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