/*-*- mode:c;indent-tabs-mode:t;c-basic-offset:8;tab-width:8;coding:utf-8 -*-│ │vi: set et ft=c ts=8 tw=8 fenc=utf-8 :vi│ ╚──────────────────────────────────────────────────────────────────────────────╝ │ │ │ Musl Libc │ │ Copyright © 2005-2014 Rich Felker, et al. │ │ │ │ Permission is hereby granted, free of charge, to any person obtaining │ │ a copy of this software and associated documentation files (the │ │ "Software"), to deal in the Software without restriction, including │ │ without limitation the rights to use, copy, modify, merge, publish, │ │ distribute, sublicense, and/or sell copies of the Software, and to │ │ permit persons to whom the Software is furnished to do so, subject to │ │ the following conditions: │ │ │ │ The above copyright notice and this permission notice shall be │ │ included in all copies or substantial portions of the Software. │ │ │ │ THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, │ │ EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF │ │ MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. │ │ IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY │ │ CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, │ │ TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE │ │ SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. │ │ │ ╚─────────────────────────────────────────────────────────────────────────────*/ #include "libc/complex.h" #include "libc/math.h" #include "libc/tinymath/complex.internal.h" #if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 asm(".ident\t\"\\n\\n\ OpenBSD libm (ISC License)\\n\ Copyright (c) 2008 Stephen L. Moshier \""); asm(".ident\t\"\\n\\n\ Musl libc (MIT License)\\n\ Copyright 2005-2014 Rich Felker, et. al.\""); asm(".include \"libc/disclaimer.inc\""); // clang-format off /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_log2l.c */ /* * Copyright (c) 2008 Stephen L. Moshier * * Permission to use, copy, modify, and distribute this software for any * purpose with or without fee is hereby granted, provided that the above * copyright notice and this permission notice appear in all copies. * * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ /* * Base 2 logarithm, long double precision * * * SYNOPSIS: * * long double x, y, log2l(); * * y = log2l( x ); * * * DESCRIPTION: * * Returns the base 2 logarithm of x. * * The argument is separated into its exponent and fractional * parts. If the exponent is between -1 and +1, the (natural) * logarithm of the fraction is approximated by * * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). * * Otherwise, setting z = 2(x-1)/x+1), * * log(x) = z + z**3 P(z)/Q(z). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0.5, 2.0 30000 9.8e-20 2.7e-20 * IEEE exp(+-10000) 70000 5.4e-20 2.3e-20 * * In the tests over the interval exp(+-10000), the logarithms * of the random arguments were uniformly distributed over * [-10000, +10000]. */ /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x) * 1/sqrt(2) <= x < sqrt(2) * Theoretical peak relative error = 6.2e-22 */ static const long double P[] = { 4.9962495940332550844739E-1L, 1.0767376367209449010438E1L, 7.7671073698359539859595E1L, 2.5620629828144409632571E2L, 4.2401812743503691187826E2L, 3.4258224542413922935104E2L, 1.0747524399916215149070E2L, }; static const long double Q[] = { /* 1.0000000000000000000000E0,*/ 2.3479774160285863271658E1L, 1.9444210022760132894510E2L, 7.7952888181207260646090E2L, 1.6911722418503949084863E3L, 2.0307734695595183428202E3L, 1.2695660352705325274404E3L, 3.2242573199748645407652E2L, }; /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), * where z = 2(x-1)/(x+1) * 1/sqrt(2) <= x < sqrt(2) * Theoretical peak relative error = 6.16e-22 */ static const long double R[4] = { 1.9757429581415468984296E-3L, -7.1990767473014147232598E-1L, 1.0777257190312272158094E1L, -3.5717684488096787370998E1L, }; static const long double S[4] = { /* 1.00000000000000000000E0L,*/ -2.6201045551331104417768E1L, 1.9361891836232102174846E2L, -4.2861221385716144629696E2L, }; /* log2(e) - 1 */ #define LOG2EA 4.4269504088896340735992e-1L #define SQRTH 0.70710678118654752440L /** * Calculates log₂𝑥. */ long double log2l(long double x) { #ifdef __x86__ // asm improves performance 39ns → 21ns // measurement made on an intel core i9 long double one; asm("fld1" : "=t"(one)); asm("fyl2x" : "=t"(x) : "0"(x), "u"(one) : "st(1)"); return x; #else long double y, z; int e; if (isnan(x)) return x; if (x == INFINITY) return x; if (x <= 0.0) { if (x == 0.0) return -1/(x*x); /* -inf with divbyzero */ return 0/0.0f; /* nan with invalid */ } /* separate mantissa from exponent */ /* Note, frexp is used so that denormal numbers * will be handled properly. */ x = frexpl(x, &e); /* logarithm using log(x) = z + z**3 P(z)/Q(z), * where z = 2(x-1)/x+1) */ if (e > 2 || e < -2) { if (x < SQRTH) { /* 2(2x-1)/(2x+1) */ e -= 1; z = x - 0.5; y = 0.5 * z + 0.5; } else { /* 2 (x-1)/(x+1) */ z = x - 0.5; z -= 0.5; y = 0.5 * x + 0.5; } x = z / y; z = x*x; y = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3)); goto done; } /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ if (x < SQRTH) { e -= 1; x = 2.0*x - 1.0; } else { x = x - 1.0; } z = x*x; y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 7)); y = y - 0.5*z; done: /* Multiply log of fraction by log2(e) * and base 2 exponent by 1 * * ***CAUTION*** * * This sequence of operations is critical and it may * be horribly defeated by some compiler optimizers. */ z = y * LOG2EA; z += x * LOG2EA; z += y; z += x; z += e; return z; #endif } #endif /* 80-bit floating point */