131914882SAlex Richardson// polynomial for approximating log(1+x) 231914882SAlex Richardson// 331914882SAlex Richardson// Copyright (c) 2019, Arm Limited. 4*072a4ba8SAndrew Turner// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 531914882SAlex Richardson 631914882SAlex Richardsondeg = 12; // poly degree 731914882SAlex Richardson// |log(1+x)| > 0x1p-4 outside the interval 831914882SAlex Richardsona = -0x1p-4; 931914882SAlex Richardsonb = 0x1.09p-4; 1031914882SAlex Richardson 1131914882SAlex Richardson// find log(1+x)/x polynomial with minimal relative error 1231914882SAlex Richardson// (minimal relative error polynomial for log(1+x) is the same * x) 1331914882SAlex Richardsondeg = deg-1; // because of /x 1431914882SAlex Richardson 1531914882SAlex Richardson// f = log(1+x)/x; using taylor series 1631914882SAlex Richardsonf = 0; 1731914882SAlex Richardsonfor i from 0 to 60 do { f = f + (-x)^i/(i+1); }; 1831914882SAlex Richardson 1931914882SAlex Richardson// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 2031914882SAlex Richardsonapprox = proc(poly,d) { 2131914882SAlex Richardson return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 2231914882SAlex Richardson}; 2331914882SAlex Richardson 2431914882SAlex Richardson// first coeff is fixed, iteratively find optimal double prec coeffs 2531914882SAlex Richardsonpoly = 1; 2631914882SAlex Richardsonfor i from 1 to deg do { 2731914882SAlex Richardson p = roundcoefficients(approx(poly,i), [|D ...|]); 2831914882SAlex Richardson poly = poly + x^i*coeff(p,0); 2931914882SAlex Richardson}; 3031914882SAlex Richardson 3131914882SAlex Richardsondisplay = hexadecimal; 3231914882SAlex Richardsonprint("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30)); 3331914882SAlex Richardsonprint("in [",a,b,"]"); 3431914882SAlex Richardsonprint("coeffs:"); 3531914882SAlex Richardsonfor i from 0 to deg do coeff(poly,i); 36