1// polynomial used for __v_log(x) 2// 3// Copyright (c) 2019, Arm Limited. 4// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 5 6deg = 6; // poly degree 7a = -0x1.fc1p-9; 8b = 0x1.009p-8; 9 10// find log(1+x)/x polynomial with minimal relative error 11// (minimal relative error polynomial for log(1+x) is the same * x) 12deg = deg-1; // because of /x 13 14// f = log(1+x)/x; using taylor series 15f = 0; 16for i from 0 to 60 do { f = f + (-x)^i/(i+1); }; 17 18// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 19approx = proc(poly,d) { 20 return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 21}; 22 23// first coeff is fixed, iteratively find optimal double prec coeffs 24poly = 1; 25for i from 1 to deg do { 26 p = roundcoefficients(approx(poly,i), [|D ...|]); 27 poly = poly + x^i*coeff(p,0); 28}; 29 30display = hexadecimal; 31print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30)); 32print("in [",a,b,"]"); 33print("coeffs:"); 34for i from 0 to deg do coeff(poly,i); 35