| /* expm1l.c |
| * |
| * Exponential function, minus 1 |
| * 128-bit long double precision |
| * |
| * |
| * |
| * SYNOPSIS: |
| * |
| * long double x, y, expm1l(); |
| * |
| * y = expm1l( 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 |
| <http://www.gnu.org/licenses/>. */ |
| |
| |
| |
| #include <errno.h> |
| #include <float.h> |
| #include <math.h> |
| #include <math_private.h> |
| #include <libm-alias-ldouble.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 = L(2.943520915569954073888921213330863757240E8), |
| P1 = L(-5.722847283900608941516165725053359168840E7), |
| P2 = L(8.944630806357575461578107295909719817253E6), |
| P3 = L(-7.212432713558031519943281748462837065308E5), |
| P4 = L(4.578962475841642634225390068461943438441E4), |
| P5 = L(-1.716772506388927649032068540558788106762E3), |
| P6 = L(4.401308817383362136048032038528753151144E1), |
| P7 = L(-4.888737542888633647784737721812546636240E-1), |
| Q0 = L(1.766112549341972444333352727998584753865E9), |
| Q1 = L(-7.848989743695296475743081255027098295771E8), |
| Q2 = L(1.615869009634292424463780387327037251069E8), |
| Q3 = L(-2.019684072836541751428967854947019415698E7), |
| Q4 = L(1.682912729190313538934190635536631941751E6), |
| Q5 = L(-9.615511549171441430850103489315371768998E4), |
| Q6 = L(3.697714952261803935521187272204485251835E3), |
| Q7 = L(-8.802340681794263968892934703309274564037E1), |
| /* Q8 = 1.000000000000000000000000000000000000000E0 */ |
| /* C1 + C2 = ln 2 */ |
| |
| C1 = L(6.93145751953125E-1), |
| C2 = L(1.428606820309417232121458176568075500134E-6), |
| /* ln 2^-114 */ |
| minarg = L(-7.9018778583833765273564461846232128760607E1), big = L(1e4932); |
| |
| |
| _Float128 |
| __expm1l (_Float128 x) |
| { |
| _Float128 px, qx, xx; |
| int32_t ix, sign; |
| ieee854_long_double_shape_type u; |
| int k; |
| |
| /* Detect infinity and NaN. */ |
| u.value = x; |
| ix = u.parts32.w0; |
| sign = ix & 0x80000000; |
| ix &= 0x7fffffff; |
| if (!sign && ix >= 0x40060000) |
| { |
| /* If num is positive and exp >= 6 use plain exp. */ |
| return __expl (x); |
| } |
| if (ix >= 0x7fff0000) |
| { |
| /* Infinity (which must be negative infinity). */ |
| if (((ix & 0xffff) | u.parts32.w1 | u.parts32.w2 | u.parts32.w3) == 0) |
| return -1; |
| /* NaN. Invalid exception if signaling. */ |
| return x + x; |
| } |
| |
| /* expm1(+- 0) = +- 0. */ |
| if ((ix == 0) && (u.parts32.w1 | u.parts32.w2 | u.parts32.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 (fabsl (x) < L(0x1p-113)) |
| { |
| math_check_force_underflow (x); |
| return x; |
| } |
| |
| /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */ |
| xx = C1 + C2; /* ln 2. */ |
| px = __floorl (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 = __ldexpl (1, k); |
| x = px * qx + (px - 1.0); |
| return x; |
| } |
| libm_hidden_def (__expm1l) |
| libm_alias_ldouble (__expm1, expm1) |