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4acf9396 GCI |
1 | /* @(#)s_expm1.c 5.1 93/09/24 */ |
2 | /* | |
3 | * ==================================================== | |
4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. | |
5 | * | |
6 | * Developed at SunPro, a Sun Microsystems, Inc. business. | |
7 | * Permission to use, copy, modify, and distribute this | |
8 | * software is freely granted, provided that this notice | |
9 | * is preserved. | |
10 | * ==================================================== | |
11 | */ | |
12 | ||
13 | #ifndef lint | |
14 | static char rcsid[] = "$Id: s_expm1.c,v 1.4 1994/03/03 17:04:33 jtc Exp $"; | |
15 | #endif | |
16 | ||
17 | /* expm1(x) | |
18 | * Returns exp(x)-1, the exponential of x minus 1. | |
19 | * | |
20 | * Method | |
21 | * 1. Argument reduction: | |
22 | * Given x, find r and integer k such that | |
23 | * | |
24 | * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 | |
25 | * | |
26 | * Here a correction term c will be computed to compensate | |
27 | * the error in r when rounded to a floating-point number. | |
28 | * | |
29 | * 2. Approximating expm1(r) by a special rational function on | |
30 | * the interval [0,0.34658]: | |
31 | * Since | |
32 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... | |
33 | * we define R1(r*r) by | |
34 | * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) | |
35 | * That is, | |
36 | * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) | |
37 | * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) | |
38 | * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... | |
39 | * We use a special Reme algorithm on [0,0.347] to generate | |
40 | * a polynomial of degree 5 in r*r to approximate R1. The | |
41 | * maximum error of this polynomial approximation is bounded | |
42 | * by 2**-61. In other words, | |
43 | * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 | |
44 | * where Q1 = -1.6666666666666567384E-2, | |
45 | * Q2 = 3.9682539681370365873E-4, | |
46 | * Q3 = -9.9206344733435987357E-6, | |
47 | * Q4 = 2.5051361420808517002E-7, | |
48 | * Q5 = -6.2843505682382617102E-9; | |
49 | * (where z=r*r, and the values of Q1 to Q5 are listed below) | |
50 | * with error bounded by | |
51 | * | 5 | -61 | |
52 | * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 | |
53 | * | | | |
54 | * | |
55 | * expm1(r) = exp(r)-1 is then computed by the following | |
56 | * specific way which minimize the accumulation rounding error: | |
57 | * 2 3 | |
58 | * r r [ 3 - (R1 + R1*r/2) ] | |
59 | * expm1(r) = r + --- + --- * [--------------------] | |
60 | * 2 2 [ 6 - r*(3 - R1*r/2) ] | |
61 | * | |
62 | * To compensate the error in the argument reduction, we use | |
63 | * expm1(r+c) = expm1(r) + c + expm1(r)*c | |
64 | * ~ expm1(r) + c + r*c | |
65 | * Thus c+r*c will be added in as the correction terms for | |
66 | * expm1(r+c). Now rearrange the term to avoid optimization | |
67 | * screw up: | |
68 | * ( 2 2 ) | |
69 | * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) | |
70 | * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) | |
71 | * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) | |
72 | * ( ) | |
73 | * | |
74 | * = r - E | |
75 | * 3. Scale back to obtain expm1(x): | |
76 | * From step 1, we have | |
77 | * expm1(x) = either 2^k*[expm1(r)+1] - 1 | |
78 | * = or 2^k*[expm1(r) + (1-2^-k)] | |
79 | * 4. Implementation notes: | |
80 | * (A). To save one multiplication, we scale the coefficient Qi | |
81 | * to Qi*2^i, and replace z by (x^2)/2. | |
82 | * (B). To achieve maximum accuracy, we compute expm1(x) by | |
83 | * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) | |
84 | * (ii) if k=0, return r-E | |
85 | * (iii) if k=-1, return 0.5*(r-E)-0.5 | |
86 | * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) | |
87 | * else return 1.0+2.0*(r-E); | |
88 | * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) | |
89 | * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else | |
90 | * (vii) return 2^k(1-((E+2^-k)-r)) | |
91 | * | |
92 | * Special cases: | |
93 | * expm1(INF) is INF, expm1(NaN) is NaN; | |
94 | * expm1(-INF) is -1, and | |
95 | * for finite argument, only expm1(0)=0 is exact. | |
96 | * | |
97 | * Accuracy: | |
98 | * according to an error analysis, the error is always less than | |
99 | * 1 ulp (unit in the last place). | |
100 | * | |
101 | * Misc. info. | |
102 | * For IEEE double | |
103 | * if x > 7.09782712893383973096e+02 then expm1(x) overflow | |
104 | * | |
105 | * Constants: | |
106 | * The hexadecimal values are the intended ones for the following | |
107 | * constants. The decimal values may be used, provided that the | |
108 | * compiler will convert from decimal to binary accurately enough | |
109 | * to produce the hexadecimal values shown. | |
110 | */ | |
111 | ||
112 | #include "math.h" | |
113 | #include <machine/endian.h> | |
114 | ||
115 | #if BYTE_ORDER == LITTLE_ENDIAN | |
116 | #define n0 1 | |
117 | #else | |
118 | #define n0 0 | |
119 | #endif | |
120 | ||
121 | #ifdef __STDC__ | |
122 | static const double | |
123 | #else | |
124 | static double | |
125 | #endif | |
126 | one = 1.0, | |
127 | huge = 1.0e+300, | |
128 | tiny = 1.0e-300, | |
129 | o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ | |
130 | ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ | |
131 | ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ | |
132 | invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ | |
133 | /* scaled coefficients related to expm1 */ | |
134 | Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ | |
135 | Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ | |
136 | Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ | |
137 | Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ | |
138 | Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ | |
139 | ||
140 | #ifdef __STDC__ | |
141 | double expm1(double x) | |
142 | #else | |
143 | double expm1(x) | |
144 | double x; | |
145 | #endif | |
146 | { | |
147 | double y,hi,lo,c,t,e,hxs,hfx,r1; | |
148 | int k,xsb; | |
149 | unsigned hx; | |
150 | ||
151 | hx = *(n0+(unsigned*)&x); /* high word of x */ | |
152 | xsb = hx&0x80000000; /* sign bit of x */ | |
153 | if(xsb==0) y=x; else y= -x; /* y = |x| */ | |
154 | hx &= 0x7fffffff; /* high word of |x| */ | |
155 | ||
156 | /* filter out huge and non-finite argument */ | |
157 | if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ | |
158 | if(hx >= 0x40862E42) { /* if |x|>=709.78... */ | |
159 | if(hx>=0x7ff00000) { | |
160 | if(((hx&0xfffff)|*(1-n0+(int*)&x))!=0) | |
161 | return x+x; /* NaN */ | |
162 | else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ | |
163 | } | |
164 | if(x > o_threshold) return huge*huge; /* overflow */ | |
165 | } | |
166 | if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ | |
167 | if(x+tiny<0.0) /* raise inexact */ | |
168 | return tiny-one; /* return -1 */ | |
169 | } | |
170 | } | |
171 | ||
172 | /* argument reduction */ | |
173 | if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ | |
174 | if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ | |
175 | if(xsb==0) | |
176 | {hi = x - ln2_hi; lo = ln2_lo; k = 1;} | |
177 | else | |
178 | {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} | |
179 | } else { | |
180 | k = invln2*x+((xsb==0)?0.5:-0.5); | |
181 | t = k; | |
182 | hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ | |
183 | lo = t*ln2_lo; | |
184 | } | |
185 | x = hi - lo; | |
186 | c = (hi-x)-lo; | |
187 | } | |
188 | else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ | |
189 | t = huge+x; /* return x with inexact flags when x!=0 */ | |
190 | return x - (t-(huge+x)); | |
191 | } | |
192 | else k = 0; | |
193 | ||
194 | /* x is now in primary range */ | |
195 | hfx = 0.5*x; | |
196 | hxs = x*hfx; | |
197 | r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); | |
198 | t = 3.0-r1*hfx; | |
199 | e = hxs*((r1-t)/(6.0 - x*t)); | |
200 | if(k==0) return x - (x*e-hxs); /* c is 0 */ | |
201 | else { | |
202 | e = (x*(e-c)-c); | |
203 | e -= hxs; | |
204 | if(k== -1) return 0.5*(x-e)-0.5; | |
205 | if(k==1) | |
206 | if(x < -0.25) return -2.0*(e-(x+0.5)); | |
207 | else return one+2.0*(x-e); | |
208 | if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ | |
209 | y = one-(e-x); | |
210 | *(n0+(int*)&y) += (k<<20); /* add k to y's exponent */ | |
211 | return y-one; | |
212 | } | |
213 | t = one; | |
214 | if(k<20) { | |
215 | *(n0+(int*)&t) = 0x3ff00000 - (0x200000>>k); /* t=1-2^-k */ | |
216 | y = t-(e-x); | |
217 | *(n0+(int*)&y) += (k<<20); /* add k to y's exponent */ | |
218 | } else { | |
219 | *(n0+(int*)&t) = ((0x3ff-k)<<20); /* 2^-k */ | |
220 | y = x-(e+t); | |
221 | y += one; | |
222 | *(n0+(int*)&y) += (k<<20); /* add k to y's exponent */ | |
223 | } | |
224 | } | |
225 | return y; | |
226 | } |