| 1 | /*
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| 2 | * ====================================================
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| 3 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 4 | *
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| 5 | * Developed at SunSoft, a Sun Microsystems, Inc. business.
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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 |
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| 12 | /** @addtogroup libmath
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| 13 | * @{
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| 14 | */
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| 15 | /** @file log mathematical function
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| 16 | */
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| 17 |
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| 18 | /* log(x)
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| 19 | * Return the logarithm of x
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| 20 | *
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| 21 | * Method :
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| 22 | * 1. Argument Reduction: find k and f such that
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| 23 | * x = 2^k * (1+f),
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| 24 | * where sqrt(2)/2 < 1+f < sqrt(2) .
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| 25 | *
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| 26 | * 2. Approximation of log(1+f).
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| 27 | * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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| 28 | * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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| 29 | * = 2s + s*R
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| 30 | * We use a special Remez algorithm on [0,0.1716] to generate
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| 31 | * a polynomial of degree 14 to approximate R The maximum error
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| 32 | * of this polynomial approximation is bounded by 2**-58.45. In
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| 33 | * other words,
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| 34 | * 2 4 6 8 10 12 14
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| 35 | * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
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| 36 | * (the values of Lg1 to Lg7 are listed in the program)
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| 37 | * and
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| 38 | * | 2 14 | -58.45
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| 39 | * | Lg1*s +...+Lg7*s - R(z) | <= 2
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| 40 | * | |
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| 41 | * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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| 42 | * In order to guarantee error in log below 1ulp, we compute log
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| 43 | * by
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| 44 | * log(1+f) = f - s*(f - R) (if f is not too large)
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| 45 | * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
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| 46 | *
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| 47 | * 3. Finally, log(x) = k*ln2 + log(1+f).
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| 48 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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| 49 | * Here ln2 is split into two floating point number:
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| 50 | * ln2_hi + ln2_lo,
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| 51 | * where n*ln2_hi is always exact for |n| < 2000.
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| 52 | *
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| 53 | * Special cases:
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| 54 | * log(x) is NaN with signal if x < 0 (including -INF) ;
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| 55 | * log(+INF) is +INF; log(0) is -INF with signal;
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| 56 | * log(NaN) is that NaN with no signal.
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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 | * Constants:
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| 63 | * The hexadecimal values are the intended ones for the following
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| 64 | * constants. The decimal values may be used, provided that the
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| 65 | * compiler will convert from decimal to binary accurately enough
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| 66 | * to produce the hexadecimal values shown.
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| 67 | */
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| 68 |
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| 69 | #include <math.h>
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| 70 | #include <stdint.h>
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| 71 | static const double
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| 72 | ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
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| 73 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
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| 74 | Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
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| 75 | Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
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| 76 | Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
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| 77 | Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
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| 78 | Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
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| 79 | Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
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| 80 | Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
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| 81 |
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| 82 | double log(double x)
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| 83 | {
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| 84 | union {double f; uint64_t i;} u = {x};
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| 85 | double_t hfsq,f,s,z,R,w,t1,t2,dk;
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| 86 | uint32_t hx;
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| 87 | int k;
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| 88 | hx = u.i>>32;
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| 89 | k = 0;
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| 90 | if (hx < 0x00100000 || hx>>31) {
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| 91 | if (u.i<<1 == 0)
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| 92 | return -1/(x*x); /* log(+-0)=-inf */
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| 93 | if (hx>>31)
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| 94 | return (x-x)/0.0; /* log(-#) = NaN */
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| 95 | /* subnormal number, scale x up */
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| 96 | k -= 54;
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| 97 | x *= 0x1p54;
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| 98 | u.f = x;
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| 99 | hx = u.i>>32;
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| 100 | } else if (hx >= 0x7ff00000) {
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| 101 | return x;
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| 102 | } else if (hx == 0x3ff00000 && u.i<<32 == 0)
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| 103 | return 0;
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| 104 |
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| 105 | /* reduce x into [sqrt(2)/2, sqrt(2)] */
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| 106 | hx += 0x3ff00000 - 0x3fe6a09e;
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| 107 | k += (int)(hx>>20) - 0x3ff;
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| 108 | hx = (hx&0x000fffff) + 0x3fe6a09e;
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| 109 | u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
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| 110 | x = u.f;
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| 111 | f = x - 1.0;
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| 112 | hfsq = 0.5*f*f;
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| 113 | s = f/(2.0+f);
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| 114 | z = s*s;
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| 115 | w = z*z;
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| 116 | t1 = w*(Lg2+w*(Lg4+w*Lg6));
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| 117 | t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
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| 118 | R = t2 + t1;
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| 119 | dk = k;
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| 120 |
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| 121 | return s*(hfsq+R) + dk*ln2_lo - hfsq + f + dk*ln2_hi;
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| 122 | }
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| 123 |
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| 124 | /** @}
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| 125 | */
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