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1 /* @(#)e_exp.c 5.1 93/09/24 */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
12 #include <LibConfig.h>
13 #include <sys/EfiCdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: e_exp.c,v 1.11 2002/05/26 22:01:49 wiz Exp $");
18 #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */
19 // C4756: overflow in constant arithmetic
20 #pragma warning ( disable : 4756 )
24 * Returns the exponential of x.
27 * 1. Argument reduction:
28 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
29 * Given x, find r and integer k such that
31 * x = k*ln2 + r, |r| <= 0.5*ln2.
33 * Here r will be represented as r = hi-lo for better
36 * 2. Approximation of exp(r) by a special rational function on
37 * the interval [0,0.34658]:
39 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
40 * We use a special Reme algorithm on [0,0.34658] to generate
41 * a polynomial of degree 5 to approximate R. The maximum error
42 * of this polynomial approximation is bounded by 2**-59. In
44 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
45 * (where z=r*r, and the values of P1 to P5 are listed below)
48 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
50 * The computation of exp(r) thus becomes
52 * exp(r) = 1 + -------
55 * = 1 + r + ----------- (for better accuracy)
59 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
61 * 3. Scale back to obtain exp(x):
62 * From step 1, we have
63 * exp(x) = 2^k * exp(r)
66 * exp(INF) is INF, exp(NaN) is NaN;
68 * for finite argument, only exp(0)=1 is exact.
71 * according to an error analysis, the error is always less than
72 * 1 ulp (unit in the last place).
76 * if x > 7.09782712893383973096e+02 then exp(x) overflow
77 * if x < -7.45133219101941108420e+02 then exp(x) underflow
80 * The hexadecimal values are the intended ones for the following
81 * constants. The decimal values may be used, provided that the
82 * compiler will convert from decimal to binary accurately enough
83 * to produce the hexadecimal values shown.
87 #include "math_private.h"
91 halF
[2] = {0.5,-0.5,},
93 twom1000
= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
94 o_threshold
= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
95 u_threshold
= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
96 ln2HI
[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
97 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
98 ln2LO
[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
99 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
100 invln2
= 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
101 P1
= 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
102 P2
= -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
103 P3
= 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
104 P4
= -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
105 P5
= 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
109 __ieee754_exp(double x
) /* default IEEE double exp */
118 xsb
= (hx
>>31)&1; /* sign bit of x */
119 hx
&= 0x7fffffff; /* high word of |x| */
121 /* filter out non-finite argument */
122 if(hx
>= 0x40862E42) { /* if |x|>=709.78... */
126 if(((hx
&0xfffff)|lx
)!=0)
127 return x
+x
; /* NaN */
128 else return (xsb
==0)? x
:0.0; /* exp(+-inf)={inf,0} */
130 if(x
> o_threshold
) return huge
*huge
; /* overflow */
131 if(x
< u_threshold
) return twom1000
*twom1000
; /* underflow */
134 /* argument reduction */
135 if(hx
> 0x3fd62e42) { /* if |x| > 0.5 ln2 */
136 if(hx
< 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
137 hi
= x
-ln2HI
[xsb
]; lo
=ln2LO
[xsb
]; k
= 1-xsb
-xsb
;
139 k
= (int32_t)(invln2
*x
+halF
[xsb
]);
141 hi
= x
- t
*ln2HI
[0]; /* t*ln2HI is exact here */
146 else if(hx
< 0x3e300000) { /* when |x|<2**-28 */
147 if(huge
+x
>one
) return one
+x
;/* trigger inexact */
151 /* x is now in primary range */
153 c
= x
- t
*(P1
+t
*(P2
+t
*(P3
+t
*(P4
+t
*P5
))));
154 if(k
==0) return one
-((x
*c
)/(c
-2.0)-x
);
155 else y
= one
-((lo
-(x
*c
)/(2.0-c
))-hi
);
159 SET_HIGH_WORD(y
,hy
+(k
<<20)); /* add k to y's exponent */
164 SET_HIGH_WORD(y
,hy
+((k
+1000)<<20)); /* add k to y's exponent */