src/fdlibm/e_log.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_log.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_log(x)
 * Return the logrithm of x
 *
 * Method :                  
 *   1. Argument Reduction: find k and f such that 
 *                      x = 2^k * (1+f), 
 *         where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *   2. Approximation of log(1+f).
 *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *               = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate 
 *      a polynomial of degree 14 to approximate R The maximum error 
 *      of this polynomial approximation is bounded by 2**-58.45. In
 *      other words,
 *                      2      4      6      8      10      12      14
 *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
 *      (the values of Lg1 to Lg7 are listed in the program)
 *      and
 *          |      2          14          |     -58.45
 *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2 
 *          |                             |
 *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *      In order to guarantee error in log below 1ulp, we compute log
 *      by
 *              log(1+f) = f - s*(f - R)        (if f is not too large)
 *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
 *      
 *      3. Finally,  log(x) = k*ln2 + log(1+f).  
 *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *         Here ln2 is split into two floating point number: 
 *                      ln2_hi + ln2_lo,
 *         where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *      log(x) is NaN with signal if x < 0 (including -INF) ; 
 *      log(+INF) is +INF; log(0) is -INF with signal;
 *      log(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * 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
ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */

static double zero   =  0.0;

#ifdef __STDC__
        double __ieee754_log(double x)
#else
        double __ieee754_log(x)
        double x;
#endif
{
        fd_twoints u;
        double hfsq,f,s,z,R,w,t1,t2,dk;
        int k,hx,i,j;
        unsigned lx;

        u.d = x;
        hx = __HI(u);           /* high word of x */
        lx = __LO(u);           /* low  word of x */

        k=0;
        if (hx < 0x00100000) {                  /* x < 2**-1022  */
            if (((hx&0x7fffffff)|lx)==0) 
                return -two54/zero;             /* log(+-0)=-inf */
            if (hx<0) return (x-x)/zero;        /* log(-#) = NaN */
            k -= 54; x *= two54; /* subnormal number, scale up x */
            u.d = x;
            hx = __HI(u);               /* high word of x */
        } 
        if (hx >= 0x7ff00000) return x+x;
        k += (hx>>20)-1023;
        hx &= 0x000fffff;
        i = (hx+0x95f64)&0x100000;
        u.d = x;
        __HI(u) = hx|(i^0x3ff00000);    /* normalize x or x/2 */
        x = u.d;
        k += (i>>20);
        f = x-1.0;
        if((0x000fffff&(2+hx))<3) {     /* |f| < 2**-20 */
            if(f==zero) { 
                if(k==0) return zero; else {dk=(double)k;
                                            return dk*ln2_hi+dk*ln2_lo;} 
            }
            R = f*f*(0.5-0.33333333333333333*f);
            if(k==0) return f-R; else {dk=(double)k;
                     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
        }
        s = f/(2.0+f); 
        dk = (double)k;
        z = s*s;
        i = hx-0x6147a;
        w = z*z;
        j = 0x6b851-hx;
        t1= w*(Lg2+w*(Lg4+w*Lg6)); 
        t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7))); 
        i |= j;
        R = t2+t1;
        if(i>0) {
            hfsq=0.5*f*f;
            if(k==0) return f-(hfsq-s*(hfsq+R)); else
                     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
        } else {
            if(k==0) return f-s*(f-R); else
                     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
        }
}
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_log.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_log(x)
53	* Return the logrithm of x
54	*
55	* Method :
56	* 1. Argument Reduction: find k and f such that
57	* x = 2^k * (1+f),
58	* where sqrt(2)/2 < 1+f < sqrt(2) .
59	*
60	* 2. Approximation of log(1+f).
61	* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
62	* = 2s + 2/3 s3 + 2/5 s5 + .....,
63	* = 2s + s*R
64	* We use a special Reme algorithm on [0,0.1716] to generate
65	* a polynomial of degree 14 to approximate R The maximum error
66	* of this polynomial approximation is bounded by 2**-58.45. In
67	* other words,
68	* 2 4 6 8 10 12 14
69	* R(z) ~ Lg1s +Lg2s +Lg3s +Lg4s +Lg5s +Lg6s +Lg7*s
70	* (the values of Lg1 to Lg7 are listed in the program)
71	* and
72	* \| 2 14 \| -58.45
73	* \| Lg1s +...+Lg7s - R(z) \| <= 2
74	* \| \|
75	* Note that 2s = f - sf = f - hfsq + shfsq, where hfsq = f*f/2.
76	* In order to guarantee error in log below 1ulp, we compute log
77	* by
78	* log(1+f) = f - s*(f - R) (if f is not too large)
79	* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
80	*
81	* 3. Finally, log(x) = k*ln2 + log(1+f).
82	* = kln2_hi+(f-(hfsq-(s(hfsq+R)+k*ln2_lo)))
83	* Here ln2 is split into two floating point number:
84	* ln2_hi + ln2_lo,
85	* where n*ln2_hi is always exact for \|n\| < 2000.
86	*
87	* Special cases:
88	* log(x) is NaN with signal if x < 0 (including -INF) ;
89	* log(+INF) is +INF; log(0) is -INF with signal;
90	* log(NaN) is that NaN with no signal.
91	*
92	* Accuracy:
93	* according to an error analysis, the error is always less than
94	* 1 ulp (unit in the last place).
95	*
96	* Constants:
97	* The hexadecimal values are the intended ones for the following
98	* constants. The decimal values may be used, provided that the
99	* compiler will convert from decimal to binary accurately enough
100	* to produce the hexadecimal values shown.
101	*/
102
103	#include "fdlibm.h"
104
105	#ifdef __STDC__
106	static const double
107	#else
108	static double
109	#endif
110	ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
111	ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
112	two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
113	Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
114	Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
115	Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
116	Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
117	Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
118	Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
119	Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
120
121	static double zero = 0.0;
122
123	#ifdef __STDC__
124	double __ieee754_log(double x)
125	#else
126	double __ieee754_log(x)
127	double x;
128	#endif
129	{
130	fd_twoints u;
131	double hfsq,f,s,z,R,w,t1,t2,dk;
132	int k,hx,i,j;
133	unsigned lx;
134
135	u.d = x;
136	hx = __HI(u); /* high word of x */
137	lx = __LO(u); /* low word of x */
138
139	k=0;
140	if (hx < 0x00100000) { /* x < 2*-1022 /
141	if (((hx&0x7fffffff)\|lx)==0)
142	return -two54/zero; /* log(+-0)=-inf */
143	if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
144	k -= 54; x = two54; / subnormal number, scale up x */
145	u.d = x;
146	hx = __HI(u); /* high word of x */
147	}
148	if (hx >= 0x7ff00000) return x+x;
149	k += (hx>>20)-1023;
150	hx &= 0x000fffff;
151	i = (hx+0x95f64)&0x100000;
152	u.d = x;
153	__HI(u) = hx\|(i^0x3ff00000); /* normalize x or x/2 */
154	x = u.d;
155	k += (i>>20);
156	f = x-1.0;
157	if((0x000fffff&(2+hx))<3) { /* \|f\| < 2*-20 /
158	if(f==zero) {
159	if(k==0) return zero; else {dk=(double)k;
160	return dkln2_hi+dkln2_lo;}
161	}
162	R = ff(0.5-0.33333333333333333*f);
163	if(k==0) return f-R; else {dk=(double)k;
164	return dkln2_hi-((R-dkln2_lo)-f);}
165	}
166	s = f/(2.0+f);
167	dk = (double)k;
168	z = s*s;
169	i = hx-0x6147a;
170	w = z*z;
171	j = 0x6b851-hx;
172	t1= w(Lg2+w(Lg4+w*Lg6));
173	t2= z(Lg1+w(Lg3+w(Lg5+wLg7)));
174	i \|= j;
175	R = t2+t1;
176	if(i>0) {
177	hfsq=0.5ff;
178	if(k==0) return f-(hfsq-s*(hfsq+R)); else
179	return dkln2_hi-((hfsq-(s(hfsq+R)+dk*ln2_lo))-f);
180	} else {
181	if(k==0) return f-s*(f-R); else
182	return dkln2_hi-((s(f-R)-dk*ln2_lo)-f);
183	}
184	}