const dint64_t *a, const dint64_t *b) { - if (!(a->hi | a->lo)) { - cp_dint (r, b); - return; - } - - switch (cmp_dint_abs (a, b)) { - case 0: - if (a->sgn ^ b->sgn) { - cp_dint (r, &ZERO); - return; - } - - cp_dint (r, a); - r->ex++; - return; - - case -1: // |A| < |B| - { - // swap operands - const dint64_t *tmp = a; a = b; b = tmp; - break; // fall through the case |A| > |B| - } - } - - // From now on, |A| > |B| thus a->ex >= b->ex - - u128 A = a->r, B = b->r; - uint64_t k = a->ex - b->ex; - - if (k > 0) { - /* Warning: the right shift x >> k is only defined for 0 <= k < n - where n is the bit-width of x. See for example - https://developer.arm.com/documentation/den0024/a/The-A64-instruction-set/Data-processing-instructions/Shift-operations - where it is said that k is interpreted modulo n. */ - B = (k < 128) ? B >> k : 0; - } - - u128 C; - unsigned char sgn = a->sgn; - - r->ex = a->ex; /* tentative exponent for the result */ - - if (a->sgn ^ b->sgn) { - /* a and b have different signs C = A + (-B) - Sterbenz case |a|/2 <= |b| <= |a| can occur only when: - * k=0: then B is not truncated, and C is exact below - * k=1 and ex>0 below: then we ensure C is exact - */ - C = A - B; - uint64_t ch = absl::Uint128High64(C); - /* We can't have C=0 here since we excluded the case |A| = |B|, - thus __builtin_clzl(C) is well-defined below. */ - uint64_t ex = ch ? __builtin_clzl(ch) : 64 + __builtin_clzl(absl::Uint128Low64(C)); - /* The error from the truncated part of B (1 ulp) is multiplied by 2^ex, - thus by 2 ulps when ex <= 1. */ - if (ex > 0) - { - if (k == 1) /* Sterbenz case */ - C = (A << ex) - (b->r << (ex - 1)); - else - C = (A << ex) - (B << ex); - /* If C0 is the previous value of C, we have: - (C0-1)*2^ex < A*2^ex-B*2^ex <= C0*2^ex - since some neglected bits from B might appear which contribute - a value less than ulp(C0)=1. - As a consequence since 2^(127-ex) <= C0 < 2^(128-ex), because C0 had - ex leading zero bits, we have 2^127-2^ex <= A*2^ex-B*2^ex < 2^128. - Thus the value of C, which is truncated to 128 bits, is the right - one (as if no truncation); moreover in some rare cases we need to - shift by 1 bit to the left. */ - r->ex -= ex; - ex = __builtin_clzl (absl::Uint128High64(C)); - /* Fall through with the code for ex = 0. */ - } - C = C << ex; - r->ex -= ex; - /* The neglected part of B is bounded by 2 ulp(C) when ex=0, 1 ulp - when ex > 0 but ex=0 at the end, and by 2*ulp(C) when ex > 0 and there - is an extra shift at the end (in that case necessarily ex=1). */ - } else { - C = A + B; - if (C < A) - { - C = ((u128) 1 << 127) | (C >> 1); - r->ex ++; - } - } - - /* In the addition case, we loose the truncated part of B, which - contributes to at most 1 ulp. If there is an exponent shift, we - might also loose the least significant bit of C, which counts as - 1/2 ulp, but the truncated part of B is now less than 1/2 ulp too, - thus in all cases the error is less than 1 ulp(r). */ - - r->sgn = sgn; - r->r = C; -} - -// Multiply two dint64_t numbers, with error bounded by 6 ulps -// on the 128-bit floating-point numbers. -// Overlap between r and a is allowed -static inline void -mul_dint (dint64_t *r, const dint64_t *a, const dint64_t *b) { - u128 bh = b->hi, bl = b->lo; - - /* compute the two middle terms */ - u128 m1 = (u128)(a->hi) * bl; - u128 m2 = (u128)(a->lo) * bh; - - /* put the 128-bit product of the high terms in r */ - r->r = (u128)(a->hi) * bh; - - /* there can be no overflow in the following addition since r <= (B-1)^2 - with B=2^64, (m1>>64) <= B-1 and (m2>>64) <= B-1, thus the sum is - bounded by (B-1)^2+2*(B-1) = B^2-1 */ - r->r += (m1 >> 64) + (m2 >> 64); - - // Ensure that r->hi starts with a 1 - uint64_t ex = r->hi >> 63; - r->r = r->r << (1 - ex); - - // Exponent and sign - // if ex=1, then ex(r) = ex(a) + ex(b) - // if ex=0, then ex(r) = ex(a) + ex(b) - 1 - r->ex = a->ex + b->ex + ex - 1; - r->sgn = a->sgn ^ b->sgn; - - /* The ignored part can be as large as 3 ulps before the shift (one - for the low part of a->hi * bl, one for the low part of a->lo * bh, - and one for the neglected a->lo * bl term). After the shift this can - be as large as 6 ulps. */ -} - -// Multiply two dint64_t numbers, assuming the low part of b is zero -// with error bounded by 2 ulps -static inline void -mul_dint_21 (dint64_t *r, const dint64_t *a, const dint64_t *b) { - u128 bh = b->hi; - u128 hi = (u128) (a->hi) * bh; - u128 lo = (u128) (a->lo) * bh; - - /* put the 128-bit product of the high terms in r */ - r->r = hi; - - /* add the middle term */ - r->r += lo >> 64; - - // Ensure that r->hi starts with a 1 - uint64_t ex = r->hi >> 63; - r->r = r->r << (1 - ex); - - // Exponent and sign - r->ex = a->ex + b->ex + ex - 1; - r->sgn = a->sgn ^ b->sgn; - - /* The ignored part can be as large as 1 ulp before the shift (truncated - part of lo). After the shift this can be as large as 2 ulps. */ -} - -// Convert a non-zero double to the corresponding dint64_t value -static inline void dint_fromd (dint64_t *a, double b) { - fast_extract (&a->ex, &a->hi, b); - - /* |b| = 2^(ex-52)*hi */ - - uint32_t t = __builtin_clzl (a->hi); - - a->sgn = b < 0.0; - a->hi = a->hi << t; - a->ex = a->ex - (t > 11 ? t - 12 : 0); - /* b = 2^ex*hi/2^64 where 1/2 <= hi/2^64 < 1 */ - a->lo = 0; -} - -static inline void subnormalize_dint(dint64_t *a) { - if (a->ex > -1023) - return; - - uint64_t ex = -(1011 + a->ex); - - uint64_t hi = a->hi >> ex; - uint64_t md = (a->hi >> (ex - 1)) & 0x1; - uint64_t lo = (a->hi & (~0ull >> ex)) || a->lo; - - switch (fegetround()) { - case FE_TONEAREST: - hi += lo ? md : hi & md; - break; - case FE_DOWNWARD: - hi += a->sgn & (md | lo); - break; - case FE_UPWARD: - hi += (!a->sgn) & (md | lo); - break; - } - - a->hi = hi << ex; - a->lo = 0; - - if (!a->hi) { - a->ex++; - a->hi = (1ll << 63); - } -} - -// Convert a dint64_t value to a double -static inline double dint_tod(dint64_t *a) { - subnormalize_dint (a); - - f64_u r = {.u = (a->hi >> 11) | (0x3ffll << 52)}; - - double rd = 0.0; - if ((a->hi >> 10) & 0x1) - rd += 0x1p-53; - - if (a->hi & 0x3ff || a->lo) - rd += 0x1p-54; - - if (a->sgn) - rd = -rd; - - r.u = r.u | a->sgn << 63; - r.f += rd; - - f64_u e; - - if (a->ex > -1022) { // The result is a normal double - if (a->ex > 1024) - if (a->ex == 1025) { - r.f = r.f * 0x1p+1; - e.f = 0x1p+1023; - } else { - r.f = 0x1.fffffffffffffp+1023; - e.f = 0x1.fffffffffffffp+1023; - } - else - e.u = ((a->ex + 1022) & 0x7ff) << 52; - } else { - if (a->ex < -1073) { - if (a->ex == -1074) { - r.f = r.f * 0x1p-1; - e.f = 0x1p-1074; - } else { - r.f = 0x0.0000000000001p-1022; - e.f = 0x0.0000000000001p-1022; - } - } else { - e.u = 1ll << (a->ex + 1073); - } - } - - return r.f * e.f; -} - -/**************** end of code copied from dint.h and pow.[ch] ****************/ - -typedef union {double f; uint64_t u;} b64u64_u; - -/* This table approximates 1/(2pi) downwards with precision 1280: - 1/(2*pi) ~ T[0]/2^64 + T[1]/2^128 + ... + T[i]/2^((i+1)*64) + ... - Computed with computeT() from sin.sage. */ -static const uint64_t T[20] = { - 0x28be60db9391054a, // i=0 - 0x7f09d5f47d4d3770, - 0x36d8a5664f10e410, - 0x7f9458eaf7aef158, - 0x6dc91b8e909374b8, - 0x1924bba82746487, // i=5 - 0x3f877ac72c4a69cf, - 0xba208d7d4baed121, - 0x3a671c09ad17df90, - 0x4e64758e60d4ce7d, - 0x272117e2ef7e4a0e, // i=10 - 0xc7fe25fff7816603, - 0xfbcbc462d6829b47, - 0xdb4d9fb3c9f2c26d, - 0xd3d18fd9a797fa8b, - 0x5d49eeb1faf97c5e, // i=15 - 0xcf41ce7de294a4ba, - 0x9afed7ec47e35742, - 0x1580cc11bf1edaea, - 0xfc33ef0826bd0d87, // i=19 -}; - -/* Table containing 128-bit approximations of sin2pi(i/2^11) for 0 <= i < 256 - (to nearest). - Each entry is to be interpreted as (hi/2^64+lo/2^128)*2^ex*(-1)*sgn. - Generated with computeS() from sin.sage. */ -static const dint64_t S[256] = { - {.lo = 0x0, .hi = 0x0, .ex = 128, .sgn=0}, - {.lo = 0x480f7956b6470765, .hi = 0xc90fc5f66525d257, .ex = -8, .sgn=0}, - {.lo = 0xcb3ff35bd4d81baa, .hi = 0xc90f87f3380388d5, .ex = -7, .sgn=0}, - {.lo = 0xb767005691b9d9d1, .hi = 0x96cb587284b81770, .ex = -6, .sgn=0}, - {.lo = 0xf1d7d06db39ea9fc, .hi = 0xc90e8fe6f63c2330, .ex = -6, .sgn=0}, - {.lo = 0xd784e031f9af76d6, .hi = 0xfb514b55ccbe541a, .ex = -6, .sgn=0}, - {.lo = 0xf91ee371d6467dca, .hi = 0x96c9b5df1877e9b5, .ex = -5, .sgn=0}, - {.lo = 0xf56e3c87ae3c56df, .hi = 0xafea690fd5912ef3, .ex = -5, .sgn=0}, - {.lo = 0xc539edcbfda0cf2c, .hi = 0xc90aafbd1b33efc9, .ex = -5, .sgn=0}, - {.lo = 0x850021e392744a4f, .hi = 0xe22a7a6729d8e453, .ex = -5, .sgn=0}, - {.lo = 0xb21ccebc9caac3, .hi = 0xfb49b98e8e7807f6, .ex = -5, .sgn=0}, - {.lo = 0xde5b1068d174be9c, .hi = 0x8a342eda160bf5ae, .ex = -4, .sgn=0}, - {.lo = 0x37b2dd49d5fca3c0, .hi = 0x96c32baca2ae68b4, .ex = -4, .sgn=0}, - {.lo = 0xb56007d16d4ad5a3, .hi = 0xa351cb7fc30bc889, .ex = -4, .sgn=0}, - {.lo = 0xcd34d2751c2e1da7, .hi = 0xafe00694866a1b44, .ex = -4, .sgn=0}, - {.lo = 0xf10bfca3d6464012, .hi = 0xbc6dd52c3a342eb5, .ex = -4, .sgn=0}, - {.lo = 0x6a17954b2b7c5171, .hi = 0xc8fb2f886ec09f37, .ex = -4, .sgn=0}, - {.lo = 0x73d1472472f4a390, .hi = 0xd5880deafc18b534, .ex = -4, .sgn=0}, - {.lo = 0x438b4a73aecd2541, .hi = 0xe214689606bf1676, .ex = -4, .sgn=0}, - {.lo = 0xc4e92d01a2f42935, .hi = 0xeea037cc04764844, .ex = -4, .sgn=0}, - {.lo = 0xf0a0e36a000c7350, .hi = 0xfb2b73cfc106ff68, .ex = -4, .sgn=0}, - {.lo = 0x60e782313f6161af, .hi = 0x83db0a7231831d8f, .ex = -3, .sgn=0}, - {.lo = 0x77724a2b2a669bc4, .hi = 0x8a2009a6b84d9402, .ex = -3, .sgn=0}, - {.lo = 0x56e0a8b0d177b55d, .hi = 0x9064b3a76a22640c, .ex = -3, .sgn=0}, - {.lo = 0xf77574094d3c35c4, .hi = 0x96a9049670cfae65, .ex = -3, .sgn=0}, - {.lo = 0x50ffe4f5caa7f1fa, .hi = 0x9cecf8962d14c822, .ex = -3, .sgn=0}, - {.lo = 0xdec1b7f2768bdafa, .hi = 0xa3308bc93904ad69, .ex = -3, .sgn=0}, - {.lo = 0x76f8c63986598c79, .hi = 0xa973ba526a6850d9, .ex = -3, .sgn=0}, - {.lo = 0xfdd2fc0936594c2d, .hi = 0xafb68054d520c60b, .ex = -3, .sgn=0}, - {.lo = 0x924bef13600f9852, .hi = 0xb5f8d9f3cd8945d6, .ex = -3, .sgn=0}, - {.lo = 0xeb13e106732687f1, .hi = 0xbc3ac352ead90abe, .ex = -3, .sgn=0}, - {.lo = 0xb228a03916371f6f, .hi = 0xc27c389609850433, .ex = -3, .sgn=0}, - {.lo = 0xc7396c894bbf7389, .hi = 0xc8bd35e14da15f0e, .ex = -3, .sgn=0}, - {.lo = 0x6b47b8c44e5b037e, .hi = 0xcefdb7592542e1e9, .ex = -3, .sgn=0}, - {.lo = 0x7337412cf70716cb, .hi = 0xd53db9224ae01bca, .ex = -3, .sgn=0}, - {.lo = 0xbb286d23e11c8337, .hi = 0xdb7d3761c7b263b6, .ex = -3, .sgn=0}, - {.lo = 0x31883b30137c6e62, .hi = 0xe1bc2e3cf616a7ac, .ex = -3, .sgn=0}, - {.lo = 0xeeb8f9c33340a2f2, .hi = 0xe7fa99d983ee098f, .ex = -3, .sgn=0}, - {.lo = 0xed16b994af6c18ae, .hi = 0xee38765d74fe4897, .ex = -3, .sgn=0}, - {.lo = 0x14e1a5488eaeab96, .hi = 0xf475bfef2551f5b9, .ex = -3, .sgn=0}, - {.lo = 0x704729ae56d78a37, .hi = 0xfab272b54b9871a2, .ex = -3, .sgn=0}, - {.lo = 0x3eac8308f1113e5e, .hi = 0x8077456b7dc2d967, .ex = -2, .sgn=0}, - {.lo = 0xdb1f70118c9c2198, .hi = 0x8395023dd418e919, .ex = -2, .sgn=0}, - {.lo = 0xc5a9decdfaad4db5, .hi = 0x86b26de5933c2e8e, .ex = -2, .sgn=0}, - {.lo = 0x97965c9860c34e44, .hi = 0x89cf8676d7abb55b, .ex = -2, .sgn=0}, - {.lo = 0xdcdca90cc73b116a, .hi = 0x8cec4a05f12739e8, .ex = -2, .sgn=0}, - {.lo = 0xa6e3df5975cca9da, .hi = 0x9008b6a763de75b7, .ex = -2, .sgn=0}, - {.lo = 0x899c4de737feec22, .hi = 0x9324ca6fe9a04b4e, .ex = -2, .sgn=0}, - {.lo = 0xa89a11e07c1fe, .hi = 0x964083747309d113, .ex = -2, .sgn=0}, - {.lo = 0x49c4863de522b217, .hi = 0x995bdfca28b53a54, .ex = -2, .sgn=0}, - {.lo = 0xe7bc08111d0bfca4, .hi = 0x9c76dd866c689dcc, .ex = -2, .sgn=0}, - {.lo = 0xf3ff913a4aadb85e, .hi = 0x9f917abeda4498df, .ex = -2, .sgn=0}, - {.lo = 0xa5dbee6084ee1260, .hi = 0xa2abb58949f2ced7, .ex = -2, .sgn=0}, - {.lo = 0x69fcb11e19f58619, .hi = 0xa5c58bfbcfd4436a, .ex = -2, .sgn=0}, - {.lo = 0xcd12a1f6ab6b095, .hi = 0xa8defc2cbe2f8fcc, .ex = -2, .sgn=0}, - {.lo = 0x8c95c4c91179176b, .hi = 0xabf80432a65ef190, .ex = -2, .sgn=0}, - {.lo = 0x3feef3bb58b1f10d, .hi = 0xaf10a22459fe32a6, .ex = -2, .sgn=0}, - {.lo = 0x16031a34d4fc855d, .hi = 0xb228d418ec1869ad, .ex = -2, .sgn=0}, - {.lo = 0xcd73fb5d8d45d302, .hi = 0xb5409827b25591f0, .ex = -2, .sgn=0}, - {.lo = 0x187e26d290714d70, .hi = 0xb857ec684627fa4c, .ex = -2, .sgn=0}, - {.lo = 0xbddd8a0365d6b1d3, .hi = 0xbb6ecef285f98a3a, .ex = -2, .sgn=0}, - {.lo = 0xdfe1b074e22fc666, .hi = 0xbe853dde9658dc60, .ex = -2, .sgn=0}, - {.lo = 0xad5a41de48f6b26f, .hi = 0xc19b3744e3262dcd, .ex = -2, .sgn=0}, - {.lo = 0xdab4e426409b23a0, .hi = 0xc4b0b93e20c0213f, .ex = -2, .sgn=0}, - {.lo = 0x5cc8c00e4fccd850, .hi = 0xc7c5c1e34d3055b2, .ex = -2, .sgn=0}, - {.lo = 0xfa6171200ab2efc3, .hi = 0xcada4f4db157cf77, .ex = -2, .sgn=0}, - {.lo = 0x65a3132adfb7dfd5, .hi = 0xcdee5f96e21b332c, .ex = -2, .sgn=0}, - {.lo = 0xaadb580a1eba209f, .hi = 0xd101f0d8c18ed1c1, .ex = -2, .sgn=0}, - {.lo = 0xdf4005ef6a64aa02, .hi = 0xd415012d802284f0, .ex = -2, .sgn=0}, - {.lo = 0x1779df36d1cc8912, .hi = 0xd7278eaf9dcd5b55, .ex = -2, .sgn=0}, - {.lo = 0xcbabaeb97af8e8aa, .hi = 0xda399779eb391377, .ex = -2, .sgn=0}, - {.lo = 0xece7f445cecf1e28, .hi = 0xdd4b19a78aed6515, .ex = -2, .sgn=0}, - {.lo = 0xebc61ade6ca83cd, .hi = 0xe05c1353f27b17e5, .ex = -2, .sgn=0}, - {.lo = 0x26a0eecdb4f16266, .hi = 0xe36c829aeba6e720, .ex = -2, .sgn=0}, - {.lo = 0x82b0aecadf808123, .hi = 0xe67c659895943123, .ex = -2, .sgn=0}, - {.lo = 0xb91caf23416e7e80, .hi = 0xe98bba6965ef725f, .ex = -2, .sgn=0}, - {.lo = 0x7244ee20f591983b, .hi = 0xec9a7f2a2a188aeb, .ex = -2, .sgn=0}, - {.lo = 0x1050cdf22f34182f, .hi = 0xefa8b1f8084ccdfc, .ex = -2, .sgn=0}, - {.lo = 0x587f3fa044e2d27d, .hi = 0xf2b650f080d0da8d, .ex = -2, .sgn=0}, - {.lo = 0x643720de93ba81bd, .hi = 0xf5c35a316f1a3c80, .ex = -2, .sgn=0}, - {.lo = 0x4221dc4ba772598d, .hi = 0xf8cfcbd90af8d57a, .ex = -2, .sgn=0}, - {.lo = 0xd24d3023da491920, .hi = 0xfbdba405e9c00cca, .ex = -2, .sgn=0}, - {.lo = 0x8b74fe2508ab8fc2, .hi = 0xfee6e0d6ff6fc5a4, .ex = -2, .sgn=0}, - {.lo = 0xfd958d68e8b49e6b, .hi = 0x80f8c035cfee8d76, .ex = -1, .sgn=0}, - {.lo = 0xfb4c92369f0cf008, .hi = 0x827dc071bfed6ffa, .ex = -1, .sgn=0}, - {.lo = 0xcb07b25a7b0372a7, .hi = 0x8402702f5b30f2a9, .ex = -1, .sgn=0}, - {.lo = 0x9d3dc689006896f4, .hi = 0x8586ce7ededc809d, .ex = -1, .sgn=0}, - {.lo = 0x9d52755ece3f70, .hi = 0x870ada70ba4e6d49, .ex = -1, .sgn=0}, - {.lo = 0x984156f553344306, .hi = 0x888e93158fb3bb04, .ex = -1, .sgn=0}, - {.lo = 0xa66d1d936c38c329, .hi = 0x8a11f77e349bc245, .ex = -1, .sgn=0}, - {.lo = 0x575f33366be0afef, .hi = 0x8b9506bbb28bb922, .ex = -1, .sgn=0}, - {.lo = 0xcb590d74f64e77c9, .hi = 0x8d17bfdf47921ac8, .ex = -1, .sgn=0}, - {.lo = 0xf2be3ecae62789d4, .hi = 0x8e9a21fa66d9ee8d, .ex = -1, .sgn=0}, - {.lo = 0x632b9cff5cfee724, .hi = 0x901c2c1eb93dee39, .ex = -1, .sgn=0}, - {.lo = 0x609c464b3dd676ec, .hi = 0x919ddd5e1ddb8b33, .ex = -1, .sgn=0}, - {.lo = 0x6a1ff8bfe6396e28, .hi = 0x931f34caaaa5d23a, .ex = -1, .sgn=0}, - {.lo = 0xae4ba773da6bf754, .hi = 0x94a03176acf82d45, .ex = -1, .sgn=0}, - {.lo = 0xe06a955a5b8e301d, .hi = 0x9620d274aa290339, .ex = -1, .sgn=0}, - {.lo = 0xfc8b7184b21f2d50, .hi = 0x97a116d7601c3515, .ex = -1, .sgn=0}, - {.lo = 0x9dd1eedf18a2e4df, .hi = 0x9920fdb1c5d5783d, .ex = -1, .sgn=0}, - {.lo = 0x9ffa0d23f3c26c62, .hi = 0x9aa086170c0a8d86, .ex = -1, .sgn=0}, - {.lo = 0xdab6b478577e7be5, .hi = 0x9c1faf1a9db554af, .ex = -1, .sgn=0}, - {.lo = 0xdb895384528d0d60, .hi = 0x9d9e77d020a5bbe6, .ex = -1, .sgn=0}, - {.lo = 0x98dbd3555ebcdefe, .hi = 0x9f1cdf4b76138b02, .ex = -1, .sgn=0}, - {.lo = 0x2f895f44a303cc0b, .hi = 0xa09ae4a0bb300a19, .ex = -1, .sgn=0}, - {.lo = 0xd29d23a624acd00c, .hi = 0xa21886e449b78316, .ex = -1, .sgn=0}, - {.lo = 0x2be036401ba87cc2, .hi = 0xa395c52ab8829dfc, .ex = -1, .sgn=0}, - {.lo = 0x82d9495ead5be348, .hi = 0xa5129e88dc17976a, .ex = -1, .sgn=0}, - {.lo = 0x17218792857f4c5a, .hi = 0xa68f1213c73b5124, .ex = -1, .sgn=0}, - {.lo = 0x3269f4702b88324a, .hi = 0xa80b1ee0cb823c27, .ex = -1, .sgn=0}, - {.lo = 0x8e3bdf8085321556, .hi = 0xa986c40579e11c0a, .ex = -1, .sgn=0}, - {.lo = 0xc1654b64a0081b46, .hi = 0xab020097a33da341, .ex = -1, .sgn=0}, - {.lo = 0x811f953984eff83e, .hi = 0xac7cd3ad58fee7f0, .ex = -1, .sgn=0}, - {.lo = 0x9a5318ac6fe94e4d, .hi = 0xadf73c5ced9db0f3, .ex = -1, .sgn=0}, - {.lo = 0x9fe5f4ea48965e2c, .hi = 0xaf7139bcf5349ac6, .ex = -1, .sgn=0}, - {.lo = 0x63c66682bae74898, .hi = 0xb0eacae4461013ed, .ex = -1, .sgn=0}, - {.lo = 0x695a5332090bb09b, .hi = 0xb263eee9f93e3088, .ex = -1, .sgn=0}, - {.lo = 0x992d96e5021e3c37, .hi = 0xb3dca4e56b1e54bb, .ex = -1, .sgn=0}, - {.lo = 0x971f4da709ad4378, .hi = 0xb554ebee3bf0b58e, .ex = -1, .sgn=0}, - {.lo = 0x35ebacd79f209137, .hi = 0xb6ccc31c5065afee, .ex = -1, .sgn=0}, - {.lo = 0x9cc3ef36746de3b8, .hi = 0xb8442987d22cf576, .ex = -1, .sgn=0}, - {.lo = 0xcdb0531c4e58484b, .hi = 0xb9bb1e4930848ead, .ex = -1, .sgn=0}, - {.lo = 0x55b92083658bb897, .hi = 0xbb31a07920c7b256, .ex = -1, .sgn=0}, - {.lo = 0xa4b0d21fc5036a5, .hi = 0xbca7af309efd7182, .ex = -1, .sgn=0}, - {.lo = 0xd1f90f79f46c7e01, .hi = 0xbe1d4988ee67380c, .ex = -1, .sgn=0}, - {.lo = 0x91a1b5eb79658c67, .hi = 0xbf926e9b9a0f2127, .ex = -1, .sgn=0}, - {.lo = 0x721853f8e528a934, .hi = 0xc1071d8275561f9b, .ex = -1, .sgn=0}, - {.lo = 0xcdc2bd470675104d, .hi = 0xc27b55579c81f96d, .ex = -1, .sgn=0}, - {.lo = 0x3122c2a59efddc37, .hi = 0xc3ef1535754b168d, .ex = -1, .sgn=0}, - {.lo = 0xf4ff2895ab6ebe89, .hi = 0xc5625c36af6a222f, .ex = -1, .sgn=0}, - {.lo = 0x14d24739de27e2e9, .hi = 0xc6d5297645257e8d, .ex = -1, .sgn=0}, - {.lo = 0x4ce0246ad4fa74, .hi = 0xc8477c0f7bde8a98, .ex = -1, .sgn=0}, - {.lo = 0x4319e5ad5b0dcb84, .hi = 0xc9b9531de49eb968, .ex = -1, .sgn=0}, - {.lo = 0xfaa3dfe675a65ee2, .hi = 0xcb2aadbd5ca47af5, .ex = -1, .sgn=0}, - {.lo = 0x2e663b3c7555a6c3, .hi = 0xcc9b8b0a0deff5d4, .ex = -1, .sgn=0}, - {.lo = 0x3c540a9eec47af38, .hi = 0xce0bea206fcf9192, .ex = -1, .sgn=0}, - {.lo = 0xa81290bdbaad62e4, .hi = 0xcf7bca1d476c516d, .ex = -1, .sgn=0}, - {.lo = 0xb9302788604e88f1, .hi = 0xd0eb2a1da855fefd, .ex = -1, .sgn=0}, - {.lo = 0x721fc87ba1d42456, .hi = 0xd25a093ef50f2482, .ex = -1, .sgn=0}, - {.lo = 0x87967926fdcecec4, .hi = 0xd3c8669edf98d680, .ex = -1, .sgn=0}, - {.lo = 0x1df22346611c6b4b, .hi = 0xd536415b69fe4c54, .ex = -1, .sgn=0}, - {.lo = 0x3090d44db12c418c, .hi = 0xd6a39892e6e04764, .ex = -1, .sgn=0}, - {.lo = 0xa573f2aa90434ba5, .hi = 0xd8106b63fa0048a0, .ex = -1, .sgn=0}, - {.lo = 0x2e349483e3fb2a6a, .hi = 0xd97cb8ed98cb93f5, .ex = -1, .sgn=0}, - {.lo = 0x362cb974182e3030, .hi = 0xdae8804f0ae6015b, .ex = -1, .sgn=0}, - {.lo = 0x3ccca3982328ed8b, .hi = 0xdc53c0a7eab49b35, .ex = -1, .sgn=0}, - {.lo = 0x1a5bd9269d408d7e, .hi = 0xddbe791825e8099e, .ex = -1, .sgn=0}, - {.lo = 0xcce2634be2bf54df, .hi = 0xdf28a8bffe06ca56, .ex = -1, .sgn=0}, - {.lo = 0x8aa895d5bf3e84ea, .hi = 0xe0924ec008f734fd, .ex = -1, .sgn=0}, - {.lo = 0xf7a1f9bd9ba13b6b, .hi = 0xe1fb6a3931894b38, .ex = -1, .sgn=0}, - {.lo = 0x7b32c72e31824e51, .hi = 0xe363fa4cb8005482, .ex = -1, .sgn=0}, - {.lo = 0xd40e9e6b989f89e5, .hi = 0xe4cbfe1c329c453a, .ex = -1, .sgn=0}, - {.lo = 0x2872ce1bfc7ad1cd, .hi = 0xe63374c98e22f0b4, .ex = -1, .sgn=0}, - {.lo = 0xf1b65cc5fd780262, .hi = 0xe79a5d770e6905dc, .ex = -1, .sgn=0}, - {.lo = 0x431626c10485bdda, .hi = 0xe900b7474edad637, .ex = -1, .sgn=0}, - {.lo = 0xcc39cfcc29960b1, .hi = 0xea66815d4304e6c8, .ex = -1, .sgn=0}, - {.lo = 0x1d90f780ae951140, .hi = 0xebcbbadc371c4aaa, .ex = -1, .sgn=0}, - {.lo = 0xc71debc372b6f9d4, .hi = 0xed3062e7d086c6f0, .ex = -1, .sgn=0}, - {.lo = 0x2a24164daec85ccb, .hi = 0xee9478a40e62bf86, .ex = -1, .sgn=0}, - {.lo = 0x527233b40d3432bb, .hi = 0xeff7fb354a0eecb1, .ex = -1, .sgn=0}, - {.lo = 0x6c48e9e3420b0f1e, .hi = 0xf15ae9c037b1d8f0, .ex = -1, .sgn=0}, - {.lo = 0x7f232aee178c6323, .hi = 0xf2bd4369e6c126d3, .ex = -1, .sgn=0}, - {.lo = 0x3c7f10db458c337c, .hi = 0xf41f0757c2889e84, .ex = -1, .sgn=0}, - {.lo = 0x93fa6107c4327527, .hi = 0xf58034af92b102a7, .ex = -1, .sgn=0}, - {.lo = 0xe1079824233fef46, .hi = 0xf6e0ca977bc6ac45, .ex = -1, .sgn=0}, - {.lo = 0xa9a56012067c570c, .hi = 0xf840c835ffbfed66, .ex = -1, .sgn=0}, - {.lo = 0x8da894471de1a18, .hi = 0xf9a02cb1fe833a0d, .ex = -1, .sgn=0}, - {.lo = 0x343fbf4a7d42af3, .hi = 0xfafef732b66d1742, .ex = -1, .sgn=0}, - {.lo = 0x27c07c911290b8d1, .hi = 0xfc5d26dfc4d5cfda, .ex = -1, .sgn=0}, - {.lo = 0x2377c3799c052fa, .hi = 0xfdbabae12696eea4, .ex = -1, .sgn=0}, - {.lo = 0xa9c6ba50490539f, .hi = 0xff17b25f38907dad, .ex = -1, .sgn=0}, - {.lo = 0x6f53873e2f1477ff, .hi = 0x803a06415c170525, .ex = 0, .sgn=0}, - {.lo = 0x5ca183dc973abc22, .hi = 0x80e7e43a61f5b6cb, .ex = 0, .sgn=0}, - {.lo = 0x9fba97fdf0c4d24c, .hi = 0x819572af6decac84, .ex = 0, .sgn=0}, - {.lo = 0x6fb2123fedfa6e22, .hi = 0x8242b1357110d372, .ex = 0, .sgn=0}, - {.lo = 0x91a965931f1a200a, .hi = 0x82ef9f618dc5b70e, .ex = 0, .sgn=0}, - {.lo = 0xbfd79717f2880abf, .hi = 0x839c3cc917ff6cb4, .ex = 0, .sgn=0}, - {.lo = 0x246efcff30cb064a, .hi = 0x8448890195846099, .ex = 0, .sgn=0}, - {.lo = 0x51917cac857fd5f5, .hi = 0x84f483a0be2f0403, .ex = 0, .sgn=0}, - {.lo = 0x327888fe4b62687b, .hi = 0x85a02c3c7c2f5ca5, .ex = 0, .sgn=0}, - {.lo = 0x85043222c9bdd18d, .hi = 0x864b826aec4c74e5, .ex = 0, .sgn=0}, - {.lo = 0x7e0b9b07548471a2, .hi = 0x86f685c25e25acf5, .ex = 0, .sgn=0}, - {.lo = 0x4e091160e2430712, .hi = 0x87a135d95473ec89, .ex = 0, .sgn=0}, - {.lo = 0x4f14c8afe4560291, .hi = 0x884b9246854ab50b, .ex = 0, .sgn=0}, - {.lo = 0xb892ca8361d8c84c, .hi = 0x88f59aa0da591421, .ex = 0, .sgn=0}, - {.lo = 0xc88302a31afce54a, .hi = 0x899f4e7f712a765e, .ex = 0, .sgn=0}, - {.lo = 0x660558a02136130a, .hi = 0x8a48ad799b6759f3, .ex = 0, .sgn=0}, - {.lo = 0x545f7d79ead8fa19, .hi = 0x8af1b726df15e13c, .ex = 0, .sgn=0}, - {.lo = 0x21a6675f51580bc4, .hi = 0x8b9a6b1ef6da4502, .ex = 0, .sgn=0}, - {.lo = 0x101a5adbcb9ffb43, .hi = 0x8c42c8f9d2372644, .ex = 0, .sgn=0}, - {.lo = 0x4d49cbaf15aecd80, .hi = 0x8cead04f95cdbf66, .ex = 0, .sgn=0}, - {.lo = 0xde2d43c6b67a7cbe, .hi = 0x8d9280b89b9df49b, .ex = 0, .sgn=0}, - {.lo = 0xbba4cfecbff54867, .hi = 0x8e39d9cd73464364, .ex = 0, .sgn=0}, - {.lo = 0xaf0e2345f3bd24b4, .hi = 0x8ee0db26e24390f8, .ex = 0, .sgn=0}, - {.lo = 0x9311a82459aa0f72, .hi = 0x8f87845de430d777, .ex = 0, .sgn=0}, - {.lo = 0xb144016c7a30b39a, .hi = 0x902dd50bab06b1b7, .ex = 0, .sgn=0}, - {.lo = 0x9d1072e09b72292, .hi = 0x90d3ccc99f5ac58b, .ex = 0, .sgn=0}, - {.lo = 0x6714fe6925b78cc4, .hi = 0x91796b31609f0c54, .ex = 0, .sgn=0}, - {.lo = 0x33d0a284a8c954ad, .hi = 0x921eafdcc560f9c5, .ex = 0, .sgn=0}, - {.lo = 0x1f8481e704e4a767, .hi = 0x92c39a65db88809d, .ex = 0, .sgn=0}, - {.lo = 0xb17821911e71c16e, .hi = 0x93682a66e896f544, .ex = 0, .sgn=0}, - {.lo = 0x1489a97671a42, .hi = 0x940c5f7a69e5ce1c, .ex = 0, .sgn=0}, - {.lo = 0xd6c7af02d5c16fd9, .hi = 0x94b0393b14e54156, .ex = 0, .sgn=0}, - {.lo = 0xac0106650f4ef023, .hi = 0x9553b743d75ac03f, .ex = 0, .sgn=0}, - {.lo = 0xd9f8e1a446e973b9, .hi = 0x95f6d92fd79f4fba, .ex = 0, .sgn=0}, - {.lo = 0xa7a7556c3b33abc1, .hi = 0x96999e9a74ddbde3, .ex = 0, .sgn=0}, - {.lo = 0xc0a03934f0cce19b, .hi = 0x973c071f4750b49c, .ex = 0, .sgn=0}, - {.lo = 0xd243aa0843a2c144, .hi = 0x97de125a2080a8ed, .ex = 0, .sgn=0}, - {.lo = 0x19cec845ac87a5c6, .hi = 0x987fbfe70b81a708, .ex = 0, .sgn=0}, - {.lo = 0xc4b992a37fb9b9bd, .hi = 0x99210f624d30facb, .ex = 0, .sgn=0}, - {.lo = 0x1ab42d43235757b6, .hi = 0x99c200686472b4a8, .ex = 0, .sgn=0}, - {.lo = 0x7e92c655656e6b85, .hi = 0x9a6292960a6f0ab0, .ex = 0, .sgn=0}, - {.lo = 0x698b94f50326a043, .hi = 0x9b02c58832cf95c0, .ex = 0, .sgn=0}, - {.lo = 0x9a5614e8ffbeac6f, .hi = 0x9ba298dc0bfc6a88, .ex = 0, .sgn=0}, - {.lo = 0xc7fd954194e6d8aa, .hi = 0x9c420c2eff590e5f, .ex = 0, .sgn=0}, - {.lo = 0x3e93627de8fd5779, .hi = 0x9ce11f1eb18147b1, .ex = 0, .sgn=0}, - {.lo = 0xe25e39549638ae68, .hi = 0x9d7fd1490285c9e3, .ex = 0, .sgn=0}, - {.lo = 0x2cad377d5c9c35d8, .hi = 0x9e1e224c0e28bc94, .ex = 0, .sgn=0}, - {.lo = 0xcc141e10c6460c8b, .hi = 0x9ebc11c62c1a1dfb, .ex = 0, .sgn=0}, - {.lo = 0xa88d5f46834bbf8d, .hi = 0x9f599f55f0340061, .ex = 0, .sgn=0}, - {.lo = 0x22cc118a0c118aa0, .hi = 0x9ff6ca9a2ab6a26d, .ex = 0, .sgn=0}, - {.lo = 0x7cec6df5bea167cf, .hi = 0xa0939331e8846237, .ex = 0, .sgn=0}, - {.lo = 0x71acea2819360c35, .hi = 0xa12ff8bc735d8af6, .ex = 0, .sgn=0}, - {.lo = 0x166c36e7bb3c402f, .hi = 0xa1cbfad9521bfd1b, .ex = 0, .sgn=0}, - {.lo = 0x3b5167ee359a234e, .hi = 0xa267992848eeb0c0, .ex = 0, .sgn=0}, - {.lo = 0x9443372e20d4377c, .hi = 0xa302d34959951243, .ex = 0, .sgn=0}, - {.lo = 0xca9a8a720d4c69c, .hi = 0xa39da8dcc39a38e5, .ex = 0, .sgn=0}, - {.lo = 0xbf623cf5301a2dde, .hi = 0xa4381983048ff747, .ex = 0, .sgn=0}, - {.lo = 0x23d251cc8d7975cc, .hi = 0xa4d224dcd849c5b0, .ex = 0, .sgn=0}, - {.lo = 0x189d39ffe11aaa2b, .hi = 0xa56bca8b391785db, .ex = 0, .sgn=0}, - {.lo = 0x8c33ebf3aa8501fb, .hi = 0xa6050a2f60002049, .ex = 0, .sgn=0}, - {.lo = 0x9b3ad6e4022183d9, .hi = 0xa69de36ac4fbfadc, .ex = 0, .sgn=0}, - {.lo = 0x149f6e75993468a3, .hi = 0xa73655df1f2f489e, .ex = 0, .sgn=0}, - {.lo = 0x6b2a39f856a69781, .hi = 0xa7ce612e65243291, .ex = 0, .sgn=0}, - {.lo = 0x3463a2c2e6e9cc55, .hi = 0xa86604facd04d969, .ex = 0, .sgn=0}, - {.lo = 0x6cc14c4f53e2e82d, .hi = 0xa8fd40e6ccd52ffd, .ex = 0, .sgn=0}, - {.lo = 0xd147625fda929af8, .hi = 0xa99414951aacae5e, .ex = 0, .sgn=0}, - {.lo = 0xb714ee81b53b4b9d, .hi = 0xaa2a7fa8acefdd63, .ex = 0, .sgn=0}, - {.lo = 0xe1b3dfc4dbda9bfd, .hi = 0xaac081c4ba89ba8a, .ex = 0, .sgn=0}, - {.lo = 0xf17cee69b0d2ecde, .hi = 0xab561a8cbb24f410, .ex = 0, .sgn=0}, - {.lo = 0x1becda8089c1a94c, .hi = 0xabeb49a46764fd15, .ex = 0, .sgn=0}, - {.lo = 0xf86ba0dde982fb59, .hi = 0xac800eafb91ef9a9, .ex = 0, .sgn=0}, - {.lo = 0x44bf16268608db96, .hi = 0xad146952eb9282af, .ex = 0, .sgn=0}, - {.lo = 0x9d30d4cfeb04f1fb, .hi = 0xada859327ba24151, .ex = 0, .sgn=0}, - {.lo = 0x3d53817865422565, .hi = 0xae3bddf3280c620d, .ex = 0, .sgn=0}, - {.lo = 0xf74d099042e8f326, .hi = 0xaecef739f1a2df10, .ex = 0, .sgn=0}, - {.lo = 0xa89a9b8f726b95bf, .hi = 0xaf61a4ac1b83a1de, .ex = 0, .sgn=0}, - {.lo = 0x8c679e67fc462d51, .hi = 0xaff3e5ef2b507c06, .ex = 0, .sgn=0}, - {.lo = 0xe4cad00d5c94bcd2, .hi = 0xb085baa8e966f6da, .ex = 0, .sgn=0}, - {.lo = 0x8d8be132d576e614, .hi = 0xb117227f6117f9f9, .ex = 0, .sgn=0}, - {.lo = 0x24784f32c3e3e5bd, .hi = 0xb1a81d18e0df4889, .ex = 0, .sgn=0}, - {.lo = 0x8cc7d4bd05ffd5ae, .hi = 0xb238aa1bfa9ad507, .ex = 0, .sgn=0}, - {.lo = 0xac9f7ebbc469ef59, .hi = 0xb2c8c92f83c1eb87, .ex = 0, .sgn=0}, - {.lo = 0x5d6635109164f740, .hi = 0xb35879fa959c323c, .ex = 0, .sgn=0}, - {.lo = 0xa156468ef6c18c60, .hi = 0xb3e7bc248d78802e, .ex = 0, .sgn=0}, - {.lo = 0x4a85350f69018c55, .hi = 0xb4768f550ce389fd, .ex = 0, .sgn=0}, -}; - -/* Table containing 128-bit approximations of cos2pi(i/2^11) for 0 <= i < 256 - (to nearest). - Each entry is to be interpreted as (hi/2^64+lo/2^128)*2^ex*(-1)*sgn. - Generated with computeC() from sin.sage. */ -static const dint64_t C[256] = { - {.lo = 0x0, .hi = 0x8000000000000000, .ex = 1, .sgn=0}, - {.lo = 0x3031437d7eccb9df, .hi = 0xffffb10b10e80e95, .ex = 0, .sgn=0}, - {.lo = 0x38e310779edfec68, .hi = 0xfffec42c7454926b, .ex = 0, .sgn=0}, - {.lo = 0x69fff9ae0dedb047, .hi = 0xfffd3964bc6275ba, .ex = 0, .sgn=0}, - {.lo = 0xb47903f7a19f8ee2, .hi = 0xfffb10b4dc96dabb, .ex = 0, .sgn=0}, - {.lo = 0x8cc193c5d508e13f, .hi = 0xfff84a1e29de8571, .ex = 0, .sgn=0}, - {.lo = 0x43366df666fd54ff, .hi = 0xfff4e5a25a8d095b, .ex = 0, .sgn=0}, - {.lo = 0x5428ed0647c9e5d1, .hi = 0xfff0e343865bbb13, .ex = 0, .sgn=0}, - {.lo = 0x5657552366961732, .hi = 0xffec4304266865d9, .ex = 0, .sgn=0}, - {.lo = 0x53aa9423bb0adc21, .hi = 0xffe704e71533c508, .ex = 0, .sgn=0}, - {.lo = 0x7d209f32d42d864e, .hi = 0xffe128ef8e9fc17a, .ex = 0, .sgn=0}, - {.lo = 0x4fd8f038449ec436, .hi = 0xffdaaf212fed72db, .ex = 0, .sgn=0}, - {.lo = 0x664649b4d541b9c5, .hi = 0xffd3977ff7bae4e9, .ex = 0, .sgn=0}, - {.lo = 0x5595ca3f421ae09c, .hi = 0xffcbe2104600a0a9, .ex = 0, .sgn=0}, - {.lo = 0x1c676208aa3be545, .hi = 0xffc38ed6dc0ef98b, .ex = 0, .sgn=0}, - {.lo = 0xccfed60a91097c48, .hi = 0xffba9dd8dc8b1e83, .ex = 0, .sgn=0}, - {.lo = 0x421e8edaaf59453e, .hi = 0xffb10f1bcb6bef1d, .ex = 0, .sgn=0}, - {.lo = 0xd2c665c2da3e7844, .hi = 0xffa6e2a58df6947d, .ex = 0, .sgn=0}, - {.lo = 0x1e1862cca089938b, .hi = 0xff9c187c6abade6a, .ex = 0, .sgn=0}, - {.lo = 0x2dabd3195a05710f, .hi = 0xff90b0a7098f6443, .ex = 0, .sgn=0}, - {.lo = 0x519c314973ccae6b, .hi = 0xff84ab2c738d6a03, .ex = 0, .sgn=0}, - {.lo = 0x3ea4f30adda3016f, .hi = 0xff780814130c893c, .ex = 0, .sgn=0}, - {.lo = 0x1b9d5851979f28fb, .hi = 0xff6ac765b39e1e19, .ex = 0, .sgn=0}, - {.lo = 0x50a7bb6a6ee3b0f1, .hi = 0xff5ce92982087867, .ex = 0, .sgn=0}, - {.lo = 0xf668633f1ab858a, .hi = 0xff4e6d680c41d0a9, .ex = 0, .sgn=0}, - {.lo = 0xb085c1828f69296a, .hi = 0xff3f542a416b0134, .ex = 0, .sgn=0}, - {.lo = 0x27e31939e2eec09c, .hi = 0xff2f9d7971ca0364, .ex = 0, .sgn=0}, - {.lo = 0xf5971326a3540ea9, .hi = 0xff1f495f4ec430d7, .ex = 0, .sgn=0}, - {.lo = 0x1f1901544271c3f8, .hi = 0xff0e57e5ead848d1, .ex = 0, .sgn=0}, - {.lo = 0xe0abd3a9b64df725, .hi = 0xfefcc917b99839a5, .ex = 0, .sgn=0}, - {.lo = 0xec34413e87ef2740, .hi = 0xfeea9cff8fa2ae54, .ex = 0, .sgn=0}, - {.lo = 0x2f88b949a72ff96c, .hi = 0xfed7d3a8a29c603b, .ex = 0, .sgn=0}, - {.lo = 0x41390efdc726e9ef, .hi = 0xfec46d1e89292cf0, .ex = 0, .sgn=0}, - {.lo = 0xb7b6cc53c3abc817, .hi = 0xfeb0696d3ae4f04d, .ex = 0, .sgn=0}, - {.lo = 0xd3af6ee4f2101c20, .hi = 0xfe9bc8a1105c22a5, .ex = 0, .sgn=0}, - {.lo = 0xb4f70c910505e10, .hi = 0xfe868ac6c3043b2e, .ex = 0, .sgn=0}, - {.lo = 0x2907cf2b3f6feac2, .hi = 0xfe70afeb6d33d6a2, .ex = 0, .sgn=0}, - {.lo = 0xd54faa364b7da8f6, .hi = 0xfe5a381c8a1aa224, .ex = 0, .sgn=0}, - {.lo = 0x87b8875373a818a4, .hi = 0xfe432367f5b90a62, .ex = 0, .sgn=0}, - {.lo = 0x8598c2c429caf7, .hi = 0xfe2b71dbecd7aefc, .ex = 0, .sgn=0}, - {.lo = 0x90cd1d959db674ef, .hi = 0xfe1323870cfe9a3d, .ex = 0, .sgn=0}, - {.lo = 0x9bfe5c51e91cbdcd, .hi = 0xfdfa3878546c3d28, .ex = 0, .sgn=0}, - {.lo = 0xe276d247626a23fd, .hi = 0xfde0b0bf220c2fd4, .ex = 0, .sgn=0}, - {.lo = 0x499ddb331d19539d, .hi = 0xfdc68c6b356db62f, .ex = 0, .sgn=0}, - {.lo = 0xfac7397cc07a6470, .hi = 0xfdabcb8caeba091b, .ex = 0, .sgn=0}, - {.lo = 0xd6e270740a186977, .hi = 0xfd906e340eaa6401, .ex = 0, .sgn=0}, - {.lo = 0x61beb8cd2696fc78, .hi = 0xfd747472367dd6c5, .ex = 0, .sgn=0}, - {.lo = 0x6c696582f346fd91, .hi = 0xfd57de5867eedc39, .ex = 0, .sgn=0}, - {.lo = 0xeae6bd951c1dabbe, .hi = 0xfd3aabf84528b50b, .ex = 0, .sgn=0}, - {.lo = 0x863b87258f11ad7e, .hi = 0xfd1cdd63d0bc8735, .ex = 0, .sgn=0}, - {.lo = 0xa06fab9f9d106709, .hi = 0xfcfe72ad6d9641f2, .ex = 0, .sgn=0}, - {.lo = 0xa4e064308f4999f4, .hi = 0xfcdf6be7def1464c, .ex = 0, .sgn=0}, - {.lo = 0xa3e22b4d38917e73, .hi = 0xfcbfc926484cd43a, .ex = 0, .sgn=0}, - {.lo = 0x5d582cac7cb4391c, .hi = 0xfc9f8a7c2d603c60, .ex = 0, .sgn=0}, - {.lo = 0x2880268f2e62955, .hi = 0xfc7eaffd720ed673, .ex = 0, .sgn=0}, - {.lo = 0x1c0d254b6c8da4bd, .hi = 0xfc5d39be5a5bbc4b, .ex = 0, .sgn=0}, - {.lo = 0x256778ffcb5c1769, .hi = 0xfc3b27d38a5d49ab, .ex = 0, .sgn=0}, - {.lo = 0x9433b49289417ea2, .hi = 0xfc187a52063060c2, .ex = 0, .sgn=0}, - {.lo = 0x25aafd7fdba12c5f, .hi = 0xfbf5314f31eb7375, .ex = 0, .sgn=0}, - {.lo = 0x7190c94899dff1b8, .hi = 0xfbd14ce0d191516e, .ex = 0, .sgn=0}, - {.lo = 0xe63ae8632b84473c, .hi = 0xfbaccd1d0903bb09, .ex = 0, .sgn=0}, - {.lo = 0x75df66f0ec3dd459, .hi = 0xfb87b21a5bf5b917, .ex = 0, .sgn=0}, - {.lo = 0x61ce9d5ef5a81487, .hi = 0xfb61fbefadddb985, .ex = 0, .sgn=0}, - {.lo = 0xb4b54683879c9c17, .hi = 0xfb3baab441e770f7, .ex = 0, .sgn=0}, - {.lo = 0x2172a361fd2a722f, .hi = 0xfb14be7fbae58156, .ex = 0, .sgn=0}, - {.lo = 0x2079880c450348ac, .hi = 0xfaed376a1b42e559, .ex = 0, .sgn=0}, - {.lo = 0x4a188aa367f90ab1, .hi = 0xfac5158bc4f4211f, .ex = 0, .sgn=0}, - {.lo = 0x10655ecd5cc771d8, .hi = 0xfa9c58fd796837d4, .ex = 0, .sgn=0}, - {.lo = 0x1fe196a53fb5b237, .hi = 0xfa7301d859796671, .ex = 0, .sgn=0}, - {.lo = 0xd24377c77a591e24, .hi = 0xfa491035e55da3a3, .ex = 0, .sgn=0}, - {.lo = 0x431c393c7f62da65, .hi = 0xfa1e842ffc96e4e0, .ex = 0, .sgn=0}, - {.lo = 0xba5dbf4510eddc8f, .hi = 0xf9f35de0dde328ab, .ex = 0, .sgn=0}, - {.lo = 0x4504ae08d19b2980, .hi = 0xf9c79d63272c4628, .ex = 0, .sgn=0}, - {.lo = 0x78685d850f80ecdc, .hi = 0xf99b42d1d57781eb, .ex = 0, .sgn=0}, - {.lo = 0x80e8c17bf80e8f02, .hi = 0xf96e4e4844d4e82a, .ex = 0, .sgn=0}, - {.lo = 0xc0e2a1352ed7f292, .hi = 0xf940bfe2304e6c45, .ex = 0, .sgn=0}, - {.lo = 0x68fc6e4d6a920bd2, .hi = 0xf91297bbb1d6cdbe, .ex = 0, .sgn=0}, - {.lo = 0x9701914c7f8fbcd7, .hi = 0xf8e3d5f1423842a0, .ex = 0, .sgn=0}, - {.lo = 0xac9f07f54ff5bc14, .hi = 0xf8b47a9fb902e76c, .ex = 0, .sgn=0}, - {.lo = 0xb36a9dfaadafc1e1, .hi = 0xf88485e44c7af48a, .ex = 0, .sgn=0}, - {.lo = 0xc7adc6b4988891bb, .hi = 0xf853f7dc9186b952, .ex = 0, .sgn=0}, - {.lo = 0xa776175bd284fe05, .hi = 0xf822d0a67b9c5cb5, .ex = 0, .sgn=0}, - {.lo = 0xa76f7efc19aed41c, .hi = 0xf7f110605caf6390, .ex = 0, .sgn=0}, - {.lo = 0x730785813f78aa1e, .hi = 0xf7beb728e51dfcb8, .ex = 0, .sgn=0}, - {.lo = 0x214cffcee9dd33ca, .hi = 0xf78bc51f239e12c6, .ex = 0, .sgn=0}, - {.lo = 0x4becad887680c197, .hi = 0xf7583a62852a23b2, .ex = 0, .sgn=0}, - {.lo = 0xf99107e50d631330, .hi = 0xf7241712d4edde49, .ex = 0, .sgn=0}, - {.lo = 0x50ca117eb18beed7, .hi = 0xf6ef5b503c328589, .ex = 0, .sgn=0}, - {.lo = 0x2c791f59cc1ffc23, .hi = 0xf6ba073b424b19e8, .ex = 0, .sgn=0}, - {.lo = 0xce8c455197cdf8a7, .hi = 0xf6841af4cc8048a4, .ex = 0, .sgn=0}, - {.lo = 0x119d358de0493956, .hi = 0xf64d969e1dfc2119, .ex = 0, .sgn=0}, - {.lo = 0x9dc7e5954c5a8f24, .hi = 0xf6167a58d7b59026, .ex = 0, .sgn=0}, - {.lo = 0xc8c615e72768d6b5, .hi = 0xf5dec646f85ba1c6, .ex = 0, .sgn=0}, - {.lo = 0xed0dd4bf62edd13f, .hi = 0xf5a67a8adc4088ca, .ex = 0, .sgn=0}, - {.lo = 0x275a2bbb2bab6c8a, .hi = 0xf56d97473d446cda, .ex = 0, .sgn=0}, - {.lo = 0x8da64484aaa0febc, .hi = 0xf5341c9f32bffeb9, .ex = 0, .sgn=0}, - {.lo = 0x163c5c7f03b718c5, .hi = 0xf4fa0ab6316ed2ec, .ex = 0, .sgn=0}, - {.lo = 0x890ac4aafa6a37bf, .hi = 0xf4bf61b00b5982b7, .ex = 0, .sgn=0}, - {.lo = 0xf8f9d3b87d11fd52, .hi = 0xf48421b0efbf939b, .ex = 0, .sgn=0}, - {.lo = 0x667e06866c07c369, .hi = 0xf4484add6b01254b, .ex = 0, .sgn=0}, - {.lo = 0x5019794a1f5896e5, .hi = 0xf40bdd5a6688662f, .ex = 0, .sgn=0}, - {.lo = 0x18ef535a7ffa7a3d, .hi = 0xf3ced94d28b2ce8a, .ex = 0, .sgn=0}, - {.lo = 0x50f29b4b49f31c37, .hi = 0xf3913edb54ba2242, .ex = 0, .sgn=0}, - {.lo = 0xd981acdcf6bc3e4, .hi = 0xf3530e2aea9d3966, .ex = 0, .sgn=0}, - {.lo = 0xa5486bdc455d56a2, .hi = 0xf314476247088f74, .ex = 0, .sgn=0}, - {.lo = 0x431be53f92ece9e6, .hi = 0xf2d4eaa8233e997d, .ex = 0, .sgn=0}, - {.lo = 0xebadcdbf915e8f6c, .hi = 0xf294f82394ffe320, .ex = 0, .sgn=0}, - {.lo = 0xaf0eed81e8c51e55, .hi = 0xf2546ffc0e72f286, .ex = 0, .sgn=0}, - {.lo = 0xe7112e89103cc0c7, .hi = 0xf21352595e0bf350, .ex = 0, .sgn=0}, - {.lo = 0x844e6a35ddc2b713, .hi = 0xf1d19f63ae7428a2, .ex = 0, .sgn=0}, - {.lo = 0x8f6bac72988088b0, .hi = 0xf18f574386712643, .ex = 0, .sgn=0}, - {.lo = 0x2730081c758fb42b, .hi = 0xf14c7a21c8cbd0f4, .ex = 0, .sgn=0}, - {.lo = 0x67127db35b287316, .hi = 0xf1090827b43725fd, .ex = 0, .sgn=0}, - {.lo = 0xc4e557b119ef3185, .hi = 0xf0c5017ee336ca0f, .ex = 0, .sgn=0}, - {.lo = 0x973ea9903ed5125f, .hi = 0xf08066514c055f7e, .ex = 0, .sgn=0}, - {.lo = 0x992d39ec5c561d28, .hi = 0xf03b36c9407aa3e8, .ex = 0, .sgn=0}, - {.lo = 0x62aef7b55319d1d4, .hi = 0xeff573116df1555d, .ex = 0, .sgn=0}, - {.lo = 0xf03a18a5e16ab641, .hi = 0xefaf1b54dd2cdf0f, .ex = 0, .sgn=0}, - {.lo = 0x767c0e8ad33bc085, .hi = 0xef682fbef23ecda6, .ex = 0, .sgn=0}, - {.lo = 0xe2398bf0eeb28cde, .hi = 0xef20b07b6c6c0b37, .ex = 0, .sgn=0}, - {.lo = 0x86f8c20fb664b01b, .hi = 0xeed89db66611e307, .ex = 0, .sgn=0}, - {.lo = 0xa1d2c3d018a9279f, .hi = 0xee8ff79c548acd0f, .ex = 0, .sgn=0}, - {.lo = 0x7872773830d368be, .hi = 0xee46be5a0813016b, .ex = 0, .sgn=0}, - {.lo = 0xfee6a1eebfa13b4a, .hi = 0xedfcf21cabacd3b1, .ex = 0, .sgn=0}, - {.lo = 0x11815196b9fbf5df, .hi = 0xedb29311c504d652, .ex = 0, .sgn=0}, - {.lo = 0x7289102076a125e5, .hi = 0xed67a1673455c601, .ex = 0, .sgn=0}, - {.lo = 0xddffe98c4f8aa031, .hi = 0xed1c1d4b344c3d4f, .ex = 0, .sgn=0}, - {.lo = 0xa8392eb238578ab0, .hi = 0xecd006ec59ea306f, .ex = 0, .sgn=0}, - {.lo = 0x7e610231ac1d6181, .hi = 0xec835e79946a3145, .ex = 0, .sgn=0}, - {.lo = 0x278047ae3dd0889, .hi = 0xec3624222d227bd1, .ex = 0, .sgn=0}, - {.lo = 0x1e99ccb9adc62ca6, .hi = 0xebe85815c767cb00, .ex = 0, .sgn=0}, - {.lo = 0xdae311e656e0661, .hi = 0xeb99fa84606ff5ff, .ex = 0, .sgn=0}, - {.lo = 0x39e39c6c2ab3655d, .hi = 0xeb4b0b9e4f345617, .ex = 0, .sgn=0}, - {.lo = 0x3383bbb5156bf1d7, .hi = 0xeafb8b944453f52f, .ex = 0, .sgn=0}, - {.lo = 0x24db98ad3a0647a1, .hi = 0xeaab7a9749f584fe, .ex = 0, .sgn=0}, - {.lo = 0x4a0ca5ea449b1c83, .hi = 0xea5ad8d8c3a91f05, .ex = 0, .sgn=0}, - {.lo = 0x15ad45b4a1b5e823, .hi = 0xea09a68a6e49cd62, .ex = 0, .sgn=0}, - {.lo = 0xcd24d4bd1056c826, .hi = 0xe9b7e3de5fdedc8b, .ex = 0, .sgn=0}, - {.lo = 0x89a92b199adfbafa, .hi = 0xe9659107077cf60f, .ex = 0, .sgn=0}, - {.lo = 0xacb1c26a06e5ae02, .hi = 0xe912ae372d27045d, .ex = 0, .sgn=0}, - {.lo = 0xf8972affb3d98e1f, .hi = 0xe8bf3ba1f1aedfbb, .ex = 0, .sgn=0}, - {.lo = 0x9fec1e78c4376186, .hi = 0xe86b397ace95c46f, .ex = 0, .sgn=0}, - {.lo = 0xbfe8378abfb87b6f, .hi = 0xe816a7f595ec9232, .ex = 0, .sgn=0}, - {.lo = 0xdbfb0fe56c6f80fe, .hi = 0xe7c187467233d508, .ex = 0, .sgn=0}, - {.lo = 0x125129529d48a92f, .hi = 0xe76bd7a1e63b9786, .ex = 0, .sgn=0}, - {.lo = 0xe2ba81b9ce96e02e, .hi = 0xe715993ccd02fe9c, .ex = 0, .sgn=0}, - {.lo = 0x82fcedb4c6434d76, .hi = 0xe6becc4c5997af06, .ex = 0, .sgn=0}, - {.lo = 0xdd2a3e32c3859960, .hi = 0xe667710616f4fc59, .ex = 0, .sgn=0}, - {.lo = 0x7613b68f6ab03130, .hi = 0xe60f879fe7e2e1e5, .ex = 0, .sgn=0}, - {.lo = 0x9b695cd67c93bd79, .hi = 0xe5b7105006d4c560, .ex = 0, .sgn=0}, - {.lo = 0x5a7c210a3a15e7ea, .hi = 0xe55e0b4d05c80388, .ex = 0, .sgn=0}, - {.lo = 0xe1f5a58c80292554, .hi = 0xe50478cdce2246bc, .ex = 0, .sgn=0}, - {.lo = 0x122785ae67f5515d, .hi = 0xe4aa5909a08fa7b4, .ex = 0, .sgn=0}, - {.lo = 0x20d63b5b9e3cd6ac, .hi = 0xe44fac3814e09856, .ex = 0, .sgn=0}, - {.lo = 0x56992551ae074e99, .hi = 0xe3f4729119e798d9, .ex = 0, .sgn=0}, - {.lo = 0xd1197dc12c63176, .hi = 0xe398ac4cf556b732, .ex = 0, .sgn=0}, - {.lo = 0x36563e2ffad8351a, .hi = 0xe33c59a4439cd8ec, .ex = 0, .sgn=0}, - {.lo = 0xd6fe4dd22e60a4a2, .hi = 0xe2df7acff7c2cf83, .ex = 0, .sgn=0}, - {.lo = 0xfd39138aa2d508ed, .hi = 0xe28210095b483751, .ex = 0, .sgn=0}, - {.lo = 0xe0521df01a1be6f5, .hi = 0xe224198a0e002123, .ex = 0, .sgn=0}, - {.lo = 0xf4e8a8372f8c5810, .hi = 0xe1c5978c05ed8691, .ex = 0, .sgn=0}, - {.lo = 0xe2f9d4600f4d0325, .hi = 0xe1668a498f1f892c, .ex = 0, .sgn=0}, - {.lo = 0x6ba8a9d9ba877899, .hi = 0xe106f1fd4b8d7c96, .ex = 0, .sgn=0}, - {.lo = 0x6d6c98fe79817946, .hi = 0xe0a6cee232f2bb9c, .ex = 0, .sgn=0}, - {.lo = 0x55ff6038a5197367, .hi = 0xe046213392aa486c, .ex = 0, .sgn=0}, - {.lo = 0x720588ff6547d884, .hi = 0xdfe4e92d0d8a37f5, .ex = 0, .sgn=0}, - {.lo = 0xab01350f013d78dd, .hi = 0xdf83270a9bbee890, .ex = 0, .sgn=0}, - {.lo = 0x64a58b2f103485dd, .hi = 0xdf20db088aa60404, .ex = 0, .sgn=0}, - {.lo = 0x4b19aa71fec3ae6d, .hi = 0xdebe05637ca94cfb, .ex = 0, .sgn=0}, - {.lo = 0x4248f15548f69ca, .hi = 0xde5aa65869193805, .ex = 0, .sgn=0}, - {.lo = 0xd597b10a01676659, .hi = 0xddf6be249c075037, .ex = 0, .sgn=0}, - {.lo = 0x739c45b982193b5e, .hi = 0xdd924d05b620678a, .ex = 0, .sgn=0}, - {.lo = 0x49c6e0ea76cbcaac, .hi = 0xdd2d5339ac8692fd, .ex = 0, .sgn=0}, - {.lo = 0xb2069fd0b482b4e8, .hi = 0xdcc7d0fec8aaf2aa, .ex = 0, .sgn=0}, - {.lo = 0xaca8017e375b64e5, .hi = 0xdc61c693a82745d5, .ex = 0, .sgn=0}, - {.lo = 0xccb7fd40d543f4a1, .hi = 0xdbfb34373c974b0e, .ex = 0, .sgn=0}, - {.lo = 0x2c19b63253da43fc, .hi = 0xdb941a28cb71ec87, .ex = 0, .sgn=0}, - {.lo = 0x5a98479cbef2ecbc, .hi = 0xdb2c78a7ede238a9, .ex = 0, .sgn=0}, - {.lo = 0x5b267c1bcff0ab62, .hi = 0xdac44ff490a02710, .ex = 0, .sgn=0}, - {.lo = 0xe257bde73d83dc1a, .hi = 0xda5ba04ef3c929f4, .ex = 0, .sgn=0}, - {.lo = 0x28e81dcb6dab91ac, .hi = 0xd9f269f7aab88c29, .ex = 0, .sgn=0}, - {.lo = 0xc4e4dc69fc2fff6f, .hi = 0xd988ad2f9bdf9bbb, .ex = 0, .sgn=0}, - {.lo = 0x1bb35ad6d2e74b67, .hi = 0xd91e6a38009da15a, .ex = 0, .sgn=0}, - {.lo = 0x1ed1a8ff78f1b632, .hi = 0xd8b3a1526517a48b, .ex = 0, .sgn=0}, - {.lo = 0x24b9fe00663574a4, .hi = 0xd84852c0a80ffcdb, .ex = 0, .sgn=0}, - {.lo = 0xced12d2899b803db, .hi = 0xd7dc7ec4fabdb011, .ex = 0, .sgn=0}, - {.lo = 0xcb78e80e67ba1b8, .hi = 0xd77025a1e0a39d8b, .ex = 0, .sgn=0}, - {.lo = 0x6cb3bfd65b38562b, .hi = 0xd703479a2f6776cc, .ex = 0, .sgn=0}, - {.lo = 0x83f082b570611d7, .hi = 0xd695e4f10ea88570, .ex = 0, .sgn=0}, - {.lo = 0x7afbefc05e9f7d99, .hi = 0xd627fde9f7d63e7e, .ex = 0, .sgn=0}, - {.lo = 0x7190b755535d4f18, .hi = 0xd5b992c8b606a351, .ex = 0, .sgn=0}, - {.lo = 0x7d00ae97abaa4096, .hi = 0xd54aa3d165cc7018, .ex = 0, .sgn=0}, - {.lo = 0xf630e8b6dac83e69, .hi = 0xd4db3148750d1819, .ex = 0, .sgn=0}, - {.lo = 0xdc4663a3168698d2, .hi = 0xd46b3b72a2d68fc9, .ex = 0, .sgn=0}, - {.lo = 0xb77d4f6bd0ee8591, .hi = 0xd3fac294ff34e4d0, .ex = 0, .sgn=0}, - {.lo = 0xa8faac741a6394dc, .hi = 0xd389c6f4eb07a41c, .ex = 0, .sgn=0}, - {.lo = 0xeeeaddb72f00e0dd, .hi = 0xd31848d817d70e16, .ex = 0, .sgn=0}, - {.lo = 0x4300fd1c1ce507e5, .hi = 0xd2a6488487a91918, .ex = 0, .sgn=0}, - {.lo = 0x981ba7e42537275f, .hi = 0xd233c6408cd64236, .ex = 0, .sgn=0}, - {.lo = 0xda7485a5aeffeb4c, .hi = 0xd1c0c252c9de2c86, .ex = 0, .sgn=0}, - {.lo = 0x744fea20e8abef92, .hi = 0xd14d3d02313c0eed, .ex = 0, .sgn=0}, - {.lo = 0x77a18eb13d2ecde5, .hi = 0xd0d93696053af098, .ex = 0, .sgn=0}, - {.lo = 0x6b8a685f6cb61c21, .hi = 0xd064af55d7c9b43e, .ex = 0, .sgn=0}, - {.lo = 0xdaf200dd81212d10, .hi = 0xcfefa7898a4ef23c, .ex = 0, .sgn=0}, - {.lo = 0xdfcb60445c1bf973, .hi = 0xcf7a1f794d7ca1b1, .ex = 0, .sgn=0}, - {.lo = 0x4d27090f10c454e, .hi = 0xcf04176da12390ac, .ex = 0, .sgn=0}, - {.lo = 0xf5babff66def7892, .hi = 0xce8d8faf5406ab8b, .ex = 0, .sgn=0}, - {.lo = 0x93e391861a034684, .hi = 0xce16888783ae13b3, .ex = 0, .sgn=0}, - {.lo = 0x23af31db7179a4aa, .hi = 0xcd9f023f9c3a059e, .ex = 0, .sgn=0}, - {.lo = 0x649474e36b8db9d3, .hi = 0xcd26fd2158358e7d, .ex = 0, .sgn=0}, - {.lo = 0x83e907fbd7aaf0b0, .hi = 0xccae7976c0691177, .ex = 0, .sgn=0}, - {.lo = 0xf839ce18e08bfb50, .hi = 0xcc35778a2bac9ca1, .ex = 0, .sgn=0}, - {.lo = 0x70cbb7f3343451be, .hi = 0xcbbbf7a63eba0dd5, .ex = 0, .sgn=0}, - {.lo = 0x2293661be51140ab, .hi = 0xcb41fa15ebff0777, .ex = 0, .sgn=0}, - {.lo = 0xd9944be1631846d8, .hi = 0xcac77f24736eb553, .ex = 0, .sgn=0}, - {.lo = 0x5328edeb3e6784de, .hi = 0xca4c871d625361a9, .ex = 0, .sgn=0}, - {.lo = 0x8335241be1693225, .hi = 0xc9d1124c931fda7a, .ex = 0, .sgn=0}, - {.lo = 0x83b0e96e1249c2b0, .hi = 0xc95520fe2d40a74b, .ex = 0, .sgn=0}, - {.lo = 0xb562c00b34ee771, .hi = 0xc8d8b37ea4ed0f62, .ex = 0, .sgn=0}, - {.lo = 0x65862939b83382e0, .hi = 0xc85bca1abaf7f0a7, .ex = 0, .sgn=0}, - {.lo = 0x2b31bc86877fd2c, .hi = 0xc7de651f7ca06749, .ex = 0, .sgn=0}, - {.lo = 0xd5c149509e9059f1, .hi = 0xc76084da43624634, .ex = 0, .sgn=0}, - {.lo = 0xcfe6c1b1a6b4e2a4, .hi = 0xc6e22998b4c6608e, .ex = 0, .sgn=0}, - {.lo = 0xe993503baf5afb41, .hi = 0xc66353a8c232a43c, .ex = 0, .sgn=0}, - {.lo = 0x43da25d99267326b, .hi = 0xc5e40358a8ba05a7, .ex = 0, .sgn=0}, - {.lo = 0xab4906075507e74, .hi = 0xc56438f6f0ec3cca, .ex = 0, .sgn=0}, - {.lo = 0xdd40950cf1ed92fa, .hi = 0xc4e3f4d26ea553b6, .ex = 0, .sgn=0}, - {.lo = 0x9dd768f30ca8e85c, .hi = 0xc463373a40dd06a3, .ex = 0, .sgn=0}, - {.lo = 0xa87e78136665cdb2, .hi = 0xc3e2007dd175f5a4, .ex = 0, .sgn=0}, - {.lo = 0x8ac9e1386e4cbabb, .hi = 0xc36050ecd50ca830, .ex = 0, .sgn=0}, - {.lo = 0x74c8f010d986a9e0, .hi = 0xc2de28d74ac6628b, .ex = 0, .sgn=0}, - {.lo = 0xb7041e9bc8c18b0d, .hi = 0xc25b888d7c1fcd38, .ex = 0, .sgn=0}, - {.lo = 0xbdf0715cb8b20bd7, .hi = 0xc1d8705ffcbb6e90, .ex = 0, .sgn=0}, - {.lo = 0x17858573216e0a22, .hi = 0xc154e09faa2ff69a, .ex = 0, .sgn=0}, - {.lo = 0x2bda5328933c854a, .hi = 0xc0d0d99dabd65d44, .ex = 0, .sgn=0}, - {.lo = 0x6dd06968e0ed1957, .hi = 0xc04c5bab7297d322, .ex = 0, .sgn=0}, - {.lo = 0xe4e62d86dd136e78, .hi = 0xbfc7671ab8bb84c6, .ex = 0, .sgn=0}, - {.lo = 0xd46655d6b012455, .hi = 0xbf41fc3d81b430db, .ex = 0, .sgn=0}, - {.lo = 0x2715ef03f8543355, .hi = 0xbebc1b6619ed9116, .ex = 0, .sgn=0}, - {.lo = 0x29d7f7b67d43b177, .hi = 0xbe35c4e716999630, .ex = 0, .sgn=0}, - {.lo = 0xac85320f528d6d5d, .hi = 0xbdaef913557d76f0, .ex = 0, .sgn=0}, - {.lo = 0x2ea36923d5d8e213, .hi = 0xbd27b83dfcbe9279, .ex = 0, .sgn=0}, - {.lo = 0x4a48496734be336d, .hi = 0xbca002ba7aaf25ea, .ex = 0, .sgn=0}, - {.lo = 0x727c405ffc73af56, .hi = 0xbc17d8dc859ad583, .ex = 0, .sgn=0}, - {.lo = 0xfce8d84068e825b6, .hi = 0xbb8f3af81b93095c, .ex = 0, .sgn=0}, - {.lo = 0x5120e35e1c1a250c, .hi = 0xbb062961823b1ddc, .ex = 0, .sgn=0}, - {.lo = 0x33201477347447d8, .hi = 0xba7ca46d46946802, .ex = 0, .sgn=0}, - {.lo = 0x39db32d014440024, .hi = 0xb9f2ac703cca0db3, .ex = 0, .sgn=0}, - {.lo = 0x9de1e3b22b8bf4db, .hi = 0xb96841bf7ffcb21a, .ex = 0, .sgn=0}, - {.lo = 0xa726f4f0828585c9, .hi = 0xb8dd64b0720df647, .ex = 0, .sgn=0}, - {.lo = 0x1c041d1ea5fb3fdb, .hi = 0xb8521598bb6bce26, .ex = 0, .sgn=0}, - {.lo = 0x2e7a35723f3ed035, .hi = 0xb7c654ce4adba9f2, .ex = 0, .sgn=0}, - {.lo = 0x7f86f63bb23f496a, .hi = 0xb73a22a755457448, .ex = 0, .sgn=0}, - {.lo = 0xeb2d28ef943dc88c, .hi = 0xb6ad7f7a557e64f2, .ex = 0, .sgn=0}, - {.lo = 0xea7c015f12b987f7, .hi = 0xb6206b9e0c13a892, .ex = 0, .sgn=0}, - {.lo = 0x737dd2824b608d13, .hi = 0xb592e7697f14dd4a, .ex = 0, .sgn=0}, -}; - -/* The following is a degree-7 polynomial with odd coefficients - approximating sin2pi(x) for -2^-24 < x < 2^-11+2^-24 - with relative error 2^-77.306. - Generated with sin_fast.sollya. */ -static const double PSfast[] = { - 0x1.921fb54442d18p+2, 0x1.1a62645446203p-52, // degree 1 (h+l) - -0x1.4abbce625be53p5, // degree 3 - 0x1.466bc678d8d63p6, // degree 5 - -0x1.331554ca19669p6, // degree 7 -}; - -/* The following is a degree-6 polynomial with even coefficients - approximating cos2pi(x) for -2^-24 < x < 2^-11+2^-24 - with relative error 2^-75.188. - Generated with cos_fast.sollya. */ -static const double PCfast[] = { - 0x1p+0, -0x1.923015cp-77, // degree 0 - -0x1.3bd3cc9be45dep4, // degree 2 - 0x1.03c1f080ad892p6, // degree 4 - -0x1.55a5c590f9e6ap6, // degree 6 -}; - -/* The following is a degree-11 polynomial with odd coefficients - approximating sin2pi(x) for 0 <= x < 2^-11 with relative error 2^-127.75. - Generated with sin_accurate.sollya. */ -static const dint64_t PS[] = { - {.lo = 0xc4c6628b80dc1cd1, .hi = 0xc90fdaa22168c234, .ex = 3, .sgn=0}, // 1 - {.lo = 0x5dc72f712aa57db4, .hi = 0xa55de7312df295f5, .ex = 6, .sgn=1}, // 3 - {.lo = 0x3f33be0021aa54d2, .hi = 0xa335e33bad570e92, .ex = 7, .sgn=0}, // 5 - {.lo = 0xe59d6ab8509a2025, .hi = 0x9969667315ec2d9d, .ex = 7, .sgn=1}, // 7 - {.lo = 0x7d5f8f76fa7d74ed, .hi = 0xa83c1a43bf1c6485, .ex = 6, .sgn=0}, // 9 - {.lo = 0xa7f0339113b8b3c5, .hi = 0xf16ab2898eae62f9, .ex = 4, .sgn=1}, // 11 -}; - -/* The following is a degree-10 polynomial with even coefficients - approximating cos2pi(x) for 0 <= x < 2^-11 with relative error 2^-137.246. - Generated with cos_accurate.sollya. */ -static const dint64_t PC[] = { - {.lo = 0x0, .hi = 0x8000000000000000, .ex = 1, .sgn=0}, // degree 0 - {.lo = 0x56e26cd9808c1949, .hi = 0x9de9e64df22ef2d2, .ex = 5, .sgn=1}, // 2 - {.lo = 0x9980f00630cb655e, .hi = 0x81e0f840dad61d9a, .ex = 7, .sgn=0}, // 4 - {.lo = 0xa508509534006249, .hi = 0xaae9e3f1e5ffcfe2, .ex = 7, .sgn=1}, // 6 - {.lo = 0xe0603ce7044eeba, .hi = 0xf0fa83448dd1e094, .ex = 6, .sgn=0}, // 8 - {.lo = 0xec63157807ebffa, .hi = 0xd368f6f4207cfe49, .ex = 5, .sgn=1}, // 10 -}; - -/* Table generated with ./buildSC 15 using accompanying buildSC.c. - For each i, 0 <= i < 256, xi=i/2^11+SC[i][0], with - SC[i][1] and SC[i][2] approximating sin2pi(xi) and cos2pi(xi) - respectively, both with 53+15 bits of accuracy. */ -static const double SC[256][3] = { - {0x0p+0, 0x0p+0, 0x1p+0}, /* 0 */ - {-0x1.c0f6cp-35, 0x1.921f892b900fep-9, 0x1.ffff621623fap-1}, /* 1 */ - {-0x1.9c7935ep-35, 0x1.921f0ea27ce01p-8, 0x1.fffd8858eca2ep-1}, /* 2 */ - {-0x1.d14d1acp-34, 0x1.2d96af779b0bbp-7, 0x1.fffa72c986392p-1}, /* 3 */ - {-0x1.dba8f6a8p-33, 0x1.921d1ce2d0a1cp-7, 0x1.fff62169dddaap-1}, /* 4 */ - {0x1.a6b7cdfp-32, 0x1.f6a29bdb7377p-7, 0x1.fff0943c02419p-1}, /* 5 */ - {0x1.b49618dp-33, 0x1.2d936d1506f3dp-6, 0x1.ffe9cb44829cp-1}, /* 6 */ - {-0x1.398d6fcp-35, 0x1.5fd4d1e21de6dp-6, 0x1.ffe1c687174b1p-1}, /* 7 */ - {-0x1.e9e9a8c8p-31, 0x1.9215597791e0ap-6, 0x1.ffd886097afcfp-1}, /* 8 */ - {-0x1.34e844cp-32, 0x1.c454f2e9480c7p-6, 0x1.ffce09ce95933p-1}, /* 9 */ - {-0x1.989a8a4p-32, 0x1.f693709b94f92p-6, 0x1.ffc251dfbac0cp-1}, /* 10 */ - {0x1.04a9b99p-30, 0x1.146860e69a571p-5, 0x1.ffb55e40a5c43p-1}, /* 11 */ - {-0x1.56947cp-36, 0x1.2d865748774adp-5, 0x1.ffa72efff95d1p-1}, /* 12 */ - {-0x1.c348768p-35, 0x1.46a396d34121ap-5, 0x1.ff97c420a8451p-1}, /* 13 */ - {0x1.9e80552p-32, 0x1.5fc00e6e4c65cp-5, 0x1.ff871dacd8761p-1}, /* 14 */ - {0x1.3f11d74p-34, 0x1.78dbaa97099ebp-5, 0x1.ff753bb18af95p-1}, /* 15 */ - {0x1.c039af4p-33, 0x1.91f65fc0abc0ap-5, 0x1.ff621e370ca7ap-1}, /* 16 */ - {0x1.53e1f8p-35, 0x1.ab101bf74ac2ep-5, 0x1.ff4dc54b00181p-1}, /* 17 */ - {0x1.114a649p-29, 0x1.c428d7de920e9p-5, 0x1.ff3830f2e9043p-1}, /* 18 */ - {0x1.adf0ef4p-31, 0x1.dd40723a3cdfbp-5, 0x1.ff21614b9d9adp-1}, /* 19 */ - {-0x1.d21f5918p-30, 0x1.f656e1e9e59cdp-5, 0x1.ff09565e83d77p-1}, /* 20 */ - {-0x1.4f54d708p-30, 0x1.07b612d6be078p-4, 0x1.fef0102c634e3p-1}, /* 21 */ - {-0x1.1efec9ap-30, 0x1.1440118ba7bdp-4, 0x1.fed58ecf342dap-1}, /* 22 */ - {0x1.cc17ba88p-29, 0x1.20c96cf0a7eedp-4, 0x1.feb9d24646fa6p-1}, /* 23 */ - {0x1.121dbe4p-33, 0x1.2d5209628edfp-4, 0x1.fe9cdacf99cffp-1}, /* 24 */ - {-0x1.9ecf61p-34, 0x1.39d9f103bf7f7p-4, 0x1.fe7ea854e6b08p-1}, /* 25 */ - {-0x1.04ede8ep-31, 0x1.466116c629e5cp-4, 0x1.fe5f3af4ee201p-1}, /* 26 */ - {-0x1.1821cecp-31, 0x1.52e773c9920c7p-4, 0x1.fe3e92c0e4108p-1}, /* 27 */ - {0x1.cdec726p-31, 0x1.5f6d02131f0b2p-4, 0x1.fe1cafc7f1a24p-1}, /* 28 */ - {-0x1.edece4dp-31, 0x1.6bf1b2653648cp-4, 0x1.fdf99233c230cp-1}, /* 29 */ - {-0x1.2aa4d1cp-31, 0x1.787585bc45f0fp-4, 0x1.fdd53a01d11d9p-1}, /* 30 */ - {0x1.d461592p-32, 0x1.84f871e32cf68p-4, 0x1.fdafa74f16482p-1}, /* 31 */ - {0x1.f0cbd728p-29, 0x1.917a71d3d2956p-4, 0x1.fd88da29f302ep-1}, /* 32 */ - {-0x1.583247p-30, 0x1.9dfb6c9865b06p-4, 0x1.fd60d2e14a6b1p-1}, /* 33 */ - {-0x1.2e81bf4p-30, 0x1.aa7b706bfdbbap-4, 0x1.fd3791484ff5p-1}, /* 34 */ - {-0x1.13941418p-28, 0x1.b6fa680a05c27p-4, 0x1.fd0d15a4b8471p-1}, /* 35 */ - {0x1.71098ffp-30, 0x1.c3785eba12b42p-4, 0x1.fce15fceddccfp-1}, /* 36 */ - {-0x1.c3519e8p-32, 0x1.cff53302f059p-4, 0x1.fcb4703b969e1p-1}, /* 37 */ - {0x1.2f522a5p-27, 0x1.dc70fb84af16ep-4, 0x1.fc8646987fc1dp-1}, /* 38 */ - {-0x1.ae9bed8p-33, 0x1.e8eb7f8a589e2p-4, 0x1.fc56e3b91ca3ap-1}, /* 39 */ - {0x1.f8868b2p-30, 0x1.f564e87d2330fp-4, 0x1.fc264701f9a09p-1}, /* 40 */ - {-0x1.b07985f8p-29, 0x1.00ee8835051f4p-3, 0x1.fbf47105f7439p-1}, /* 41 */ - {0x1.cbdaa94p-30, 0x1.072a05e1d4d8ep-3, 0x1.fbc16172a9e36p-1}, /* 42 */ - {0x1.37c5b908p-28, 0x1.0d64df9619f0dp-3, 0x1.fb8d18b635327p-1}, /* 43 */ - {-0x1.068b5fc8p-28, 0x1.139f09bc617f5p-3, 0x1.fb5797351da85p-1}, /* 44 */ - {-0x1.8ea66818p-29, 0x1.19d8919fa4ec8p-3, 0x1.fb20dc7da8affp-1}, /* 45 */ - {0x1.6278ceb8p-28, 0x1.2011719d50b87p-3, 0x1.fae8e8bd4427fp-1}, /* 46 */ - {-0x1.096df84p-29, 0x1.264993433763ap-3, 0x1.faafbcbfca356p-1}, /* 47 */ - {0x1.9b2534fp-29, 0x1.2c810967bbf7p-3, 0x1.fa7557d8d987ep-1}, /* 48 */ - {0x1.215b4ep-34, 0x1.32b7bfa25c91bp-3, 0x1.fa39bac71954bp-1}, /* 49 */ - {-0x1.94db891p-30, 0x1.38edb9d29b39dp-3, 0x1.f9fce56700a6dp-1}, /* 50 */ - {0x1.7727f7b8p-29, 0x1.3f22f7c3cce3ap-3, 0x1.f9bed7b8c8d8cp-1}, /* 51 */ - {-0x1.0cb33038p-29, 0x1.45576971dd53p-3, 0x1.f97f925d53c83p-1}, /* 52 */ - {-0x1.9071106p-31, 0x1.4b8b175c71e22p-3, 0x1.f93f14feb8022p-1}, /* 53 */ - {0x1.62741e78p-29, 0x1.51bdfa7ea30d5p-3, 0x1.f8fd5fe3efac8p-1}, /* 54 */ - {0x1.f8e16d0cp-28, 0x1.57f00e80e6e12p-3, 0x1.f8ba733a1ceb1p-1}, /* 55 */ - {-0x1.76acbcap-31, 0x1.5e2143b7bc1c2p-3, 0x1.f8764fad5e9bfp-1}, /* 56 */ - {-0x1.0a0f73ap-30, 0x1.6451a76411746p-3, 0x1.f830f4ad232d8p-1}, /* 57 */ - {0x1.ca11d1bcp-28, 0x1.6a8135d7bd143p-3, 0x1.f7ea625eb5af7p-1}, /* 58 */ - {-0x1.02f23628p-29, 0x1.70afd74071191p-3, 0x1.f7a299d3f182ap-1}, /* 59 */ - {0x1.b34dcb8p-29, 0x1.76dda08544b5cp-3, 0x1.f7599a1ac7ecdp-1}, /* 60 */ - {0x1.161ff4p-32, 0x1.7d0a7bf2d4abap-3, 0x1.f70f64322da74p-1}, /* 61 */ - {-0x1.c49b8b4p-31, 0x1.83366ddb3de23p-3, 0x1.f6c3f7e7c2707p-1}, /* 62 */ - {0x1.21da851p-29, 0x1.8961743b1429p-3, 0x1.f6775552a6ba2p-1}, /* 63 */ - {0x1.ac63edap-30, 0x1.8f8b851098588p-3, 0x1.f6297cef0cdd6p-1}, /* 64 */ - {0x1.27ef489cp-27, 0x1.95b4a5b9f2cebp-3, 0x1.f5da6e7820551p-1}, /* 65 */ - {0x1.ae8937p-30, 0x1.9bdcc07900146p-3, 0x1.f58a2b0689c82p-1}, /* 66 */ - {0x1.eb48c7ep-29, 0x1.a203e4a4f950ep-3, 0x1.f538b1d392049p-1}, /* 67 */ - {-0x1.bfd282fp-29, 0x1.a829ffaad0d79p-3, 0x1.f4e603d51f1aap-1}, /* 68 */ - {0x1.7ccf638p-29, 0x1.ae4f1fa80e1b5p-3, 0x1.f492204c5ef9ep-1}, /* 69 */ - {-0x1.2435c578p-28, 0x1.b4732b72ebc86p-3, 0x1.f43d0890e1e72p-1}, /* 70 */ - {0x1.0293fecp-30, 0x1.ba9634155f866p-3, 0x1.f3e6bbb6c2ea4p-1}, /* 71 */ - {-0x1.7bb1f92p-29, 0x1.c0b82461f65ep-3, 0x1.f38f3ae6f9afcp-1}, /* 72 */ - {0x1.27aaebcp-29, 0x1.c6d906faacf65p-3, 0x1.f3368589e17a2p-1}, /* 73 */ - {-0x1.2e2bcd5p-27, 0x1.ccf8c3f74a6c9p-3, 0x1.f2dc9cfb5fa74p-1}, /* 74 */ - {-0x1.6f070acp-30, 0x1.d31773ba218a8p-3, 0x1.f2817fd4d045bp-1}, /* 75 */ - {0x1.469adfcp-29, 0x1.d935004779e57p-3, 0x1.f2252f59c122dp-1}, /* 76 */ - {0x1.4f51c18p-32, 0x1.df5164301377ap-3, 0x1.f1c7abdeaa3efp-1}, /* 77 */ - {0x1.78e44dap-29, 0x1.e56ca4202807cp-3, 0x1.f168f51c5d5d5p-1}, /* 78 */ - {0x1.49bb5f8p-32, 0x1.eb86b4a1b7e9bp-3, 0x1.f1090bc4b68p-1}, /* 79 */ - {-0x1.67ba541p-28, 0x1.f19f9369d5e93p-3, 0x1.f0a7effdc937fp-1}, /* 80 */ - {0x1.c0cab95p-29, 0x1.f7b74ab7219d2p-3, 0x1.f045a1219e594p-1}, /* 81 */ - {-0x1.2b77e32p-30, 0x1.fdcdc0ca3288dp-3, 0x1.efe220cf5c751p-1}, /* 82 */ - {-0x1.e0d8cbp-33, 0x1.01f18054c8362p-2, 0x1.ef7d6e54c347dp-1}, /* 83 */ - {-0x1.ecd5b9cp-29, 0x1.04fb7f6d35d68p-2, 0x1.ef178a6f9a987p-1}, /* 84 */ - {0x1.eb24de5p-29, 0x1.0804e1d369ff2p-2, 0x1.eeb074934fdfp-1}, /* 85 */ - {0x1.4a897c4p-30, 0x1.0b0d9d7b0d042p-2, 0x1.ee482e14bcdep-1}, /* 86 */ - {0x1.336c376p-30, 0x1.0e15b555e7becp-2, 0x1.eddeb6908ca8cp-1}, /* 87 */ - {-0x1.3952d9p-31, 0x1.111d25efd48b8p-2, 0x1.ed740e7eb8dd6p-1}, /* 88 */ - {0x1.fc2a5d4p-31, 0x1.1423ef5c7e1bdp-2, 0x1.ed0835dc24e89p-1}, /* 89 */ - {0x1.a88ed37p-29, 0x1.172a0eb8361dap-2, 0x1.ec9b2d0ec8288p-1}, /* 90 */ - {-0x1.8ca4cb94p-27, 0x1.1a2f7b10b6d7p-2, 0x1.ec2cf55d6117cp-1}, /* 91 */ - {0x1.0144524p-27, 0x1.1d3446fd0cd3fp-2, 0x1.ebbd8c1d62f96p-1}, /* 92 */ - {-0x1.abf810cp-28, 0x1.203855b85f89ap-2, 0x1.eb4cf57454132p-1}, /* 93 */ - {0x1.5d4c5d58p-28, 0x1.233bbcca40561p-2, 0x1.eadb2e40746cap-1}, /* 94 */ - {-0x1.a1b0c58p-29, 0x1.263e685b1d714p-2, 0x1.ea68396d87754p-1}, /* 95 */ - {-0x1.77c8dacp-29, 0x1.294061d2eb611p-2, 0x1.e9f41597393c8p-1}, /* 96 */ - {0x1.915540ep-30, 0x1.2c41a580014cfp-2, 0x1.e97ec348fb87fp-1}, /* 97 */ - {-0x1.abb6d9bp-28, 0x1.2f422b2d0990cp-2, 0x1.e90843c55b996p-1}, /* 98 */ - {-0x1.b8ee5d58p-28, 0x1.3241f8cea2836p-2, 0x1.e890962268c49p-1}, /* 99 */ - {-0x1.1cd29828p-28, 0x1.35410a8396266p-2, 0x1.e817baf85c094p-1}, /* 100 */ - {-0x1.e216afp-32, 0x1.383f5e08283e2p-2, 0x1.e79db2a188b0ap-1}, /* 101 */ - {-0x1.24afc3p-31, 0x1.3b3cef6993c0bp-2, 0x1.e7227dbf82004p-1}, /* 102 */ - {-0x1.aa1657cp-31, 0x1.3e39be4767224p-2, 0x1.e6a61c62d5274p-1}, /* 103 */ - {-0x1.c5b65fap-30, 0x1.4135c898485bbp-2, 0x1.e6288ee07fea5p-1}, /* 104 */ - {0x1.23e8978p-32, 0x1.44310de3c284bp-2, 0x1.e5a9d54bbd26cp-1}, /* 105 */ - {-0x1.2b1d77ap-29, 0x1.472b8976d498dp-2, 0x1.e529f06cb187dp-1}, /* 106 */ - {-0x1.daaa348p-31, 0x1.4a253cb97efd1p-2, 0x1.e4a8e007231a2p-1}, /* 107 */ - {-0x1.322f5708p-28, 0x1.4d1e2260c3422p-2, 0x1.e426a500f6e33p-1}, /* 108 */ - {0x1.64758e8p-29, 0x1.50163eca0b337p-2, 0x1.e3a33e996b722p-1}, /* 109 */ - {0x1.12486278p-28, 0x1.530d89a17e007p-2, 0x1.e31eae3fb917bp-1}, /* 110 */ - {-0x1.6c3416ccp-27, 0x1.5603fcf8cd8a3p-2, 0x1.e298f502a579bp-1}, /* 111 */ - {0x1.ab481ffp-29, 0x1.58f9a896aa209p-2, 0x1.e2121016e14fcp-1}, /* 112 */ - {-0x1.6eb838bp-29, 0x1.5bee77aaf890bp-2, 0x1.e18a032eb4df5p-1}, /* 113 */ - {-0x1.d159b8p-32, 0x1.5ee2734efeef5p-2, 0x1.e100ccaa6bd78p-1}, /* 114 */ - {-0x1.a42e4ap-34, 0x1.61d595bedeabcp-2, 0x1.e0766d944915ep-1}, /* 115 */ - {-0x1.43d0dcp-30, 0x1.64c7dd5cc0cd1p-2, 0x1.dfeae63903034p-1}, /* 116 */ - {-0x1.8c7bdb7p-27, 0x1.67b9453ca2122p-2, 0x1.df5e378482eaep-1}, /* 117 */ - {0x1.1c0ead6p-30, 0x1.6aa9d844c980ap-2, 0x1.ded05f6a23a52p-1}, /* 118 */ - {0x1.7d526p-31, 0x1.6d99867e90d92p-2, 0x1.de4160e97b2e2p-1}, /* 119 */ - {0x1.924e0368p-28, 0x1.7088555d3c816p-2, 0x1.ddb13afb14e37p-1}, /* 120 */ - {-0x1.74b7c3ep-30, 0x1.73763c09fba09p-2, 0x1.dd1fef5335416p-1}, /* 121 */ - {-0x1.7943adp-30, 0x1.766340685c982p-2, 0x1.dc8d7ccf2567ap-1}, /* 122 */ - {0x1.79dd614p-29, 0x1.794f5f7522b88p-2, 0x1.dbf9e402aa5c3p-1}, /* 123 */ - {0x1.7b64f32p-30, 0x1.7c3a939c32d81p-2, 0x1.db652607e0db1p-1}, /* 124 */ - {-0x1.2bea5ce8p-28, 0x1.7f24db825141cp-2, 0x1.dacf43268b5bp-1}, /* 125 */ - {0x1.733c024p-30, 0x1.820e3b8bf15ap-2, 0x1.da383a7aed887p-1}, /* 126 */ - {-0x1.eac0fc94p-27, 0x1.84f6a51d077b3p-2, 0x1.d9a00efd84537p-1}, /* 127 */ - {0x1.aca37338p-27, 0x1.87de2f4704f98p-2, 0x1.d906bbf17f4dap-1}, /* 128 */ - {-0x1.910c4fp-30, 0x1.8ac4b7dc0d986p-2, 0x1.d86c4862b5d6ep-1}, /* 129 */ - {-0x1.33bb86p-31, 0x1.8daa52b4dc041p-2, 0x1.d7d0b0374a559p-1}, /* 130 */ - {-0x1.69e1507p-27, 0x1.908ef408ad22p-2, 0x1.d733f5e71c3bcp-1}, /* 131 */ - {0x1.cffacf08p-27, 0x1.9372ab7784d36p-2, 0x1.d696161d786c9p-1}, /* 132 */ - {-0x1.8629d9fp-26, 0x1.965552b0849abp-2, 0x1.d5f7190eeae23p-1}, /* 133 */ - {0x1.415p-30, 0x1.99371687c64f3p-2, 0x1.d556f5155d9ddp-1}, /* 134 */ - {-0x1.bd37aad8p-27, 0x1.9c17cf40715cbp-2, 0x1.d4b5b2caf8386p-1}, /* 135 */ - {0x1.d02cde7p-26, 0x1.9ef79ea4d995dp-2, 0x1.d4134ac5eb246p-1}, /* 136 */ - {-0x1.10547acp-30, 0x1.a1d653d9adf5ep-2, 0x1.d36fc7d291602p-1}, /* 137 */ - {-0x1.01a1a228p-27, 0x1.a4b40f9c0120bp-2, 0x1.d2cb22b45236bp-1}, /* 138 */ - {0x1.3ce2bacp-29, 0x1.a790ce2056b9ap-2, 0x1.d2255c3ae11a5p-1}, /* 139 */ - {-0x1.ccb4a6p-32, 0x1.aa6c828db4ea8p-2, 0x1.d17e774d4e3e2p-1}, /* 140 */ - {0x1.5db4bp-29, 0x1.ad47321f29847p-2, 0x1.d0d672bc0b122p-1}, /* 141 */ - {0x1.32f6a6ep-29, 0x1.b020d7a285e23p-2, 0x1.d02d4fb84d334p-1}, /* 142 */ - {0x1.cf8e39bcp-26, 0x1.b2f97c27f7494p-2, 0x1.cf830c2248c5ep-1}, /* 143 */ - {0x1.8927bbp-30, 0x1.b5d10129a750ap-2, 0x1.ced7af22cb105p-1}, /* 144 */ - {-0x1.3dec3c1p-28, 0x1.b8a77f8d0bbc5p-2, 0x1.ce2b32e50d6cdp-1}, /* 145 */ - {-0x1.26ba536p-28, 0x1.bb7cf08f0290dp-2, 0x1.cd7d98fcf3b1ep-1}, /* 146 */ - {0x1.23c568ep-29, 0x1.be51524e3aa53p-2, 0x1.cccee1da3d56ep-1}, /* 147 */ - {-0x1.f3b3afp-29, 0x1.c1249c1f5f2f6p-2, 0x1.cc1f0f95e1e24p-1}, /* 148 */ - {-0x1.1286a47p-28, 0x1.c3f6d2ef7054bp-2, 0x1.cb6e20ff37e81p-1}, /* 149 */ - {0x1.641214ep-29, 0x1.c6c7f594003d9p-2, 0x1.cabc165bf1b6p-1}, /* 150 */ - {0x1.0cda7c9p-27, 0x1.c997ff2bffccbp-2, 0x1.ca08f0dee434cp-1}, /* 151 */ - {-0x1.5557ac9p-28, 0x1.cc66e7b42e8f1p-2, 0x1.c954b28bca62ep-1}, /* 152 */ - {0x1.555eb62p-28, 0x1.cf34bccc567a1p-2, 0x1.c89f57f6e20f3p-1}, /* 153 */ - {-0x1.4e0e361p-28, 0x1.d2016cbb5e39ap-2, 0x1.c7e8e59999e1fp-1}, /* 154 */ - {0x1.446da1ep-29, 0x1.d4cd039d0ed05p-2, 0x1.c731585f970ebp-1}, /* 155 */ - {0x1.103d328p-29, 0x1.d797767638decp-2, 0x1.c678b3174afe1p-1}, /* 156 */ - {0x1.5814d6p-28, 0x1.da60c7ae9dc22p-2, 0x1.c5bef522be6fbp-1}, /* 157 */ - {-0x1.5e2321ep-29, 0x1.dd28f054cbb3fp-2, 0x1.c5042052c8c42p-1}, /* 158 */ - {-0x1.a259ffep-29, 0x1.dfeff54854631p-2, 0x1.c44833611bc7dp-1}, /* 159 */ - {-0x1.4f28d8p-31, 0x1.e2b5d34665b35p-2, 0x1.c38b2f278ea7ep-1}, /* 160 */ - {-0x1.de571p-36, 0x1.e57a86d137f2p-2, 0x1.c2cd1493d05c2p-1}, /* 161 */ - {0x1.e0d8d14p-29, 0x1.e83e0ffb7bfb4p-2, 0x1.c20de3a08ea07p-1}, /* 162 */ - {-0x1.12a858ep-28, 0x1.eb0067e48baf4p-2, 0x1.c14d9e2bd511ep-1}, /* 163 */ - {0x1.9a17403p-27, 0x1.edc19997a4431p-2, 0x1.c08c413089b2ep-1}, /* 164 */ - {0x1.68c8636p-29, 0x1.f0819163d1bcp-2, 0x1.bfc9d21568f32p-1}, /* 165 */ - {0x1.4cc5eb8p-29, 0x1.f3405a482e11dp-2, 0x1.bf064dd580fc9p-1}, /* 166 */ - {-0x1.fce7cd8p-27, 0x1.f5fde8f3f11d4p-2, 0x1.be41b798f6b97p-1}, /* 167 */ - {-0x1.af8169p-29, 0x1.f8ba4c98a9816p-2, 0x1.bd7c0b1a7f14bp-1}, /* 168 */ - {0x1.6e39e2p-33, 0x1.fb7575d1ea75p-2, 0x1.bcb54cac5dde5p-1}, /* 169 */ - {0x1.30f9256p-28, 0x1.fe2f665dcd168p-2, 0x1.bbed7bd1e17bp-1}, /* 170 */ - {0x1.626de2p-31, 0x1.00740ca0d5fbbp-1, 0x1.bb2499f9fe7a3p-1}, /* 171 */ - {0x1.5cc703p-30, 0x1.01cfc8afeea0ep-1, 0x1.ba5aa650dd495p-1}, /* 172 */ - {-0x1.6191e6p-32, 0x1.032ae54fe4057p-1, 0x1.b98fa2065a5e6p-1}, /* 173 */ - {-0x1.6b1485p-31, 0x1.0485624c328c8p-1, 0x1.b8c38d39737bcp-1}, /* 174 */ - {-0x1.11fbc3ap-29, 0x1.05df3e66a716dp-1, 0x1.b7f668a580fdp-1}, /* 175 */ - {-0x1.0eca7fp-27, 0x1.07387825589ecp-1, 0x1.b728352c44517p-1}, /* 176 */ - {-0x1.8073bc9ep-25, 0x1.089109ef1284dp-1, 0x1.b658f630112edp-1}, /* 177 */ - {-0x1.9dcf0adp-27, 0x1.09e9051603e29p-1, 0x1.b588a13ab750fp-1}, /* 178 */ - {-0x1.06ea9fp-29, 0x1.0b405820e78e7p-1, 0x1.b4b740d3cc07bp-1}, /* 179 */ - {-0x1.36a8d0cp-30, 0x1.0c9704a1ea4e5p-1, 0x1.b3e4d40f5524dp-1}, /* 180 */ - {0x1.63d1f3p-30, 0x1.0ded0bc01a533p-1, 0x1.b3115a3a628afp-1}, /* 181 */ - {0x1.f3181f14p-26, 0x1.0f4270e4787bfp-1, 0x1.b23cd1314c779p-1}, /* 182 */ - {-0x1.f269b78p-29, 0x1.109723e75c5cfp-1, 0x1.b167430cfebdbp-1}, /* 183 */ - {0x1.1d84dc08p-27, 0x1.11eb36bc9db52p-1, 0x1.b090a4915ee88p-1}, /* 184 */ - {-0x1.08e60068p-27, 0x1.133e9ba0061d8p-1, 0x1.afb8fe69a6527p-1}, /* 185 */ - {0x1.cda72abp-27, 0x1.14915d557a7c9p-1, 0x1.aee049bc0aeep-1}, /* 186 */ - {-0x1.f32f95p-30, 0x1.15e36dfb6bb55p-1, 0x1.ae068f6991699p-1}, /* 187 */ - {0x1.138092dp-28, 0x1.1734d6f34d7fp-1, 0x1.ad2bc96c1e1f5p-1}, /* 188 */ - {0x1.6b382dd4p-26, 0x1.188595ae376a5p-1, 0x1.ac4ff962bdb6dp-1}, /* 189 */ - {-0x1.f12fafap-28, 0x1.19d59f592a587p-1, 0x1.ab7326685eb57p-1}, /* 190 */ - {-0x1.2909e5ap-28, 0x1.1b2500aed7ac6p-1, 0x1.aa954823cf815p-1}, /* 191 */ - {-0x1.d66a8978p-25, 0x1.1c73aa0150cf9p-1, 0x1.a9b668fb0503fp-1}, /* 192 */ - {0x1.311ea86p-27, 0x1.1dc1b7db74db1p-1, 0x1.a8d675d9c6cc8p-1}, /* 193 */ - {-0x1.41c02b8p-31, 0x1.1f0f08a1a06a4p-1, 0x1.a7f5853bb4309p-1}, /* 194 */ - {-0x1.ca1f4edp-26, 0x1.205ba57211271p-1, 0x1.a71391146958fp-1}, /* 195 */ - {-0x1.910ce77p-28, 0x1.21a7988f8326bp-1, 0x1.a63092626202fp-1}, /* 196 */ - {0x1.2bfadbeep-25, 0x1.22f2dc71afab6p-1, 0x1.a54c8cd9fd0d9p-1}, /* 197 */ - {-0x1.5f1c02a8p-27, 0x1.243d5df4afb93p-1, 0x1.a4678dbbe5e73p-1}, /* 198 */ - {-0x1.db12b9p-30, 0x1.2587347f493a4p-1, 0x1.a38184db0df23p-1}, /* 199 */ - {-0x1.7b29ep-30, 0x1.26d05490f2f61p-1, 0x1.a29a7a2f40b49p-1}, /* 200 */ - {-0x1.b3ddca4p-29, 0x1.2818be6930629p-1, 0x1.a1b26d8f070d7p-1}, /* 201 */ - {0x1.e112744p-29, 0x1.2960730ff2bcdp-1, 0x1.a0c95e3df5e0ep-1}, /* 202 */ - {-0x1.5269766p-28, 0x1.2aa76dafcbbf4p-1, 0x1.9fdf4fae1df6fp-1}, /* 203 */ - {-0x1.09777e1p-28, 0x1.2bedb1b6b4e15p-1, 0x1.9ef43f6cbe162p-1}, /* 204 */ - {0x1.ae2051fp-28, 0x1.2d333e4617f25p-1, 0x1.9e082e148680ep-1}, /* 205 */ - {-0x1.36f6ced8p-27, 0x1.2e780cb47180ep-1, 0x1.9d1b207f383c3p-1}, /* 206 */ - {-0x1.23fdc6bp-28, 0x1.2fbc23fba2f44p-1, 0x1.9c2d1197130a7p-1}, /* 207 */ - {0x1.bc540ep-33, 0x1.30ff7fd6d967dp-1, 0x1.9b3e0478b961bp-1}, /* 208 */ - {-0x1.cfb4ed7p-28, 0x1.32421da0bf0e9p-1, 0x1.9a4dfb1c89326p-1}, /* 209 */ - {0x1.55802aecp-26, 0x1.3384042a92b1dp-1, 0x1.995cf06920d11p-1}, /* 210 */ - {0x1.60719e4p-28, 0x1.34c52608e3a92p-1, 0x1.986aee6d6837ep-1}, /* 211 */ - {-0x1.cbf2e48p-30, 0x1.36058ac8863b6p-1, 0x1.9777ef832c986p-1}, /* 212 */ - {0x1.9061c32p-27, 0x1.374533ab707dp-1, 0x1.9683f2ad7e2ecp-1}, /* 213 */ - {-0x1.da84dfep-27, 0x1.3884160f9488fp-1, 0x1.958f000fdd50ap-1}, /* 214 */ - {0x1.92e8a74p-29, 0x1.39c23eba6b22ap-1, 0x1.94990dd9cee51p-1}, /* 215 */ - {-0x1.bff5d9ap-29, 0x1.3affa20756bddp-1, 0x1.93a225056084ap-1}, /* 216 */ - {0x1.4c462p-36, 0x1.3c3c4498e98ebp-1, 0x1.92aa41fbb951cp-1}, /* 217 */ - {-0x1.e4613e9p-28, 0x1.3d782261dff62p-1, 0x1.91b167e92d706p-1}, /* 218 */ - {0x1.0eb2964p-30, 0x1.3eb33ed579bbep-1, 0x1.90b794146043cp-1}, /* 219 */ - {-0x1.60abec2p-29, 0x1.3fed94c834d8ap-1, 0x1.8fbcca9583479p-1}, /* 220 */ - {0x1.6954977p-27, 0x1.4127281ddac03p-1, 0x1.8ec1085083553p-1}, /* 221 */ - {0x1.a16fec2p-29, 0x1.425ff1f841235p-1, 0x1.8dc452ca328d3p-1}, /* 222 */ - {-0x1.27bcdd3p-27, 0x1.4397f44aa44f2p-1, 0x1.8cc6a8771e165p-1}, /* 223 */ - {-0x1.60dded4p-28, 0x1.44cf317a563dbp-1, 0x1.8bc8076122736p-1}, /* 224 */ - {-0x1.9a8f405cp-26, 0x1.4605a2b02d705p-1, 0x1.8ac875232f3efp-1}, /* 225 */ - {0x1.32777dcp-27, 0x1.473b532bc5a67p-1, 0x1.89c7e8713120cp-1}, /* 226 */ - {-0x1.1418a7bp-26, 0x1.4870306ca20e2p-1, 0x1.88c670a0ea774p-1}, /* 227 */ - {-0x1.fed182ep-28, 0x1.49a44886b534p-1, 0x1.87c401fdf05e5p-1}, /* 228 */ - {0x1.86144d8p-27, 0x1.4ad796ea1410cp-1, 0x1.86c0a04dbacc5p-1}, /* 229 */ - {0x1.1bc2e6p-33, 0x1.4c0a14640d2afp-1, 0x1.85bc51aa114c2p-1}, /* 230 */ - {-0x1.f53d2fep-28, 0x1.4d3bc5aaa8cd5p-1, 0x1.84b7121b30a13p-1}, /* 231 */ - {-0x1.2e100ap-30, 0x1.4e6cab91556bep-1, 0x1.83b0e0e6b6cccp-1}, /* 232 */ - {-0x1.fa58c62p-29, 0x1.4f9cc1c69fddep-1, 0x1.82a9c1c1ab463p-1}, /* 233 */ - {0x1.bb491ep-33, 0x1.50cc09fdcbd92p-1, 0x1.81a1b3342f858p-1}, /* 234 */ - {0x1.a11541p-28, 0x1.51fa82c3aa029p-1, 0x1.8098b67ea8509p-1}, /* 235 */ - {0x1.ab0a5d3p-27, 0x1.53282b20b96b6p-1, 0x1.7f8ecc791953p-1}, /* 236 */ - {-0x1.cba0438p-28, 0x1.5454fe43a7d7cp-1, 0x1.7e83f96af78ap-1}, /* 237 */ - {-0x1.0dd83a4p-29, 0x1.5581033a81573p-1, 0x1.7d783712e20ecp-1}, /* 238 */ - {-0x1.e9a8299p-28, 0x1.56ac33fbb8253p-1, 0x1.7c6b8acf90fa6p-1}, /* 239 */ - {0x1.225c4aap-29, 0x1.57d6939d4b513p-1, 0x1.7b5df1da18065p-1}, /* 240 */ - {-0x1.82e66ep-27, 0x1.59001b9e64d79p-1, 0x1.7a4f72157cfdfp-1}, /* 241 */ - {0x1.51a6a354p-26, 0x1.5a28d5b36d597p-1, 0x1.794002a7c9023p-1}, /* 242 */ - {0x1.13917f4p-26, 0x1.5b50b4e10bec1p-1, 0x1.782faf6dc7ba2p-1}, /* 243 */ - {0x1.49310ccp-30, 0x1.5c77bc15ab4efp-1, 0x1.771e75c43942ep-1}, /* 244 */ - {0x1.24d493cp-30, 0x1.5d9dee9de49dbp-1, 0x1.760c529bc17bp-1}, /* 245 */ - {-0x1.04638f7p-26, 0x1.5ec347044e0f4p-1, 0x1.74f94b0af972p-1}, /* 246 */ - {-0x1.3f41b28p-29, 0x1.5fe7cb834600cp-1, 0x1.73e55936a516p-1}, /* 247 */ - {-0x1.a5f6f5cp-30, 0x1.610b7515d1562p-1, 0x1.72d083b8214ebp-1}, /* 248 */ - {0x1.19fb2ep-28, 0x1.622e459eafbc1p-1, 0x1.71bac8c7b0592p-1}, /* 249 */ - {-0x1.56d2c2bp-28, 0x1.6350396fe4e62p-1, 0x1.70a42bec51665p-1}, /* 250 */ - {-0x1.3c156c2p-28, 0x1.64715385bed93p-1, 0x1.6f8caa4969708p-1}, /* 251 */ - {-0x1.f23e576p-29, 0x1.659191d2fd57fp-1, 0x1.6e7445d74f711p-1}, /* 252 */ - {0x1.1e4be38p-30, 0x1.66b0f41d484c4p-1, 0x1.6d5afecd4938dp-1}, /* 253 */ - {-0x1.397cc8d8p-27, 0x1.67cf76eac73dfp-1, 0x1.6c40d89625f63p-1}, /* 254 */ - {-0x1.202f686p-28, 0x1.68ed1e0990551p-1, 0x1.6b25cf728c35p-1}, /* 255 */ -}; - -// Multiply exactly a and b, such that *hi + *lo = a * b. -static inline void a_mul(double *hi, double *lo, double a, double b) { - *hi = a * b; - *lo = __builtin_fma (a, b, -*hi); -} - -/* Multiply a double with a double double : a * (bh + bl) - with error bounded by ulp(lo) */ -static inline void s_mul (double *hi, double *lo, double a, double bh, - double bl) { - a_mul (hi, lo, a, bh); /* exact */ - *lo = __builtin_fma (a, bl, *lo); - /* the error is bounded by ulp(lo), where |lo| < |a*bl| + ulp(hi) */ -} - -// Returns (ah + al) * (bh + bl) - (al * bl) -// We can ignore al * bl when assuming al <= ulp(ah) and bl <= ulp(bh) -static inline void d_mul(double *hi, double *lo, double ah, double al, - double bh, double bl) { - double s, t; - - a_mul(hi, &s, ah, bh); - t = __builtin_fma(al, bh, s); - *lo = __builtin_fma(ah, bl, t); -} - -static inline void -fast_two_sum(double *hi, double *lo, double a, double b) -{ - double e; - - *hi = a + b; - e = *hi - a; /* exact */ - *lo = b - e; /* exact */ -} - -/* Put in h+l an approximation of sin2pi(xh+xl), - for 2^-24 <= xh+xl < 2^-11 + 2^-24, - and |xl| < 2^-52.36, with absolute error < 2^-77.09 - (see evalPSfast() in sin.sage). - Assume uh + ul approximates (xh+xl)^2. */ -static void -evalPSfast (double *h, double *l, double xh, double xl, double uh, double ul) -{ - double t; - *h = PSfast[4]; // degree 7 - *h = __builtin_fma (*h, uh, PSfast[3]); // degree 5 - *h = __builtin_fma (*h, uh, PSfast[2]); // degree 3 - s_mul (h, l, *h, uh, ul); - fast_two_sum (h, &t, PSfast[0], *h); - *l += PSfast[1] + t; - // multiply by xh+xl - d_mul (h, l, *h, *l, xh, xl); -} - -/* Put in h+l an approximation of cos2pi(xh+xl), - for 2^-24 <= xh+xl < 2^-11 + 2^-24, - and |xl| < 2^-52.36, with relative error < 2^-69.96 - (see evalPCfast() in sin.sage). - Assume uh + ul approximates (xh+xl)^2. */ -static void -evalPCfast (double *h, double *l, double uh, double ul) -{ - double t; - *h = PCfast[4]; // degree 6 - *h = __builtin_fma (*h, uh, PCfast[3]); // degree 4 - *h = __builtin_fma (*h, uh, PCfast[2]); // degree 2 - s_mul (h, l, *h, uh, ul); - fast_two_sum (h, &t, PCfast[0], *h); - *l += PCfast[1] + t; -} - -/* Put in Y an approximation of sin2pi(X), for 0 <= X < 2^-11, - where X2 approximates X^2. - Absolute error bounded by 2^-132.999 with 0 <= Y < 0.003068 - (see evalPS() in sin.sage), and relative error bounded by - 2^-124.648 (see evalPSrel(K=8) in sin.sage). */ -static void -evalPS (dint64_t *Y, dint64_t *X, dint64_t *X2) -{ - mul_dint_21 (Y, X2, PS+5); // degree 11 - add_dint (Y, Y, PS+4); // degree 9 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PS+3); // degree 7 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PS+2); // degree 5 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PS+1); // degree 3 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PS+0); // degree 1 - mul_dint (Y, Y, X); // multiply by X -} - -/* Put in Y an approximation of cos2pi(X), for 0 <= X < 2^-11, - where X2 approximates X^2. - Absolute/relative error bounded by 2^-125.999 with 0.999995 < Y <= 1 - (see evalPC() in sin.sage). */ -static void -evalPC (dint64_t *Y, dint64_t *X2) -{ - mul_dint_21 (Y, X2, PC+5); // degree 10 - add_dint (Y, Y, PC+4); // degree 8 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PC+3); // degree 6 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PC+2); // degree 4 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PC+1); // degree 2 - mul_dint (Y, Y, X2); - add_dint (Y, Y, PC+0); // degree 0 -} - -// normalize X such that X->hi has its most significant bit set (if X <> 0) -static void -normalize (dint64_t *X) -{ - int cnt; - if (X->hi != 0) - { - cnt = __builtin_clzl (X->hi); - if (cnt) - { - X->hi = (X->hi << cnt) | (X->lo >> (64 - cnt)); - X->lo = X->lo << cnt; - } - X->ex -= cnt; - } - else if (X->lo != 0) - { - cnt = __builtin_clzl (X->lo); - X->hi = X->lo << cnt; - X->lo = 0; - X->ex -= 64 + cnt; - } -} - -/* Approximate X/(2pi) mod 1. If Xin is the input value, and Xout the - output value, we have: - |Xout - (Xin/(2pi) mod 1)| < 2^-126.67*|Xout| - Assert X is normalized at input, and normalize X at output. -*/ -static void -reduce (dint64_t *X) -{ - int e = X->ex; - u128 u; - - if (e <= 1) // |X| < 2 - { - /* multiply by T[0]/2^64 + T[1]/2^128, where - |T[0]/2^64 + T[1]/2^128 - 1/(2pi)| < 2^-130.22 */ - u = (u128) X->hi * (u128) T[1]; - uint64_t tiny = absl::Uint128Low64(u); - X->lo = absl::Uint128High64(u); - u = (u128) X->hi * (u128) T[0]; - X->lo += absl::Uint128Low64(u); - X->hi = absl::Uint128High64(u) + (X->lo < (uint64_t) u); - /* hi + lo/2^64 + tiny/2^128 = hi_in * (T[0]/2^64 + T[1]/2^128) thus - |hi + lo/2^64 + tiny/2^128 - hi_in/(2*pi)| < hi_in * 2^-130.22 - Since X is normalized at input, hi_in >= 2^63, and since T[0] >= 2^61, - we have hi >= 2^(63+61-64) = 2^60, thus the normalize() below - perform a left shift by at most 3 bits */ - int e = X->ex; - normalize (X); - e = e - X->ex; - // put the upper e bits of tiny into X->lo - if (e) - X->lo |= tiny >> (64 - e); - /* The error is bounded by 2^-130.22 (relative) + ulp(lo) (absolute). - Since now X->hi >= 2^63, the absolute error of ulp(lo) converts into - a relative error of less than 2^-127. - This yields a maximal relative error of: - (1 + 2^-130.22) * (1 + 2^-127) - 1 < 2^-126.852. - */ - return; - } - - // now 2 <= e <= 1024 - - /* The upper 64-bit word X->hi corresponds to hi/2^64*2^e, if multiplied by - T[i]/2^((i+1)*64) it yields hi*T[i]/2^128 * 2^(e-i*64). - If e-64i <= -128, it contributes to less than 2^-128; - if e-64i >= 128, it yields an integer, which is 0 modulo 1. - We thus only consider the values of i such that -127 <= e-64i <= 127, - i.e., (-127+e)/64 <= i <= (127+e)/64. - Up to 4 consecutive values of T[i] can contribute (only 3 when e is a - multiple of 64). */ - int i = (e < 127) ? 0 : (e - 127 + 64 - 1) / 64; // ceil((e-127)/64) - // 0 <= i <= 15 - uint64_t c[5]; - u = (u128) X->hi * (u128) T[i+3]; // i+3 <= 18 - c[0] = Uint128Low64(u); - c[1] = Uint128High64(u); - u = (u128) X->hi * (u128) T[i+2]; - c[1] += Uint128Low64(u); - c[2] = Uint128High64(u) + (c[1] < (uint64_t) u); - u = (u128) X->hi * (u128) T[i+1]; - c[2] += Uint128Low64(u); - c[3] = Uint128High64(u) + (c[2] < (uint64_t) u); - u = (u128) X->hi * (u128) T[i]; - c[3] += Uint128Low64(u); - c[4] = Uint128High64(u) + (c[3] < (uint64_t) u); - - /* up to here, the ignored part hi*(T[i+4]+T[i+5]+...) can contribute by - less than 2^64 in c[0], thus less than 1 in c[1] */ - - int f = e - 64 * i; // hi*T[i]/2^128 is multiplied by 2^f - /* {c, 5} = hi*(T[i]+T[i+1]/2^64+T[i+2]/2^128+T[i+3]/2^192) */ - /* now shift c[0..4] by f bits to the left */ - uint64_t tiny; - if (f < 64) - { - X->hi = (c[4] << f) | (c[3] >> (64 - f)); - X->lo = (c[3] << f) | (c[2] >> (64 - f)); - tiny = (c[2] << f) | (c[1] >> (64 - f)); - /* the ignored part was less than 1 in c[1], - thus less than 2^(f-64) <= 1/2 in tiny */ - } - else if (f == 64) - { - X->hi = c[3]; - X->lo = c[2]; - tiny = c[1]; - /* the ignored part was less than 1 in c[1], - thus less than 1 in tiny */ - } - else /* 65 <= f <= 127: this case can only occur when e >= 65 */ - { - int g = f - 64; /* 1 <= g <= 63 */ - /* we compute an extra term */ - u = (u128) X->hi * (u128) T[i+4]; // i+4 <= 19 - u = u >> 64; - c[0] += Uint128Low64(u); - c[1] += (c[0] < u); - c[2] += (c[0] < u) && c[1] == 0; - c[3] += (c[0] < u) && c[1] == 0 && c[2] == 0; - c[4] += (c[0] < u) && c[1] == 0 && c[2] == 0 && c[3] == 0; - X->hi = (c[3] << g) | (c[2] >> (64 - g)); - X->lo = (c[2] << g) | (c[1] >> (64 - g)); - tiny = (c[1] << g) | (c[0] >> (64 - g)); - /* the ignored part was less than 1 in c[0], - thus less than 1/2 in tiny */ - } - /* The approximation error between X/in(2pi) mod 1 and - X->hi/2^64 + X->lo/2^128 + tiny/2^192 is: - (a) the ignored part in tiny, which is less than ulp(tiny), - thus less than 1/2^192; - (b) the ignored terms hi*T[i+4] + ... or hi*T[i+5] + ..., - which accumulate to less than ulp(tiny) too, thus - less than 1/2^192. - Thus the approximation error is less than 2^-191 (absolute). - */ - X->ex = 0; - normalize (X); - /* the worst case (for 2^25 <= x < 2^1024) is X->ex = -61, attained - for |x| = 0x1.6ac5b262ca1ffp+851 */ - if (X->ex < 0) // put the upper -ex bits of tiny into low bits of lo - X->lo |= tiny >> (64 + X->ex); - /* Since X->ex >= -61, it means X >= 2^-62 before the normalization, - thus the maximal absolute error of 2^-191 yields a relative error - bounded by 2^-191/2^-62 = 2^-129. - There is an additional truncation error (for tiny) of at most 1 ulp - of X->lo, thus at most 2^-127. - The relative error is thus bounded by 2^-126.67. */ -} - -/* Given Xin:=X with 0 <= Xin < 1, return i and modify X such that - Xin = i/2^11 + Xout, with 0 <= Xout < 2^-11. - This operation is exact. */ -static int -reduce2 (dint64_t *X) -{ - if (X->ex <= -11) - return 0; - int sh = 64 - 11 - X->ex; - int i = X->hi >> sh; - X->hi = X->hi & ((1ull << sh) - 1); - normalize (X); - return i; -} - -/* h+l <- c1/2^64 + c0/2^128 */ -static void -set_dd (double *h, double *l, uint64_t c1, uint64_t c0) -{ - uint64_t e, f, g; - b64u64_u t; - if (c1) - { - e = __builtin_clzl (c1); - if (e) - { - c1 = (c1 << e) | (c0 >> (64 - e)); - c0 = c0 << e; - } - f = 0x3fe - e; - t.u = (f << 52) | ((c1 << 1) >> 12); - *h = t.f; - c0 = (c1 << 53) | (c0 >> 11); - if (c0) - { - g = __builtin_clzl (c0); - if (g) - c0 = c0 << g; - t.u = ((f - 53 - g) << 52) | ((c0 << 1) >> 12); - *l = t.f; - } - else - *l = 0; - } - else if (c0) - { - e = __builtin_clzl (c0); - f = 0x3fe - 64 - e; - c0 = c0 << (e+1); // most significant bit shifted out - /* put the upper 52 bits of c0 into h */ - t.u = (f << 52) | (c0 >> 12); - *h = t.f; - /* put the lower 12 bits of c0 into l */ - c0 = c0 << 52; - if (c0) - { - int g = __builtin_clzl (c0); - c0 = c0 << (g+1); - t.u = ((f - 64 - g) << 52) | (c0 >> 12); - *l = t.f; - } - else - *l = 0; - } - else - *h = *l = 0; - /* Since we truncate from two 64-bit words to a double-double, - we have another truncation error of less than 2^-106, thus - the absolute error is bounded as follows: - | h + l - frac(x/(2pi)) | < 2^-75.999 + 2^-106 < 2^-75.998 */ -} - -/* Assuming 0x1.7137449123ef6p-26 < x < +Inf, - return i and set h,l such that i/2^11+h+l approximates frac(x/(2pi)). - If x <= 0x1.921fb54442d18p+2: - | i/2^11 + h + l - frac(x/(2pi)) | < 2^-104.116 * |i/2^11 + h + l| - with |h| < 2^-11 and |l| < 2^-52.36. - - Otherwise only the absolute error is bounded: - | i/2^11 + h + l - frac(x/(2pi)) | < 2^-75.998 - with 0 <= h < 2^-11 and |l| < 2^-53. - - In both cases we have |l| < 2^-51.64*|i/2^11 + h|. - - Put in err1 a bound for the absolute error: - | i/2^11 + h + l - frac(x/(2pi)) |. -*/ -static int -reduce_fast (double *h, double *l, double x, double *err1) -{ - if (__builtin_expect(x <= 0x1.921fb54442d17p+2, 1)) [[likely]] // x < 2*pi - { - /* | CH+CL - 1/(2pi) | < 2^-110.523 */ -#define CH 0x1.45f306dc9c883p-3 -#define CL -0x1.6b01ec5417056p-57 - a_mul (h, l, CH, x); // exact - *l = __builtin_fma (CL, x, *l); - /* The error in the above fma() is at most ulp(l), - where |l| <= CL*|x|+|l_in|. - Assume 2^(e-1) <= x < 2^e. - Then |h| < 2^(e-2) and |l_in| <= 1/2 ulp(2^(e-2)) = 2^(e-55), - where l_in is the value of l after a_mul. - Then |l| <= CL*x + 2^(e-55) <= 2^e*(CL+2-55) < 2^e * 2^-55.6. - The rounding error of the fma() is bounded by - ulp(l) <= 2^e * ulp(2^-55.6) = 2^(e-108). - The error due to the approximation of 1/(2pi) - is bounded by 2^-110.523*x <= 2^(e-110.523). - Adding both errors yields: - |h + l - x/(2pi)| < 2^e * (2^-108 + 2^-110.523) < 2^e * 2^-107.768. - Since |x/(2pi)| > 2^(e-1)/(2pi), the relative error is bounded by: - 2^e * 2^-107.768 / (2^(e-1)/(2pi)) = 4pi * 2^-107.768 < 2^-104.116. - - Bound on l: since |h| < 1, we have after |l| <= ulp(h) <= 2^-53 - after a_mul(), and then |l| <= |CL|*0x1.921fb54442d17p+2 + 2^-53 - < 2^-52.36. - - Bound on l relative to h: after a_mul() we have |l| <= ulp(h) - <= 2^-52*h. After fma() we have |l| <= CL*x + 2^-52*h - <= 2^-53.84*CH*x + 2^-52*h <= (2^-53.84+2^-52)*h < 2^-51.64*h. - */ - *err1 = 0x1.d9p-105 * *h; // error < 2^-104.116 * h - } - else // x > 0x1.921fb54442d17p+2 - { - b64u64_u t = {.f = x}; - int e = (t.u >> 52) & 0x7ff; /* 1025 <= e <= 2046 */ - /* We have 2^(e-1023) <= x < 2^(e-1022), thus - ulp(x) is a multiple of 2^(e-1075), for example - if x is just above 2*pi, e=1025, 2^2 <= x < 2^e, - and ulp(x) is a multiple of 2^-50. - On the other side 1/(2pi) ~ T[0]/2^64 + T[1]/2^128 + T[2]/2^192 + ... - Let i be the smallest integer such that 2^(e-1075)/2^(64*(i+1)) - is not an integer, i.e., e - 1139 - 64i < 0, i.e., - i >= (e-1138)/64. */ - uint64_t m = (1ull << 52) | (t.u & 0xffffffffffffful); - uint64_t c[3]; - u128 u; - // x = m/2^53 * 2^(e-1022) - if (e <= 1074) // 1025 <= e <= 1074: 2^2 <= x < 2^52 - { - /* In that case the contribution of x*T[2]/2^192 is less than - 2^(52+64-192) <= 2^-76. */ - u = (u128) m * (u128) T[1]; - c[0] = absl::Uint128Low64(u); - c[1] = absl::Uint128High64(u); - u = (u128) m * (u128) T[0]; - c[1] += absl::Uint128Low64(u); - c[2] = absl::Uint128High64(u) + (c[1] < (uint64_t) u); - /* | c[2]*2^128+c[1]*2^64+c[0] - m/(2pi)*2^128 | < m*T[2]/2^64 < 2^53 - thus: - | (c[2]*2^128+c[1]*2^64+c[0])*2^(e-1203) - x/(2pi) | < 2^(e-1150) - The low 1075-e bits of c[2] contribute to frac(x/(2pi)). - */ - e = 1075 - e; // 1 <= e <= 50 - // e is the number of low bits of C[2] contributing to frac(x/(2pi)) - } - else // 1075 <= e <= 2046, 2^52 <= x < 2^1024 - { - int i = (e - 1138 + 63) / 64; // i = ceil((e-1138)/64), 0 <= i <= 15 - /* m*T[i] contributes to f = 1139 + 64*i - e bits to frac(x/(2pi)) - with 1 <= f <= 64 - m*T[i+1] contributes a multiple of 2^(-f-64), - and at most to 2^(53-f) - m*T[i+2] contributes a multiple of 2^(-f-128), - and at most to 2^(-11-f) - m*T[i+3] contributes a multiple of 2^(-f-192), - and at most to 2^(-75-f) <= 2^-76 - */ - u = (u128) m * (u128) T[i+2]; - c[0] = absl::Uint128Low64(u); - c[1] = absl::Uint128High64(u); - u = (u128) m * (u128) T[i+1]; - c[1] += absl::Uint128Low64(u); - c[2] = absl::Uint128High64(u) + (c[1] < (uint64_t) u); - u = (u128) m * (u128) T[i]; - c[2] += absl::Uint128Low64(u); - e = 1139 + (i<<6) - e; // 1 <= e <= 64 - // e is the number of low bits of C[2] contributing to frac(x/(2pi)) - } - if (e == 64) - { - c[0] = c[1]; - c[1] = c[2]; - } - else - { - c[0] = (c[1] << (64 - e)) | c[0] >> e; - c[1] = (c[2] << (64 - e)) | c[1] >> e; - } - /* In all cases the ignored contribution from x*T[2] or x*T[i+3] - is less than 2^-76, - and the truncated part from the above shift is less than 2^-128 thus: - | c[1]/2^64 + c[0]/2^128 - frac(x/(2pi)) | < 2^-76+2^-128 < 2^-75.999 - */ - set_dd (h, l, c[1], c[0]); - /* set_dd() ensures |h| < 1 and |l| < ulp(h) <= 2^-53 */ - *err1 = 0x1.01p-76; - } - - double i = __builtin_floor (*h * 0x1p11); - *h = __builtin_fma (i, -0x1p-11, *h); - return i; -} - -/* return the maximal absolute error */ -static double -sin_fast (double *h, double *l, double x) -{ - int neg = x < 0, is_sin = 1; - double absx = neg ? -x : x; - - /* now x > 0x1.7137449123ef6p-26 */ - double err1; - int i = reduce_fast (h, l, absx, &err1); - /* err1 is an absolute bound for | i/2^11 + h + l - frac(x/(2pi)) |: - | i/2^11 + h + l - frac(x/(2pi)) | < err1 */ - - // if i >= 2^10: 1/2 <= frac(x/(2pi)) < 1 thus pi <= x <= 2pi - // we use sin(pi+x) = -sin(x) - neg = neg ^ (i >> 10); - i = i & 0x3ff; - // | i/2^11 + h + l - frac(x/(2pi)) | mod 1/2 < err1 - - // now i < 2^10 - // if i >= 2^9: 1/4 <= frac(x/(2pi)) < 1/2 thus pi/2 <= x <= pi - // we use sin(pi/2+x) = cos(x) - is_sin = is_sin ^ (i >> 9); - i = i & 0x1ff; - // | i/2^11 + h + l - frac(x/(2pi)) | mod 1/4 < err1 - - // now 0 <= i < 2^9 - // if i >= 2^8: 1/8 <= frac(x/(2pi)) < 1/4 - // we use sin(pi/2-x) = cos(x) - if (i & 0x100) // case pi/4 <= x_red <= pi/2 - { - is_sin = !is_sin; - i = 0x1ff - i; - /* 0x1p-11 - h is exact below: indeed, reduce_fast first computes - a first value of h (say h0, with 0 <= h0 < 1), then i = floor(h0*2^11) - and h1 = h0 - 2^11*i with 0 <= h1 < 2^-11. - If i >= 2^8 here, this implies h0 >= 1/2^3, thus ulp(h0) >= 2^-55: - h0 and h1 are integer multiples of 2^-55. - Thus h1 = k*2^-55 with 0 <= k < 2^44 (since 0 <= h1 < 2^-11). - Then 0x1p-11 - h = (2^44-k)*2^-55 is exactly representable. - We can have a huge cancellation in 0x1p-11 - h, for example for - x = 0x1.61a3db8c8d129p+1023 where we have before this operation - h = 0x1.ffffffffff8p-12, and h = 0x1p-53 afterwards. But this - does not hurt since we bound the absolute error and not the - relative error at the end. */ - *h = 0x1p-11 - *h; - *l = -*l; - } - - /* Now 0 <= i < 256 and 0 <= h+l < 2^-11 - with | i/2^11 + h + l - frac(x/(2pi)) | cmod 1/4 < err1 - If is_sin=1, sin |x| = sin2pi (R + err1); - if is_sin=0, sin |x| = cos2pi (R + err1). - In both cases R = i/2^11 + h + l, 0 <= R < 1/4. - */ - double sh, sl, ch, cl; - /* since the SC[] table evaluates at i/2^11 + SC[i][0] and not at i/2^11, - we must subtract SC[i][0] from h+l */ - /* Here h = k*2^-55 with 0 <= k < 2^44, and SC[i][0] is an integer - multiple of 2^-62, with |SC[i][0]| < 2^-24, thus SC[i][0] = m*2^-62 - with |m| < 2^38. It follows h-SC[i][0] = (k*2^7 + m)*2^-62 with - 2^51 - 2^38 < k*2^7 + m < 2^51 + 2^38, thus h-SC[i][0] is exact. - Now |h| < 2^-11 + 2^-24. */ - *h -= SC[i][0]; - // now -2^-24 < h < 2^-11+2^-24 - // from reduce_fast() we have |l| < 2^-52.36 - double uh, ul; - a_mul (&uh, &ul, *h, *h); - ul = __builtin_fma (*h + *h, *l, ul); - // uh+ul approximates (h+l)^2 - evalPSfast (&sh, &sl, *h, *l, uh, ul); - /* the absolute error of evalPSfast() is less than 2^-77.09 from - routine evalPSfast() in sin.sage: - | sh + sh - sin2pi(h+l) | < 2^-77.09 */ - evalPCfast (&ch, &cl, uh, ul); - /* the relative error of evalPCfast() is less than 2^-69.96 from - routine evalPCfast(rel=true) in sin.sage: - | ch + cl - cos2pi(h+l) | < 2^-69.96 * |ch + cl| */ - double err; - if (is_sin) - { - s_mul (&sh, &sl, SC[i][2], sh, sl); - s_mul (&ch, &cl, SC[i][1], ch, cl); - fast_two_sum (h, l, ch, sh); - *l += sl + cl; - /* absolute error bounded by 2^-68.588 - from global_error(is_sin=true,rel=false) in sin.sage: - | h + l - sin2pi (R) | < 2^-68.588 - thus: - | h + l - sin |x| | < 2^-68.588 + | sin2pi (R) - sin |x| | - < 2^-68.588 + err1 */ - err = 0x1.55p-69; // 2^-66.588 < 0x1.55p-69 - } - else - { - s_mul (&ch, &cl, SC[i][2], ch, cl); - s_mul (&sh, &sl, SC[i][1], sh, sl); - fast_two_sum (h, l, ch, -sh); - *l += cl - sl; - /* absolute error bounded by 2^-68.414 - from global_error(is_sin=false,rel=false) in sin.sage: - | h + l - cos2pi (R) | < 2^-68.414 - thus: - | h + l - sin |x| | < 2^-68.414 + | cos2pi (R) - sin |x| | - < 2^-68.414 + err1 */ - err = 0x1.81p-69; // 2^-68.414 < 0x1.81p-69 - } - static const double sgn[2] = {1.0, -1.0}; - *h *= sgn[neg]; - *l *= sgn[neg]; - return err + err1; -} - -/* Assume x is a regular number, and |x| > 0x1.7137449123ef6p-26. */ -static double -sin_accurate (double x) -{ - double absx = (x > 0) ? x : -x; - - dint64_t X[1]; - dint_fromd (X, absx); - - /* reduce argument */ - reduce (X); - - // now |X - x/(2pi) mod 1| < 2^-126.67*X, with 0 <= X < 1. - - int neg = x < 0, is_sin = 1; - - // Write X = i/2^11 + r with 0 <= r < 2^11. - int i = reduce2 (X); // exact - - if (i & 0x400) // pi <= x < 2*pi: sin(x) = -sin(x-pi) - { - neg = !neg; - i = i & 0x3ff; - } - - // now i < 2^10 - - if (i & 0x200) // pi/2 <= x < pi: sin(x) = cos(x-pi/2) - { - is_sin = 0; - i = i & 0x1ff; - } - - // now 0 <= i < 2^9 - - if (i & 0x100) - // pi/4 <= x < pi/2: sin(x) = cos(pi/2-x), cos(x) = sin(pi/2-x) - { - is_sin = !is_sin; - X->sgn = 1; // negate X - add_dint (X, &MAGIC, X); // X -> 2^-11 - X - // here: 256 <= i <= 511 - i = 0x1ff - i; - // now 0 <= i < 256 - } - - // now 0 <= i < 256 and 0 <= X < 2^-11 - - /* If is_sin=1, sin |x| = sin2pi (R * (1 + eps)) - (cases 0 <= x < pi/4 and 3pi/4 <= x < pi) - if is_sin=0, sin |x| = cos2pi (R * (1 + eps)) - (case pi/4 <= x < 3pi/4) - In both cases R = i/2^11 + X, 0 <= R < 1/4, and |eps| < 2^-126.67. - */ - - dint64_t U[1], V[1], X2[1]; - mul_dint (X2, X, X); // X2 approximates X^2 - evalPC (U, X2); // cos2pi(X) - /* since 0 <= X < 2^-11, we have 0.999 < U <= 1 */ - evalPS (V, X, X2); // sin2pi(X) - /* since 0 <= X < 2^-11, we have 0 <= V < 0.0005 */ - if (is_sin) - { - // sin2pi(R) ~ sin2pi(i/2^11)*cos2pi(X)+cos2pi(i/2^11)*sin2pi(X) - mul_dint (U, S+i, U); - /* since 0 <= S[i] < 0.705 and 0.999 < Uin <= 1, we have - 0 <= U < 0.705 */ - mul_dint (V, C+i, V); - /* For the error analysis, we distinguish the case i=0. - For i=0, we have S[i]=0 and C[1]=1, thus V is the value computed - by evalPS() above, with relative error < 2^-124.648. - - For 1 <= i < 256, analyze_sin_case1(rel=true) from sin.sage gives a - relative error bound of -122.797 (obtained for i=1). - In all cases, the relative error for the computation of - sin2pi(i/2^11)*cos2pi(X)+cos2pi(i/2^11)*sin2pi(X) is bounded by -122.797 - not taking into account the approximation error in R: - |U - sin2pi(R)| < |U| * 2^-122.797, with U the value computed - after add_dint (U, U, V) below. - - For the approximation error in R, we have: - sin |x| = sin2pi (R * (1 + eps)) - R = i/2^11 + X, 0 <= R < 1/4, and |eps| < 2^-126.67. - Thus sin|x| = sin2pi(R+R*eps) - = sin2pi(R)+R*eps*2*pi*cos2pi(theta), theta in [R,R+R*eps] - Since 2*pi*R/sin(2*pi*R) < pi/2 for R < 1/4, it follows: - | sin|x| - sin2pi(R) | < pi/2*R*|sin(2*pi*R)| - | sin|x| - sin2pi(R) | < 2^-126.018 * |sin2pi(R)|. - - Adding both errors we get: - | sin|x| - U | < |U| * 2^-122.797 + 2^-126.018 * |sin2pi(R)| - < |U| * 2^-122.797 + 2^-126.018 * |U| * (1 + 2^-122.797) - < |U| * 2^-122.650. - */ - } - else - { - // cos2pi(R) ~ cos2pi(i/2^11)*cos2pi(X)-sin2pi(i/2^11)*sin2pi(X) - mul_dint (U, C+i, U); - mul_dint (V, S+i, V); - V->sgn = 1 - V->sgn; // negate V - /* For 0 <= i < 256, analyze_sin_case2(rel=true) from sin.sage gives a - relative error bound of -123.540 (obtained for i=0): - |U - cos2pi(R)| < |U| * 2^-123.540, with U the value computed - after add_dint (U, U, V) below. - - For the approximation error in R, we have: - sin |x| = cos2pi (R * (1 + eps)) - R = i/2^11 + X, 0 <= R < 1/4, and |eps| < 2^-126.67. - Thus sin|x| = cos2pi(R+R*eps) - = cos2pi(R)-R*eps*2*pi*sin2pi(theta), theta in [R,R+R*eps] - Since we have R < 1/4, we have cos2pi(R) >= sqrt(2)/2, - and it follows: - | sin|x|/cos2pi(R) - 1 | < 2*pi*R*eps/(sqrt(2)/2) - < pi/2*eps/sqrt(2) [since R < 1/4] - < 2^-126.518. - Adding both errors we get: - | sin|x| - U | < |U| * 2^-123.540 + 2^-126.518 * |cos2pi(R)| - < |U| * 2^-123.540 + 2^-126.518 * |U| * (1 + 2^-123.540) - < |U| * 2^-123.367. - */ - } - add_dint (U, U, V); - /* If is_sin=1: - | sin|x| - U | < |U| * 2^-122.650 - If is_sin=0: - | cos|x| - U | < |U| * 2^-123.367. - In all cases the total error is bounded by |U| * 2^-122.650. - The term |U| * 2^-122.650 contributes to at most 2^(128-122.650) < 41 ulps - relatively to U->lo. - */ - uint64_t err = 41; - uint64_t hi0, hi1, lo0, lo1; - lo0 = U->lo - err; - hi0 = U->hi - (lo0 > U->lo); - lo1 = U->lo + err; - hi1 = U->hi + (lo1 < U->lo); - /* check the upper 54 bits are equal */ - if ((hi0 >> 10) != (hi1 >> 10)) - { - static const double exceptions[][3] = { - {0x1.e0000000001c2p-20, 0x1.dfffffffff02ep-20, 0x1.dcba692492527p-146}, - }; - for (int i = 0; i < 1; i++) - { - if (__builtin_fabs (x) == exceptions[i][0]) - return (x > 0) ? exceptions[i][1] + exceptions[i][2] - : -exceptions[i][1] - exceptions[i][2]; - } - printf ("Rounding test of accurate path failed for sin(%la)\n", x); - printf ("Please report the above to core-math@inria.fr\n"); - exit (1); - } - - if (neg) - U->sgn = 1 - U->sgn; - - double y = dint_tod (U); - - return y; -} - -double __cdecl -cr_sin (double x) -{ - b64u64_u t = {.f = x}; - int e = (t.u >> 52) & 0x7ff; - - if (__builtin_expect (e == 0x7ff, 0)) [[unlikely]] /* NaN, +Inf and -Inf. */ - { - t.u = ~0ull; - return t.f; - } - - /* now x is a regular number */ - - /* For |x| <= 0x1.7137449123ef6p-26, sin(x) rounds to x (to nearest): - we can assume x >= 0 without loss of generality since sin(-x) = -sin(x), - we have x - x^3/6 < sin(x) < x for say 0 < x <= 1 thus - |sin(x) - x| < x^3/6. - Write x = c*2^e with 1/2 <= c < 1. - Then ulp(x)/2 = 2^(e-54), and x^3/6 = c^3/6*2^(3e), thus - x^3/6 < ulp(x)/2 rewrites as c^3/6*2^(3e) < 2^(e-54), - or c^3*2^(2e+53) < 3 (1). - For e <= -26, since c^3 < 1, we have c^3*2^(2e+53) < 2 < 3. - For e=-25, (1) rewrites 8*c^3 < 3 which yields c <= 0x1.7137449123ef6p-1. - */ - uint64_t ux = t.u & 0x7fffffffffffffff; - // 0x3e57137449123ef6 = 0x1.7137449123ef6p-26 - if (ux <= 0x3e57137449123ef6) - // Taylor expansion of sin(x) is x - x^3/6 around zero - // for x=-0, fma (x, -0x1p-54, x) returns +0 - return (x == 0) ? x :__builtin_fma (x, -0x1p-54, x); - - double h, l, err; - err = sin_fast (&h, &l, x); - double left = h + (l - err), right = h + (l + err); - /* With SC[] from ./buildSC 15 we get 1100 failures out of 50000000 - random tests, i.e., about 0.002%. */ - if (left == right) - return left; - - return sin_accurate (x); -} - -} // namespace internal -} // namespace _sin -} // namespace functions -} // namespace principia - -#if PRINCIPIA_COMPILER_MSVC -#undef __builtin_clzl -#undef __builtin_fma -#endif -#undef __builtin_expect -#undef __builtin_fabs -#undef __builtin_floor diff --git a/functions/sin.hpp b/functions/sin.hpp deleted file mode 100644 index c5a78e48b6..0000000000 --- a/functions/sin.hpp +++ /dev/null @@ -1,16 +0,0 @@ -#pragma once - -namespace principia { -namespace functions { -namespace _sin { -namespace internal { - -double __cdecl cr_sin(double x); - -} // namespace internal - -using internal::cr_sin; - -} // namespace _sin -} // namespace functions -} // namespace principia diff --git a/inria_core-math.props b/inria_core-math.props new file mode 100644 index 0000000000..1dca014502 --- /dev/null +++ b/inria_core-math.props @@ -0,0 +1,16 @@ + + + + + + + + $(SolutionDir)..\Inria\core-math\include;%(AdditionalIncludeDirectories) + + + $(SolutionDir)..\Inria\core-math\msvc\$(PrincipiaDependencyConfiguration)\$(Platform);%(AdditionalLibraryDirectories) + core-math.lib;%(AdditionalDependencies) + + + + \ No newline at end of file diff --git a/principia.props b/principia.props index d3507578f7..679aef9cb0 100644 --- a/principia.props +++ b/principia.props @@ -176,6 +176,7 @@ + diff --git a/rebuild_all_solutions.ps1 b/rebuild_all_solutions.ps1 index 6669caf430..fb552f060c 100644 --- a/rebuild_all_solutions.ps1 +++ b/rebuild_all_solutions.ps1 @@ -9,12 +9,14 @@ $dependencies = @(".\Google\glog\msvc\glog.sln", ".\Google\benchmark\msvc\google-benchmark.sln", ".\Google\gipfeli\msvc\gipfeli.sln", ".\Google\abseil-cpp\msvc\abseil-cpp.sln", + ".\Inria\core-math\msvc\core-math.sln", ".\LLNL\zfp\msvc\zfp.sln") foreach ($directory_and_repositories in @( @("Boost", @("config", "multiprecision")), @("Google", @("glog", "googletest", "protobuf", "benchmark", "gipfeli", "abseil-cpp", "chromium")), + @("Inria", @("core-math")), @("LLNL", @("zfp")))) { $directory, $repositories = $directory_and_repositories New-Item -ItemType Directory -Force -Path $directory