1*412f47f9SXin Li// polynomial for approximating log(1+x) 2*412f47f9SXin Li// 3*412f47f9SXin Li// Copyright (c) 2019, Arm Limited. 4*412f47f9SXin Li// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 5*412f47f9SXin Li 6*412f47f9SXin Lideg = 6; // poly degree 7*412f47f9SXin Li// interval ~= 1/(2*N), where N is the table entries 8*412f47f9SXin Lia = -0x1.fp-9; 9*412f47f9SXin Lib = 0x1.fp-9; 10*412f47f9SXin Li 11*412f47f9SXin Li// find log(1+x) polynomial with minimal absolute error 12*412f47f9SXin Lif = log(1+x); 13*412f47f9SXin Li 14*412f47f9SXin Li// return p that minimizes |f(x) - poly(x) - x^d*p(x)| 15*412f47f9SXin Liapprox = proc(poly,d) { 16*412f47f9SXin Li return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10); 17*412f47f9SXin Li}; 18*412f47f9SXin Li 19*412f47f9SXin Li// first coeff is fixed, iteratively find optimal double prec coeffs 20*412f47f9SXin Lipoly = x; 21*412f47f9SXin Lifor i from 2 to deg do { 22*412f47f9SXin Li p = roundcoefficients(approx(poly,i), [|D ...|]); 23*412f47f9SXin Li poly = poly + x^i*coeff(p,0); 24*412f47f9SXin Li}; 25*412f47f9SXin Li 26*412f47f9SXin Lidisplay = hexadecimal; 27*412f47f9SXin Liprint("abs error:", accurateinfnorm(f(x)-poly(x), [a;b], 30)); 28*412f47f9SXin Li// relative error computation fails if f(0)==0 29*412f47f9SXin Li// g = f(x)/x = log(1+x)/x; using taylor series 30*412f47f9SXin Lig = 0; 31*412f47f9SXin Lifor i from 0 to 60 do { g = g + (-x)^i/(i+1); }; 32*412f47f9SXin Liprint("rel error:", accurateinfnorm(1-poly(x)/x/g(x), [a;b], 30)); 33*412f47f9SXin Liprint("in [",a,b,"]"); 34*412f47f9SXin Liprint("coeffs:"); 35*412f47f9SXin Lifor i from 0 to deg do coeff(poly,i); 36