1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */
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2 | /*
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3 | * ====================================================
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4 | * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
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5 | *
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6 | * Permission to use, copy, modify, and distribute this
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7 | * software is freely granted, provided that this notice
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8 | * is preserved.
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9 | * ====================================================
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10 | */
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11 | /* exp(x)
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12 | * Returns the exponential of x.
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13 | *
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14 | * Method
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15 | * 1. Argument reduction:
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16 | * Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
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17 | * Given x, find r and integer k such that
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18 | *
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19 | * x = k*ln2 + r, |r| <= 0.5*ln2.
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20 | *
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21 | * Here r will be represented as r = hi-lo for better
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22 | * accuracy.
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23 | *
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24 | * 2. Approximation of exp(r) by a special rational function on
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25 | * the interval [0,0.34658]:
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26 | * Write
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27 | * R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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28 | * We use a special Remez algorithm on [0,0.34658] to generate
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29 | * a polynomial of degree 5 to approximate R. The maximum error
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30 | * of this polynomial approximation is bounded by 2**-59. In
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31 | * other words,
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32 | * R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
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33 | * (where z=r*r, and the values of P1 to P5 are listed below)
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34 | * and
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35 | * | 5 | -59
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36 | * | 2.0+P1*z+...+P5*z - R(z) | <= 2
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37 | * | |
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38 | * The computation of exp(r) thus becomes
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39 | * 2*r
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40 | * exp(r) = 1 + ----------
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41 | * R(r) - r
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42 | * r*c(r)
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43 | * = 1 + r + ----------- (for better accuracy)
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44 | * 2 - c(r)
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45 | * where
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46 | * 2 4 10
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47 | * c(r) = r - (P1*r + P2*r + ... + P5*r ).
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48 | *
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49 | * 3. Scale back to obtain exp(x):
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50 | * From step 1, we have
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51 | * exp(x) = 2^k * exp(r)
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52 | *
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53 | * Special cases:
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54 | * exp(INF) is INF, exp(NaN) is NaN;
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55 | * exp(-INF) is 0, and
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56 | * for finite argument, only exp(0)=1 is exact.
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57 | *
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58 | * Accuracy:
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59 | * according to an error analysis, the error is always less than
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60 | * 1 ulp (unit in the last place).
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61 | *
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62 | * Misc. info.
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63 | * For IEEE double
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64 | * if x > 709.782712893383973096 then exp(x) overflows
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65 | * if x < -745.133219101941108420 then exp(x) underflows
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66 | */
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67 |
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68 | #include "libm.h"
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69 |
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70 | static const double
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71 | half[2] = {0.5,-0.5},
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72 | ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
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73 | ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
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74 | invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
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75 | P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
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76 | P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
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77 | P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
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78 | P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
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79 | P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
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80 |
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81 | double exp(double x)
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82 | {
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83 | double_t hi, lo, c, xx, y;
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84 | int k, sign;
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85 | uint32_t hx;
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86 |
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87 | GET_HIGH_WORD(hx, x);
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88 | sign = hx>>31;
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89 | hx &= 0x7fffffff; /* high word of |x| */
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90 |
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91 | /* special cases */
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92 | if (hx >= 0x4086232b) { /* if |x| >= 708.39... */
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93 | if (isnan(x))
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94 | return x;
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95 | if (x > 709.782712893383973096) {
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96 | /* overflow if x!=inf */
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97 | x *= 0x1p1023;
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98 | return x;
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99 | }
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100 | if (x < -708.39641853226410622) {
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101 | /* underflow if x!=-inf */
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102 | FORCE_EVAL((float)(-0x1p-149/x));
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103 | if (x < -745.13321910194110842)
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104 | return 0;
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105 | }
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106 | }
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107 |
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108 | /* argument reduction */
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109 | if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
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110 | if (hx >= 0x3ff0a2b2) /* if |x| >= 1.5 ln2 */
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111 | k = (int)(invln2*x + half[sign]);
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112 | else
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113 | k = 1 - sign - sign;
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114 | hi = x - k*ln2hi; /* k*ln2hi is exact here */
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115 | lo = k*ln2lo;
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116 | x = hi - lo;
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117 | } else if (hx > 0x3e300000) { /* if |x| > 2**-28 */
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118 | k = 0;
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119 | hi = x;
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120 | lo = 0;
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121 | } else {
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122 | /* inexact if x!=0 */
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123 | FORCE_EVAL(0x1p1023 + x);
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124 | return 1 + x;
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125 | }
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126 |
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127 | /* x is now in primary range */
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128 | xx = x*x;
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129 | c = x - xx*(P1+xx*(P2+xx*(P3+xx*(P4+xx*P5))));
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130 | y = 1 + (x*c/(2-c) - lo + hi);
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131 | if (k == 0)
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132 | return y;
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133 | return scalbn(y, k);
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134 | }
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