[388] | 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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