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44 lines
1.3 KiB
44 lines
1.3 KiB
// polynomial for approximating log2(1+x)
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//
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// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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// See https://llvm.org/LICENSE.txt for license information.
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// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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deg = 11; // poly degree
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// |log2(1+x)| > 0x1p-4 outside the interval
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a = -0x1.5b51p-5;
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b = 0x1.6ab2p-5;
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ln2 = evaluate(log(2),0);
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invln2hi = double(1/ln2 + 0x1p21) - 0x1p21; // round away last 21 bits
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invln2lo = double(1/ln2 - invln2hi);
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// find log2(1+x)/x polynomial with minimal relative error
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// (minimal relative error polynomial for log2(1+x) is the same * x)
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deg = deg-1; // because of /x
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// f = log(1+x)/x; using taylor series
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f = 0;
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for i from 0 to 60 do { f = f + (-x)^i/(i+1); };
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f = f/ln2;
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// return p that minimizes |f(x) - poly(x) - x^d*p(x)|/|f(x)|
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approx = proc(poly,d) {
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return remez(1 - poly(x)/f(x), deg-d, [a;b], x^d/f(x), 1e-10);
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};
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// first coeff is fixed, iteratively find optimal double prec coeffs
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poly = invln2hi + invln2lo;
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for i from 1 to deg do {
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p = roundcoefficients(approx(poly,i), [|D ...|]);
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poly = poly + x^i*coeff(p,0);
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};
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display = hexadecimal;
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print("invln2hi:", invln2hi);
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print("invln2lo:", invln2lo);
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print("rel error:", accurateinfnorm(1-poly(x)/f(x), [a;b], 30));
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print("in [",a,b,"]");
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print("coeffs:");
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for i from 0 to deg do coeff(poly,i);
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