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