1*412f47f9SXin Li// polynomial for approximating log10(1+x) 2*412f47f9SXin Li// 3*412f47f9SXin Li// Copyright (c) 2019-2023, 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// |log10(1+x)| > 0x1p-5 outside the interval 8*412f47f9SXin Lia = -0x1.p-5; 9*412f47f9SXin Lib = 0x1.p-5; 10*412f47f9SXin Li 11*412f47f9SXin Liln10 = evaluate(log(10),0); 12*412f47f9SXin Liinvln10hi = double(1/ln10 + 0x1p21) - 0x1p21; // round away last 21 bits 13*412f47f9SXin Liinvln10lo = double(1/ln10 - invln10hi); 14*412f47f9SXin Li 15*412f47f9SXin Li// find log10(1+x)/x polynomial with minimal relative error 16*412f47f9SXin Li// (minimal relative error polynomial for log10(1+x) is the same * x) 17*412f47f9SXin Lideg = deg-1; // because of /x 18*412f47f9SXin Li 19*412f47f9SXin Li// f = log(1+x)/x; using taylor series 20*412f47f9SXin Lif = 0; 21*412f47f9SXin Lifor i from 0 to 60 do { f = f + (-x)^i/(i+1); }; 22*412f47f9SXin Lif = f/ln10; 23*412f47f9SXin Li 24*412f47f9SXin Li// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 25*412f47f9SXin Liapprox = proc(poly,d) { 26*412f47f9SXin Li return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 27*412f47f9SXin Li}; 28*412f47f9SXin Li 29*412f47f9SXin Li// first coeff is fixed, iteratively find optimal double prec coeffs 30*412f47f9SXin Lipoly = invln10hi + invln10lo; 31*412f47f9SXin Lifor i from 1 to deg do { 32*412f47f9SXin Li p = roundcoefficients(approx(poly,i), [|D ...|]); 33*412f47f9SXin Li poly = poly + x^i*coeff(p,0); 34*412f47f9SXin Li}; 35*412f47f9SXin Lidisplay = hexadecimal; 36*412f47f9SXin Liprint("invln10hi:", invln10hi); 37*412f47f9SXin Liprint("invln10lo:", invln10lo); 38*412f47f9SXin Liprint("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30)); 39*412f47f9SXin Liprint("in [",a,b,"]"); 40*412f47f9SXin Liprint("coeffs:"); 41*412f47f9SXin Lifor i from 0 to deg do coeff(poly,i); 42*412f47f9SXin Li 43*412f47f9SXin Lidisplay = decimal; 44*412f47f9SXin Liprint("in [",a,b,"]"); 45