1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /* __ieee754_exp(x)
13 * Returns the exponential of x.
14 *
15 * Method
16 * 1. Argument reduction:
17 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
18 * Given x, find r and integer k such that
19 *
20 * x = k*ln2 + r, |r| <= 0.5*ln2.
21 *
22 * Here r will be represented as r = hi-lo for better
23 * accuracy.
24 *
25 * 2. Approximation of exp(r) by a special rational function on
26 * the interval [0,0.34658]:
27 * Write
28 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
29 * We use a special Reme algorithm on [0,0.34658] to generate
30 * a polynomial of degree 5 to approximate R. The maximum error
31 * of this polynomial approximation is bounded by 2**-59. In
32 * other words,
33 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
34 * (where z=r*r, and the values of P1 to P5 are listed below)
35 * and
36 * | 5 | -59
37 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
38 * | |
39 * The computation of exp(r) thus becomes
40 * 2*r
41 * exp(r) = 1 + -------
42 * R - r
43 * r*R1(r)
44 * = 1 + r + ----------- (for better accuracy)
45 * 2 - R1(r)
46 * where
47 * 2 4 10
48 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
49 *
50 * 3. Scale back to obtain exp(x):
51 * From step 1, we have
52 * exp(x) = 2^k * exp(r)
53 *
54 * Special cases:
55 * exp(INF) is INF, exp(NaN) is NaN;
56 * exp(-INF) is 0, and
57 * for finite argument, only exp(0)=1 is exact.
58 *
59 * Accuracy:
60 * according to an error analysis, the error is always less than
61 * 1 ulp (unit in the last place).
62 *
63 * Misc. info.
64 * For IEEE double
65 * if x > 7.09782712893383973096e+02 then exp(x) overflow
66 * if x < -7.45133219101941108420e+02 then exp(x) underflow
67 *
68 * Constants:
69 * The hexadecimal values are the intended ones for the following
70 * constants. The decimal values may be used, provided that the
71 * compiler will convert from decimal to binary accurately enough
72 * to produce the hexadecimal values shown.
73 */
74
75 #include "math_libm.h"
76 #include "math_private.h"
77
78 #ifdef __WATCOMC__ /* Watcom defines huge=__huge */
79 #undef huge
80 #endif
81
82 static const double
83 one = 1.0,
84 halF[2] = {0.5,-0.5,},
85 huge = 1.0e+300,
86 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
87 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
88 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
89 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
90 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
91 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
92 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
93 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
94 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
95 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
96 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
97 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
98 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
99
__ieee754_exp(double x)100 double __ieee754_exp(double x) /* default IEEE double exp */
101 {
102 double y;
103 double hi = 0.0;
104 double lo = 0.0;
105 double c;
106 double t;
107 int32_t k=0;
108 int32_t xsb;
109 u_int32_t hx;
110
111 GET_HIGH_WORD(hx,x);
112 xsb = (hx>>31)&1; /* sign bit of x */
113 hx &= 0x7fffffff; /* high word of |x| */
114
115 /* filter out non-finite argument */
116 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
117 if(hx>=0x7ff00000) {
118 u_int32_t lx;
119 GET_LOW_WORD(lx,x);
120 if(((hx&0xfffff)|lx)!=0)
121 return x+x; /* NaN */
122 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
123 }
124 #if 1
125 if(x > o_threshold) return huge*huge; /* overflow */
126 #else /* !!! FIXME: check this: "huge * huge" is a compiler warning, maybe they wanted +Inf? */
127 if(x > o_threshold) return INFINITY; /* overflow */
128 #endif
129
130 if(x < u_threshold) return twom1000*twom1000; /* underflow */
131 }
132
133 /* argument reduction */
134 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
135 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
136 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
137 } else {
138 k = (int32_t) (invln2*x+halF[xsb]);
139 t = k;
140 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
141 lo = t*ln2LO[0];
142 }
143 x = hi - lo;
144 }
145 else if(hx < 0x3e300000) { /* when |x|<2**-28 */
146 if(huge+x>one) return one+x;/* trigger inexact */
147 }
148 else k = 0;
149
150 /* x is now in primary range */
151 t = x*x;
152 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
153 if(k==0) return one-((x*c)/(c-2.0)-x);
154 else y = one-((lo-(x*c)/(2.0-c))-hi);
155 if(k >= -1021) {
156 u_int32_t hy;
157 GET_HIGH_WORD(hy,y);
158 SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
159 return y;
160 } else {
161 u_int32_t hy;
162 GET_HIGH_WORD(hy,y);
163 SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
164 return y*twom1000;
165 }
166 }
167
168 /*
169 * wrapper exp(x)
170 */
171 #ifndef _IEEE_LIBM
exp(double x)172 double exp(double x)
173 {
174 static const double o_threshold = 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */
175 static const double u_threshold = -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */
176
177 double z = __ieee754_exp(x);
178 if (_LIB_VERSION == _IEEE_)
179 return z;
180 if (isfinite(x)) {
181 if (x > o_threshold)
182 return __kernel_standard(x, x, 6); /* exp overflow */
183 if (x < u_threshold)
184 return __kernel_standard(x, x, 7); /* exp underflow */
185 }
186 return z;
187 }
188 #else
189 strong_alias(__ieee754_exp, exp)
190 #endif
191 libm_hidden_def(exp)
192