/* $OpenBSD: e_log2l.c,v 1.1 2011/07/06 00:02:42 martynas Exp $ */ /* * 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. */ /* log2l.c * Base 2 logarithm, 128-bit 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, 2.0 100,000 2.6e-34 4.9e-35 * IEEE exp(+-10000) 100,000 9.6e-35 4.0e-35 * * 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 = 5.3e-37, * relative peak error spread = 2.3e-14 */ static const long double P[13] = { 1.313572404063446165910279910527789794488E4L, 7.771154681358524243729929227226708890930E4L, 2.014652742082537582487669938141683759923E5L, 3.007007295140399532324943111654767187848E5L, 2.854829159639697837788887080758954924001E5L, 1.797628303815655343403735250238293741397E5L, 7.594356839258970405033155585486712125861E4L, 2.128857716871515081352991964243375186031E4L, 3.824952356185897735160588078446136783779E3L, 4.114517881637811823002128927449878962058E2L, 2.321125933898420063925789532045674660756E1L, 4.998469661968096229986658302195402690910E-1L, 1.538612243596254322971797716843006400388E-6L }; static const long double Q[12] = { 3.940717212190338497730839731583397586124E4L, 2.626900195321832660448791748036714883242E5L, 7.777690340007566932935753241556479363645E5L, 1.347518538384329112529391120390701166528E6L, 1.514882452993549494932585972882995548426E6L, 1.158019977462989115839826904108208787040E6L, 6.132189329546557743179177159925690841200E5L, 2.248234257620569139969141618556349415120E5L, 5.605842085972455027590989944010492125825E4L, 9.147150349299596453976674231612674085381E3L, 9.104928120962988414618126155557301584078E2L, 4.839208193348159620282142911143429644326E1L /* 1.000000000000000000000000000000000000000E0L, */ }; /* 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 = 1.1e-35, * relative peak error spread 1.1e-9 */ static const long double R[6] = { 1.418134209872192732479751274970992665513E5L, -8.977257995689735303686582344659576526998E4L, 2.048819892795278657810231591630928516206E4L, -2.024301798136027039250415126250455056397E3L, 8.057002716646055371965756206836056074715E1L, -8.828896441624934385266096344596648080902E-1L }; static const long double S[6] = { 1.701761051846631278975701529965589676574E6L, -1.332535117259762928288745111081235577029E6L, 4.001557694070773974936904547424676279307E5L, -5.748542087379434595104154610899551484314E4L, 3.998526750980007367835804959888064681098E3L, -1.186359407982897997337150403816839480438E2L /* 1.000000000000000000000000000000000000000E0L, */ }; static const long double /* log2(e) - 1 */ LOG2EA = 4.4269504088896340735992468100189213742664595E-1L, /* sqrt(2)/2 */ SQRTH = 7.071067811865475244008443621048490392848359E-1L; /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ static long double neval (long double x, const long double *p, int n) { long double y; p += n; y = *p--; do { y = y * x + *p--; } while (--n > 0); return y; } /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ static long double deval (long double x, const long double *p, int n) { long double y; p += n; y = x + *p--; do { y = y * x + *p--; } while (--n > 0); return y; } long double log2l(long double x) { long double z; long double y; int e; int64_t hx, lx; /* Test for domain */ GET_LDOUBLE_WORDS64 (hx, lx, x); if (((hx & 0x7fffffffffffffffLL) | lx) == 0) return __math_divzerol(1); if (hx < 0) return __math_invalidl(x); if (hx >= 0x7fff000000000000LL) return (x + x); /* 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 - 0.5L; y = 0.5L * z + 0.5L; } else { /* 2 (x-1)/(x+1) */ z = x - 0.5L; z -= 0.5L; y = 0.5L * x + 0.5L; } x = z / y; z = x * x; y = x * (z * neval (z, R, 5) / deval (z, S, 5)); goto done; } /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ if (x < SQRTH) { e -= 1; x = 2.0L * x - 1.0L; /* 2x - 1 */ } else { x = x - 1.0L; } z = x * x; y = x * (z * neval (x, P, 12) / deval (x, Q, 11)); y = y - 0.5L * z; done: /* Multiply log of fraction by log2(e) * and base 2 exponent by 1 */ z = y * LOG2EA; z += x * LOG2EA; z += y; z += x; z += e; return (z); }