1// polynomial for approximating log(1+x) 2// 3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. 4// See https://llvm.org/LICENSE.txt for license information. 5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception 6 7deg = 6; // poly degree 8// interval ~= 1/(2*N), where N is the table entries 9a = -0x1.fp-9; 10b = 0x1.fp-9; 11 12// find log(1+x) polynomial with minimal absolute error 13f = log(1+x); 14 15// return p that minimizes |f(x) - poly(x) - x^d*p(x)| 16approx = proc(poly,d) { 17 return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10); 18}; 19 20// first coeff is fixed, iteratively find optimal double prec coeffs 21poly = x; 22for i from 2 to deg do { 23 p = roundcoefficients(approx(poly,i), [|D ...|]); 24 poly = poly + x^i*coeff(p,0); 25}; 26 27display = hexadecimal; 28print("abs error:", accurateinfnorm(f(x)-poly(x), [a;b], 30)); 29// relative error computation fails if f(0)==0 30// g = f(x)/x = log(1+x)/x; using taylor series 31g = 0; 32for i from 0 to 60 do { g = g + (-x)^i/(i+1); }; 33print("rel error:", accurateinfnorm(1-poly(x)/x/g(x), [a;b], 30)); 34print("in [",a,b,"]"); 35print("coeffs:"); 36for i from 0 to deg do coeff(poly,i); 37