1*f3087befSAndrew Turner// polynomial for approximating v_log10f(1+x) 2*f3087befSAndrew Turner// 3*f3087befSAndrew Turner// Copyright (c) 2019-2024, Arm Limited. 4*f3087befSAndrew Turner// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 5*f3087befSAndrew Turner 6*f3087befSAndrew Turnerdeg = 9; // poly degree 7*f3087befSAndrew Turner// |log10(1+x)| > 0x1p-4 outside the interval 8*f3087befSAndrew Turnera = -1/3; 9*f3087befSAndrew Turnerb = 1/3; 10*f3087befSAndrew Turner 11*f3087befSAndrew Turnerdisplay = hexadecimal; 12*f3087befSAndrew Turnerprint("log10(2) = ", single(log10(2))); 13*f3087befSAndrew Turner 14*f3087befSAndrew Turnerln10 = evaluate(log(10),0); 15*f3087befSAndrew Turnerinvln10 = single(1/ln10); 16*f3087befSAndrew Turner 17*f3087befSAndrew Turner// find log10(1+x)/x polynomial with minimal relative error 18*f3087befSAndrew Turner// (minimal relative error polynomial for log10(1+x) is the same * x) 19*f3087befSAndrew Turnerdeg = deg-1; // because of /x 20*f3087befSAndrew Turner 21*f3087befSAndrew Turner// f = log(1+x)/x; using taylor series 22*f3087befSAndrew Turnerf = 0; 23*f3087befSAndrew Turnerfor i from 0 to 60 do { f = f + (-x)^i/(i+1); }; 24*f3087befSAndrew Turnerf = f/ln10; 25*f3087befSAndrew Turner 26*f3087befSAndrew Turner// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 27*f3087befSAndrew Turnerapprox = proc(poly,d) { 28*f3087befSAndrew Turner return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 29*f3087befSAndrew Turner}; 30*f3087befSAndrew Turner 31*f3087befSAndrew Turner// first coeff is fixed, iteratively find optimal double prec coeffs 32*f3087befSAndrew Turnerpoly = invln10; 33*f3087befSAndrew Turnerfor i from 1 to deg do { 34*f3087befSAndrew Turner p = roundcoefficients(approx(poly,i), [|SG ...|]); 35*f3087befSAndrew Turner poly = poly + x^i*coeff(p,0); 36*f3087befSAndrew Turner}; 37*f3087befSAndrew Turnerdisplay = hexadecimal; 38*f3087befSAndrew Turnerprint("invln10:", invln10); 39*f3087befSAndrew Turnerprint("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30)); 40*f3087befSAndrew Turnerprint("in [",a,b,"]"); 41*f3087befSAndrew Turnerprint("coeffs:"); 42*f3087befSAndrew Turnerfor i from 0 to deg do single(coeff(poly,i)); 43*f3087befSAndrew Turner 44*f3087befSAndrew Turnerdisplay = decimal; 45*f3087befSAndrew Turnerprint("in [",a,b,"]"); 46