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

double attribute_hidden __ieee754_pow (double x, double y)
 

Variables

static const double bp [] = {1.0, 1.5,}
 
static const double dp_h [] = { 0.0, 5.84962487220764160156e-01,}
 
static const double dp_l [] = { 0.0, 1.35003920212974897128e-08,}
 
static const double zero = 0.0
 
static const double one = 1.0
 
static const double two = 2.0
 
static const double two53 = 9007199254740992.0
 
static const double huge = 1.0e300
 
static const double tiny = 1.0e-300
 
static const double L1 = 5.99999999999994648725e-01
 
static const double L2 = 4.28571428578550184252e-01
 
static const double L3 = 3.33333329818377432918e-01
 
static const double L4 = 2.72728123808534006489e-01
 
static const double L5 = 2.30660745775561754067e-01
 
static const double L6 = 2.06975017800338417784e-01
 
static const double P1 = 1.66666666666666019037e-01
 
static const double P2 = -2.77777777770155933842e-03
 
static const double P3 = 6.61375632143793436117e-05
 
static const double P4 = -1.65339022054652515390e-06
 
static const double P5 = 4.13813679705723846039e-08
 
static const double lg2 = 6.93147180559945286227e-01
 
static const double lg2_h = 6.93147182464599609375e-01
 
static const double lg2_l = -1.90465429995776804525e-09
 
static const double ovt = 8.0085662595372944372e-0017
 
static const double cp = 9.61796693925975554329e-01
 
static const double cp_h = 9.61796700954437255859e-01
 
static const double cp_l = -7.02846165095275826516e-09
 
static const double ivln2 = 1.44269504088896338700e+00
 
static const double ivln2_h = 1.44269502162933349609e+00
 
static const double ivln2_l = 1.92596299112661746887e-08
 

Function Documentation

◆ __ieee754_pow()

double attribute_hidden __ieee754_pow ( double  x,
double  y 
)

Definition at line 99 of file e_pow.c.

References __ieee754_pow(), __ieee754_sqrt, bp, cp, cp_h, cp_l, dp_h, dp_l, EXTRACT_WORDS, fabs(), GET_HIGH_WORD, huge, i, ivln2, ivln2_h, ivln2_l, j, k, L1, L2, L3, L4, L5, L6, lg2, lg2_h, lg2_l, libm_hidden_def(), one, ovt, P1, P2, P3, P4, P5, scalbn, SET_HIGH_WORD, SET_LOW_WORD, strong_alias, tiny, two, two53, and zero.

Referenced by __ieee754_pow().

100 {
101  double z,ax,z_h,z_l,p_h,p_l;
102  double y1,t1,t2,r,s,t,u,v,w;
103  int32_t i,j,k,yisint,n;
104  int32_t hx,hy,ix,iy;
105  u_int32_t lx,ly;
106 
107  EXTRACT_WORDS(hx,lx,x);
108  /* x==1: 1**y = 1 (even if y is NaN) */
109  if (hx==0x3ff00000 && lx==0) {
110  return x;
111  }
112  ix = hx&0x7fffffff;
113 
114  EXTRACT_WORDS(hy,ly,y);
115  iy = hy&0x7fffffff;
116 
117  /* y==zero: x**0 = 1 */
118  if((iy|ly)==0) return one;
119 
120  /* +-NaN return x+y */
121  if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
122  iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
123  return x+y;
124 
125  /* determine if y is an odd int when x < 0
126  * yisint = 0 ... y is not an integer
127  * yisint = 1 ... y is an odd int
128  * yisint = 2 ... y is an even int
129  */
130  yisint = 0;
131  if(hx<0) {
132  if(iy>=0x43400000) yisint = 2; /* even integer y */
133  else if(iy>=0x3ff00000) {
134  k = (iy>>20)-0x3ff; /* exponent */
135  if(k>20) {
136  j = ly>>(52-k);
137  if((j<<(52-k))==ly) yisint = 2-(j&1);
138  } else if(ly==0) {
139  j = iy>>(20-k);
140  if((j<<(20-k))==iy) yisint = 2-(j&1);
141  }
142  }
143  }
144 
145  /* special value of y */
146  if(ly==0) {
147  if (iy==0x7ff00000) { /* y is +-inf */
148  if (((ix-0x3ff00000)|lx)==0)
149  return one; /* +-1**+-inf is 1 (yes, weird rule) */
150  if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
151  return (hy>=0) ? y : zero;
152  /* (|x|<1)**-,+inf = inf,0 */
153  return (hy<0) ? -y : zero;
154  }
155  if(iy==0x3ff00000) { /* y is +-1 */
156  if(hy<0) return one/x; else return x;
157  }
158  if(hy==0x40000000) return x*x; /* y is 2 */
159  if(hy==0x3fe00000) { /* y is 0.5 */
160  if(hx>=0) /* x >= +0 */
161  return __ieee754_sqrt(x);
162  }
163  }
164 
165  ax = fabs(x);
166  /* special value of x */
167  if(lx==0) {
168  if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
169  z = ax; /*x is +-0,+-inf,+-1*/
170  if(hy<0) z = one/z; /* z = (1/|x|) */
171  if(hx<0) {
172  if(((ix-0x3ff00000)|yisint)==0) {
173  z = (z-z)/(z-z); /* (-1)**non-int is NaN */
174  } else if(yisint==1)
175  z = -z; /* (x<0)**odd = -(|x|**odd) */
176  }
177  return z;
178  }
179  }
180 
181  /* (x<0)**(non-int) is NaN */
182  if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
183 
184  /* |y| is huge */
185  if(iy>0x41e00000) { /* if |y| > 2**31 */
186  if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
187  if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
188  if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
189  }
190  /* over/underflow if x is not close to one */
191  if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
192  if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
193  /* now |1-x| is tiny <= 2**-20, suffice to compute
194  log(x) by x-x^2/2+x^3/3-x^4/4 */
195  t = x-1; /* t has 20 trailing zeros */
196  w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
197  u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
198  v = t*ivln2_l-w*ivln2;
199  t1 = u+v;
200  SET_LOW_WORD(t1,0);
201  t2 = v-(t1-u);
202  } else {
203  double s2,s_h,s_l,t_h,t_l;
204  n = 0;
205  /* take care subnormal number */
206  if(ix<0x00100000)
207  {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
208  n += ((ix)>>20)-0x3ff;
209  j = ix&0x000fffff;
210  /* determine interval */
211  ix = j|0x3ff00000; /* normalize ix */
212  if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
213  else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
214  else {k=0;n+=1;ix -= 0x00100000;}
215  SET_HIGH_WORD(ax,ix);
216 
217  /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
218  u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
219  v = one/(ax+bp[k]);
220  s = u*v;
221  s_h = s;
222  SET_LOW_WORD(s_h,0);
223  /* t_h=ax+bp[k] High */
224  t_h = zero;
225  SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
226  t_l = ax - (t_h-bp[k]);
227  s_l = v*((u-s_h*t_h)-s_h*t_l);
228  /* compute log(ax) */
229  s2 = s*s;
230  r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
231  r += s_l*(s_h+s);
232  s2 = s_h*s_h;
233  t_h = 3.0+s2+r;
234  SET_LOW_WORD(t_h,0);
235  t_l = r-((t_h-3.0)-s2);
236  /* u+v = s*(1+...) */
237  u = s_h*t_h;
238  v = s_l*t_h+t_l*s;
239  /* 2/(3log2)*(s+...) */
240  p_h = u+v;
241  SET_LOW_WORD(p_h,0);
242  p_l = v-(p_h-u);
243  z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
244  z_l = cp_l*p_h+p_l*cp+dp_l[k];
245  /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
246  t = (double)n;
247  t1 = (((z_h+z_l)+dp_h[k])+t);
248  SET_LOW_WORD(t1,0);
249  t2 = z_l-(((t1-t)-dp_h[k])-z_h);
250  }
251 
252  s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
253  if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
254  s = -one;/* (-ve)**(odd int) */
255 
256  /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
257  y1 = y;
258  SET_LOW_WORD(y1,0);
259  p_l = (y-y1)*t1+y*t2;
260  p_h = y1*t1;
261  z = p_l+p_h;
262  EXTRACT_WORDS(j,i,z);
263  if (j>=0x40900000) { /* z >= 1024 */
264  if(((j-0x40900000)|i)!=0) /* if z > 1024 */
265  return s*huge*huge; /* overflow */
266  else {
267  if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
268  }
269  } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
270  if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
271  return s*tiny*tiny; /* underflow */
272  else {
273  if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
274  }
275  }
276  /*
277  * compute 2**(p_h+p_l)
278  */
279  i = j&0x7fffffff;
280  k = (i>>20)-0x3ff;
281  n = 0;
282  if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
283  n = j+(0x00100000>>(k+1));
284  k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
285  t = zero;
286  SET_HIGH_WORD(t,n&~(0x000fffff>>k));
287  n = ((n&0x000fffff)|0x00100000)>>(20-k);
288  if(j<0) n = -n;
289  p_h -= t;
290  }
291  t = p_l+p_h;
292  SET_LOW_WORD(t,0);
293  u = t*lg2_h;
294  v = (p_l-(t-p_h))*lg2+t*lg2_l;
295  z = u+v;
296  w = v-(z-u);
297  t = z*z;
298  t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
299  r = (z*t1)/(t1-two)-(w+z*w);
300  z = one-(r-z);
301  GET_HIGH_WORD(j,z);
302  j += (n<<20);
303  if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
304  else SET_HIGH_WORD(z,j);
305  return s*z;
306 }
#define GET_HIGH_WORD(i, d)
Definition: math_private.h:108
GLdouble GLdouble GLdouble r
Definition: SDL_opengl.h:2079
static const double two53
Definition: e_pow.c:73
GLdouble GLdouble z
static const double dp_l[]
Definition: e_pow.c:69
GLdouble s
Definition: SDL_opengl.h:2063
const GLdouble * v
Definition: SDL_opengl.h:2064
GLint GLint GLint GLint GLint x
Definition: SDL_opengl.h:1574
signed int int32_t
static const double ivln2_h
Definition: e_pow.c:96
static const double P3
Definition: e_pow.c:85
static const double L1
Definition: e_pow.c:77
static const double L6
Definition: e_pow.c:82
static const double lg2
Definition: e_pow.c:88
static const double lg2_l
Definition: e_pow.c:90
static const double ovt
Definition: e_pow.c:91
static const double tiny
Definition: e_pow.c:75
static const double bp[]
Definition: e_pow.c:67
#define __ieee754_sqrt
Definition: math_private.h:47
#define SET_HIGH_WORD(d, v)
Definition: math_private.h:136
GLuint GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat t1
#define scalbn
Definition: math_private.h:45
static const double dp_h[]
Definition: e_pow.c:68
GLfixed y1
static const double L3
Definition: e_pow.c:79
static const double P1
Definition: e_pow.c:83
static const double P4
Definition: e_pow.c:86
unsigned int u_int32_t
Definition: math_private.h:31
static const double L5
Definition: e_pow.c:81
static const double two
Definition: e_pow.c:72
#define SET_LOW_WORD(d, v)
Definition: math_private.h:146
#define EXTRACT_WORDS(ix0, ix1, d)
Definition: math_private.h:98
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:50
static const double one
Definition: e_pow.c:71
GLubyte GLubyte GLubyte GLubyte w
static const double lg2_h
Definition: e_pow.c:89
GLint GLint GLint GLint GLint GLint y
Definition: SDL_opengl.h:1574
static const double cp_l
Definition: e_pow.c:94
static const double L2
Definition: e_pow.c:78
static const double zero
Definition: e_pow.c:70
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:50
static const double ivln2_l
Definition: e_pow.c:97
static const double huge
Definition: e_pow.c:74
static const double L4
Definition: e_pow.c:80
GLdouble n
static const double cp
Definition: e_pow.c:92
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 int return Display Window Cursor return Display Window return Display Drawable GC int int unsigned int unsigned int return Display Drawable GC int int _Xconst char int return Display Drawable GC int int unsigned int unsigned int return Display return Display Cursor return Display GC return XModifierKeymap return char Display Window int return Display return Display Atom return Display Window XWindowAttributes return Display Window return Display XEvent Bool(*) XPointer return Display Window Bool unsigned int int int Window Cursor Time return Display Window int return KeySym return Display _Xconst char Bool return Display _Xconst char return XKeyEvent char int KeySym XComposeStatus return Display int int int XVisualInfo return Display Window int int return _Xconst char return Display XEvent return Display Drawable GC XImage int int int int unsigned int unsigned int return Display Window Window Window int int int int unsigned int return Display Window Window int int return Display Window unsigned int unsigned int return Display Window Bool long XEvent return Display GC unsigned long return Display Window int Time return Display Window Window return Display Window unsigned long return Display Window XSizeHints Display Colormap XColor int return char int XTextProperty return XFontStruct _Xconst char int int int int XCharStruct return Display Window return Display Time return Display Colormap return Display Window Window int int unsigned int unsigned int int int return Display Window int return XExtensionInfo Display char XExtensionHooks int XPointer return XExtensionInfo XExtensionInfo Display return Display return Display unsigned long Display GC Display char long Display xReply int Bool return Display Bool return Display int SDL_X11_XESetEventToWireRetType return Display Window Window Window Window unsigned int return Display XShmSegmentInfo return Display Drawable GC XImage int int int int unsigned int unsigned int Boo k)
Definition: SDL_x11sym.h:213
static const double cp_h
Definition: e_pow.c:93
static const double ivln2
Definition: e_pow.c:95
static const double P5
Definition: e_pow.c:87
static const double P2
Definition: e_pow.c:84
double fabs(double x)
Definition: s_fabs.c:22
GLdouble GLdouble t
Definition: SDL_opengl.h:2071

Variable Documentation

◆ bp

const double bp[] = {1.0, 1.5,}
static

Definition at line 67 of file e_pow.c.

Referenced by __ieee754_pow().

◆ cp

const double cp = 9.61796693925975554329e-01
static

Definition at line 92 of file e_pow.c.

Referenced by __ieee754_pow(), and mmap_resize().

◆ cp_h

const double cp_h = 9.61796700954437255859e-01
static

Definition at line 93 of file e_pow.c.

Referenced by __ieee754_pow().

◆ cp_l

const double cp_l = -7.02846165095275826516e-09
static

Definition at line 94 of file e_pow.c.

Referenced by __ieee754_pow().

◆ dp_h

const double dp_h[] = { 0.0, 5.84962487220764160156e-01,}
static

Definition at line 68 of file e_pow.c.

Referenced by __ieee754_pow().

◆ dp_l

const double dp_l[] = { 0.0, 1.35003920212974897128e-08,}
static

Definition at line 69 of file e_pow.c.

Referenced by __ieee754_pow().

◆ huge

const double huge = 1.0e300
static

Definition at line 74 of file e_pow.c.

Referenced by __ieee754_pow().

◆ ivln2

const double ivln2 = 1.44269504088896338700e+00
static

Definition at line 95 of file e_pow.c.

Referenced by __ieee754_pow().

◆ ivln2_h

const double ivln2_h = 1.44269502162933349609e+00
static

Definition at line 96 of file e_pow.c.

Referenced by __ieee754_pow().

◆ ivln2_l

const double ivln2_l = 1.92596299112661746887e-08
static

Definition at line 97 of file e_pow.c.

Referenced by __ieee754_pow().

◆ L1

const double L1 = 5.99999999999994648725e-01
static

Definition at line 77 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

◆ L2

const double L2 = 4.28571428578550184252e-01
static

Definition at line 78 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

◆ L3

const double L3 = 3.33333329818377432918e-01
static

Definition at line 79 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

◆ L4

const double L4 = 2.72728123808534006489e-01
static

Definition at line 80 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

◆ L5

const double L5 = 2.30660745775561754067e-01
static

Definition at line 81 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

◆ L6

const double L6 = 2.06975017800338417784e-01
static

Definition at line 82 of file e_pow.c.

Referenced by __ieee754_pow(), and SDL_tolower().

◆ lg2

const double lg2 = 6.93147180559945286227e-01
static

Definition at line 88 of file e_pow.c.

Referenced by __ieee754_pow().

◆ lg2_h

const double lg2_h = 6.93147182464599609375e-01
static

Definition at line 89 of file e_pow.c.

Referenced by __ieee754_pow().

◆ lg2_l

const double lg2_l = -1.90465429995776804525e-09
static

Definition at line 90 of file e_pow.c.

Referenced by __ieee754_pow().

◆ one

const double one = 1.0
static

Definition at line 71 of file e_pow.c.

Referenced by __ieee754_pow().

◆ ovt

const double ovt = 8.0085662595372944372e-0017
static

Definition at line 91 of file e_pow.c.

Referenced by __ieee754_pow().

◆ P1

const double P1 = 1.66666666666666019037e-01
static

Definition at line 83 of file e_pow.c.

Referenced by __ieee754_pow().

◆ P2

const double P2 = -2.77777777770155933842e-03
static

Definition at line 84 of file e_pow.c.

Referenced by __ieee754_pow().

◆ P3

const double P3 = 6.61375632143793436117e-05
static

Definition at line 85 of file e_pow.c.

Referenced by __ieee754_pow().

◆ P4

const double P4 = -1.65339022054652515390e-06
static

Definition at line 86 of file e_pow.c.

Referenced by __ieee754_pow().

◆ P5

const double P5 = 4.13813679705723846039e-08
static

Definition at line 87 of file e_pow.c.

Referenced by __ieee754_pow().

◆ tiny

const double tiny = 1.0e-300
static

Definition at line 75 of file e_pow.c.

Referenced by __ieee754_pow().

◆ two

const double two = 2.0
static

Definition at line 72 of file e_pow.c.

Referenced by __ieee754_pow().

◆ two53

const double two53 = 9007199254740992.0
static

Definition at line 73 of file e_pow.c.

Referenced by __ieee754_pow().

◆ zero

const double zero = 0.0
static

Definition at line 70 of file e_pow.c.

Referenced by __ieee754_pow().