summaryrefslogtreecommitdiff
path: root/src/math/exp.c
blob: 2b182acdeace1763cce73ab9b637d81cdcf48df9 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */
/*
 * ====================================================
 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
 *
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */
/* exp(x)
 * Returns the exponential of x.
 *
 * Method
 *   1. Argument reduction:
 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
 *      Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
 *
 *      Here r will be represented as r = hi-lo for better
 *      accuracy.
 *
 *   2. Approximation of exp(r) by a special rational function on
 *      the interval [0,0.34658]:
 *      Write
 *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
 *      We use a special Remez algorithm on [0,0.34658] to generate
 *      a polynomial of degree 5 to approximate R. The maximum error
 *      of this polynomial approximation is bounded by 2**-59. In
 *      other words,
 *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
 *      (where z=r*r, and the values of P1 to P5 are listed below)
 *      and
 *          |                  5          |     -59
 *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
 *          |                             |
 *      The computation of exp(r) thus becomes
 *                              2*r
 *              exp(r) = 1 + ----------
 *                            R(r) - r
 *                                 r*c(r)
 *                     = 1 + r + ----------- (for better accuracy)
 *                                2 - c(r)
 *      where
 *                              2       4             10
 *              c(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
 *
 *   3. Scale back to obtain exp(x):
 *      From step 1, we have
 *         exp(x) = 2^k * exp(r)
 *
 * Special cases:
 *      exp(INF) is INF, exp(NaN) is NaN;
 *      exp(-INF) is 0, and
 *      for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * Misc. info.
 *      For IEEE double
 *          if x >  709.782712893383973096 then exp(x) overflows
 *          if x < -745.133219101941108420 then exp(x) underflows
 */

#include "libm.h"

static const double
half[2] = {0.5,-0.5},
ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */

double exp(double x)
{
	double_t hi, lo, c, xx, y;
	int k, sign;
	uint32_t hx;

	GET_HIGH_WORD(hx, x);
	sign = hx>>31;
	hx &= 0x7fffffff;  /* high word of |x| */

	/* special cases */
	if (hx >= 0x4086232b) {  /* if |x| >= 708.39... */
		if (isnan(x))
			return x;
		if (x > 709.782712893383973096) {
			/* overflow if x!=inf */
			STRICT_ASSIGN(double, x, 0x1p1023 * x);
			return x;
		}
		if (x < -708.39641853226410622) {
			/* underflow if x!=-inf */
			FORCE_EVAL((float)(-0x1p-149/x));
			if (x < -745.13321910194110842)
				return 0;
		}
	}

	/* argument reduction */
	if (hx > 0x3fd62e42) {  /* if |x| > 0.5 ln2 */
		if (hx >= 0x3ff0a2b2)  /* if |x| >= 1.5 ln2 */
			k = (int)(invln2*x + half[sign]);
		else
			k = 1 - sign - sign;
		hi = x - k*ln2hi;  /* k*ln2hi is exact here */
		lo = k*ln2lo;
		STRICT_ASSIGN(double, x, hi - lo);
	} else if (hx > 0x3e300000)  {  /* if |x| > 2**-28 */
		k = 0;
		hi = x;
		lo = 0;
	} else {
		/* inexact if x!=0 */
		FORCE_EVAL(0x1p1023 + x);
		return 1 + x;
	}

	/* x is now in primary range */
	xx = x*x;
	c = x - xx*(P1+xx*(P2+xx*(P3+xx*(P4+xx*P5))));
	y = 1 + (x*c/(2-c) - lo + hi);
	if (k == 0)
		return y;
	return scalbn(y, k);
}