1/*
2 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved.
3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
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5 * This code is free software; you can redistribute it and/or modify it
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7 * published by the Free Software Foundation. Oracle designates this
8 * particular file as subject to the "Classpath" exception as provided
9 * by Oracle in the LICENSE file that accompanied this code.
10 *
11 * This code is distributed in the hope that it will be useful, but WITHOUT
12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
14 * version 2 for more details (a copy is included in the LICENSE file that
15 * accompanied this code).
16 *
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19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
20 *
21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
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24 */
25
26/* expm1(x)
27 * Returns exp(x)-1, the exponential of x minus 1.
28 *
29 * Method
30 * 1. Argument reduction:
31 * Given x, find r and integer k such that
32 *
33 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
34 *
35 * Here a correction term c will be computed to compensate
36 * the error in r when rounded to a floating-point number.
37 *
38 * 2. Approximating expm1(r) by a special rational function on
39 * the interval [0,0.34658]:
40 * Since
41 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
42 * we define R1(r*r) by
43 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
44 * That is,
45 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
46 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
47 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
48 * We use a special Reme algorithm on [0,0.347] to generate
49 * a polynomial of degree 5 in r*r to approximate R1. The
50 * maximum error of this polynomial approximation is bounded
51 * by 2**-61. In other words,
52 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
53 * where Q1 = -1.6666666666666567384E-2,
54 * Q2 = 3.9682539681370365873E-4,
55 * Q3 = -9.9206344733435987357E-6,
56 * Q4 = 2.5051361420808517002E-7,
57 * Q5 = -6.2843505682382617102E-9;
58 * (where z=r*r, and the values of Q1 to Q5 are listed below)
59 * with error bounded by
60 * | 5 | -61
61 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
62 * | |
63 *
64 * expm1(r) = exp(r)-1 is then computed by the following
65 * specific way which minimize the accumulation rounding error:
66 * 2 3
67 * r r [ 3 - (R1 + R1*r/2) ]
68 * expm1(r) = r + --- + --- * [--------------------]
69 * 2 2 [ 6 - r*(3 - R1*r/2) ]
70 *
71 * To compensate the error in the argument reduction, we use
72 * expm1(r+c) = expm1(r) + c + expm1(r)*c
73 * ~ expm1(r) + c + r*c
74 * Thus c+r*c will be added in as the correction terms for
75 * expm1(r+c). Now rearrange the term to avoid optimization
76 * screw up:
77 * ( 2 2 )
78 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
79 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
80 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
81 * ( )
82 *
83 * = r - E
84 * 3. Scale back to obtain expm1(x):
85 * From step 1, we have
86 * expm1(x) = either 2^k*[expm1(r)+1] - 1
87 * = or 2^k*[expm1(r) + (1-2^-k)]
88 * 4. Implementation notes:
89 * (A). To save one multiplication, we scale the coefficient Qi
90 * to Qi*2^i, and replace z by (x^2)/2.
91 * (B). To achieve maximum accuracy, we compute expm1(x) by
92 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
93 * (ii) if k=0, return r-E
94 * (iii) if k=-1, return 0.5*(r-E)-0.5
95 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
96 * else return 1.0+2.0*(r-E);
97 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
98 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
99 * (vii) return 2^k(1-((E+2^-k)-r))
100 *
101 * Special cases:
102 * expm1(INF) is INF, expm1(NaN) is NaN;
103 * expm1(-INF) is -1, and
104 * for finite argument, only expm1(0)=0 is exact.
105 *
106 * Accuracy:
107 * according to an error analysis, the error is always less than
108 * 1 ulp (unit in the last place).
109 *
110 * Misc. info.
111 * For IEEE double
112 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
113 *
114 * Constants:
115 * The hexadecimal values are the intended ones for the following
116 * constants. The decimal values may be used, provided that the
117 * compiler will convert from decimal to binary accurately enough
118 * to produce the hexadecimal values shown.
119 */
120
121#include "fdlibm.h"
122
123#ifdef __STDC__
124static const double
125#else
126static double
127#endif
128one = 1.0,
129huge = 1.0e+300,
130tiny = 1.0e-300,
131o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
132ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
133ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
134invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
135 /* scaled coefficients related to expm1 */
136Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
137Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
138Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
139Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
140Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
141
142#ifdef __STDC__
143 double expm1(double x)
144#else
145 double expm1(x)
146 double x;
147#endif
148{
149 double y,hi,lo,c=0,t,e,hxs,hfx,r1;
150 int k,xsb;
151 unsigned hx;
152
153 hx = __HI(x); /* high word of x */
154 xsb = hx&0x80000000; /* sign bit of x */
155 if(xsb==0) y=x; else y= -x; /* y = |x| */
156 hx &= 0x7fffffff; /* high word of |x| */
157
158 /* filter out huge and non-finite argument */
159 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
160 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
161 if(hx>=0x7ff00000) {
162 if(((hx&0xfffff)|__LO(x))!=0)
163 return x+x; /* NaN */
164 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
165 }
166 if(x > o_threshold) return huge*huge; /* overflow */
167 }
168 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
169 if(x+tiny<0.0) /* raise inexact */
170 return tiny-one; /* return -1 */
171 }
172 }
173
174 /* argument reduction */
175 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
176 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
177 if(xsb==0)
178 {hi = x - ln2_hi; lo = ln2_lo; k = 1;}
179 else
180 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
181 } else {
182 k = invln2*x+((xsb==0)?0.5:-0.5);
183 t = k;
184 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
185 lo = t*ln2_lo;
186 }
187 x = hi - lo;
188 c = (hi-x)-lo;
189 }
190 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
191 t = huge+x; /* return x with inexact flags when x!=0 */
192 return x - (t-(huge+x));
193 }
194 else k = 0;
195
196 /* x is now in primary range */
197 hfx = 0.5*x;
198 hxs = x*hfx;
199 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
200 t = 3.0-r1*hfx;
201 e = hxs*((r1-t)/(6.0 - x*t));
202 if(k==0) return x - (x*e-hxs); /* c is 0 */
203 else {
204 e = (x*(e-c)-c);
205 e -= hxs;
206 if(k== -1) return 0.5*(x-e)-0.5;
207 if(k==1) {
208 if(x < -0.25) return -2.0*(e-(x+0.5));
209 else return one+2.0*(x-e);
210 }
211 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
212 y = one-(e-x);
213 __HI(y) += (k<<20); /* add k to y's exponent */
214 return y-one;
215 }
216 t = one;
217 if(k<20) {
218 __HI(t) = 0x3ff00000 - (0x200000>>k); /* t=1-2^-k */
219 y = t-(e-x);
220 __HI(y) += (k<<20); /* add k to y's exponent */
221 } else {
222 __HI(t) = ((0x3ff-k)<<20); /* 2^-k */
223 y = x-(e+t);
224 y += one;
225 __HI(y) += (k<<20); /* add k to y's exponent */
226 }
227 }
228 return y;
229}
230