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// polynomial for approximating 2^x
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
// exp2f parameters
deg = 3; // poly degree
N = 32; // table entries
b = 1/(2*N); // interval
a = -b;
//// exp2 parameters
//deg = 5; // poly degree
//N = 128; // table entries
//b = 1/(2*N); // interval
//a = -b;
// find polynomial with minimal relative error
f = 2^x;
// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)|
approx = proc(poly,d) {
return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10);
};
// return p that minimizes |f(x) - poly(x) - x^d*p(x)|
approx_abs = proc(poly,d) {
return remez(f(x) - poly(x), deg-d, [a;b], x^d, 1e-10);
};
// first coeff is fixed, iteratively find optimal double prec coeffs
poly = 1;
for i from 1 to deg do {
p = roundcoefficients(approx(poly,i), [|D ...|]);
// p = roundcoefficients(approx_abs(poly,i), [|D ...|]);
poly = poly + x^i*coeff(p,0);
};
display = hexadecimal;
print("rel error:", accurateinfnorm(1-poly(x)/2^x, [a;b], 30));
print("abs error:", accurateinfnorm(2^x-poly(x), [a;b], 30));
print("in [",a,b,"]");
// double interval error for non-nearest rounding:
print("rel2 error:", accurateinfnorm(1-poly(x)/2^x, [2*a;2*b], 30));
print("abs2 error:", accurateinfnorm(2^x-poly(x), [2*a;2*b], 30));
print("in [",2*a,2*b,"]");
print("coeffs:");
for i from 0 to deg do coeff(poly,i);
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