1 /*- 2 * Copyright (c) 2011 David Schultz <das (at) FreeBSD.ORG> 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice, this list of conditions and the following disclaimer. 10 * 2. Redistributions in binary form must reproduce the above copyright 11 * notice, this list of conditions and the following disclaimer in the 12 * documentation and/or other materials provided with the distribution. 13 * 14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 24 * SUCH DAMAGE. 25 */ 26 27 #include <sys/cdefs.h> 28 __FBSDID("$FreeBSD$"); 29 30 #include <complex.h> 31 32 #include "math.h" 33 #include "math_private.h" 34 35 static const uint32_t k = 1799; /* constant for reduction */ 36 static const double kln2 = 1246.97177782734161156; /* k * ln2 */ 37 38 /* 39 * Compute exp(x), scaled to avoid spurious overflow. An exponent is 40 * returned separately in 'expt'. 41 * 42 * Input: ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91 43 * Output: 2**1023 <= y < 2**1024 44 */ 45 static double 46 __frexp_exp(double x, int *expt) 47 { 48 double exp_x; 49 uint32_t hx; 50 51 /* 52 * We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to 53 * minimize |exp(kln2) - 2**k|. We also scale the exponent of 54 * exp_x to MAX_EXP so that the result can be multiplied by 55 * a tiny number without losing accuracy due to denormalization. 56 */ 57 exp_x = exp(x - kln2); 58 GET_HIGH_WORD(hx, exp_x); 59 *expt = (hx >> 20) - (0x3ff + 1023) + k; 60 SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20)); 61 return (exp_x); 62 } 63 64 /* 65 * __ldexp_exp(x, expt) and __ldexp_cexp(x, expt) compute exp(x) * 2**expt. 66 * They are intended for large arguments (real part >= ln(DBL_MAX)) 67 * where care is needed to avoid overflow. 68 * 69 * The present implementation is narrowly tailored for our hyperbolic and 70 * exponential functions. We assume expt is small (0 or -1), and the caller 71 * has filtered out very large x, for which overflow would be inevitable. 72 */ 73 74 double 75 __ldexp_exp(double x, int expt) 76 { 77 double exp_x, scale; 78 int ex_expt; 79 80 exp_x = __frexp_exp(x, &ex_expt); 81 expt += ex_expt; 82 INSERT_WORDS(scale, (0x3ff + expt) << 20, 0); 83 return (exp_x * scale); 84 } 85 86 double complex 87 __ldexp_cexp(double complex z, int expt) 88 { 89 double x, y, exp_x, scale1, scale2; 90 int ex_expt, half_expt; 91 92 x = creal(z); 93 y = cimag(z); 94 exp_x = __frexp_exp(x, &ex_expt); 95 expt += ex_expt; 96 97 /* 98 * Arrange so that scale1 * scale2 == 2**expt. We use this to 99 * compensate for scalbn being horrendously slow. 100 */ 101 half_expt = expt / 2; 102 INSERT_WORDS(scale1, (0x3ff + half_expt) << 20, 0); 103 half_expt = expt - half_expt; 104 INSERT_WORDS(scale2, (0x3ff + half_expt) << 20, 0); 105 106 return (cpack(cos(y) * exp_x * scale1 * scale2, 107 sin(y) * exp_x * scale1 * scale2)); 108 } 109
