1 | /* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.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 SunPro, 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 | /* double log1p(double x)
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13 | * Return the natural logarithm of 1+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 | * 1+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 | * Note. If k=0, then f=x is exact. However, if k!=0, then f
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21 | * may not be representable exactly. In that case, a correction
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22 | * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
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23 | * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
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24 | * and add back the correction term c/u.
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25 | * (Note: when x > 2**53, one can simply return log(x))
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26 | *
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27 | * 2. Approximation of log(1+f): See log.c
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28 | *
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29 | * 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
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30 | *
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31 | * Special cases:
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32 | * log1p(x) is NaN with signal if x < -1 (including -INF) ;
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33 | * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
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34 | * log1p(NaN) is that NaN with no signal.
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35 | *
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36 | * Accuracy:
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37 | * according to an error analysis, the error is always less than
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38 | * 1 ulp (unit in the last place).
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39 | *
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40 | * Constants:
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41 | * The hexadecimal values are the intended ones for the following
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42 | * constants. The decimal values may be used, provided that the
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43 | * compiler will convert from decimal to binary accurately enough
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44 | * to produce the hexadecimal values shown.
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45 | *
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46 | * Note: Assuming log() return accurate answer, the following
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47 | * algorithm can be used to compute log1p(x) to within a few ULP:
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48 | *
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49 | * u = 1+x;
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50 | * if(u==1.0) return x ; else
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51 | * return log(u)*(x/(u-1.0));
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52 | *
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53 | * See HP-15C Advanced Functions Handbook, p.193.
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54 | */
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55 |
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56 | #include "libm.h"
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57 |
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58 | static const double
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59 | ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
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60 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
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61 | Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
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62 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
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63 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
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64 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
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65 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
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66 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
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67 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
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68 |
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69 | double log1p(double x)
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70 | {
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71 | union {double f; uint64_t i;} u = {x};
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72 | double_t hfsq,f,c,s,z,R,w,t1,t2,dk;
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73 | uint32_t hx,hu;
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74 | int k;
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75 |
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76 | hx = u.i>>32;
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77 | k = 1;
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78 | if (hx < 0x3fda827a || hx>>31) { /* 1+x < sqrt(2)+ */
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79 | if (hx >= 0xbff00000) { /* x <= -1.0 */
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80 | if (x == -1)
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81 | return x/0.0; /* log1p(-1) = -inf */
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82 | return (x-x)/0.0; /* log1p(x<-1) = NaN */
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83 | }
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84 | if (hx<<1 < 0x3ca00000<<1) { /* |x| < 2**-53 */
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85 | /* underflow if subnormal */
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86 | if ((hx&0x7ff00000) == 0)
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87 | FORCE_EVAL((float)x);
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88 | return x;
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89 | }
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90 | if (hx <= 0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
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91 | k = 0;
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92 | c = 0;
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93 | f = x;
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94 | }
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95 | } else if (hx >= 0x7ff00000)
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96 | return x;
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97 | if (k) {
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98 | u.f = 1 + x;
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99 | hu = u.i>>32;
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100 | hu += 0x3ff00000 - 0x3fe6a09e;
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101 | k = (int)(hu>>20) - 0x3ff;
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102 | /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
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103 | if (k < 54) {
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104 | c = k >= 2 ? 1-(u.f-x) : x-(u.f-1);
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105 | c /= u.f;
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106 | } else
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107 | c = 0;
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108 | /* reduce u into [sqrt(2)/2, sqrt(2)] */
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109 | hu = (hu&0x000fffff) + 0x3fe6a09e;
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110 | u.i = (uint64_t)hu<<32 | (u.i&0xffffffff);
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111 | f = u.f - 1;
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112 | }
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113 | hfsq = 0.5*f*f;
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114 | s = f/(2.0+f);
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115 | z = s*s;
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116 | w = z*z;
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117 | t1 = w*(Lg2+w*(Lg4+w*Lg6));
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118 | t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
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119 | R = t2 + t1;
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120 | dk = k;
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121 | return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi;
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122 | }
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