2 * Copyright (c) 2011 David Schultz <das@FreeBSD.ORG>
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26 * $FreeBSD: head/lib/msun/src/k_exp.c 226597 2011-10-21 06:27:56Z das $
32 #include "math_private.h"
34 static const uint32_t k = 1799; /* constant for reduction */
35 static const double kln2 = 1246.97177782734161156; /* k * ln2 */
38 * Compute exp(x), scaled to avoid spurious overflow. An exponent is
39 * returned separately in 'expt'.
41 * Input: ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91
42 * Output: 2**1023 <= y < 2**1024
45 __frexp_exp(double x, int *expt)
51 * We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to
52 * minimize |exp(kln2) - 2**k|. We also scale the exponent of
53 * exp_x to MAX_EXP so that the result can be multiplied by
54 * a tiny number without losing accuracy due to denormalization.
56 exp_x = exp(x - kln2);
57 GET_HIGH_WORD(hx, exp_x);
58 *expt = (hx >> 20) - (0x3ff + 1023) + k;
59 SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20));
64 * __ldexp_exp(x, expt) and __ldexp_cexp(x, expt) compute exp(x) * 2**expt.
65 * They are intended for large arguments (real part >= ln(DBL_MAX))
66 * where care is needed to avoid overflow.
68 * The present implementation is narrowly tailored for our hyperbolic and
69 * exponential functions. We assume expt is small (0 or -1), and the caller
70 * has filtered out very large x, for which overflow would be inevitable.
74 __ldexp_exp(double x, int expt)
79 exp_x = __frexp_exp(x, &ex_expt);
81 INSERT_WORDS(scale, (0x3ff + expt) << 20, 0);
82 return (exp_x * scale);
86 __ldexp_cexp(double complex z, int expt)
88 double x, y, exp_x, scale1, scale2;
89 int ex_expt, half_expt;
93 exp_x = __frexp_exp(x, &ex_expt);
97 * Arrange so that scale1 * scale2 == 2**expt. We use this to
98 * compensate for scalbn being horrendously slow.
100 half_expt = expt / 2;
101 INSERT_WORDS(scale1, (0x3ff + half_expt) << 20, 0);
102 half_expt = expt - half_expt;
103 INSERT_WORDS(scale2, (0x3ff + half_expt) << 20, 0);
105 return (cpack(cos(y) * exp_x * scale1 * scale2,
106 sin(y) * exp_x * scale1 * scale2));