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 /* expm1q.c * * Exponential function, minus 1 * 128-bit long double precision * * * * SYNOPSIS: * * long double x, y, expm1q(); * * y = expm1q( x ); * * * * DESCRIPTION: * * Returns e (2.71828...) raised to the x power, minus one. * * Range reduction is accomplished by separating the argument * into an integer k and fraction f such that * * x k f * e = 2 e. * * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1 * in the basic range [-0.5 ln 2, 0.5 ln 2]. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -79,+MAXLOG 100,000 1.7e-34 4.5e-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" /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x) -.5 ln 2 < x < .5 ln 2 Theoretical peak relative error = 8.1e-36 */ static const __float128 P0 = 2.943520915569954073888921213330863757240E8Q, P1 = -5.722847283900608941516165725053359168840E7Q, P2 = 8.944630806357575461578107295909719817253E6Q, P3 = -7.212432713558031519943281748462837065308E5Q, P4 = 4.578962475841642634225390068461943438441E4Q, P5 = -1.716772506388927649032068540558788106762E3Q, P6 = 4.401308817383362136048032038528753151144E1Q, P7 = -4.888737542888633647784737721812546636240E-1Q, Q0 = 1.766112549341972444333352727998584753865E9Q, Q1 = -7.848989743695296475743081255027098295771E8Q, Q2 = 1.615869009634292424463780387327037251069E8Q, Q3 = -2.019684072836541751428967854947019415698E7Q, Q4 = 1.682912729190313538934190635536631941751E6Q, Q5 = -9.615511549171441430850103489315371768998E4Q, Q6 = 3.697714952261803935521187272204485251835E3Q, Q7 = -8.802340681794263968892934703309274564037E1Q, /* Q8 = 1.000000000000000000000000000000000000000E0 */ /* C1 + C2 = ln 2 */ C1 = 6.93145751953125E-1Q, C2 = 1.428606820309417232121458176568075500134E-6Q, /* ln 2^-114 */ minarg = -7.9018778583833765273564461846232128760607E1Q, big = 1e4932Q; __float128 expm1q (__float128 x) { __float128 px, qx, xx; int32_t ix, sign; ieee854_float128 u; int k; /* Detect infinity and NaN. */ u.value = x; ix = u.words32.w0; sign = ix & 0x80000000; ix &= 0x7fffffff; if (!sign && ix >= 0x40060000) { /* If num is positive and exp >= 6 use plain exp. */ return expq (x); } if (ix >= 0x7fff0000) { /* Infinity (which must be negative infinity). */ if (((ix & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) return -1; /* NaN. Invalid exception if signaling. */ return x + x; } /* expm1(+- 0) = +- 0. */ if ((ix == 0) && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0) return x; /* Minimum value. */ if (x < minarg) return (4.0/big - 1); /* Avoid internal underflow when result does not underflow, while ensuring underflow (without returning a zero of the wrong sign) when the result does underflow. */ if (fabsq (x) < 0x1p-113Q) { math_check_force_underflow (x); return x; } /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */ xx = C1 + C2; /* ln 2. */ px = floorq (0.5 + x / xx); k = px; /* remainder times ln 2 */ x -= px * C1; x -= px * C2; /* Approximate exp(remainder ln 2). */ px = (((((((P7 * x + P6) * x + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x; qx = (((((((x + Q7) * x + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0; xx = x * x; qx = x + (0.5 * xx + xx * px / qx); /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2). We have qx = exp(remainder ln 2) - 1, so exp(x) - 1 = 2^k (qx + 1) - 1 = 2^k qx + 2^k - 1. */ px = ldexpq (1, k); x = px * qx + (px - 1.0); return x; }