2 /* @(#)e_exp.c 5.1 93/09/24 */
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
11 * ====================================================
15 * Returns the exponential of x.
18 * 1. Argument reduction:
19 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
20 * Given x, find r and integer k such that
22 * x = k*ln2 + r, |r| <= 0.5*ln2.
24 * Here r will be represented as r = hi-lo for better
27 * 2. Approximation of exp(r) by a special rational function on
28 * the interval [0,0.34658]:
30 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
31 * We use a special Reme algorithm on [0,0.34658] to generate
32 * a polynomial of degree 5 to approximate R. The maximum error
33 * of this polynomial approximation is bounded by 2**-59. In
35 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
36 * (where z=r*r, and the values of P1 to P5 are listed below)
39 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
41 * The computation of exp(r) thus becomes
43 * exp(r) = 1 + -------
46 * = 1 + r + ----------- (for better accuracy)
50 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
52 * 3. Scale back to obtain exp(x):
53 * From step 1, we have
54 * exp(x) = 2^k * exp(r)
57 * exp(INF) is INF, exp(NaN) is NaN;
59 * for finite argument, only exp(0)=1 is exact.
62 * according to an error analysis, the error is always less than
63 * 1 ulp (unit in the last place).
67 * if x > 7.09782712893383973096e+02 then exp(x) overflow
68 * if x < -7.45133219101941108420e+02 then exp(x) underflow
71 * The hexadecimal values are the intended ones for the following
72 * constants. The decimal values may be used, provided that the
73 * compiler will convert from decimal to binary accurately enough
74 * to produce the hexadecimal values shown.
79 #ifndef _DOUBLE_IS_32BITS
87 halF
[2] = {0.5,-0.5,},
89 twom1000
= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
90 o_threshold
= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
91 u_threshold
= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
92 ln2HI
[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
93 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
94 ln2LO
[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
95 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
96 invln2
= 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
97 P1
= 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
98 P2
= -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
99 P3
= 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
100 P4
= -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
101 P5
= 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
105 double __ieee754_exp(double x
) /* default IEEE double exp */
107 double __ieee754_exp(x
) /* default IEEE double exp */
116 xsb
= (hx
>>31)&1; /* sign bit of x */
117 hx
&= 0x7fffffff; /* high word of |x| */
119 /* filter out non-finite argument */
120 if(hx
>= 0x40862E42) { /* if |x|>=709.78... */
124 if(((hx
&0xfffff)|lx
)!=0)
125 return x
+x
; /* NaN */
126 else return (xsb
==0)? x
:0.0; /* exp(+-inf)={inf,0} */
128 if(x
> o_threshold
) return huge
*huge
; /* overflow */
129 if(x
< u_threshold
) return twom1000
*twom1000
; /* underflow */
132 /* argument reduction */
133 if(hx
> 0x3fd62e42) { /* if |x| > 0.5 ln2 */
134 if(hx
< 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
135 hi
= x
-ln2HI
[xsb
]; lo
=ln2LO
[xsb
]; k
= 1-xsb
-xsb
;
137 k
= invln2
*x
+halF
[xsb
];
139 hi
= x
- t
*ln2HI
[0]; /* t*ln2HI is exact here */
144 else if(hx
< 0x3e300000) { /* when |x|<2**-28 */
145 if(huge
+x
>one
) return one
+x
;/* trigger inexact */
149 /* x is now in primary range */
151 c
= x
- t
*(P1
+t
*(P2
+t
*(P3
+t
*(P4
+t
*P5
))));
152 if(k
==0) return one
-((x
*c
)/(c
-2.0)-x
);
153 else y
= one
-((lo
-(x
*c
)/(2.0-c
))-hi
);
157 SET_HIGH_WORD(y
,hy
+(k
<<20)); /* add k to y's exponent */
162 SET_HIGH_WORD(y
,hy
+((k
+1000)<<20)); /* add k to y's exponent */
167 #endif /* defined(_DOUBLE_IS_32BITS) */