1 /* @(#)e_exp.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13 /* exp(x)
14 * Returns the exponential of x.
15 *
16 * Method
17 * 1. Argument reduction:
18 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
19 * Given x, find r and integer k such that
20 *
21 * x = k*ln2 + r, |r| <= 0.5*ln2.
22 *
23 * Here r will be represented as r = hi-lo for better
24 * accuracy.
25 *
26 * 2. Approximation of exp(r) by a special rational function on
27 * the interval [0,0.34658]:
28 * Write
29 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
30 * We use a special Remes algorithm on [0,0.34658] to generate
31 * a polynomial of degree 5 to approximate R. The maximum error
32 * of this polynomial approximation is bounded by 2**-59. In
33 * other words,
34 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
35 * (where z=r*r, and the values of P1 to P5 are listed below)
36 * and
37 * | 5 | -59
38 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
39 * | |
40 * The computation of exp(r) thus becomes
41 * 2*r
42 * exp(r) = 1 + -------
43 * R - r
44 * r*R1(r)
45 * = 1 + r + ----------- (for better accuracy)
46 * 2 - R1(r)
47 * where
48 * 2 4 10
49 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
50 *
51 * 3. Scale back to obtain exp(x):
52 * From step 1, we have
53 * exp(x) = 2^k * exp(r)
54 *
55 * Special cases:
56 * exp(INF) is INF, exp(NaN) is NaN;
57 * exp(-INF) is 0, and
58 * for finite argument, only exp(0)=1 is exact.
59 *
60 * Accuracy:
61 * according to an error analysis, the error is always less than
62 * 1 ulp (unit in the last place).
63 *
64 * Misc. info.
65 * For IEEE double
66 * if x > 7.09782712893383973096e+02 then exp(x) overflow
67 * if x < -7.45133219101941108420e+02 then exp(x) underflow
68 *
69 * Constants:
70 * The hexadecimal values are the intended ones for the following
71 * constants. The decimal values may be used, provided that the
72 * compiler will convert from decimal to binary accurately enough
73 * to produce the hexadecimal values shown.
74 */
75
76 #include <float.h>
77 #include <math.h>
78
79 #include "math_private.h"
80
81 static const double
82 one = 1.0,
83 halF[2] = {0.5,-0.5,},
84 huge = 1.0e+300,
85 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
86 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
87 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
88 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
89 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
90 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
91 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
92 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
93 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
94 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
95 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
96 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
97 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
98
99
100 double
exp(double x)101 exp(double x) /* default IEEE double exp */
102 {
103 double y,hi,lo,c,t;
104 int32_t k,xsb;
105 u_int32_t hx;
106
107 GET_HIGH_WORD(hx,x);
108 xsb = (hx>>31)&1; /* sign bit of x */
109 hx &= 0x7fffffff; /* high word of |x| */
110
111 /* filter out non-finite argument */
112 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
113 if(hx>=0x7ff00000) {
114 u_int32_t lx;
115 GET_LOW_WORD(lx,x);
116 if(((hx&0xfffff)|lx)!=0)
117 return x+x; /* NaN */
118 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
119 }
120 if(x > o_threshold) return huge*huge; /* overflow */
121 if(x < u_threshold) return twom1000*twom1000; /* underflow */
122 }
123
124 /* argument reduction */
125 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
126 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
127 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
128 } else {
129 k = invln2*x+halF[xsb];
130 t = k;
131 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
132 lo = t*ln2LO[0];
133 }
134 x = hi - lo;
135 }
136 else if(hx < 0x3e300000) { /* when |x|<2**-28 */
137 if(huge+x>one) return one+x;/* trigger inexact */
138 }
139 else k = 0;
140
141 /* x is now in primary range */
142 t = x*x;
143 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
144 if(k==0) return one-((x*c)/(c-2.0)-x);
145 else y = one-((lo-(x*c)/(2.0-c))-hi);
146 if(k >= -1021) {
147 u_int32_t hy;
148 GET_HIGH_WORD(hy,y);
149 SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
150 return y;
151 } else {
152 u_int32_t hy;
153 GET_HIGH_WORD(hy,y);
154 SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
155 return y*twom1000;
156 }
157 }
158
159 #if LDBL_MANT_DIG == DBL_MANT_DIG
160 __strong_alias(expl, exp);
161 #endif /* LDBL_MANT_DIG == DBL_MANT_DIG */
162