1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25 /*
26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 * Use is subject to license terms.
28 */
29
30 #pragma weak __expm1 = expm1
31
32 /* INDENT OFF */
33 /*
34 * expm1(x)
35 * Returns exp(x)-1, the exponential of x minus 1.
36 *
37 * Method
38 * 1. Arugment reduction:
39 * Given x, find r and integer k such that
40 *
41 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
42 *
43 * Here a correction term c will be computed to compensate
44 * the error in r when rounded to a floating-point number.
45 *
46 * 2. Approximating expm1(r) by a special rational function on
47 * the interval [0,0.34658]:
48 * Since
49 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
50 * we define R1(r*r) by
51 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
52 * That is,
53 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
54 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
55 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
56 * We use a special Reme algorithm on [0,0.347] to generate
57 * a polynomial of degree 5 in r*r to approximate R1. The
58 * maximum error of this polynomial approximation is bounded
59 * by 2**-61. In other words,
60 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
61 * where Q1 = -1.6666666666666567384E-2,
62 * Q2 = 3.9682539681370365873E-4,
63 * Q3 = -9.9206344733435987357E-6,
64 * Q4 = 2.5051361420808517002E-7,
65 * Q5 = -6.2843505682382617102E-9;
66 * (where z=r*r, and the values of Q1 to Q5 are listed below)
67 * with error bounded by
68 * | 5 | -61
69 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
70 * | |
71 *
72 * expm1(r) = exp(r)-1 is then computed by the following
73 * specific way which minimize the accumulation rounding error:
74 * 2 3
75 * r r [ 3 - (R1 + R1*r/2) ]
76 * expm1(r) = r + --- + --- * [--------------------]
77 * 2 2 [ 6 - r*(3 - R1*r/2) ]
78 *
79 * To compensate the error in the argument reduction, we use
80 * expm1(r+c) = expm1(r) + c + expm1(r)*c
81 * ~ expm1(r) + c + r*c
82 * Thus c+r*c will be added in as the correction terms for
83 * expm1(r+c). Now rearrange the term to avoid optimization
84 * screw up:
85 * ( 2 2 )
86 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
87 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
88 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
89 * ( )
90 *
91 * = r - E
92 * 3. Scale back to obtain expm1(x):
93 * From step 1, we have
94 * expm1(x) = either 2^k*[expm1(r)+1] - 1
95 * = or 2^k*[expm1(r) + (1-2^-k)]
96 * 4. Implementation notes:
97 * (A). To save one multiplication, we scale the coefficient Qi
98 * to Qi*2^i, and replace z by (x^2)/2.
99 * (B). To achieve maximum accuracy, we compute expm1(x) by
100 * (i) if x < -56*ln2, return -1.0, (raise inexact if x != inf)
101 * (ii) if k=0, return r-E
102 * (iii) if k=-1, return 0.5*(r-E)-0.5
103 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
104 * else return 1.0+2.0*(r-E);
105 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
106 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
107 * (vii) return 2^k(1-((E+2^-k)-r))
108 *
109 * Special cases:
110 * expm1(INF) is INF, expm1(NaN) is NaN;
111 * expm1(-INF) is -1, and
112 * for finite argument, only expm1(0)=0 is exact.
113 *
114 * Accuracy:
115 * according to an error analysis, the error is always less than
116 * 1 ulp (unit in the last place).
117 *
118 * Misc. info.
119 * For IEEE double
120 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
121 *
122 * Constants:
123 * The hexadecimal values are the intended ones for the following
124 * constants. The decimal values may be used, provided that the
125 * compiler will convert from decimal to binary accurately enough
126 * to produce the hexadecimal values shown.
127 */
128 /* INDENT ON */
129
130 #include "libm_macros.h"
131 #include <math.h>
132
133 static const double xxx[] = {
134 /* one */ 1.0,
135 /* huge */ 1.0e+300,
136 /* tiny */ 1.0e-300,
137 /* o_threshold */ 7.09782712893383973096e+02, /* 40862E42 FEFA39EF */
138 /* ln2_hi */ 6.93147180369123816490e-01, /* 3FE62E42 FEE00000 */
139 /* ln2_lo */ 1.90821492927058770002e-10, /* 3DEA39EF 35793C76 */
140 /* invln2 */ 1.44269504088896338700e+00, /* 3FF71547 652B82FE */
141 /* scaled coefficients related to expm1 */
142 /* Q1 */ -3.33333333333331316428e-02, /* BFA11111 111110F4 */
143 /* Q2 */ 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
144 /* Q3 */ -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
145 /* Q4 */ 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
146 /* Q5 */ -2.01099218183624371326e-07 /* BE8AFDB7 6E09C32D */
147 };
148 #define one xxx[0]
149 #define huge xxx[1]
150 #define tiny xxx[2]
151 #define o_threshold xxx[3]
152 #define ln2_hi xxx[4]
153 #define ln2_lo xxx[5]
154 #define invln2 xxx[6]
155 #define Q1 xxx[7]
156 #define Q2 xxx[8]
157 #define Q3 xxx[9]
158 #define Q4 xxx[10]
159 #define Q5 xxx[11]
160
161 double
expm1(double x)162 expm1(double x) {
163 double y, hi, lo, c = 0.0L, t, e, hxs, hfx, r1;
164 int k, xsb;
165 unsigned hx;
166
167 hx = ((unsigned *) &x)[HIWORD]; /* high word of x */
168 xsb = hx & 0x80000000; /* sign bit of x */
169 if (xsb == 0)
170 y = x;
171 else
172 y = -x; /* y = |x| */
173 hx &= 0x7fffffff; /* high word of |x| */
174
175 /* filter out huge and non-finite argument */
176 /* for example exp(38)-1 is approximately 3.1855932e+16 */
177 if (hx >= 0x4043687A) {
178 /* if |x|>=56*ln2 (~38.8162...) */
179 if (hx >= 0x40862E42) { /* if |x|>=709.78... -> inf */
180 if (hx >= 0x7ff00000) {
181 if (((hx & 0xfffff) | ((int *) &x)[LOWORD])
182 != 0)
183 return (x * x); /* + -> * for Cheetah */
184 else
185 /* exp(+-inf)={inf,-1} */
186 return (xsb == 0 ? x : -1.0);
187 }
188 if (x > o_threshold)
189 return (huge * huge); /* overflow */
190 }
191 if (xsb != 0) { /* x < -56*ln2, return -1.0 w/inexact */
192 if (x + tiny < 0.0) /* raise inexact */
193 return (tiny - one); /* return -1 */
194 }
195 }
196
197 /* argument reduction */
198 if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
199 if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
200 if (xsb == 0) { /* positive number */
201 hi = x - ln2_hi;
202 lo = ln2_lo;
203 k = 1;
204 } else {
205 /* negative number */
206 hi = x + ln2_hi;
207 lo = -ln2_lo;
208 k = -1;
209 }
210 } else {
211 /* |x| > 1.5 ln2 */
212 k = (int) (invln2 * x + (xsb == 0 ? 0.5 : -0.5));
213 t = k;
214 hi = x - t * ln2_hi; /* t*ln2_hi is exact here */
215 lo = t * ln2_lo;
216 }
217 x = hi - lo;
218 c = (hi - x) - lo; /* still at |x| > 0.5 ln2 */
219 } else if (hx < 0x3c900000) {
220 /* when |x|<2**-54, return x */
221 t = huge + x; /* return x w/inexact when x != 0 */
222 return (x - (t - (huge + x)));
223 } else
224 /* |x| <= 0.5 ln2 */
225 k = 0;
226
227 /* x is now in primary range */
228 hfx = 0.5 * x;
229 hxs = x * hfx;
230 r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
231 t = 3.0 - r1 * hfx;
232 e = hxs * ((r1 - t) / (6.0 - x * t));
233 if (k == 0) /* |x| <= 0.5 ln2 */
234 return (x - (x * e - hxs));
235 else { /* |x| > 0.5 ln2 */
236 e = (x * (e - c) - c);
237 e -= hxs;
238 if (k == -1)
239 return (0.5 * (x - e) - 0.5);
240 if (k == 1) {
241 if (x < -0.25)
242 return (-2.0 * (e - (x + 0.5)));
243 else
244 return (one + 2.0 * (x - e));
245 }
246 if (k <= -2 || k > 56) { /* suffice to return exp(x)-1 */
247 y = one - (e - x);
248 ((int *) &y)[HIWORD] += k << 20;
249 return (y - one);
250 }
251 t = one;
252 if (k < 20) {
253 ((int *) &t)[HIWORD] = 0x3ff00000 - (0x200000 >> k);
254 /* t = 1 - 2^-k */
255 y = t - (e - x);
256 ((int *) &y)[HIWORD] += k << 20;
257 } else {
258 ((int *) &t)[HIWORD] = (0x3ff - k) << 20; /* 2^-k */
259 y = x - (e + t);
260 y += one;
261 ((int *) &y)[HIWORD] += k << 20;
262 }
263 }
264 return (y);
265 }
266