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

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Wed Aug 1 13:51:53 2007 UTC (14 years, 9 months ago) by siliconforks
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Initial import.

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 /* @(#)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 }

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