1*31914882SAlex Richardson// polynomial for approximating 2^x 2*31914882SAlex Richardson// 3*31914882SAlex Richardson// Copyright (c) 2019, Arm Limited. 4*31914882SAlex Richardson// SPDX-License-Identifier: MIT 5*31914882SAlex Richardson 6*31914882SAlex Richardson// exp2f parameters 7*31914882SAlex Richardsondeg = 3; // poly degree 8*31914882SAlex RichardsonN = 32; // table entries 9*31914882SAlex Richardsonb = 1/(2*N); // interval 10*31914882SAlex Richardsona = -b; 11*31914882SAlex Richardson 12*31914882SAlex Richardson//// exp2 parameters 13*31914882SAlex Richardson//deg = 5; // poly degree 14*31914882SAlex Richardson//N = 128; // table entries 15*31914882SAlex Richardson//b = 1/(2*N); // interval 16*31914882SAlex Richardson//a = -b; 17*31914882SAlex Richardson 18*31914882SAlex Richardson// find polynomial with minimal relative error 19*31914882SAlex Richardson 20*31914882SAlex Richardsonf = 2^x; 21*31914882SAlex Richardson 22*31914882SAlex Richardson// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 23*31914882SAlex Richardsonapprox = proc(poly,d) { 24*31914882SAlex Richardson return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 25*31914882SAlex Richardson}; 26*31914882SAlex Richardson// return p that minimizes |f(x) - poly(x) - x^d*p(x)| 27*31914882SAlex Richardsonapprox_abs = proc(poly,d) { 28*31914882SAlex Richardson return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10); 29*31914882SAlex Richardson}; 30*31914882SAlex Richardson 31*31914882SAlex Richardson// first coeff is fixed, iteratively find optimal double prec coeffs 32*31914882SAlex Richardsonpoly = 1; 33*31914882SAlex Richardsonfor i from 1 to deg do { 34*31914882SAlex Richardson p = roundcoefficients(approx(poly,i), [|D ...|]); 35*31914882SAlex Richardson// p = roundcoefficients(approx_abs(poly,i), [|D ...|]); 36*31914882SAlex Richardson poly = poly + x^i*coeff(p,0); 37*31914882SAlex Richardson}; 38*31914882SAlex Richardson 39*31914882SAlex Richardsondisplay = hexadecimal; 40*31914882SAlex Richardsonprint("rel error:", accurateinfnorm(1-poly(x)/2^x, [a;b], 30)); 41*31914882SAlex Richardsonprint("abs error:", accurateinfnorm(2^x-poly(x), [a;b], 30)); 42*31914882SAlex Richardsonprint("in [",a,b,"]"); 43*31914882SAlex Richardson// double interval error for non-nearest rounding: 44*31914882SAlex Richardsonprint("rel2 error:", accurateinfnorm(1-poly(x)/2^x, [2*a;2*b], 30)); 45*31914882SAlex Richardsonprint("abs2 error:", accurateinfnorm(2^x-poly(x), [2*a;2*b], 30)); 46*31914882SAlex Richardsonprint("in [",2*a,2*b,"]"); 47*31914882SAlex Richardsonprint("coeffs:"); 48*31914882SAlex Richardsonfor i from 0 to deg do coeff(poly,i); 49