WIP: Implement multi-word division for BigInt

[ellie-idb]

This PR implements a version of Algorithm D (specified by Donald Knuth in TOACP Vol 2. 4.3, C implementation by Henry S. Warren Jr.), which aims to implement traditional "school-book" division for arbitrarily long numbers.

This is part of the process of getting #373 merged.

Status

• Basic un-typed implementation
• Implement in UInt
• Implement in BigInt

[codecov-commenter]

Codecov Report

Merging #387 (d1ef009) into master (05abcc5) will increase coverage by 0.10%.
The diff coverage is 96.82%.

❗ Current head d1ef009 differs from pull request most recent head b9a6aa8. Consider uploading reports for the commit b9a6aa8 to get more accurate results

@@            Coverage Diff             @@
##           master     #387      +/-   ##
==========================================
+ Coverage   92.86%   92.97%   +0.10%     
==========================================
  Files          65       66       +1     
  Lines       15970    16303     +333     
==========================================
+ Hits        14830    15157     +327     
- Misses       1140     1146       +6     

Impacted Files	Coverage Δ
source/mir/serde.d	72.50% <ø> (ø)
source/mir/series.d	94.69% <ø> (ø)
source/mir/numeric.d	92.89% <89.47%> (-0.48%)	⬇️
source/mir/format.d	95.83% <92.30%> (+0.12%)	⬆️
source/mir/bignum/integer.d	91.71% <97.67%> (+4.01%)	⬆️
source/mir/algebraic_alias/transform.d	100.00% <100.00%> (ø)
source/mir/bignum/fixed.d	95.90% <100.00%> (+1.65%)	⬆️
source/mir/bignum/low_level_view.d	94.10% <100.00%> (+0.13%)	⬆️
source/mir/math/stat.d	100.00% <100.00%> (ø)
source/mir/ndslice/fuse.d	86.07% <100.00%> (+0.54%)	⬆️
... and 8 more

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[ellie-idb]

@9il opCmp for BigUIntView fails if the coefficients don't exactly match, which means that a line like this will fail:

auto b = BigInt!4.fromHexString("c39b18a9f06fd8e962d99935cea0707f79a222050aaeaaaed17feb7aa76999d7");

auto rem = b /= UInt!128.fromHexString("f79a222050aaeaaa417fa25a2ac93291"); 

// fails
// b is BigIntView!(ulong, WordEndian.little)(BigUIntView!(ulong, WordEndian.little)([8989662390899871995, 14572942669551396987, 0, 0]), false)
// rhs is BigIntView!(ulong, WordEndian.little)(BigUIntView!(ulong, WordEndian.little)([8989662390899871995, 14572942669551396987]), false)
assert(b == BigInt!4.fromHexString("ca3d7e25aebe687b7cc1b250b44690fb"));
assert(rem == UInt!128.fromHexString("bf4c87424431d21563f23b1fc00d75ac"));

Should I automatically normalize the quotient view after dividing? Or would you be fine with replacing BigUIntView.opCmp with:

    ///
    bool opEquals(scope BigUIntView!(const W, endian) rhs)
        const @safe pure nothrow @nogc scope
    {
        return this.normalized.coefficients == rhs.normalized.coefficients;
    }

[9il]

just qhat > uint.max and remove b

[ellie-idb]

Simplifying the while statement to just while (qhat > uint.max) causes test failure, but I'll pull out the b variable.

[9il]

I mean qhat >= b => qhat > uint.max . The other part of condition is OK

[9il]

Should I automatically normalize the quotient view after dividing? Or would you be fine with replacing BigUIntView.opCmp with:

Yes, it should be normalized inside divm. You can pass the views by reference if needed. The same is true for the remainder.

[9il]

Also we need %=, /, and % operations. And standalone divRem method that performs both division and remainder for Uints. All /=, %=, /, and % can call it instead of the divm.

[ellie-idb]

OK. I got this implemented in UInt -- should it also be implemented in BigInt?

[9il]

No

[ellie-idb]

@9il able to review?

[9il]

No

[9il]

There are still pending comments.

[ellie-idb]

FYI: This is ready for further review @9il