1df930be7Sderaadt /* @(#)e_exp.c 5.1 93/09/24 */
2df930be7Sderaadt /*
3df930be7Sderaadt * ====================================================
4df930be7Sderaadt * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5df930be7Sderaadt *
6df930be7Sderaadt * Developed at SunPro, a Sun Microsystems, Inc. business.
7df930be7Sderaadt * Permission to use, copy, modify, and distribute this
8df930be7Sderaadt * software is freely granted, provided that this notice
9df930be7Sderaadt * is preserved.
10df930be7Sderaadt * ====================================================
11df930be7Sderaadt */
12df930be7Sderaadt
137b36286aSmartynas /* exp(x)
14df930be7Sderaadt * Returns the exponential of x.
15df930be7Sderaadt *
16df930be7Sderaadt * Method
17df930be7Sderaadt * 1. Argument reduction:
18df930be7Sderaadt * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
19df930be7Sderaadt * Given x, find r and integer k such that
20df930be7Sderaadt *
21df930be7Sderaadt * x = k*ln2 + r, |r| <= 0.5*ln2.
22df930be7Sderaadt *
23df930be7Sderaadt * Here r will be represented as r = hi-lo for better
24df930be7Sderaadt * accuracy.
25df930be7Sderaadt *
26df930be7Sderaadt * 2. Approximation of exp(r) by a special rational function on
27df930be7Sderaadt * the interval [0,0.34658]:
28df930be7Sderaadt * Write
29df930be7Sderaadt * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
3071a22d32Smartynas * We use a special Remes algorithm on [0,0.34658] to generate
31df930be7Sderaadt * a polynomial of degree 5 to approximate R. The maximum error
32df930be7Sderaadt * of this polynomial approximation is bounded by 2**-59. In
33df930be7Sderaadt * other words,
34df930be7Sderaadt * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
35df930be7Sderaadt * (where z=r*r, and the values of P1 to P5 are listed below)
36df930be7Sderaadt * and
37df930be7Sderaadt * | 5 | -59
38df930be7Sderaadt * | 2.0+P1*z+...+P5*z - R(z) | <= 2
39df930be7Sderaadt * | |
40df930be7Sderaadt * The computation of exp(r) thus becomes
41df930be7Sderaadt * 2*r
42df930be7Sderaadt * exp(r) = 1 + -------
43df930be7Sderaadt * R - r
44df930be7Sderaadt * r*R1(r)
45df930be7Sderaadt * = 1 + r + ----------- (for better accuracy)
46df930be7Sderaadt * 2 - R1(r)
47df930be7Sderaadt * where
48df930be7Sderaadt * 2 4 10
49df930be7Sderaadt * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
50df930be7Sderaadt *
51df930be7Sderaadt * 3. Scale back to obtain exp(x):
52df930be7Sderaadt * From step 1, we have
53df930be7Sderaadt * exp(x) = 2^k * exp(r)
54df930be7Sderaadt *
55df930be7Sderaadt * Special cases:
56df930be7Sderaadt * exp(INF) is INF, exp(NaN) is NaN;
57df930be7Sderaadt * exp(-INF) is 0, and
58df930be7Sderaadt * for finite argument, only exp(0)=1 is exact.
59df930be7Sderaadt *
60df930be7Sderaadt * Accuracy:
61df930be7Sderaadt * according to an error analysis, the error is always less than
62df930be7Sderaadt * 1 ulp (unit in the last place).
63df930be7Sderaadt *
64df930be7Sderaadt * Misc. info.
65df930be7Sderaadt * For IEEE double
66df930be7Sderaadt * if x > 7.09782712893383973096e+02 then exp(x) overflow
67df930be7Sderaadt * if x < -7.45133219101941108420e+02 then exp(x) underflow
68df930be7Sderaadt *
69df930be7Sderaadt * Constants:
70df930be7Sderaadt * The hexadecimal values are the intended ones for the following
71df930be7Sderaadt * constants. The decimal values may be used, provided that the
72df930be7Sderaadt * compiler will convert from decimal to binary accurately enough
73df930be7Sderaadt * to produce the hexadecimal values shown.
74df930be7Sderaadt */
75df930be7Sderaadt
7649393c00Smartynas #include <float.h>
7749393c00Smartynas #include <math.h>
7849393c00Smartynas
79df930be7Sderaadt #include "math_private.h"
80df930be7Sderaadt
81df930be7Sderaadt static const double
82df930be7Sderaadt one = 1.0,
83df930be7Sderaadt halF[2] = {0.5,-0.5,},
84df930be7Sderaadt huge = 1.0e+300,
85df930be7Sderaadt twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
86df930be7Sderaadt o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
87df930be7Sderaadt u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
88df930be7Sderaadt ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
89df930be7Sderaadt -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
90df930be7Sderaadt ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
91df930be7Sderaadt -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
92df930be7Sderaadt invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
93df930be7Sderaadt P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
94df930be7Sderaadt P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
95df930be7Sderaadt P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
96df930be7Sderaadt P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
97df930be7Sderaadt P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
98df930be7Sderaadt
99df930be7Sderaadt
100e7beb4a7Smillert double
exp(double x)1017b36286aSmartynas exp(double x) /* default IEEE double exp */
102df930be7Sderaadt {
103df930be7Sderaadt double y,hi,lo,c,t;
104df930be7Sderaadt int32_t k,xsb;
105df930be7Sderaadt u_int32_t hx;
106df930be7Sderaadt
107df930be7Sderaadt GET_HIGH_WORD(hx,x);
108df930be7Sderaadt xsb = (hx>>31)&1; /* sign bit of x */
109df930be7Sderaadt hx &= 0x7fffffff; /* high word of |x| */
110df930be7Sderaadt
111df930be7Sderaadt /* filter out non-finite argument */
112df930be7Sderaadt if(hx >= 0x40862E42) { /* if |x|>=709.78... */
113df930be7Sderaadt if(hx>=0x7ff00000) {
114df930be7Sderaadt u_int32_t lx;
115df930be7Sderaadt GET_LOW_WORD(lx,x);
116df930be7Sderaadt if(((hx&0xfffff)|lx)!=0)
117df930be7Sderaadt return x+x; /* NaN */
118df930be7Sderaadt else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
119df930be7Sderaadt }
120df930be7Sderaadt if(x > o_threshold) return huge*huge; /* overflow */
121df930be7Sderaadt if(x < u_threshold) return twom1000*twom1000; /* underflow */
122df930be7Sderaadt }
123df930be7Sderaadt
124df930be7Sderaadt /* argument reduction */
125df930be7Sderaadt if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
126df930be7Sderaadt if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
127df930be7Sderaadt hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
128df930be7Sderaadt } else {
129df930be7Sderaadt k = invln2*x+halF[xsb];
130df930be7Sderaadt t = k;
131df930be7Sderaadt hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
132df930be7Sderaadt lo = t*ln2LO[0];
133df930be7Sderaadt }
134df930be7Sderaadt x = hi - lo;
135df930be7Sderaadt }
136df930be7Sderaadt else if(hx < 0x3e300000) { /* when |x|<2**-28 */
137df930be7Sderaadt if(huge+x>one) return one+x;/* trigger inexact */
138df930be7Sderaadt }
139df930be7Sderaadt else k = 0;
140df930be7Sderaadt
141df930be7Sderaadt /* x is now in primary range */
142df930be7Sderaadt t = x*x;
143df930be7Sderaadt c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
144df930be7Sderaadt if(k==0) return one-((x*c)/(c-2.0)-x);
145df930be7Sderaadt else y = one-((lo-(x*c)/(2.0-c))-hi);
146df930be7Sderaadt if(k >= -1021) {
147df930be7Sderaadt u_int32_t hy;
148df930be7Sderaadt GET_HIGH_WORD(hy,y);
149df930be7Sderaadt SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
150df930be7Sderaadt return y;
151df930be7Sderaadt } else {
152df930be7Sderaadt u_int32_t hy;
153df930be7Sderaadt GET_HIGH_WORD(hy,y);
154df930be7Sderaadt SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
155df930be7Sderaadt return y*twom1000;
156df930be7Sderaadt }
157df930be7Sderaadt }
158*2f2c0062Sguenther DEF_STD(exp);
159*2f2c0062Sguenther LDBL_MAYBE_CLONE(exp);
160