1// polynomial for approximating log10(1+x)
2//
3// Copyright (c) 2019-2023, Arm Limited.
4// SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
5
6deg = 6; // poly degree
7// |log10(1+x)| > 0x1p-5 outside the interval
8a = -0x1.p-5;
9b = 0x1.p-5;
10
11ln10 = evaluate(log(10),0);
12invln10hi = double(1/ln10 + 0x1p21) - 0x1p21; // round away last 21 bits
13invln10lo = double(1/ln10 - invln10hi);
14
15// find log10(1+x)/x polynomial with minimal relative error
16// (minimal relative error polynomial for log10(1+x) is the same * x)
17deg = deg-1; // because of /x
18
19// f = log(1+x)/x; using taylor series
20f = 0;
21for i from 0 to 60 do { f = f + (-x)^i/(i+1); };
22f = f/ln10;
23
24// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)|
25approx = proc(poly,d) {
26  return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10);
27};
28
29// first coeff is fixed, iteratively find optimal double prec coeffs
30poly = invln10hi + invln10lo;
31for i from 1 to deg do {
32  p = roundcoefficients(approx(poly,i), [|D ...|]);
33  poly = poly + x^i*coeff(p,0);
34};
35display = hexadecimal;
36print("invln10hi:", invln10hi);
37print("invln10lo:", invln10lo);
38print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30));
39print("in [",a,b,"]");
40print("coeffs:");
41for i from 0 to deg do coeff(poly,i);
42
43display = decimal;
44print("in [",a,b,"]");
45