1 /* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */
3 * ====================================================
4 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
9 * ====================================================
12 * Returns the exponential of x.
15 * 1. Argument reduction:
16 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
17 * Given x, find r and integer k such that
19 * x = k*ln2 + r, |r| <= 0.5*ln2.
21 * Here r will be represented as r = hi-lo for better
24 * 2. Approximation of exp(r) by a special rational function on
25 * the interval [0,0.34658]:
27 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
28 * We use a special Remes algorithm on [0,0.34658] to generate
29 * a polynomial of degree 5 to approximate R. The maximum error
30 * of this polynomial approximation is bounded by 2**-59. In
32 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
33 * (where z=r*r, and the values of P1 to P5 are listed below)
36 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
38 * The computation of exp(r) thus becomes
40 * exp(r) = 1 + -------
43 * = 1 + r + ----------- (for better accuracy)
47 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
49 * 3. Scale back to obtain exp(x):
50 * From step 1, we have
51 * exp(x) = 2^k * exp(r)
54 * exp(INF) is INF, exp(NaN) is NaN;
56 * for finite argument, only exp(0)=1 is exact.
59 * according to an error analysis, the error is always less than
60 * 1 ulp (unit in the last place).
64 * if x > 7.09782712893383973096e+02 then exp(x) overflow
65 * if x < -7.45133219101941108420e+02 then exp(x) underflow
68 * The hexadecimal values are the intended ones for the following
69 * constants. The decimal values may be used, provided that the
70 * compiler will convert from decimal to binary accurately enough
71 * to produce the hexadecimal values shown.
78 halF[2] = {0.5,-0.5,},
80 o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
81 u_threshold = -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
82 ln2HI[2] = { 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
83 -6.93147180369123816490e-01},/* 0xbfe62e42, 0xfee00000 */
84 ln2LO[2] = { 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
85 -1.90821492927058770002e-10},/* 0xbdea39ef, 0x35793c76 */
86 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
87 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
88 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
89 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
90 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
91 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
93 static const volatile double
94 twom1000 = 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0 */
98 double y,hi=0.0,lo=0.0,c,t,twopk;
102 GET_HIGH_WORD(hx, x);
103 xsb = (hx>>31)&1; /* sign bit of x */
104 hx &= 0x7fffffff; /* high word of |x| */
106 /* filter out non-finite argument */
107 if (hx >= 0x40862E42) { /* if |x| >= 709.78... */
108 if (hx >= 0x7ff00000) {
112 if (((hx&0xfffff)|lx) != 0) /* NaN */
114 return xsb==0 ? x : 0.0; /* exp(+-inf)={inf,0} */
117 return huge*huge; /* overflow */
119 return twom1000*twom1000; /* underflow */
122 /* argument reduction */
123 if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
124 if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
129 k = (int)(invln2*x+halF[xsb]);
131 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
134 STRICT_ASSIGN(double, x, hi - lo);
135 } else if(hx < 0x3e300000) { /* |x| < 2**-28 */
142 /* x is now in primary range */
145 INSERT_WORDS(twopk, 0x3ff00000+(k<<20), 0);
147 INSERT_WORDS(twopk, 0x3ff00000+((k+1000)<<20), 0);
148 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
150 return one - ((x*c)/(c-2.0) - x);
151 y = one-((lo-(x*c)/(2.0-c))-hi);
153 return y*twopk*twom1000;
155 return y*2.0*0x1p1023;