src/fdlibm/e_exp.c

/* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
 *
 * ***** BEGIN LICENSE BLOCK *****
 * Version: MPL 1.1/GPL 2.0/LGPL 2.1
 *
 * The contents of this file are subject to the Mozilla Public License Version
 * 1.1 (the "License"); you may not use this file except in compliance with
 * the License. You may obtain a copy of the License at
 * http://www.mozilla.org/MPL/
 *
 * Software distributed under the License is distributed on an "AS IS" basis,
 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
 * for the specific language governing rights and limitations under the
 * License.
 *
 * The Original Code is Mozilla Communicator client code, released
 * March 31, 1998.
 *
 * The Initial Developer of the Original Code is
 * Sun Microsystems, Inc.
 * Portions created by the Initial Developer are Copyright (C) 1998
 * the Initial Developer. All Rights Reserved.
 *
 * Contributor(s):
 *
 * Alternatively, the contents of this file may be used under the terms of
 * either of the GNU General Public License Version 2 or later (the "GPL"),
 * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
 * in which case the provisions of the GPL or the LGPL are applicable instead
 * of those above. If you wish to allow use of your version of this file only
 * under the terms of either the GPL or the LGPL, and not to allow others to
 * use your version of this file under the terms of the MPL, indicate your
 * decision by deleting the provisions above and replace them with the notice
 * and other provisions required by the GPL or the LGPL. If you do not delete
 * the provisions above, a recipient may use your version of this file under
 * the terms of any one of the MPL, the GPL or the LGPL.
 *
 * ***** END LICENSE BLOCK ***** */

/* @(#)e_exp.c 1.3 95/01/18 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice 
 * is preserved.
 * ====================================================
 */

/* __ieee754_exp(x)
 * Returns the exponential of x.
 *
 * Method
 *   1. Argument reduction:
 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
 *      Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2.  
 *
 *      Here r will be represented as r = hi-lo for better 
 *      accuracy.
 *
 *   2. Approximation of exp(r) by a special rational function on
 *      the interval [0,0.34658]:
 *      Write
 *          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
 *      We use a special Reme algorithm on [0,0.34658] to generate 
 *      a polynomial of degree 5 to approximate R. The maximum error 
 *      of this polynomial approximation is bounded by 2**-59. In
 *      other words,
 *          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
 *      (where z=r*r, and the values of P1 to P5 are listed below)
 *      and
 *          |                  5          |     -59
 *          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2 
 *          |                             |
 *      The computation of exp(r) thus becomes
 *                             2*r
 *              exp(r) = 1 + -------
 *                            R - r
 *                                 r*R1(r)      
 *                     = 1 + r + ----------- (for better accuracy)
 *                                2 - R1(r)
 *      where
 *                               2       4             10
 *              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
 *      
 *   3. Scale back to obtain exp(x):
 *      From step 1, we have
 *         exp(x) = 2^k * exp(r)
 *
 * Special cases:
 *      exp(INF) is INF, exp(NaN) is NaN;
 *      exp(-INF) is 0, and
 *      for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * Misc. info.
 *      For IEEE double 
 *          if x >  7.09782712893383973096e+02 then exp(x) overflow
 *          if x < -7.45133219101941108420e+02 then exp(x) underflow
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following 
 * constants. The decimal values may be used, provided that the 
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

#include "fdlibm.h"

#ifdef __STDC__
static const double
#else
static double
#endif
one     = 1.0,
halF[2] = {0.5,-0.5,},
really_big      = 1.0e+300,
twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
             -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
             -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */


#ifdef __STDC__
        double __ieee754_exp(double x)  /* default IEEE double exp */
#else
        double __ieee754_exp(x) /* default IEEE double exp */
        double x;
#endif
{
        fd_twoints u;
        double y,hi,lo,c,t;
        int k, xsb;
        unsigned hx;

        u.d = x;
        hx  = __HI(u);  /* high word of x */
        xsb = (hx>>31)&1;               /* sign bit of x */
        hx &= 0x7fffffff;               /* high word of |x| */

    /* filter out non-finite argument */
        if(hx >= 0x40862E42) {                  /* if |x|>=709.78... */
            if(hx>=0x7ff00000) {
                u.d = x;
                if(((hx&0xfffff)|__LO(u))!=0)
                     return x+x;                /* NaN */
                else return (xsb==0)? x:0.0;    /* exp(+-inf)={inf,0} */
            }
            if(x > o_threshold) return really_big*really_big; /* overflow */
            if(x < u_threshold) return twom1000*twom1000; /* underflow */
        }

    /* argument reduction */
        if(hx > 0x3fd62e42) {           /* if  |x| > 0.5 ln2 */ 
            if(hx < 0x3FF0A2B2) {       /* and |x| < 1.5 ln2 */
                hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
            } else {
                k  = (int)(invln2*x+halF[xsb]);
                t  = k;
                hi = x - t*ln2HI[0];    /* t*ln2HI is exact here */
                lo = t*ln2LO[0];
            }
            x  = hi - lo;
        } 
        else if(hx < 0x3e300000)  {     /* when |x|<2**-28 */
            if(really_big+x>one) return one+x;/* trigger inexact */
        }
        else k = 0;

    /* x is now in primary range */
        t  = x*x;
        c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
        if(k==0)        return one-((x*c)/(c-2.0)-x); 
        else            y = one-((lo-(x*c)/(2.0-c))-hi);
        if(k >= -1021) {
            u.d = y;
            __HI(u) += (k<<20); /* add k to y's exponent */
            y = u.d;
            return y;
        } else {
            u.d = y;
            __HI(u) += ((k+1000)<<20);/* add k to y's exponent */
            y = u.d;
            return y*twom1000;
        }
}
1	siliconforks	2	/* -- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 --
2			*
3			* *** BEGIN LICENSE BLOCK ***
4			* Version: MPL 1.1/GPL 2.0/LGPL 2.1
5			*
6			* The contents of this file are subject to the Mozilla Public License Version
7			* 1.1 (the "License"); you may not use this file except in compliance with
8			* the License. You may obtain a copy of the License at
9			* http://www.mozilla.org/MPL/
10			*
11			* Software distributed under the License is distributed on an "AS IS" basis,
12			* WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
13			* for the specific language governing rights and limitations under the
14			* License.
15			*
16			* The Original Code is Mozilla Communicator client code, released
17			* March 31, 1998.
18			*
19			* The Initial Developer of the Original Code is
20			* Sun Microsystems, Inc.
21			* Portions created by the Initial Developer are Copyright (C) 1998
22			* the Initial Developer. All Rights Reserved.
23			*
24			* Contributor(s):
25			*
26			* Alternatively, the contents of this file may be used under the terms of
27			* either of the GNU General Public License Version 2 or later (the "GPL"),
28			* or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
29			* in which case the provisions of the GPL or the LGPL are applicable instead
30			* of those above. If you wish to allow use of your version of this file only
31			* under the terms of either the GPL or the LGPL, and not to allow others to
32			* use your version of this file under the terms of the MPL, indicate your
33			* decision by deleting the provisions above and replace them with the notice
34			* and other provisions required by the GPL or the LGPL. If you do not delete
35			* the provisions above, a recipient may use your version of this file under
36			* the terms of any one of the MPL, the GPL or the LGPL.
37			*
38			* *** END LICENSE BLOCK *** */
39
40			/* @(#)e_exp.c 1.3 95/01/18 */
41			/*
42			* ====================================================
43			* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
44			*
45			* Developed at SunSoft, a Sun Microsystems, Inc. business.
46			* Permission to use, copy, modify, and distribute this
47			* software is freely granted, provided that this notice
48			* is preserved.
49			* ====================================================
50			*/
51
52			/* __ieee754_exp(x)
53			* Returns the exponential of x.
54			*
55			* Method
56			* 1. Argument reduction:
57			* Reduce x to an r so that \|r\| <= 0.5*ln2 ~ 0.34658.
58			* Given x, find r and integer k such that
59			*
60			* x = kln2 + r, \|r\| <= 0.5ln2.
61			*
62			* Here r will be represented as r = hi-lo for better
63			* accuracy.
64			*
65			* 2. Approximation of exp(r) by a special rational function on
66			* the interval [0,0.34658]:
67			* Write
68			* R(r*2) = r(exp(r)+1)/(exp(r)-1) = 2 + rr/6 - r*4/360 + ...
69			* We use a special Reme algorithm on [0,0.34658] to generate
70			* a polynomial of degree 5 to approximate R. The maximum error
71			* of this polynomial approximation is bounded by 2**-59. In
72			* other words,
73			* R(z) ~ 2.0 + P1z + P2z*2 + P3z*3 + P4z*4 + P5z**5
74			* (where z=r*r, and the values of P1 to P5 are listed below)
75			* and
76			* \| 5 \| -59
77			* \| 2.0+P1z+...+P5z - R(z) \| <= 2
78			* \| \|
79			* The computation of exp(r) thus becomes
80			* 2*r
81			* exp(r) = 1 + -------
82			* R - r
83			* r*R1(r)
84			* = 1 + r + ----------- (for better accuracy)
85			* 2 - R1(r)
86			* where
87			* 2 4 10
88			* R1(r) = r - (P1r + P2r + ... + P5*r ).
89			*
90			* 3. Scale back to obtain exp(x):
91			* From step 1, we have
92			* exp(x) = 2^k * exp(r)
93			*
94			* Special cases:
95			* exp(INF) is INF, exp(NaN) is NaN;
96			* exp(-INF) is 0, and
97			* for finite argument, only exp(0)=1 is exact.
98			*
99			* Accuracy:
100			* according to an error analysis, the error is always less than
101			* 1 ulp (unit in the last place).
102			*
103			* Misc. info.
104			* For IEEE double
105			* if x > 7.09782712893383973096e+02 then exp(x) overflow
106			* if x < -7.45133219101941108420e+02 then exp(x) underflow
107			*
108			* Constants:
109			* The hexadecimal values are the intended ones for the following
110			* constants. The decimal values may be used, provided that the
111			* compiler will convert from decimal to binary accurately enough
112			* to produce the hexadecimal values shown.
113			*/
114
115			#include "fdlibm.h"
116
117			#ifdef __STDC__
118			static const double
119			#else
120			static double
121			#endif
122			one = 1.0,
123			halF[2] = {0.5,-0.5,},
124			really_big = 1.0e+300,
125			twom1000= 9.33263618503218878990e-302, /* 2*-1000=0x01700000,0/
126			o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
127			u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
128			ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
129			-6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
130			ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
131			-1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
132			invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
133			P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
134			P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
135			P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
136			P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
137			P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
138
139
140			#ifdef __STDC__
141			double __ieee754_exp(double x) /* default IEEE double exp */
142			#else
143			double __ieee754_exp(x) /* default IEEE double exp */
144			double x;
145			#endif
146			{
147			fd_twoints u;
148			double y,hi,lo,c,t;
149			int k, xsb;
150			unsigned hx;
151
152			u.d = x;
153			hx = __HI(u); /* high word of x */
154			xsb = (hx>>31)&1; /* sign bit of x */
155			hx &= 0x7fffffff; /* high word of \|x\| */
156
157			/* filter out non-finite argument */
158			if(hx >= 0x40862E42) { /* if \|x\|>=709.78... */
159			if(hx>=0x7ff00000) {
160			u.d = x;
161			if(((hx&0xfffff)\|__LO(u))!=0)
162			return x+x; /* NaN */
163			else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
164			}
165			if(x > o_threshold) return really_bigreally_big; / overflow */
166			if(x < u_threshold) return twom1000twom1000; / underflow */
167			}
168
169			/* argument reduction */
170			if(hx > 0x3fd62e42) { /* if \|x\| > 0.5 ln2 */
171			if(hx < 0x3FF0A2B2) { /* and \|x\| < 1.5 ln2 */
172			hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
173			} else {
174			k = (int)(invln2*x+halF[xsb]);
175			t = k;
176			hi = x - tln2HI[0]; / tln2HI is exact here /
177			lo = t*ln2LO[0];
178			}
179			x = hi - lo;
180			}
181			else if(hx < 0x3e300000) { /* when \|x\|<2*-28 /
182			if(really_big+x>one) return one+x;/* trigger inexact */
183			}
184			else k = 0;
185
186			/* x is now in primary range */
187			t = x*x;
188			c = x - t(P1+t(P2+t(P3+t(P4+t*P5))));
189			if(k==0) return one-((x*c)/(c-2.0)-x);
190			else y = one-((lo-(x*c)/(2.0-c))-hi);
191			if(k >= -1021) {
192			u.d = y;
193			__HI(u) += (k<<20); /* add k to y's exponent */
194			y = u.d;
195			return y;
196			} else {
197			u.d = y;
198			__HI(u) += ((k+1000)<<20);/* add k to y's exponent */
199			y = u.d;
200			return y*twom1000;
201			}
202			}