1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_log2.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 | /*
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13 | * Return the base 2 logarithm of x. See log.c for most comments.
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14 | *
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15 | * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2
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16 | * as in log.c, then combine and scale in extra precision:
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17 | * log2(x) = (f - f*f/2 + r)/log(2) + k
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18 | */
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19 |
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20 | #include <math.h>
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21 | #include <stdint.h>
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22 |
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23 | static const double
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24 | ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
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25 | ivln2lo = 1.67517131648865118353e-10, /* 0x3de705fc, 0x2eefa200 */
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26 | Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
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27 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
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28 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
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29 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
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30 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
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31 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
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32 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
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33 |
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34 | double log2(double x)
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35 | {
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36 | union {double f; uint64_t i;} u = {x};
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37 | double_t hfsq,f,s,z,R,w,t1,t2,y,hi,lo,val_hi,val_lo;
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38 | uint32_t hx;
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39 | int k;
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40 |
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41 | hx = u.i>>32;
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42 | k = 0;
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43 | if (hx < 0x00100000 || hx>>31) {
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44 | if (u.i<<1 == 0)
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45 | return -1/(x*x); /* log(+-0)=-inf */
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46 | if (hx>>31)
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47 | return (x-x)/0.0; /* log(-#) = NaN */
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48 | /* subnormal number, scale x up */
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49 | k -= 54;
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50 | x *= 0x1p54;
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51 | u.f = x;
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52 | hx = u.i>>32;
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53 | } else if (hx >= 0x7ff00000) {
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54 | return x;
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55 | } else if (hx == 0x3ff00000 && u.i<<32 == 0)
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56 | return 0;
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57 |
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58 | /* reduce x into [sqrt(2)/2, sqrt(2)] */
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59 | hx += 0x3ff00000 - 0x3fe6a09e;
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60 | k += (int)(hx>>20) - 0x3ff;
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61 | hx = (hx&0x000fffff) + 0x3fe6a09e;
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62 | u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
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63 | x = u.f;
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64 |
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65 | f = x - 1.0;
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66 | hfsq = 0.5*f*f;
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67 | s = f/(2.0+f);
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68 | z = s*s;
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69 | w = z*z;
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70 | t1 = w*(Lg2+w*(Lg4+w*Lg6));
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71 | t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
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72 | R = t2 + t1;
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73 |
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74 | /*
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75 | * f-hfsq must (for args near 1) be evaluated in extra precision
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76 | * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
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77 | * This is fairly efficient since f-hfsq only depends on f, so can
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78 | * be evaluated in parallel with R. Not combining hfsq with R also
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79 | * keeps R small (though not as small as a true `lo' term would be),
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80 | * so that extra precision is not needed for terms involving R.
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81 | *
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82 | * Compiler bugs involving extra precision used to break Dekker's
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83 | * theorem for spitting f-hfsq as hi+lo, unless double_t was used
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84 | * or the multi-precision calculations were avoided when double_t
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85 | * has extra precision. These problems are now automatically
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86 | * avoided as a side effect of the optimization of combining the
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87 | * Dekker splitting step with the clear-low-bits step.
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88 | *
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89 | * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
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90 | * precision to avoid a very large cancellation when x is very near
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91 | * these values. Unlike the above cancellations, this problem is
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92 | * specific to base 2. It is strange that adding +-1 is so much
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93 | * harder than adding +-ln2 or +-log10_2.
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94 | *
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95 | * This uses Dekker's theorem to normalize y+val_hi, so the
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96 | * compiler bugs are back in some configurations, sigh. And I
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97 | * don't want to used double_t to avoid them, since that gives a
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98 | * pessimization and the support for avoiding the pessimization
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99 | * is not yet available.
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100 | *
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101 | * The multi-precision calculations for the multiplications are
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102 | * routine.
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103 | */
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104 |
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105 | /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */
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106 | hi = f - hfsq;
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107 | u.f = hi;
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108 | u.i &= (uint64_t)-1<<32;
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109 | hi = u.f;
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110 | lo = f - hi - hfsq + s*(hfsq+R);
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111 |
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112 | val_hi = hi*ivln2hi;
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113 | val_lo = (lo+hi)*ivln2lo + lo*ivln2hi;
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114 |
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115 | /* spadd(val_hi, val_lo, y), except for not using double_t: */
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116 | y = k;
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117 | w = y + val_hi;
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118 | val_lo += (y - w) + val_hi;
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119 | val_hi = w;
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120 |
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121 | return val_lo + val_hi;
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122 | }
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