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	/* -- 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	}