/* log1pl.c * * Relative error logarithm * Natural logarithm of 1+x, 128-bit long double precision * * * * SYNOPSIS: * * long double x, y, log1pl(); * * y = log1pl( x ); * * * * DESCRIPTION: * * Returns the base e (2.718...) logarithm of 1+x. * * The argument 1+x is separated into its exponent and fractional * parts. If the exponent is between -1 and +1, the 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(w-1)/(w+1), * * log(w) = z + z^3 P(z)/Q(z). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -1, 8 100000 1.9e-34 4.3e-35 */ /* Copyright 2001 by Stephen L. Moshier This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with this library; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ #include "math.h" #include "math_private.h" /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x) * 1/sqrt(2) <= 1+x < sqrt(2) * Theoretical peak relative error = 5.3e-37, * relative peak error spread = 2.3e-14 */ static const long double P12 = 1.538612243596254322971797716843006400388E-6L, P11 = 4.998469661968096229986658302195402690910E-1L, P10 = 2.321125933898420063925789532045674660756E1L, P9 = 4.114517881637811823002128927449878962058E2L, P8 = 3.824952356185897735160588078446136783779E3L, P7 = 2.128857716871515081352991964243375186031E4L, P6 = 7.594356839258970405033155585486712125861E4L, P5 = 1.797628303815655343403735250238293741397E5L, P4 = 2.854829159639697837788887080758954924001E5L, P3 = 3.007007295140399532324943111654767187848E5L, P2 = 2.014652742082537582487669938141683759923E5L, P1 = 7.771154681358524243729929227226708890930E4L, P0 = 1.313572404063446165910279910527789794488E4L, /* Q12 = 1.000000000000000000000000000000000000000E0L, */ Q11 = 4.839208193348159620282142911143429644326E1L, Q10 = 9.104928120962988414618126155557301584078E2L, Q9 = 9.147150349299596453976674231612674085381E3L, Q8 = 5.605842085972455027590989944010492125825E4L, Q7 = 2.248234257620569139969141618556349415120E5L, Q6 = 6.132189329546557743179177159925690841200E5L, Q5 = 1.158019977462989115839826904108208787040E6L, Q4 = 1.514882452993549494932585972882995548426E6L, Q3 = 1.347518538384329112529391120390701166528E6L, Q2 = 7.777690340007566932935753241556479363645E5L, Q1 = 2.626900195321832660448791748036714883242E5L, Q0 = 3.940717212190338497730839731583397586124E4L; /* 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 R5 = -8.828896441624934385266096344596648080902E-1L, R4 = 8.057002716646055371965756206836056074715E1L, R3 = -2.024301798136027039250415126250455056397E3L, R2 = 2.048819892795278657810231591630928516206E4L, R1 = -8.977257995689735303686582344659576526998E4L, R0 = 1.418134209872192732479751274970992665513E5L, /* S6 = 1.000000000000000000000000000000000000000E0L, */ S5 = -1.186359407982897997337150403816839480438E2L, S4 = 3.998526750980007367835804959888064681098E3L, S3 = -5.748542087379434595104154610899551484314E4L, S2 = 4.001557694070773974936904547424676279307E5L, S1 = -1.332535117259762928288745111081235577029E6L, S0 = 1.701761051846631278975701529965589676574E6L; /* C1 + C2 = ln 2 */ static const long double C1 = 6.93145751953125E-1L; static const long double C2 = 1.428606820309417232121458176568075500134E-6L; static const long double sqrth = 0.7071067811865475244008443621048490392848L; /* ln (2^16384 * (1 - 2^-113)) */ static const long double maxlog = 1.1356523406294143949491931077970764891253E4L; static const long double big = 2e4932L; static const long double zero = 0.0L; #if 1 /* Make sure these are prototyped. */ long double frexpl (long double, int *); long double ldexpl (long double, int); #endif long double __log1pl (long double xm1) { long double x, y, z, r, s; ieee854_long_double_shape_type u; int32_t ix; int e; /* Test for NaN or infinity input. */ u.value = xm1; ix = u.parts32.w0 & 0x7fffffff; if (ix >= 0x7fff0000) return xm1; /* log1p(+- 0) = +- 0. */ if ((ix == 0) && (u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0) return xm1; x = xm1 + 1.0L; /* log1p(-1) = -inf */ if (x <= 0.0L) { if (x == 0.0L) return (-1.0L / zero); else return (zero / zero); } /* Separate mantissa from exponent. */ /* Use frexp used so that denormal numbers will be handled properly. */ x = frexpl (x, &e); /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2), 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; r = ((((R5 * z + R4) * z + R3) * z + R2) * z + R1) * z + R0; s = (((((z + S5) * z + S4) * z + S3) * z + S2) * z + S1) * z + S0; z = x * (z * r / s); z = z + e * C2; z = z + x; z = z + e * C1; return (z); } /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */ if (x < sqrth) { e -= 1; if (e != 0) x = 2.0L * x - 1.0L; /* 2x - 1 */ else x = xm1; } else { if (e != 0) x = x - 1.0L; else x = xm1; } z = x * x; r = (((((((((((P12 * x + P11) * x + P10) * x + P9) * x + P8) * x + P7) * x + P6) * x + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0; s = (((((((((((x + Q11) * x + Q10) * x + Q9) * x + Q8) * x + Q7) * x + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0; y = x * (z * r / s); y = y + e * C2; z = y - 0.5L * z; z = z + x; z = z + e * C1; return (z); } weak_alias (__log1pl, log1pl) #ifdef NO_LONG_DOUBLE strong_alias (__log1p, __log1pl) weak_alias (__log1p, log1pl) #endif