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42 lines
1.2 KiB
42 lines
1.2 KiB
// polynomial for approximating log2(1+x)
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//
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// Copyright (c) 2019, Arm Limited.
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// SPDX-License-Identifier: MIT
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deg = 7; // poly degree
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// interval ~= 1/(2*N), where N is the table entries
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a= -0x1.f45p-8;
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b= 0x1.f45p-8;
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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) polynomial with minimal absolute error
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f = log(1+x)/ln2;
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// return p that minimizes |f(x) - poly(x) - x^d*p(x)|
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approx = proc(poly,d) {
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return remez(f(x) - poly(x), deg-d, [a;b], x^d, 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 = x*(invln2lo + invln2hi);
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for i from 2 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("abs error:", accurateinfnorm(f(x)-poly(x), [a;b], 30));
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//// relative error computation fails if f(0)==0
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//// g = f(x)/x = log2(1+x)/x; using taylor series
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//g = 0;
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//for i from 0 to 60 do { g = g + (-x)^i/(i+1)/ln2; };
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//print("rel error:", accurateinfnorm(1-(poly(x)/x)/g(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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