1 /* Copyright JS Foundation and other contributors, http://js.foundation
2 *
3 * Licensed under the Apache License, Version 2.0 (the "License");
4 * you may not use this file except in compliance with the License.
5 * You may obtain a copy of the License at
6 *
7 * http://www.apache.org/licenses/LICENSE-2.0
8 *
9 * Unless required by applicable law or agreed to in writing, software
10 * distributed under the License is distributed on an "AS IS" BASIS
11 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12 * See the License for the specific language governing permissions and
13 * limitations under the License.
14 *
15 * This file is based on work under the following copyright and permission
16 * notice:
17 *
18 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
19 *
20 * Permission to use, copy, modify, and distribute this
21 * software is freely granted, provided that this notice
22 * is preserved.
23 *
24 * @(#)s_expm1.c 5.1 93/09/24
25 */
26
27 #include "jerry-libm-internal.h"
28
29 /* expm1(x)
30 * Returns exp(x)-1, the exponential of x minus 1.
31 *
32 * Method
33 * 1. Argument reduction:
34 * Given x, find r and integer k such that
35 *
36 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
37 *
38 * Here a correction term c will be computed to compensate
39 * the error in r when rounded to a floating-point number.
40 *
41 * 2. Approximating expm1(r) by a special rational function on
42 * the interval [0,0.34658]:
43 * Since
44 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
45 * we define R1(r*r) by
46 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
47 * That is,
48 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
49 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
50 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
51 * We use a special Reme algorithm on [0,0.347] to generate
52 * a polynomial of degree 5 in r*r to approximate R1. The
53 * maximum error of this polynomial approximation is bounded
54 * by 2**-61. In other words,
55 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
56 * where Q1 = -1.6666666666666567384E-2,
57 * Q2 = 3.9682539681370365873E-4,
58 * Q3 = -9.9206344733435987357E-6,
59 * Q4 = 2.5051361420808517002E-7,
60 * Q5 = -6.2843505682382617102E-9;
61 * z = r*r,
62 * with error bounded by
63 * | 5 | -61
64 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
65 * | |
66 *
67 * expm1(r) = exp(r)-1 is then computed by the following
68 * specific way which minimize the accumulation rounding error:
69 * 2 3
70 * r r [ 3 - (R1 + R1*r/2) ]
71 * expm1(r) = r + --- + --- * [--------------------]
72 * 2 2 [ 6 - r*(3 - R1*r/2) ]
73 *
74 * To compensate the error in the argument reduction, we use
75 * expm1(r+c) = expm1(r) + c + expm1(r)*c
76 * ~ expm1(r) + c + r*c
77 * Thus c+r*c will be added in as the correction terms for
78 * expm1(r+c). Now rearrange the term to avoid optimization
79 * screw up:
80 * ( 2 2 )
81 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
82 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
83 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
84 * ( )
85 *
86 * = r - E
87 * 3. Scale back to obtain expm1(x):
88 * From step 1, we have
89 * expm1(x) = either 2^k*[expm1(r)+1] - 1
90 * = or 2^k*[expm1(r) + (1-2^-k)]
91 * 4. Implementation notes:
92 * (A). To save one multiplication, we scale the coefficient Qi
93 * to Qi*2^i, and replace z by (x^2)/2.
94 * (B). To achieve maximum accuracy, we compute expm1(x) by
95 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
96 * (ii) if k=0, return r-E
97 * (iii) if k=-1, return 0.5*(r-E)-0.5
98 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
99 * else return 1.0+2.0*(r-E);
100 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
101 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
102 * (vii) return 2^k(1-((E+2^-k)-r))
103 *
104 * Special cases:
105 * expm1(INF) is INF, expm1(NaN) is NaN;
106 * expm1(-INF) is -1, and
107 * for finite argument, only expm1(0)=0 is exact.
108 *
109 * Accuracy:
110 * according to an error analysis, the error is always less than
111 * 1 ulp (unit in the last place).
112 *
113 * Misc. info.
114 * For IEEE double
115 * if x > 7.09782712893383973096e+02 then expm1(x) overflow
116 *
117 * Constants:
118 * The hexadecimal values are the intended ones for the following
119 * constants. The decimal values may be used, provided that the
120 * compiler will convert from decimal to binary accurately enough
121 * to produce the hexadecimal values shown.
122 */
123
124 #define one 1.0
125 #define huge 1.0e+300
126 #define tiny 1.0e-300
127 #define o_threshold 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */
128 #define ln2_hi 6.93147180369123816490e-01 /* 0x3fe62e42, 0xfee00000 */
129 #define ln2_lo 1.90821492927058770002e-10 /* 0x3dea39ef, 0x35793c76 */
130 #define invln2 1.44269504088896338700e+00 /* 0x3ff71547, 0x652b82fe */
131
132 /* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
133 #define Q1 -3.33333333333331316428e-02 /* BFA11111 111110F4 */
134 #define Q2 1.58730158725481460165e-03 /* 3F5A01A0 19FE5585 */
135 #define Q3 -7.93650757867487942473e-05 /* BF14CE19 9EAADBB7 */
136 #define Q4 4.00821782732936239552e-06 /* 3ED0CFCA 86E65239 */
137 #define Q5 -2.01099218183624371326e-07 /* BE8AFDB7 6E09C32D */
138
139 double
expm1(double x)140 expm1 (double x)
141 {
142 double y, hi, lo, c, e, hxs, hfx, r1;
143 double_accessor t, twopk;
144 int k, xsb;
145 unsigned int hx;
146
147 hx = __HI (x);
148 xsb = hx & 0x80000000; /* sign bit of x */
149 hx &= 0x7fffffff; /* high word of |x| */
150
151 /* filter out huge and non-finite argument */
152 if (hx >= 0x4043687A)
153 {
154 /* if |x|>=56*ln2 */
155 if (hx >= 0x40862E42)
156 {
157 /* if |x|>=709.78... */
158 if (hx >= 0x7ff00000)
159 {
160 unsigned int low;
161 low = __LO (x);
162 if (((hx & 0xfffff) | low) != 0)
163 {
164 /* NaN */
165 return x + x;
166 }
167 else
168 {
169 /* exp(+-inf)-1={inf,-1} */
170 return (xsb == 0) ? x : -1.0;
171 }
172 }
173 if (x > o_threshold)
174 {
175 /* overflow */
176 return huge * huge;
177 }
178 }
179 if (xsb != 0)
180 {
181 /* x < -56*ln2, return -1.0 with inexact */
182 if (x + tiny < 0.0) /* raise inexact */
183 {
184 /* return -1 */
185 return tiny - one;
186 }
187 }
188 }
189
190 /* argument reduction */
191 if (hx > 0x3fd62e42)
192 {
193 /* if |x| > 0.5 ln2 */
194 if (hx < 0x3FF0A2B2)
195 {
196 /* and |x| < 1.5 ln2 */
197 if (xsb == 0)
198 {
199 hi = x - ln2_hi;
200 lo = ln2_lo;
201 k = 1;
202 }
203 else
204 {
205 hi = x + ln2_hi;
206 lo = -ln2_lo;
207 k = -1;
208 }
209 }
210 else
211 {
212 k = (int) (invln2 * x + ((xsb == 0) ? 0.5 : -0.5));
213 t.dbl = k;
214 hi = x - t.dbl * ln2_hi; /* t*ln2_hi is exact here */
215 lo = t.dbl * ln2_lo;
216 }
217 x = hi - lo;
218 c = (hi - x) - lo;
219 }
220 else if (hx < 0x3c900000)
221 {
222 /* when |x|<2**-54, return x */
223 return x;
224 }
225 else
226 {
227 k = 0;
228 }
229
230 /* x is now in primary range */
231 hfx = 0.5 * x;
232 hxs = x * hfx;
233 r1 = one + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
234 t.dbl = 3.0 - r1 * hfx;
235 e = hxs * ((r1 - t.dbl) / (6.0 - x * t.dbl));
236 if (k == 0)
237 {
238 /* c is 0 */
239 return x - (x * e - hxs);
240 }
241 else
242 {
243 twopk.as_int.hi = 0x3ff00000 + ((unsigned int) k << 20); /* 2^k */
244 twopk.as_int.lo = 0;
245 e = (x * (e - c) - c);
246 e -= hxs;
247 if (k == -1)
248 {
249 return 0.5 * (x - e) - 0.5;
250 }
251 if (k == 1)
252 {
253 if (x < -0.25)
254 {
255 return -2.0 * (e - (x + 0.5));
256 }
257 else
258 {
259 return one + 2.0 * (x - e);
260 }
261 }
262 if ((k <= -2) || (k > 56))
263 {
264 /* suffice to return exp(x)-1 */
265 y = one - (e - x);
266 if (k == 1024)
267 {
268 y = y * 2.0 * 0x1p1023;
269 }
270 else
271 {
272 y = y * twopk.dbl;
273 }
274 return y - one;
275 }
276 t.dbl = one;
277 if (k < 20)
278 {
279 t.as_int.hi = (0x3ff00000 - (0x200000 >> k)); /* t=1-2^-k */
280 y = t.dbl - (e - x);
281 y = y * twopk.dbl;
282 }
283 else
284 {
285 t.as_int.hi = ((0x3ff - k) << 20); /* 2^-k */
286 y = x - (e + t.dbl);
287 y += one;
288 y = y * twopk.dbl;
289 }
290 }
291 return y;
292 } /* expm1 */
293
294 #undef one
295 #undef huge
296 #undef tiny
297 #undef o_threshold
298 #undef ln2_hi
299 #undef ln2_lo
300 #undef invln2
301 #undef Q1
302 #undef Q2
303 #undef Q3
304 #undef Q4
305 #undef Q5
306