stillwater-sc / universal / 31842405092

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Commit Message

fix(takum_log): make multiplication and division exact (#1312)

Both operators evaluated in a double: operator*= was
convert_ieee754(double(*this) * double(rhs)).  That rounds each operand to a
double before combining them, and takum_log<64,3> carries 59 fraction bits of l
against a double's 53.  Measured against a 113-bit reference, 99% of
wide-configuration products came back incorrectly rounded.

Neither operation needs a double, or any extended-precision type.
|x*y| = sqrt(e)^(lx + ly) and |x/y| = sqrt(e)^(lx - ly), and l is an integer
characteristic plus a p-bit fraction, so aligning the two on the wider
denominator and adding or subtracting is exact integer arithmetic.
encode_fraction() then performs the single rounding the result needs.

                        before        after
    takum_log<16,3>     0.00%         0.00%
    takum_log<32,3>     0.00%         0.00%
    takum_log<64,3>    99.22%         0.00%

Addition and subtraction are untouched, as is the linear takum.  Those have no
logarithmic shortcut and need the extended-precision path #1300 proposes.

The suite exists because nothing else covers the new code: removing the division
borrow is caught by neither the api, mathlib, conversion nor logic suites, nor by
identity checks like a/1 and a/a, which never trigger a borrow.  The load-bearing
check compares against a reference recomputed in dd_cascade; removing the borrow
trips it with 7,100 failures at 8 bits and 289,024 at 10, while correctly leaving
multiplication passing.

Known gap, tracked as follow-up: that reference check runs at 8 and 10 bits only,
so the 64-bit width this change is for has no committed coverage.  It is verified
against a 113-bit __float128 oracle, which needs quadmath and cannot ship.
VerifyCorrectlyRounded also uses 1ull << nbits, which would be undefined at
nbits == 64; a bounded 64-bit case has to avoid instantiating it directly.

Also adds takum_log to the conventional-commit scopes.

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