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1
2/* @(#)e_exp.c 1.6 04/04/22 */
3/* \$FreeBSD: head/lib/msun/src/e_exp.c 251024 2013-05-27 08:50:10Z das \$ */
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4/*
5 * ====================================================
b34b60bc 7 *
b34b60bc 8 * Permission to use, copy, modify, and distribute this
6ff43c94 9 * software is freely granted, provided that this notice
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10 * is preserved.
11 * ====================================================
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12 */
13
6ff43c94 14/* __ieee754_exp(x)
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15 * Returns the exponential of x.
16 *
17 * Method
18 * 1. Argument reduction:
19 * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
20 * Given x, find r and integer k such that
21 *
6ff43c94 22 * x = k*ln2 + r, |r| <= 0.5*ln2.
b34b60bc 23 *
6ff43c94 24 * Here r will be represented as r = hi-lo for better
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25 * accuracy.
26 *
27 * 2. Approximation of exp(r) by a special rational function on
28 * the interval [0,0.34658]:
29 * Write
30 * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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31 * We use a special Remes algorithm on [0,0.34658] to generate
32 * a polynomial of degree 5 to approximate R. The maximum error
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33 * of this polynomial approximation is bounded by 2**-59. In
34 * other words,
35 * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
36 * (where z=r*r, and the values of P1 to P5 are listed below)
37 * and
38 * | 5 | -59
6ff43c94 39 * | 2.0+P1*z+...+P5*z - R(z) | <= 2
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40 * | |
41 * The computation of exp(r) thus becomes
42 * 2*r
43 * exp(r) = 1 + -------
44 * R - r
6ff43c94 45 * r*R1(r)
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46 * = 1 + r + ----------- (for better accuracy)
47 * 2 - R1(r)
48 * where
49 * 2 4 10
50 * R1(r) = r - (P1*r + P2*r + ... + P5*r ).
6ff43c94 51 *
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52 * 3. Scale back to obtain exp(x):
53 * From step 1, we have
54 * exp(x) = 2^k * exp(r)
55 *
56 * Special cases:
57 * exp(INF) is INF, exp(NaN) is NaN;
58 * exp(-INF) is 0, and
59 * for finite argument, only exp(0)=1 is exact.
60 *
61 * Accuracy:
62 * according to an error analysis, the error is always less than
63 * 1 ulp (unit in the last place).
64 *
65 * Misc. info.
6ff43c94 66 * For IEEE double
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67 * if x > 7.09782712893383973096e+02 then exp(x) overflow
68 * if x < -7.45133219101941108420e+02 then exp(x) underflow
69 *
70 * Constants:
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71 * The hexadecimal values are the intended ones for the following
72 * constants. The decimal values may be used, provided that the
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73 * compiler will convert from decimal to binary accurately enough
74 * to produce the hexadecimal values shown.
75 */
76
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77#include <float.h>
78
79#include "math.h"
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80#include "math_private.h"
81
82static const double
83one = 1.0,
84halF[2] = {0.5,-0.5,},
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85o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
86u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
87ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
88 -6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
89ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
90 -1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
91invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
92P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
93P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
94P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
95P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
96P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
97
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98static volatile double
99huge = 1.0e+300,
100twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/
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101
102double
6ff43c94 103__ieee754_exp(double x) /* default IEEE double exp */
b34b60bc 104{
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105 double y,hi=0.0,lo=0.0,c,t,twopk;
106 int32_t k=0,xsb;
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107 u_int32_t hx;
108
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109 GET_HIGH_WORD(hx,x);
110 xsb = (hx>>31)&1; /* sign bit of x */
111 hx &= 0x7fffffff; /* high word of |x| */
112
113 /* filter out non-finite argument */
114 if(hx >= 0x40862E42) { /* if |x|>=709.78... */
115 if(hx>=0x7ff00000) {
116 u_int32_t lx;
117 GET_LOW_WORD(lx,x);
118 if(((hx&0xfffff)|lx)!=0)
119 return x+x; /* NaN */
120 else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
121 }
122 if(x > o_threshold) return huge*huge; /* overflow */
123 if(x < u_threshold) return twom1000*twom1000; /* underflow */
124 }
125
126 /* argument reduction */
6ff43c94 127 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
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128 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
129 hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
130 } else {
6ff43c94 131 k = (int)(invln2*x+halF[xsb]);
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132 t = k;
133 hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
134 lo = t*ln2LO[0];
135 }
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136 STRICT_ASSIGN(double, x, hi - lo);
137 }
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138 else if(hx < 0x3e300000) { /* when |x|<2**-28 */
139 if(huge+x>one) return one+x;/* trigger inexact */
140 }
141 else k = 0;
142
143 /* x is now in primary range */
144 t = x*x;
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145 if(k >= -1021)
146 INSERT_WORDS(twopk,0x3ff00000+(k<<20), 0);
147 else
148 INSERT_WORDS(twopk,0x3ff00000+((k+1000)<<20), 0);
b34b60bc 149 c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
6ff43c94 150 if(k==0) return one-((x*c)/(c-2.0)-x);
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151 else y = one-((lo-(x*c)/(2.0-c))-hi);
152 if(k >= -1021) {
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153 if (k==1024) return y*2.0*0x1p1023;
154 return y*twopk;
b34b60bc 155 } else {
6ff43c94 156 return y*twopk*twom1000;
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157 }
158}
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159
160#if (LDBL_MANT_DIG == 53)
161__weak_reference(exp, expl);
162#endif