SDL  2.0
e_exp.c
Go to the documentation of this file.
1 /*
2  * ====================================================
3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4  *
5  * Developed at SunPro, a Sun Microsystems, Inc. business.
6  * Permission to use, copy, modify, and distribute this
7  * software is freely granted, provided that this notice
8  * is preserved.
9  * ====================================================
10  */
11 
12 /* __ieee754_exp(x)
13  * Returns the exponential of x.
14  *
15  * Method
16  * 1. Argument reduction:
17  * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
18  * Given x, find r and integer k such that
19  *
20  * x = k*ln2 + r, |r| <= 0.5*ln2.
21  *
22  * Here r will be represented as r = hi-lo for better
23  * accuracy.
24  *
25  * 2. Approximation of exp(r) by a special rational function on
26  * the interval [0,0.34658]:
27  * Write
28  * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
29  * We use a special Reme algorithm on [0,0.34658] to generate
30  * a polynomial of degree 5 to approximate R. The maximum error
31  * of this polynomial approximation is bounded by 2**-59. In
32  * other words,
33  * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
34  * (where z=r*r, and the values of P1 to P5 are listed below)
35  * and
36  * | 5 | -59
37  * | 2.0+P1*z+...+P5*z - R(z) | <= 2
38  * | |
39  * The computation of exp(r) thus becomes
40  * 2*r
41  * exp(r) = 1 + -------
42  * R - r
43  * r*R1(r)
44  * = 1 + r + ----------- (for better accuracy)
45  * 2 - R1(r)
46  * where
47  * 2 4 10
48  * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
49  *
50  * 3. Scale back to obtain exp(x):
51  * From step 1, we have
52  * exp(x) = 2^k * exp(r)
53  *
54  * Special cases:
55  * exp(INF) is INF, exp(NaN) is NaN;
56  * exp(-INF) is 0, and
57  * for finite argument, only exp(0)=1 is exact.
58  *
59  * Accuracy:
60  * according to an error analysis, the error is always less than
61  * 1 ulp (unit in the last place).
62  *
63  * Misc. info.
64  * For IEEE double
65  * if x > 7.09782712893383973096e+02 then exp(x) overflow
66  * if x < -7.45133219101941108420e+02 then exp(x) underflow
67  *
68  * Constants:
69  * The hexadecimal values are the intended ones for the following
70  * constants. The decimal values may be used, provided that the
71  * compiler will convert from decimal to binary accurately enough
72  * to produce the hexadecimal values shown.
73  */
74 
75 #include "math_libm.h"
76 #include "math_private.h"
77 
78 static const double
79 one = 1.0,
80 halF[2] = {0.5,-0.5,},
81 huge = 1.0e+300,
82 twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
83 o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
84 u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
85 ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
86  -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
87 ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
88  -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
89 invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
90 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
91 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
92 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
93 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
94 P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
95 
96 double __ieee754_exp(double x) /* default IEEE double exp */
97 {
98  double y;
99  double hi = 0.0;
100  double lo = 0.0;
101  double c;
102  double t;
103  int32_t k=0;
104  int32_t xsb;
105  u_int32_t hx;
106 
107  GET_HIGH_WORD(hx,x);
108  xsb = (hx>>31)&1; /* sign bit of x */
109  hx &= 0x7fffffff; /* high word of |x| */
110 
111  /* filter out non-finite argument */
112  if(hx >= 0x40862E42) { /* if |x|>=709.78... */
113  if(hx>=0x7ff00000) {
114  u_int32_t lx;
115  GET_LOW_WORD(lx,x);
116  if(((hx&0xfffff)|lx)!=0)
117  return x+x; /* NaN */
118  else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
119  }
120  #if 1
121  if(x > o_threshold) return huge*huge; /* overflow */
122  #else /* !!! FIXME: check this: "huge * huge" is a compiler warning, maybe they wanted +Inf? */
123  if(x > o_threshold) return INFINITY; /* overflow */
124  #endif
125 
126  if(x < u_threshold) return twom1000*twom1000; /* underflow */
127  }
128 
129  /* argument reduction */
130  if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
131  if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
132  hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
133  } else {
134  k = (int32_t) (invln2*x+halF[xsb]);
135  t = k;
136  hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
137  lo = t*ln2LO[0];
138  }
139  x = hi - lo;
140  }
141  else if(hx < 0x3e300000) { /* when |x|<2**-28 */
142  if(huge+x>one) return one+x;/* trigger inexact */
143  }
144  else k = 0;
145 
146  /* x is now in primary range */
147  t = x*x;
148  c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
149  if(k==0) return one-((x*c)/(c-2.0)-x);
150  else y = one-((lo-(x*c)/(2.0-c))-hi);
151  if(k >= -1021) {
152  u_int32_t hy;
153  GET_HIGH_WORD(hy,y);
154  SET_HIGH_WORD(y,hy+(k<<20)); /* add k to y's exponent */
155  return y;
156  } else {
157  u_int32_t hy;
158  GET_HIGH_WORD(hy,y);
159  SET_HIGH_WORD(y,hy+((k+1000)<<20)); /* add k to y's exponent */
160  return y*twom1000;
161  }
162 }
163 
164 /*
165  * wrapper exp(x)
166  */
167 #ifndef _IEEE_LIBM
168 double exp(double x)
169 {
170  static const double o_threshold = 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */
171  static const double u_threshold = -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */
172 
173  double z = __ieee754_exp(x);
174  if (_LIB_VERSION == _IEEE_)
175  return z;
176  if (isfinite(x)) {
177  if (x > o_threshold)
178  return __kernel_standard(x, x, 6); /* exp overflow */
179  if (x < u_threshold)
180  return __kernel_standard(x, x, 7); /* exp underflow */
181  }
182  return z;
183 }
184 #else
186 #endif
187 libm_hidden_def(exp)
#define GET_HIGH_WORD(i, d)
Definition: math_private.h:109
static const double twom1000
Definition: e_exp.c:82
static const double P5
Definition: e_exp.c:94
GLdouble GLdouble z
GLint GLint GLint GLint GLint x
Definition: SDL_opengl.h:1574
signed int int32_t
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
#define strong_alias(x, y)
Definition: math_private.h:28
static const double ln2HI[2]
Definition: e_exp.c:85
static const double one
Definition: e_exp.c:79
#define SET_HIGH_WORD(d, v)
Definition: math_private.h:137
static const double P4
Definition: e_exp.c:93
double __ieee754_exp(double x)
Definition: e_exp.c:96
unsigned int u_int32_t
Definition: math_private.h:31
static const double halF[2]
Definition: e_exp.c:80
static const double P3
Definition: e_exp.c:92
static const double invln2
Definition: e_exp.c:89
static const double ln2LO[2]
Definition: e_exp.c:87
const GLubyte * c
GLint GLint GLint GLint GLint GLint y
Definition: SDL_opengl.h:1574
static const double u_threshold
Definition: e_exp.c:84
#define GET_LOW_WORD(i, d)
Definition: math_private.h:118
static const double P2
Definition: e_exp.c:91
libm_hidden_def(scalbln)
Definition: s_scalbn.c:62
static const double huge
Definition: e_exp.c:81
GLdouble GLdouble t
Definition: SDL_opengl.h:2071
static const double P1
Definition: e_exp.c:90
static const double o_threshold
Definition: e_exp.c:83