/[jscoverage]/trunk/js/src/fdlibm/s_expm1.c
ViewVC logotype

Annotation of /trunk/js/src/fdlibm/s_expm1.c

Parent Directory Parent Directory | Revision Log Revision Log


Revision 2 - (hide annotations)
Wed Aug 1 13:51:53 2007 UTC (14 years, 11 months ago) by siliconforks
File MIME type: text/plain
File size: 9470 byte(s)
Initial import.

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     /* @(#)s_expm1.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     /* expm1(x)
53     * Returns exp(x)-1, the exponential of x minus 1.
54     *
55     * Method
56     * 1. Argument reduction:
57     * Given x, find r and integer k such that
58     *
59     * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
60     *
61     * Here a correction term c will be computed to compensate
62     * the error in r when rounded to a floating-point number.
63     *
64     * 2. Approximating expm1(r) by a special rational function on
65     * the interval [0,0.34658]:
66     * Since
67     * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
68     * we define R1(r*r) by
69     * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
70     * That is,
71     * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
72     * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
73     * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
74     * We use a special Reme algorithm on [0,0.347] to generate
75     * a polynomial of degree 5 in r*r to approximate R1. The
76     * maximum error of this polynomial approximation is bounded
77     * by 2**-61. In other words,
78     * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
79     * where Q1 = -1.6666666666666567384E-2,
80     * Q2 = 3.9682539681370365873E-4,
81     * Q3 = -9.9206344733435987357E-6,
82     * Q4 = 2.5051361420808517002E-7,
83     * Q5 = -6.2843505682382617102E-9;
84     * (where z=r*r, and the values of Q1 to Q5 are listed below)
85     * with error bounded by
86     * | 5 | -61
87     * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
88     * | |
89     *
90     * expm1(r) = exp(r)-1 is then computed by the following
91     * specific way which minimize the accumulation rounding error:
92     * 2 3
93     * r r [ 3 - (R1 + R1*r/2) ]
94     * expm1(r) = r + --- + --- * [--------------------]
95     * 2 2 [ 6 - r*(3 - R1*r/2) ]
96     *
97     * To compensate the error in the argument reduction, we use
98     * expm1(r+c) = expm1(r) + c + expm1(r)*c
99     * ~ expm1(r) + c + r*c
100     * Thus c+r*c will be added in as the correction terms for
101     * expm1(r+c). Now rearrange the term to avoid optimization
102     * screw up:
103     * ( 2 2 )
104     * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
105     * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
106     * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
107     * ( )
108     *
109     * = r - E
110     * 3. Scale back to obtain expm1(x):
111     * From step 1, we have
112     * expm1(x) = either 2^k*[expm1(r)+1] - 1
113     * = or 2^k*[expm1(r) + (1-2^-k)]
114     * 4. Implementation notes:
115     * (A). To save one multiplication, we scale the coefficient Qi
116     * to Qi*2^i, and replace z by (x^2)/2.
117     * (B). To achieve maximum accuracy, we compute expm1(x) by
118     * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
119     * (ii) if k=0, return r-E
120     * (iii) if k=-1, return 0.5*(r-E)-0.5
121     * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
122     * else return 1.0+2.0*(r-E);
123     * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
124     * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
125     * (vii) return 2^k(1-((E+2^-k)-r))
126     *
127     * Special cases:
128     * expm1(INF) is INF, expm1(NaN) is NaN;
129     * expm1(-INF) is -1, and
130     * for finite argument, only expm1(0)=0 is exact.
131     *
132     * Accuracy:
133     * according to an error analysis, the error is always less than
134     * 1 ulp (unit in the last place).
135     *
136     * Misc. info.
137     * For IEEE double
138     * if x > 7.09782712893383973096e+02 then expm1(x) overflow
139     *
140     * Constants:
141     * The hexadecimal values are the intended ones for the following
142     * constants. The decimal values may be used, provided that the
143     * compiler will convert from decimal to binary accurately enough
144     * to produce the hexadecimal values shown.
145     */
146    
147     #include "fdlibm.h"
148    
149     #ifdef __STDC__
150     static const double
151     #else
152     static double
153     #endif
154     one = 1.0,
155     really_big = 1.0e+300,
156     tiny = 1.0e-300,
157     o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
158     ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
159     ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
160     invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
161     /* scaled coefficients related to expm1 */
162     Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
163     Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
164     Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
165     Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
166     Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
167    
168     #ifdef __STDC__
169     double fd_expm1(double x)
170     #else
171     double fd_expm1(x)
172     double x;
173     #endif
174     {
175     fd_twoints u;
176     double y,hi,lo,c,t,e,hxs,hfx,r1;
177     int k,xsb;
178     unsigned hx;
179    
180     u.d = x;
181     hx = __HI(u); /* high word of x */
182     xsb = hx&0x80000000; /* sign bit of x */
183     if(xsb==0) y=x; else y= -x; /* y = |x| */
184     hx &= 0x7fffffff; /* high word of |x| */
185    
186     /* filter out huge and non-finite argument */
187     if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
188     if(hx >= 0x40862E42) { /* if |x|>=709.78... */
189     if(hx>=0x7ff00000) {
190     u.d = x;
191     if(((hx&0xfffff)|__LO(u))!=0)
192     return x+x; /* NaN */
193     else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
194     }
195     if(x > o_threshold) return really_big*really_big; /* overflow */
196     }
197     if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
198     if(x+tiny<0.0) /* raise inexact */
199     return tiny-one; /* return -1 */
200     }
201     }
202    
203     /* argument reduction */
204     if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
205     if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
206     if(xsb==0)
207     {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
208     else
209     {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
210     } else {
211     k = (int)(invln2*x+((xsb==0)?0.5:-0.5));
212     t = k;
213     hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
214     lo = t*ln2_lo;
215     }
216     x = hi - lo;
217     c = (hi-x)-lo;
218     }
219     else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
220     t = really_big+x; /* return x with inexact flags when x!=0 */
221     return x - (t-(really_big+x));
222     }
223     else k = 0;
224    
225     /* x is now in primary range */
226     hfx = 0.5*x;
227     hxs = x*hfx;
228     r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
229     t = 3.0-r1*hfx;
230     e = hxs*((r1-t)/(6.0 - x*t));
231     if(k==0) return x - (x*e-hxs); /* c is 0 */
232     else {
233     e = (x*(e-c)-c);
234     e -= hxs;
235     if(k== -1) return 0.5*(x-e)-0.5;
236     if(k==1)
237     if(x < -0.25) return -2.0*(e-(x+0.5));
238     else return one+2.0*(x-e);
239     if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
240     y = one-(e-x);
241     u.d = y;
242     __HI(u) += (k<<20); /* add k to y's exponent */
243     y = u.d;
244     return y-one;
245     }
246     t = one;
247     if(k<20) {
248     u.d = t;
249     __HI(u) = 0x3ff00000 - (0x200000>>k); /* t=1-2^-k */
250     t = u.d;
251     y = t-(e-x);
252     u.d = y;
253     __HI(u) += (k<<20); /* add k to y's exponent */
254     y = u.d;
255     } else {
256     u.d = t;
257     __HI(u) = ((0x3ff-k)<<20); /* 2^-k */
258     t = u.d;
259     y = x-(e+t);
260     y += one;
261     u.d = y;
262     __HI(u) += (k<<20); /* add k to y's exponent */
263     y = u.d;
264     }
265     }
266     return y;
267     }

  ViewVC Help
Powered by ViewVC 1.1.24