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// polynomial for approximating log(1+x)
//
// Copyright (c) 2019, Arm Limited.
// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
deg = 12; // poly degree
// |log(1+x)| > 0x1p-4 outside the interval
a = -0x1p-4;
b = 0x1.09p-4;
// find log(1+x)/x polynomial with minimal relative error
// (minimal relative error polynomial for log(1+x) is the same * x)
deg = deg-1; // because of /x
// f = log(1+x)/x; using taylor series
f = 0;
for i from 0 to 60 do { f = f + (-x)^i/(i+1); };
// 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);
};
// 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 ...|]);
poly = poly + x^i*coeff(p,0);
};
display = hexadecimal;
print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30));
print("in [",a,b,"]");
print("coeffs:");
for i from 0 to deg do coeff(poly,i);
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