v3.0.0: Exact square roots and faster wide-type paths - #11
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A 128-bit fixed-point division costs about twelve multiplications, and forming a divisor like 182 in T overflows types with few integer bits. CordicNumber gains div_int and mul_int, which work on the raw representation, plus wrapping_mul/add/sub and checked_int_log2; round and to_i32 now saturate instead of wrapping at MAX. exp and sinh_cosh divide by their Taylor divisors with div_int. sin_cos and exp estimate their reductions with stored reciprocals when the integer bits do not outnumber the fractional bits, and correct the estimate once in either direction. ln normalises from the leading bit instead of a shift loop. Horner evaluation wraps, since every intermediate stays below 1 for the tables in use. Results are bit-identical on every layout with at least 8 integer bits except where the old code was wrong: exp above 2^31·ln 2 on 64-bit and wider types wrapped in to_i32, and angles within π of MAX saturated the reduction to sin_cos(0). I4F12, I4F60 and I3F125 gain working exp and sinh_cosh. I64F64: sin_cos 52 -> 26 ns, exp 159 -> 41 ns, sinh_cosh 175 -> 54 ns, and ln normalisation is O(1). Adds I64F64 benchmarks, instantiates the no-panic binary on three layouts, and raises the no-panic floor to 0.1.37.
pow(base, exponent) = exp(exponent · ln(base)), fallible on a negative base or a zero base with a negative exponent; 0^0 is 1 and exp's saturation applies. ln is split into its domain check and ln_positive so pow can skip the check. The logarithm's rounding error is multiplied by the exponent, so the relative error grows with |exponent|: about 1e-4 at I16F16 for exponent 1.5. 219 ns on I64F64, 29 ns on I16F16.
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sqrt refined a seed with an absolute 2^-32 convergence epsilon, so on I64F64 it was exact above 100 and returned relative errors above 1 below 10^-13. It now returns the representable value nearest the true root at every magnitude: the integer root of the raw bits, plus one Newton step and a 256-bit remainder test on 128-bit types. Verified against the fixed crate's rounded-down root on 14 layouts. I64F64 sqrt 114 -> 42 ns above 1 and 180 -> 24 ns below. CORDIC vectoring stops after ⌈frac/3⌉ shift stages and closes the residual with one division; the skipped stages only added table angles of exactly 2^-i. atan2 212 -> 66 ns and ln 240 -> 127 ns on I64F64, and the hyperbolic kernel delivers 64 bits instead of 54. asinh and acosh use the logarithmic form above 1 and 1.5, where the atanh argument reduction tripled the error at every step. exp and sinh_cosh gain a third series tier for 40 or more fractional bits (degree 19, and 21/22), taking their I64F64 relative error from 1e-13 to 2e-18; the previous comment understated the degree-12 truncation by 400x. Accuracy gate against the 2.1.0 baseline: 30 improve, 10 unchanged, none regress. asinh I16F16 mean 6.44e-4 -> 2.42e-5, sqrt I32F32 2.70e-12 -> 1.37e-12, atan I16F16 1.50e-5 -> 7.79e-6. CordicNumber gained required methods and the kernels now return the residual vector, so this is a major version.
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Codecov's ignore pattern did not match the bench file, which sits directly under benches/, and the no-panic binary is compiled only under its feature, so neither can be covered by tests; both are excluded by folder.
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Implements the findings from a downstream audit of the 128-bit types, plus the corrections they exposed.
expandsinh_cosh, reciprocal reductions insin_cosandexp, a bit-scan normalisation inln, wrapping Horner evaluation. I64F64:sin_cos2x,exp3.9x,sinh_cosh3.3x. Also fixesexpabove 2^31·ln 2 on wide types,sin_coswithin π ofMAX, andexp/sinh_coshon types with fewer than 8 integer bits.pow, fallible, asexp(exponent · ln(base)).sqrtis now correctly rounded at every magnitude (it was unusable below 10^-11 on I64F64), CORDIC vectoring closes with one division (atan23.2x,ln1.9x, hyperbolic kernel delivers 64 bits instead of 54),asinh/acoshuse the logarithmic form above 1, andexp/sinh_coshgain a series tier for 40+ fractional bits.Accuracy gate against the 2.1.0 baseline: 30 improve, 10 unchanged, none regress.
CordicNumbergained required methods and the kernels return the residual vector, hence the major version.Each commit builds and passes tests, clippy and the no-panic check on its own.