SDL  2.0
e_pow.c File Reference
#include "math_libm.h"
#include "math_private.h"
+ Include dependency graph for e_pow.c:

Go to the source code of this file.

Functions

 libm_hidden_proto (scalbn)
 
double attribute_hidden __ieee754_pow (double x, double y)
 

Variables

 dp_h []
 
 dp_l []
 
 zero = 0.0
 
 one = 1.0
 
 two = 2.0
 
 two53 = 9007199254740992.0
 
 huge_val = 1.0e300
 
 tiny = 1.0e-300
 
 L1 = 5.99999999999994648725e-01
 
 L2 = 4.28571428578550184252e-01
 
 L3 = 3.33333329818377432918e-01
 
 L4 = 2.72728123808534006489e-01
 
 L5 = 2.30660745775561754067e-01
 
 L6 = 2.06975017800338417784e-01
 
 P1 = 1.66666666666666019037e-01
 
 P2 = -2.77777777770155933842e-03
 
 P3 = 6.61375632143793436117e-05
 
 P4 = -1.65339022054652515390e-06
 
 P5 = 4.13813679705723846039e-08
 
 lg2 = 6.93147180559945286227e-01
 
 lg2_h = 6.93147182464599609375e-01
 
 lg2_l = -1.90465429995776804525e-09
 
 ovt = 8.0085662595372944372e-0017
 
 cp = 9.61796693925975554329e-01
 
 cp_h = 9.61796700954437255859e-01
 
 cp_l = -7.02846165095275826516e-09
 
 ivln2 = 1.44269504088896338700e+00
 
 ivln2_h = 1.44269502162933349609e+00
 
 ivln2_l = 1.92596299112661746887e-08
 

Function Documentation

double attribute_hidden __ieee754_pow ( double  x,
double  y 
)

Definition at line 106 of file e_pow.c.

References __ieee754_sqrt, cp, cp_h, cp_l, dp_h, dp_l, EXTRACT_WORDS, fabs, GET_HIGH_WORD, huge_val, i, ivln2, ivln2_h, ivln2_l, j, L1, L2, L3, L4, L5, L6, lg2, lg2_h, lg2_l, one, ovt, P1, P2, P3, P4, P5, scalbn, SET_HIGH_WORD, SET_LOW_WORD, tiny, two, two53, and zero.

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  }
#define GET_HIGH_WORD(i, d)
Definition: math_private.h:103
ovt
Definition: e_pow.c:95
one
Definition: e_pow.c:78
GLdouble GLdouble GLdouble r
Definition: SDL_opengl.h:2072
ivln2_l
Definition: e_pow.c:101
GLdouble n
signed int int32_t
tiny
Definition: e_pow.c:79
two
Definition: e_pow.c:78
cp_l
Definition: e_pow.c:98
P3
Definition: e_pow.c:89
GLint GLint GLint GLint GLint x
Definition: SDL_opengl.h:1567
#define __ieee754_sqrt
Definition: math_private.h:42
lg2
Definition: e_pow.c:92
#define SET_HIGH_WORD(d, v)
Definition: math_private.h:131
GLuint GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat t1
#define scalbn
Definition: math_private.h:40
GLfixed y1
const GLdouble * v
Definition: SDL_opengl.h:2057
P2
Definition: e_pow.c:88
dp_l[]
Definition: e_pow.c:75
GLdouble s
Definition: SDL_opengl.h:2056
unsigned int u_int32_t
Definition: math_private.h:29
ivln2
Definition: e_pow.c:99
cp
Definition: e_pow.c:96
#define SET_LOW_WORD(d, v)
Definition: math_private.h:141
#define EXTRACT_WORDS(ix0, ix1, d)
Definition: math_private.h:93
L3
Definition: e_pow.c:83
GLdouble GLdouble t
Definition: SDL_opengl.h:2064
P5
Definition: e_pow.c:91
lg2_h
Definition: e_pow.c:93
P4
Definition: e_pow.c:90
#define fabs
Definition: math_private.h:36
L6
Definition: e_pow.c:86
return Display return Display Bool Bool int int int return Display XEvent Bool(*) XPointer return Display return Display Drawable _Xconst char unsigned int unsigned int return Display Pixmap Pixmap XColor XColor unsigned int unsigned int return Display _Xconst char char int char return Display Visual unsigned int int int char unsigned int unsigned int in i)
Definition: SDL_x11sym.h:42
ivln2_h
Definition: e_pow.c:100
GLdouble GLdouble z
L5
Definition: e_pow.c:85
GLint GLint GLint GLint GLint GLint y
Definition: SDL_opengl.h:1567
lg2_l
Definition: e_pow.c:94
P1
Definition: e_pow.c:87
dp_h[]
Definition: e_pow.c:72
GLubyte GLubyte GLubyte GLubyte w
cp_h
Definition: e_pow.c:97
huge_val
Definition: e_pow.c:79
L4
Definition: e_pow.c:84
zero
Definition: e_pow.c:78
L1
Definition: e_pow.c:81
L2
Definition: e_pow.c:82
two53
Definition: e_pow.c:78
return Display return Display Bool Bool int int int return Display XEvent Bool(*) XPointer return Display return Display Drawable _Xconst char unsigned int unsigned int return Display Pixmap Pixmap XColor XColor unsigned int unsigned int return Display _Xconst char char int char return Display Visual unsigned int int int char unsigned int unsigned int int in j)
Definition: SDL_x11sym.h:42
libm_hidden_proto ( scalbn  )

Definition at line 65 of file e_pow.c.

72  { 1.0, 1.5, }, dp_h[] = {
dp_h[]
Definition: e_pow.c:72

Variable Documentation

cp = 9.61796693925975554329e-01

Definition at line 96 of file e_pow.c.

Referenced by __ieee754_pow(), and mmap_resize().

cp_h = 9.61796700954437255859e-01

Definition at line 97 of file e_pow.c.

Referenced by __ieee754_pow().

cp_l = -7.02846165095275826516e-09

Definition at line 98 of file e_pow.c.

Referenced by __ieee754_pow().

dp_h[]
Initial value:
= {
0.0, 5.84962487220764160156e-01,}

Definition at line 72 of file e_pow.c.

Referenced by __ieee754_pow().

dp_l[]
Initial value:
= {
0.0, 1.35003920212974897128e-08,}

Definition at line 75 of file e_pow.c.

Referenced by __ieee754_pow().

huge_val = 1.0e300

Definition at line 79 of file e_pow.c.

Referenced by __ieee754_pow(), and libm_hidden_proto().

ivln2 = 1.44269504088896338700e+00

Definition at line 99 of file e_pow.c.

Referenced by __ieee754_pow().

ivln2_h = 1.44269502162933349609e+00

Definition at line 100 of file e_pow.c.

Referenced by __ieee754_pow().

ivln2_l = 1.92596299112661746887e-08

Definition at line 101 of file e_pow.c.

Referenced by __ieee754_pow().

L1 = 5.99999999999994648725e-01

Definition at line 81 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

L2 = 4.28571428578550184252e-01

Definition at line 82 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

L3 = 3.33333329818377432918e-01

Definition at line 83 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

L4 = 2.72728123808534006489e-01

Definition at line 84 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

L5 = 2.30660745775561754067e-01

Definition at line 85 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

L6 = 2.06975017800338417784e-01

Definition at line 86 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

lg2 = 6.93147180559945286227e-01

Definition at line 92 of file e_pow.c.

Referenced by __ieee754_pow().

lg2_h = 6.93147182464599609375e-01

Definition at line 93 of file e_pow.c.

Referenced by __ieee754_pow().

lg2_l = -1.90465429995776804525e-09

Definition at line 94 of file e_pow.c.

Referenced by __ieee754_pow().

one = 1.0

Definition at line 78 of file e_pow.c.

Referenced by __ieee754_pow().

ovt = 8.0085662595372944372e-0017

Definition at line 95 of file e_pow.c.

Referenced by __ieee754_pow().

P1 = 1.66666666666666019037e-01

Definition at line 87 of file e_pow.c.

Referenced by __ieee754_pow().

P2 = -2.77777777770155933842e-03

Definition at line 88 of file e_pow.c.

Referenced by __ieee754_pow().

P3 = 6.61375632143793436117e-05

Definition at line 89 of file e_pow.c.

Referenced by __ieee754_pow().

P4 = -1.65339022054652515390e-06

Definition at line 90 of file e_pow.c.

Referenced by __ieee754_pow().

P5 = 4.13813679705723846039e-08

Definition at line 91 of file e_pow.c.

Referenced by __ieee754_pow().

tiny = 1.0e-300

Definition at line 79 of file e_pow.c.

Referenced by __ieee754_pow(), and libm_hidden_proto().

two = 2.0

Definition at line 78 of file e_pow.c.

Referenced by __ieee754_pow().

two53 = 9007199254740992.0

Definition at line 78 of file e_pow.c.

Referenced by __ieee754_pow().

zero = 0.0

Definition at line 78 of file e_pow.c.

Referenced by __ieee754_pow().