1/* origin: FreeBSD /usr/src/lib/msun/src/k_exp.c */ 2/*- 3 * Copyright (c) 2011 David Schultz <das@FreeBSD.ORG> 4 * All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 18 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 25 * SUCH DAMAGE. 26 */ 27 28#include "libm.h" 29 30static const uint32_t k = 1799; /* constant for reduction */ 31static const double kln2 = 1246.97177782734161156; /* k * ln2 */ 32 33/* 34 * Compute exp(x), scaled to avoid spurious overflow. An exponent is 35 * returned separately in 'expt'. 36 * 37 * Input: ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91 38 * Output: 2**1023 <= y < 2**1024 39 */ 40static double __frexp_exp(double x, int *expt) 41{ 42 double exp_x; 43 uint32_t hx; 44 45 /* 46 * We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to 47 * minimize |exp(kln2) - 2**k|. We also scale the exponent of 48 * exp_x to MAX_EXP so that the result can be multiplied by 49 * a tiny number without losing accuracy due to denormalization. 50 */ 51 exp_x = exp(x - kln2); 52 GET_HIGH_WORD(hx, exp_x); 53 *expt = (hx >> 20) - (0x3ff + 1023) + k; 54 SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20)); 55 return exp_x; 56} 57 58/* 59 * __ldexp_cexp(x, expt) compute exp(x) * 2**expt. 60 * It is intended for large arguments (real part >= ln(DBL_MAX)) 61 * where care is needed to avoid overflow. 62 * 63 * The present implementation is narrowly tailored for our hyperbolic and 64 * exponential functions. We assume expt is small (0 or -1), and the caller 65 * has filtered out very large x, for which overflow would be inevitable. 66 */ 67double complex __ldexp_cexp(double complex z, int expt) 68{ 69 double x, y, exp_x, scale1, scale2; 70 int ex_expt, half_expt; 71 72 x = creal(z); 73 y = cimag(z); 74 exp_x = __frexp_exp(x, &ex_expt); 75 expt += ex_expt; 76 77 /* 78 * Arrange so that scale1 * scale2 == 2**expt. We use this to 79 * compensate for scalbn being horrendously slow. 80 */ 81 half_expt = expt / 2; 82 INSERT_WORDS(scale1, (0x3ff + half_expt) << 20, 0); 83 half_expt = expt - half_expt; 84 INSERT_WORDS(scale2, (0x3ff + half_expt) << 20, 0); 85 86 return CMPLX(cos(y) * exp_x * scale1 * scale2, sin(y) * exp_x * scale1 * scale2); 87} 88