1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_log.c */
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2 | /*
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3 | * ====================================================
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4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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5 | *
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6 | * Developed at SunSoft, a Sun Microsystems, Inc. business.
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7 | * Permission to use, copy, modify, and distribute this
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8 | * software is freely granted, provided that this notice
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9 | * is preserved.
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10 | * ====================================================
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11 | */
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12 | /* log(x)
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13 | * Return the logarithm of x
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14 | *
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15 | * Method :
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16 | * 1. Argument Reduction: find k and f such that
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17 | * x = 2^k * (1+f),
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18 | * where sqrt(2)/2 < 1+f < sqrt(2) .
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19 | *
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20 | * 2. Approximation of log(1+f).
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21 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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22 | * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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23 | * = 2s + s*R
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24 | * We use a special Remez algorithm on [0,0.1716] to generate
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25 | * a polynomial of degree 14 to approximate R The maximum error
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26 | * of this polynomial approximation is bounded by 2**-58.45. In
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27 | * other words,
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28 | * 2 4 6 8 10 12 14
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29 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
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30 | * (the values of Lg1 to Lg7 are listed in the program)
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31 | * and
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32 | * | 2 14 | -58.45
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33 | * | Lg1*s +...+Lg7*s - R(z) | <= 2
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34 | * | |
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35 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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36 | * In order to guarantee error in log below 1ulp, we compute log
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37 | * by
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38 | * log(1+f) = f - s*(f - R) (if f is not too large)
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39 | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
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40 | *
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41 | * 3. Finally, log(x) = k*ln2 + log(1+f).
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42 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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43 | * Here ln2 is split into two floating point number:
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44 | * ln2_hi + ln2_lo,
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45 | * where n*ln2_hi is always exact for |n| < 2000.
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46 | *
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47 | * Special cases:
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48 | * log(x) is NaN with signal if x < 0 (including -INF) ;
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49 | * log(+INF) is +INF; log(0) is -INF with signal;
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50 | * log(NaN) is that NaN with no signal.
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51 | *
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52 | * Accuracy:
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53 | * according to an error analysis, the error is always less than
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54 | * 1 ulp (unit in the last place).
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55 | *
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56 | * Constants:
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57 | * The hexadecimal values are the intended ones for the following
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58 | * constants. The decimal values may be used, provided that the
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59 | * compiler will convert from decimal to binary accurately enough
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60 | * to produce the hexadecimal values shown.
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61 | */
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62 |
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63 | #include <math.h>
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64 | #include <stdint.h>
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65 |
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66 | static const double
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67 | ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
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68 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
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69 | Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
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70 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
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71 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
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72 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
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73 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
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74 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
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75 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
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76 |
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77 | double log(double x)
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78 | {
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79 | union {double f; uint64_t i;} u = {x};
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80 | double_t hfsq,f,s,z,R,w,t1,t2,dk;
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81 | uint32_t hx;
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82 | int k;
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83 |
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84 | hx = u.i>>32;
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85 | k = 0;
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86 | if (hx < 0x00100000 || hx>>31) {
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87 | if (u.i<<1 == 0)
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88 | return -1/(x*x); /* log(+-0)=-inf */
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89 | if (hx>>31)
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90 | return (x-x)/0.0; /* log(-#) = NaN */
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91 | /* subnormal number, scale x up */
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92 | k -= 54;
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93 | x *= 0x1p54;
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94 | u.f = x;
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95 | hx = u.i>>32;
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96 | } else if (hx >= 0x7ff00000) {
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97 | return x;
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98 | } else if (hx == 0x3ff00000 && u.i<<32 == 0)
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99 | return 0;
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100 |
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101 | /* reduce x into [sqrt(2)/2, sqrt(2)] */
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102 | hx += 0x3ff00000 - 0x3fe6a09e;
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103 | k += (int)(hx>>20) - 0x3ff;
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104 | hx = (hx&0x000fffff) + 0x3fe6a09e;
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105 | u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
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106 | x = u.f;
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107 |
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108 | f = x - 1.0;
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109 | hfsq = 0.5*f*f;
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110 | s = f/(2.0+f);
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111 | z = s*s;
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112 | w = z*z;
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113 | t1 = w*(Lg2+w*(Lg4+w*Lg6));
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114 | t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
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115 | R = t2 + t1;
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116 | dk = k;
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117 | return s*(hfsq+R) + dk*ln2_lo - hfsq + f + dk*ln2_hi;
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118 | }
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