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 /* log1pq.c * * Relative error logarithm * Natural logarithm of 1+x, 128-bit long double precision * * * * SYNOPSIS: * * long double x, y, log1pq(); * * y = log1pq( 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, see . */ #include "quadmath-imp.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 __float128 P12 = 1.538612243596254322971797716843006400388E-6Q, P11 = 4.998469661968096229986658302195402690910E-1Q, P10 = 2.321125933898420063925789532045674660756E1Q, P9 = 4.114517881637811823002128927449878962058E2Q, P8 = 3.824952356185897735160588078446136783779E3Q, P7 = 2.128857716871515081352991964243375186031E4Q, P6 = 7.594356839258970405033155585486712125861E4Q, P5 = 1.797628303815655343403735250238293741397E5Q, P4 = 2.854829159639697837788887080758954924001E5Q, P3 = 3.007007295140399532324943111654767187848E5Q, P2 = 2.014652742082537582487669938141683759923E5Q, P1 = 7.771154681358524243729929227226708890930E4Q, P0 = 1.313572404063446165910279910527789794488E4Q, /* Q12 = 1.000000000000000000000000000000000000000E0L, */ Q11 = 4.839208193348159620282142911143429644326E1Q, Q10 = 9.104928120962988414618126155557301584078E2Q, Q9 = 9.147150349299596453976674231612674085381E3Q, Q8 = 5.605842085972455027590989944010492125825E4Q, Q7 = 2.248234257620569139969141618556349415120E5Q, Q6 = 6.132189329546557743179177159925690841200E5Q, Q5 = 1.158019977462989115839826904108208787040E6Q, Q4 = 1.514882452993549494932585972882995548426E6Q, Q3 = 1.347518538384329112529391120390701166528E6Q, Q2 = 7.777690340007566932935753241556479363645E5Q, Q1 = 2.626900195321832660448791748036714883242E5Q, Q0 = 3.940717212190338497730839731583397586124E4Q; /* 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 __float128 R5 = -8.828896441624934385266096344596648080902E-1Q, R4 = 8.057002716646055371965756206836056074715E1Q, R3 = -2.024301798136027039250415126250455056397E3Q, R2 = 2.048819892795278657810231591630928516206E4Q, R1 = -8.977257995689735303686582344659576526998E4Q, R0 = 1.418134209872192732479751274970992665513E5Q, /* S6 = 1.000000000000000000000000000000000000000E0L, */ S5 = -1.186359407982897997337150403816839480438E2Q, S4 = 3.998526750980007367835804959888064681098E3Q, S3 = -5.748542087379434595104154610899551484314E4Q, S2 = 4.001557694070773974936904547424676279307E5Q, S1 = -1.332535117259762928288745111081235577029E6Q, S0 = 1.701761051846631278975701529965589676574E6Q; /* C1 + C2 = ln 2 */ static const __float128 C1 = 6.93145751953125E-1Q; static const __float128 C2 = 1.428606820309417232121458176568075500134E-6Q; static const __float128 sqrth = 0.7071067811865475244008443621048490392848Q; /* ln (2^16384 * (1 - 2^-113)) */ static const __float128 zero = 0; __float128 log1pq (__float128 xm1) { __float128 x, y, z, r, s; ieee854_float128 u; int32_t hx; int e; /* Test for NaN or infinity input. */ u.value = xm1; hx = u.words32.w0; if ((hx & 0x7fffffff) >= 0x7fff0000) return xm1 + fabsq (xm1); /* log1p(+- 0) = +- 0. */ if (((hx & 0x7fffffff) == 0) && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) return xm1; if ((hx & 0x7fffffff) < 0x3f8e0000) { math_check_force_underflow (xm1); if ((int) xm1 == 0) return xm1; } if (xm1 >= 0x1p113Q) x = xm1; else x = xm1 + 1; /* log1p(-1) = -inf */ if (x <= 0) { if (x == 0) return (-1 / zero); /* log1p(-1) = -inf */ else return (zero / (x - x)); } /* Separate mantissa from exponent. */ /* Use frexp used so that denormal numbers will be handled properly. */ x = frexpq (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 - 0.5Q; y = 0.5Q * z + 0.5Q; } else { /* 2 (x-1)/(x+1) */ z = x - 0.5Q; z -= 0.5Q; y = 0.5Q * x + 0.5Q; } 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 * x - 1; /* 2x - 1 */ else x = xm1; } else { if (e != 0) x = x - 1; 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 - 0.5Q * z; z = z + x; z = z + e * C1; return (z); }