1// polynomial for approximating log10(1+x) 2// 3// Copyright (c) 2019-2023, Arm Limited. 4// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 5 6deg = 6; // poly degree 7// |log10(1+x)| > 0x1p-5 outside the interval 8a = -0x1.p-5; 9b = 0x1.p-5; 10 11ln10 = evaluate(log(10),0); 12invln10hi = double(1/ln10 + 0x1p21) - 0x1p21; // round away last 21 bits 13invln10lo = double(1/ln10 - invln10hi); 14 15// find log10(1+x)/x polynomial with minimal relative error 16// (minimal relative error polynomial for log10(1+x) is the same * x) 17deg = deg-1; // because of /x 18 19// f = log(1+x)/x; using taylor series 20f = 0; 21for i from 0 to 60 do { f = f + (-x)^i/(i+1); }; 22f = f/ln10; 23 24// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 25approx = proc(poly,d) { 26 return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 27}; 28 29// first coeff is fixed, iteratively find optimal double prec coeffs 30poly = invln10hi + invln10lo; 31for i from 1 to deg do { 32 p = roundcoefficients(approx(poly,i), [|D ...|]); 33 poly = poly + x^i*coeff(p,0); 34}; 35display = hexadecimal; 36print("invln10hi:", invln10hi); 37print("invln10lo:", invln10lo); 38print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30)); 39print("in [",a,b,"]"); 40print("coeffs:"); 41for i from 0 to deg do coeff(poly,i); 42 43display = decimal; 44print("in [",a,b,"]"); 45