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e_pow.c
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1 /* @(#)e_pow.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 #if defined(LIBM_SCCS) && !defined(lint)
14 static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
15 #endif
16
17 /* __ieee754_pow(x,y) return x**y
18 *
19 * n
20 * Method: Let x = 2 * (1+f)
21 * 1. Compute and return log2(x) in two pieces:
22 * log2(x) = w1 + w2,
23 * where w1 has 53-24 = 29 bit trailing zeros.
24 * 2. Perform y*log2(x) = n+y' by simulating muti-precision
25 * arithmetic, where |y'|<=0.5.
26 * 3. Return x**y = 2**n*exp(y'*log2)
27 *
28 * Special cases:
29 * 1. (anything) ** 0 is 1
30 * 2. (anything) ** 1 is itself
31 * 3. (anything) ** NAN is NAN
32 * 4. NAN ** (anything except 0) is NAN
33 * 5. +-(|x| > 1) ** +INF is +INF
34 * 6. +-(|x| > 1) ** -INF is +0
35 * 7. +-(|x| < 1) ** +INF is +0
36 * 8. +-(|x| < 1) ** -INF is +INF
37 * 9. +-1 ** +-INF is NAN
38 * 10. +0 ** (+anything except 0, NAN) is +0
39 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
40 * 12. +0 ** (-anything except 0, NAN) is +INF
41 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
42 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
43 * 15. +INF ** (+anything except 0,NAN) is +INF
44 * 16. +INF ** (-anything except 0,NAN) is +0
45 * 17. -INF ** (anything) = -0 ** (-anything)
46 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
47 * 19. (-anything except 0 and inf) ** (non-integer) is NAN
48 *
49 * Accuracy:
50 * pow(x,y) returns x**y nearly rounded. In particular
51 * pow(integer,integer)
52 * always returns the correct integer provided it is
53 * representable.
54 *
55 * Constants :
56 * The hexadecimal values are the intended ones for the following
57 * constants. The decimal values may be used, provided that the
58 * compiler will convert from decimal to binary accurately enough
59 * to produce the hexadecimal values shown.
60 */
61
62 #include "math_libm.h"
63 #include "math_private.h"
64
65 libm_hidden_proto(scalbn)
66 libm_hidden_proto(fabs)
67 #ifdef __STDC__
68 static const double
69 #else
70 static double
71 #endif
72 bp[] = { 1.0, 1.5, }, dp_h[] = {
73 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
74
75 dp_l[] = {
76 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
77
78 zero = 0.0, one = 1.0, two = 2.0, two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
79 huge_val = 1.0e300, tiny = 1.0e-300,
80 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
81 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
82 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
83 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
84 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
85 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
86 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
87 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
88 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
89 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
90 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
91 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
92 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
93 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
94 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
95 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
96 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
97 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
98 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h */
99 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
100 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2 */
101 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail */
102
103 #ifdef __STDC__
104 double attribute_hidden __ieee754_pow(double x, double y)
105 #else
106 double attribute_hidden __ieee754_pow(x, y)
107 double x, y;
108 #endif
109 {
110 double z, ax, z_h, z_l, p_h, p_l;
111 double y1, t1, t2, r, s, t, u, v, w;
112 int32_t i, j, k, yisint, n;
113 int32_t hx, hy, ix, iy;
114 u_int32_t lx, ly;
115
116 EXTRACT_WORDS(hx, lx, x);
117 EXTRACT_WORDS(hy, ly, y);
118 ix = hx & 0x7fffffff;
119 iy = hy & 0x7fffffff;
120
121 /* y==zero: x**0 = 1 */
122 if ((iy | ly) == 0)
123 return one;
124
125 /* +-NaN return x+y */
126 if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
127 iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
128 return x + y;
129
130 /* determine if y is an odd int when x < 0
131 * yisint = 0 ... y is not an integer
132 * yisint = 1 ... y is an odd int
133 * yisint = 2 ... y is an even int
134 */
135 yisint = 0;
136 if (hx < 0) {
137 if (iy >= 0x43400000)
138 yisint = 2; /* even integer y */
139 else if (iy >= 0x3ff00000) {
140 k = (iy >> 20) - 0x3ff; /* exponent */
141 if (k > 20) {
142 j = ly >> (52 - k);
143 if ((j << (52 - k)) == ly)
144 yisint = 2 - (j & 1);
145 } else if (ly == 0) {
146 j = iy >> (20 - k);
147 if ((j << (20 - k)) == iy)
148 yisint = 2 - (j & 1);
149 }
150 }
151 }
152
153 /* special value of y */
154 if (ly == 0) {
155 if (iy == 0x7ff00000) { /* y is +-inf */
156 if (((ix - 0x3ff00000) | lx) == 0)
157 return y - y; /* inf**+-1 is NaN */
158 else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
159 return (hy >= 0) ? y : zero;
160 else /* (|x|<1)**-,+inf = inf,0 */
161 return (hy < 0) ? -y : zero;
162 }
163 if (iy == 0x3ff00000) { /* y is +-1 */
164 if (hy < 0)
165 return one / x;
166 else
167 return x;
168 }
169 if (hy == 0x40000000)
170 return x * x; /* y is 2 */
171 if (hy == 0x3fe00000) { /* y is 0.5 */
172 if (hx >= 0) /* x >= +0 */
173 return __ieee754_sqrt(x);
174 }
175 }
176
177 ax = fabs(x);
178 /* special value of x */
179 if (lx == 0) {
180 if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
181 z = ax; /* x is +-0,+-inf,+-1 */
182 if (hy < 0)
183 z = one / z; /* z = (1/|x|) */
184 if (hx < 0) {
185 if (((ix - 0x3ff00000) | yisint) == 0) {
186 z = (z - z) / (z - z); /* (-1)**non-int is NaN */
187 } else if (yisint == 1)
188 z = -z; /* (x<0)**odd = -(|x|**odd) */
189 }
190 return z;
191 }
192 }
193
194 /* (x<0)**(non-int) is NaN */
195 if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
196 return (x - x) / (x - x);
197
198 /* |y| is huge */
199 if (iy > 0x41e00000) { /* if |y| > 2**31 */
200 if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
201 if (ix <= 0x3fefffff)
202 return (hy < 0) ? huge_val * huge_val : tiny * tiny;
203 if (ix >= 0x3ff00000)
204 return (hy > 0) ? huge_val * huge_val : tiny * tiny;
205 }
206 /* over/underflow if x is not close to one */
207 if (ix < 0x3fefffff)
208 return (hy < 0) ? huge_val * huge_val : tiny * tiny;
209 if (ix > 0x3ff00000)
210 return (hy > 0) ? huge_val * huge_val : tiny * tiny;
211 /* now |1-x| is tiny <= 2**-20, suffice to compute
212 log(x) by x-x^2/2+x^3/3-x^4/4 */
213 t = x - 1; /* t has 20 trailing zeros */
214 w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
215 u = ivln2_h * t; /* ivln2_h has 21 sig. bits */
216 v = t * ivln2_l - w * ivln2;
217 t1 = u + v;
218 SET_LOW_WORD(t1, 0);
219 t2 = v - (t1 - u);
220 } else {
221 double s2, s_h, s_l, t_h, t_l;
222 n = 0;
223 /* take care subnormal number */
224 if (ix < 0x00100000) {
225 ax *= two53;
226 n -= 53;
227 GET_HIGH_WORD(ix, ax);
228 }
229 n += ((ix) >> 20) - 0x3ff;
230 j = ix & 0x000fffff;
231 /* determine interval */
232 ix = j | 0x3ff00000; /* normalize ix */
233 if (j <= 0x3988E)
234 k = 0; /* |x|<sqrt(3/2) */
235 else if (j < 0xBB67A)
236 k = 1; /* |x|<sqrt(3) */
237 else {
238 k = 0;
239 n += 1;
240 ix -= 0x00100000;
241 }
242 SET_HIGH_WORD(ax, ix);
243
244 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
245 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
246 v = one / (ax + bp[k]);
247 s = u * v;
248 s_h = s;
249 SET_LOW_WORD(s_h, 0);
250 /* t_h=ax+bp[k] High */
251 t_h = zero;
252 SET_HIGH_WORD(t_h,
253 ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
254 t_l = ax - (t_h - bp[k]);
255 s_l = v * ((u - s_h * t_h) - s_h * t_l);
256 /* compute log(ax) */
257 s2 = s * s;
258 r = s2 * s2 * (L1 +
259 s2 * (L2 +
260 s2 * (L3 +
261 s2 * (L4 + s2 * (L5 + s2 * L6)))));
262 r += s_l * (s_h + s);
263 s2 = s_h * s_h;
264 t_h = 3.0 + s2 + r;
265 SET_LOW_WORD(t_h, 0);
266 t_l = r - ((t_h - 3.0) - s2);
267 /* u+v = s*(1+...) */
268 u = s_h * t_h;
269 v = s_l * t_h + t_l * s;
270 /* 2/(3log2)*(s+...) */
271 p_h = u + v;
272 SET_LOW_WORD(p_h, 0);
273 p_l = v - (p_h - u);
274 z_h = cp_h * p_h; /* cp_h+cp_l = 2/(3*log2) */
275 z_l = cp_l * p_h + p_l * cp + dp_l[k];
276 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
277 t = (double) n;
278 t1 = (((z_h + z_l) + dp_h[k]) + t);
279 SET_LOW_WORD(t1, 0);
280 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
281 }
282
283 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
284 if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
285 s = -one; /* (-ve)**(odd int) */
286
287 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
288 y1 = y;
289 SET_LOW_WORD(y1, 0);
290 p_l = (y - y1) * t1 + y * t2;
291 p_h = y1 * t1;
292 z = p_l + p_h;
293 EXTRACT_WORDS(j, i, z);
294 if (j >= 0x40900000) { /* z >= 1024 */
295 if (((j - 0x40900000) | i) != 0) /* if z > 1024 */
296 return s * huge_val * huge_val; /* overflow */
297 else {
298 if (p_l + ovt > z - p_h)
299 return s * huge_val * huge_val; /* overflow */
300 }
301 } else if ((j & 0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */
302 if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */
303 return s * tiny * tiny; /* underflow */
304 else {
305 if (p_l <= z - p_h)
306 return s * tiny * tiny; /* underflow */
307 }
308 }
309 /*
310 * compute 2**(p_h+p_l)
311 */
312 i = j & 0x7fffffff;
313 k = (i >> 20) - 0x3ff;
314 n = 0;
315 if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
316 n = j + (0x00100000 >> (k + 1));
317 k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */
318 t = zero;
319 SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
320 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
321 if (j < 0)
322 n = -n;
323 p_h -= t;
324 }
325 t = p_l + p_h;
326 SET_LOW_WORD(t, 0);
327 u = t * lg2_h;
328 v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
329 z = u + v;
330 w = v - (z - u);
331 t = z * z;
332 t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
333 r = (z * t1) / (t1 - two) - (w + z * w);
334 z = one - (r - z);
335 GET_HIGH_WORD(j, z);
336 j += (n << 20);
337 if ((j >> 20) <= 0)
338 z = scalbn(z, n); /* subnormal output */
339 else
340 SET_HIGH_WORD(z, j);
341 return s * z;
342 }
two53
static double two53
Definition: e_pow.c:78
GET_HIGH_WORD
#define GET_HIGH_WORD(i, d)
Definition: math_private.h:102
__ieee754_pow
#define __ieee754_pow
Definition: math_private.h:39
dp_h
static double dp_h[]
Definition: e_pow.c:72
ivln2_h
static double ivln2_h
Definition: e_pow.c:100
s
GLdouble s
Definition: glew.h:1376
L5
static double L5
Definition: e_pow.c:85
t2
t2
Definition: e_log.c:151
k
int32_t k
Definition: e_log.c:102
zero
static double zero
Definition: e_pow.c:78
n
GLclampd n
Definition: glew.h:7287
huge_val
static double huge_val
Definition: e_pow.c:79
x
EGLSurface EGLint x
Definition: eglext.h:293
t
GLdouble GLdouble t
Definition: glew.h:1384
j
int32_t j
Definition: e_log.c:102
one
static double one
Definition: e_pow.c:78
int32_t
long int32_t
Definition: types.h:9
y1
GLuint GLfloat GLfloat GLfloat GLfloat y1
Definition: glew.h:11582
__ieee754_sqrt
#define __ieee754_sqrt
Definition: math_private.h:42
SET_HIGH_WORD
#define SET_HIGH_WORD(d, v)
Definition: math_private.h:130
ivln2
static double ivln2
Definition: e_pow.c:99
SET_LOW_WORD
SET_LOW_WORD(z, low)
L6
static double L6
Definition: e_pow.c:86
t1
GLuint GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat t1
Definition: glew.h:11582
two
static double two
Definition: e_pow.c:78
L2
static double L2
Definition: e_pow.c:82
u_int32_t
unsigned int u_int32_t
Definition: math_private.h:29
u
ALuint u
Definition: alMain.h:58
v
const GLdouble * v
Definition: glew.h:1377
P1
static double P1
Definition: e_pow.c:87
EXTRACT_WORDS
#define EXTRACT_WORDS(ix0, ix1, d)
Definition: math_private.h:92
L4
static double L4
Definition: e_pow.c:84
ivln2_l
static double ivln2_l
Definition: e_pow.c:101
math_libm.h
lg2_l
static double lg2_l
Definition: e_pow.c:94
L1
static double L1
Definition: e_pow.c:81
lg2
static double lg2
Definition: e_pow.c:92
bp
static double bp[]
Definition: e_pow.c:72
tiny
static double tiny
Definition: e_pow.c:79
y
EGLSurface EGLint EGLint y
Definition: eglext.h:293
dp_l
static double dp_l[]
Definition: e_pow.c:75
hx
int32_t hx
Definition: e_log.c:102
attribute_hidden
double attribute_hidden
ix
int32_t ix
Definition: e_rem_pio2.c:100
lx
u_int32_t lx
Definition: e_log.c:105
libm_hidden_proto
#define libm_hidden_proto(x)
Definition: math_private.h:25
P2
static double P2
Definition: e_pow.c:88
P3
static double P3
Definition: e_pow.c:89
r
GLdouble GLdouble GLdouble r
Definition: glew.h:1392
z
GLint GLint GLint GLint z
Definition: gl2ext.h:1214
L3
static double L3
Definition: e_pow.c:83
w
GLint GLint GLint GLint GLint w
Definition: gl2ext.h:1215
iy
int iy
Definition: k_sin.c:69
scalbn
double scalbn(double x, int n)
Definition: s_scalbn.c:42
cp_l
static double cp_l
Definition: e_pow.c:98
i
int i
Definition: pngrutil.c:1377
cp_h
static double cp_h
Definition: e_pow.c:97
lg2_h
static double lg2_h
Definition: e_pow.c:93
P4
static double P4
Definition: e_pow.c:90
math_private.h
fabs
double fabs(double x)
Definition: s_fabs.c:29
ovt
static double ovt
Definition: e_pow.c:95
cp
static double cp
Definition: e_pow.c:96
P5
static double P5
Definition: e_pow.c:91