2 /* @(#)e_exp.c 1.6 04/04/22 */
4 * ====================================================
5 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
13 #include <sys/cdefs.h>
14 __FBSDID("$FreeBSD$");
17 * Returns the exponential of x.
20 * 1. Argument reduction:
21 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
22 * Given x, find r and integer k such that
24 * x = k*ln2 + r, |r| <= 0.5*ln2.
26 * Here r will be represented as r = hi-lo for better
29 * 2. Approximation of exp(r) by a special rational function on
30 * the interval [0,0.34658]:
32 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
33 * We use a special Remes algorithm on [0,0.34658] to generate
34 * a polynomial of degree 5 to approximate R. The maximum error
35 * of this polynomial approximation is bounded by 2**-59. In
37 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
38 * (where z=r*r, and the values of P1 to P5 are listed below)
41 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
43 * The computation of exp(r) thus becomes
45 * exp(r) = 1 + -------
48 * = 1 + r + ----------- (for better accuracy)
52 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
54 * 3. Scale back to obtain exp(x):
55 * From step 1, we have
56 * exp(x) = 2^k * exp(r)
59 * exp(INF) is INF, exp(NaN) is NaN;
61 * for finite argument, only exp(0)=1 is exact.
64 * according to an error analysis, the error is always less than
65 * 1 ulp (unit in the last place).
69 * if x > 7.09782712893383973096e+02 then exp(x) overflow
70 * if x < -7.45133219101941108420e+02 then exp(x) underflow
73 * The hexadecimal values are the intended ones for the following
74 * constants. The decimal values may be used, provided that the
75 * compiler will convert from decimal to binary accurately enough
76 * to produce the hexadecimal values shown.
80 #include "math_private.h"
84 halF[2] = {0.5,-0.5,},
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 */
99 static volatile double
100 twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/
103 __ieee754_exp(double x) /* default IEEE double exp */
105 double y,hi=0.0,lo=0.0,c,t,twopk;
110 xsb = (hx>>31)&1; /* sign bit of x */
111 hx &= 0x7fffffff; /* high word of |x| */
113 /* filter out non-finite argument */
114 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
118 if(((hx&0xfffff)|lx)!=0)
119 return x+x; /* NaN */
120 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
122 if(x > o_threshold) return huge*huge; /* overflow */
123 if(x < u_threshold) return twom1000*twom1000; /* underflow */
126 /* argument reduction */
127 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
128 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
129 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
131 k = (int)(invln2*x+halF[xsb]);
133 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
138 else if(hx < 0x3e300000) { /* when |x|<2**-28 */
139 if(huge+x>one) return one+x;/* trigger inexact */
143 /* x is now in primary range */
146 INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0);
148 INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0);
149 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
150 if(k==0) return one-((x*c)/(c-2.0)-x);
151 else y = one-((lo-(x*c)/(2.0-c))-hi);
153 if (k==1024) return y*2.0*0x1p1023;
156 return y*twopk*twom1000;