Round exact quotients of approximations to the requested digits [patch] - #141
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Divide returns a terminating quotient exactly. Asin(±1), Atan2(y, 0), DegreesToRadians and RootN(x, -n) divided a value already rounded to a wider working precision, so when the quotient terminated they returned the guard digits too, and those digits were wrong: Asin(1, 5) gave 1.570796326794895 and RootN(1.0486, -2, 4) gave 0.9765625. Add DivideApproximation, which divides at working precision and then rounds to significantDigits, and use it on those paths and on the other approx / approx paths (Tan, TanPi, AsinPi, AcosPi, AtanPi, RadiansToDegrees). RootN(x, -n) now takes the inner root with guard digits. Fixes #121 Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014jNTPLxThrQnZyafw1CZ1p
The general paths of AsinPi and AcosPi and all of TanPi had no test, which left the new DivideApproximation call sites there uncovered. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_014jNTPLxThrQnZyafw1CZ1p
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Fixes #121
What changed
Divide(a, b, sd)returns a terminating quotient exactly and doesn't round it. Several paths divided a value that was already rounded to a wider working precision, most oftenPiTo(working). When that quotient terminated, the caller got the guard digits back as well, and those digits were wrong.A new private helper,
DivideApproximation(numerator, denominator, working, sd), divides at working precision and then callsReduceSignificance(sd). It is now used on:Asinat ±1,Atan2withx = 0,DegreesToRadians, andRootNwithn < 0. The inner root ofRootN(x, -n)is now also computed withRootGuardDigits, so the reciprocal comes from an unrounded root.TanandTanPi(sin/cos),AsinPi,AcosPi,AtanPiandRadiansToDegrees.Tests
Asin(±1, 5)1.570796326794895±1.5708Atan2(±1, 0, 5)1.570796326794895±1.5708DegreesToRadians(180, 5)3.141592653589793.1416DegreesToRadians(90, 5)1.5708RootN(1.0486, -2, 4)0.97656250.9766These cases are in
TestExactQuotientsOfApproximatePiStillRoundToTheRequestedDigitsandTestNegativeDegreeRootRoundsTheReciprocalOfAnUnroundedRoot. Both tests failed onmainand pass with the fix. The full suite passes: 436 of 436.This PR is independent of #140 (the Exp2 guard digits). Both come from the same precision-contract audit, but they touch different files.
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https://claude.ai/code/session_014jNTPLxThrQnZyafw1CZ1p
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