1*412f47f9SXin Li// polynomial for approximating 2^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 Li// exp2f parameters 7*412f47f9SXin Lideg = 3; // poly degree 8*412f47f9SXin LiN = 32; // table entries 9*412f47f9SXin Lib = 1/(2*N); // interval 10*412f47f9SXin Lia = -b; 11*412f47f9SXin Li 12*412f47f9SXin Li//// exp2 parameters 13*412f47f9SXin Li//deg = 5; // poly degree 14*412f47f9SXin Li//N = 128; // table entries 15*412f47f9SXin Li//b = 1/(2*N); // interval 16*412f47f9SXin Li//a = -b; 17*412f47f9SXin Li 18*412f47f9SXin Li// find polynomial with minimal relative error 19*412f47f9SXin Li 20*412f47f9SXin Lif = 2^x; 21*412f47f9SXin Li 22*412f47f9SXin Li// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)| 23*412f47f9SXin Liapprox = proc(poly,d) { 24*412f47f9SXin Li return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10); 25*412f47f9SXin Li}; 26*412f47f9SXin Li// return p that minimizes |f(x) - poly(x) - x^d*p(x)| 27*412f47f9SXin Liapprox_abs = proc(poly,d) { 28*412f47f9SXin Li return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10); 29*412f47f9SXin Li}; 30*412f47f9SXin Li 31*412f47f9SXin Li// first coeff is fixed, iteratively find optimal double prec coeffs 32*412f47f9SXin Lipoly = 1; 33*412f47f9SXin Lifor i from 1 to deg do { 34*412f47f9SXin Li p = roundcoefficients(approx(poly,i), [|D ...|]); 35*412f47f9SXin Li// p = roundcoefficients(approx_abs(poly,i), [|D ...|]); 36*412f47f9SXin Li poly = poly + x^i*coeff(p,0); 37*412f47f9SXin Li}; 38*412f47f9SXin Li 39*412f47f9SXin Lidisplay = hexadecimal; 40*412f47f9SXin Liprint("rel error:", accurateinfnorm(1-poly(x)/2^x, [a;b], 30)); 41*412f47f9SXin Liprint("abs error:", accurateinfnorm(2^x-poly(x), [a;b], 30)); 42*412f47f9SXin Liprint("in [",a,b,"]"); 43*412f47f9SXin Li// double interval error for non-nearest rounding: 44*412f47f9SXin Liprint("rel2 error:", accurateinfnorm(1-poly(x)/2^x, [2*a;2*b], 30)); 45*412f47f9SXin Liprint("abs2 error:", accurateinfnorm(2^x-poly(x), [2*a;2*b], 30)); 46*412f47f9SXin Liprint("in [",2*a,2*b,"]"); 47*412f47f9SXin Liprint("coeffs:"); 48*412f47f9SXin Lifor i from 0 to deg do coeff(poly,i); 49