| // polynomial for approximating log10(1+x) |
| // |
| // Copyright (c) 2019-2023, Arm Limited. |
| // SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception |
| |
| deg = 6; // poly degree |
| // |log10(1+x)| > 0x1p-5 outside the interval |
| a = -0x1.p-5; |
| b = 0x1.p-5; |
| |
| ln10 = evaluate(log(10),0); |
| invln10hi = double(1/ln10 + 0x1p21) - 0x1p21; // round away last 21 bits |
| invln10lo = double(1/ln10 - invln10hi); |
| |
| // find log10(1+x)/x polynomial with minimal relative error |
| // (minimal relative error polynomial for log10(1+x) is the same * x) |
| deg = deg-1; // because of /x |
| |
| // f = log(1+x)/x; using taylor series |
| f = 0; |
| for i from 0 to 60 do { f = f + (-x)^i/(i+1); }; |
| f = f/ln10; |
| |
| // return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| |
| approx = proc(poly,d) { |
| return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); |
| }; |
| |
| // first coeff is fixed, iteratively find optimal double prec coeffs |
| poly = invln10hi + invln10lo; |
| for i from 1 to deg do { |
| p = roundcoefficients(approx(poly,i), [|D ...|]); |
| poly = poly + x^i*coeff(p,0); |
| }; |
| display = hexadecimal; |
| print("invln10hi:", invln10hi); |
| print("invln10lo:", invln10lo); |
| print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30)); |
| print("in [",a,b,"]"); |
| print("coeffs:"); |
| for i from 0 to deg do coeff(poly,i); |
| |
| display = decimal; |
| print("in [",a,b,"]"); |