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 #include <LibConfig.h>
13 #include <sys/EfiCdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: e_exp.c,v 1.11 2002/05/26 22:01:49 wiz Exp $");
16 #endif
17
18 #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */
19 // C4756: overflow in constant arithmetic
20 #pragma warning ( disable : 4756 )
21 #endif
22
23 /* __ieee754_exp(x)
24 * Returns the exponential of x.
25 *
26 * Method
27 * 1. Argument reduction:
28 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
29 * Given x, find r and integer k such that
30 *
31 * x = k*ln2 + r, |r| <= 0.5*ln2.
32 *
33 * Here r will be represented as r = hi-lo for better
34 * accuracy.
35 *
36 * 2. Approximation of exp(r) by a special rational function on
37 * the interval [0,0.34658]:
38 * Write
39 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
40 * We use a special Reme algorithm on [0,0.34658] to generate
41 * a polynomial of degree 5 to approximate R. The maximum error
42 * of this polynomial approximation is bounded by 2**-59. In
43 * other words,
44 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
45 * (where z=r*r, and the values of P1 to P5 are listed below)
46 * and
47 * | 5 | -59
48 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
49 * | |
50 * The computation of exp(r) thus becomes
51 * 2*r
52 * exp(r) = 1 + -------
53 * R - r
54 * r*R1(r)
55 * = 1 + r + ----------- (for better accuracy)
56 * 2 - R1(r)
57 * where
58 * 2 4 10
59 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
60 *
61 * 3. Scale back to obtain exp(x):
62 * From step 1, we have
63 * exp(x) = 2^k * exp(r)
64 *
65 * Special cases:
66 * exp(INF) is INF, exp(NaN) is NaN;
67 * exp(-INF) is 0, and
68 * for finite argument, only exp(0)=1 is exact.
69 *
70 * Accuracy:
71 * according to an error analysis, the error is always less than
72 * 1 ulp (unit in the last place).
73 *
74 * Misc. info.
75 * For IEEE double
76 * if x > 7.09782712893383973096e+02 then exp(x) overflow
77 * if x < -7.45133219101941108420e+02 then exp(x) underflow
78 *
79 * Constants:
80 * The hexadecimal values are the intended ones for the following
81 * constants. The decimal values may be used, provided that the
82 * compiler will convert from decimal to binary accurately enough
83 * to produce the hexadecimal values shown.
84 */
85
86 #include "math.h"
87 #include "math_private.h"
88
89 static const double
90 one = 1.0,
91 halF[2] = {0.5,-0.5,},
92 huge = 1.0e+300,
93 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
94 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
95 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
96 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
97 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
98 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
99 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
100 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
101 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
102 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
103 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
104 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
105 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
106
107
108 double
__ieee754_exp(double x)109 __ieee754_exp(double x) /* default IEEE double exp */
110 {
111 double y,hi,lo,c,t;
112 int32_t k,xsb;
113 u_int32_t hx;
114
115 hi = lo = 0;
116 k = 0;
117 GET_HIGH_WORD(hx,x);
118 xsb = (hx>>31)&1; /* sign bit of x */
119 hx &= 0x7fffffff; /* high word of |x| */
120
121 /* filter out non-finite argument */
122 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
123 if(hx>=0x7ff00000) {
124 u_int32_t lx;
125 GET_LOW_WORD(lx,x);
126 if(((hx&0xfffff)|lx)!=0)
127 return x+x; /* NaN */
128 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
129 }
130 if(x > o_threshold) return huge*huge; /* overflow */
131 if(x < u_threshold) return twom1000*twom1000; /* underflow */
132 }
133
134 /* argument reduction */
135 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
136 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
137 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
138 } else {
139 k = (int32_t)(invln2*x+halF[xsb]);
140 t = k;
141 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
142 lo = t*ln2LO[0];
143 }
144 x = hi - lo;
145 }
146 else if(hx < 0x3e300000) { /* when |x|<2**-28 */
147 if(huge+x>one) return one+x;/* trigger inexact */
148 }
149 else k = 0;
150
151 /* x is now in primary range */
152 t = x*x;
153 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
154 if(k==0) return one-((x*c)/(c-2.0)-x);
155 else y = one-((lo-(x*c)/(2.0-c))-hi);
156 if(k >= -1021) {
157 u_int32_t hy;
158 GET_HIGH_WORD(hy,y);
159 SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
160 return y;
161 } else {
162 u_int32_t hy;
163 GET_HIGH_WORD(hy,y);
164 SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
165 return y*twom1000;
166 }
167 }
168