1bbbf1280Sopenharmony_ci// polynomial for approximating log2(1+x) 2bbbf1280Sopenharmony_ci// 3bbbf1280Sopenharmony_ci// Copyright (c) 2019, Arm Limited. 4bbbf1280Sopenharmony_ci// SPDX-License-Identifier: MIT 5bbbf1280Sopenharmony_ci 6bbbf1280Sopenharmony_cideg = 7; // poly degree 7bbbf1280Sopenharmony_ci// interval ~= 1/(2*N), where N is the table entries 8bbbf1280Sopenharmony_cia= -0x1.f45p-8; 9bbbf1280Sopenharmony_cib= 0x1.f45p-8; 10bbbf1280Sopenharmony_ci 11bbbf1280Sopenharmony_ciln2 = evaluate(log(2),0); 12bbbf1280Sopenharmony_ciinvln2hi = double(1/ln2 + 0x1p21) - 0x1p21; // round away last 21 bits 13bbbf1280Sopenharmony_ciinvln2lo = double(1/ln2 - invln2hi); 14bbbf1280Sopenharmony_ci 15bbbf1280Sopenharmony_ci// find log2(1+x) polynomial with minimal absolute error 16bbbf1280Sopenharmony_cif = log(1+x)/ln2; 17bbbf1280Sopenharmony_ci 18bbbf1280Sopenharmony_ci// return p that minimizes |f(x) - poly(x) - x^d*p(x)| 19bbbf1280Sopenharmony_ciapprox = proc(poly,d) { 20bbbf1280Sopenharmony_ci return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10); 21bbbf1280Sopenharmony_ci}; 22bbbf1280Sopenharmony_ci 23bbbf1280Sopenharmony_ci// first coeff is fixed, iteratively find optimal double prec coeffs 24bbbf1280Sopenharmony_cipoly = x*(invln2lo + invln2hi); 25bbbf1280Sopenharmony_cifor i from 2 to deg do { 26bbbf1280Sopenharmony_ci p = roundcoefficients(approx(poly,i), [|D ...|]); 27bbbf1280Sopenharmony_ci poly = poly + x^i*coeff(p,0); 28bbbf1280Sopenharmony_ci}; 29bbbf1280Sopenharmony_ci 30bbbf1280Sopenharmony_cidisplay = hexadecimal; 31bbbf1280Sopenharmony_ciprint("invln2hi:", invln2hi); 32bbbf1280Sopenharmony_ciprint("invln2lo:", invln2lo); 33bbbf1280Sopenharmony_ciprint("abs error:", accurateinfnorm(f(x)-poly(x), [a;b], 30)); 34bbbf1280Sopenharmony_ci//// relative error computation fails if f(0)==0 35bbbf1280Sopenharmony_ci//// g = f(x)/x = log2(1+x)/x; using taylor series 36bbbf1280Sopenharmony_ci//g = 0; 37bbbf1280Sopenharmony_ci//for i from 0 to 60 do { g = g + (-x)^i/(i+1)/ln2; }; 38bbbf1280Sopenharmony_ci//print("rel error:", accurateinfnorm(1-(poly(x)/x)/g(x), [a;b], 30)); 39bbbf1280Sopenharmony_ciprint("in [",a,b,"]"); 40bbbf1280Sopenharmony_ciprint("coeffs:"); 41bbbf1280Sopenharmony_cifor i from 0 to deg do coeff(poly,i); 42