SDL  2.0
e_exp.c
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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
185 strong_alias(__ieee754_exp, exp)
186 #endif
187 libm_hidden_def(exp)