math/tools/exp2.sollya

// polynomial for approximating 2^x
//
// Copyright (c) 2019, Arm Limited.
// SPDX-License-Identifier: MIT

// 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);