src/libm/e_log.c
author Holmes Futrell <hfutrell@umail.ucsb.edu>
Thu, 01 Jan 2009 23:49:28 +0000
changeset 2950 04c9f1e4c496
parent 2756 a98604b691c8
child 3162 dc1eb82ffdaa
permissions -rw-r--r--
Added target testdraw2 for running the test/testdraw2.c test.
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/* @(#)e_log.c 5.1 93/09/24 */
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/*
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 * ====================================================
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 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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 *
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 * Developed at SunPro, a Sun Microsystems, Inc. business.
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 * Permission to use, copy, modify, and distribute this
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 * software is freely granted, provided that this notice
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 * is preserved.
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 * ====================================================
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 */
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#if defined(LIBM_SCCS) && !defined(lint)
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static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
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#endif
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/* __ieee754_log(x)
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 * Return the logrithm of x
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 *
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 * Method :
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 *   1. Argument Reduction: find k and f such that
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 *			x = 2^k * (1+f),
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 *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
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 *
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 *   2. Approximation of log(1+f).
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 *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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 *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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 *	     	 = 2s + s*R
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 *      We use a special Reme algorithm on [0,0.1716] to generate
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 * 	a polynomial of degree 14 to approximate R The maximum error
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 *	of this polynomial approximation is bounded by 2**-58.45. In
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 *	other words,
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 *		        2      4      6      8      10      12      14
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 *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
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 *  	(the values of Lg1 to Lg7 are listed in the program)
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 *	and
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 *	    |      2          14          |     -58.45
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 *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
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 *	    |                             |
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 *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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 *	In order to guarantee error in log below 1ulp, we compute log
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 *	by
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 *		log(1+f) = f - s*(f - R)	(if f is not too large)
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 *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
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 *
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 *	3. Finally,  log(x) = k*ln2 + log(1+f).
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 *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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 *	   Here ln2 is split into two floating point number:
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 *			ln2_hi + ln2_lo,
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 *	   where n*ln2_hi is always exact for |n| < 2000.
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 *
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 * Special cases:
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 *	log(x) is NaN with signal if x < 0 (including -INF) ;
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 *	log(+INF) is +INF; log(0) is -INF with signal;
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 *	log(NaN) is that NaN with no signal.
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 *
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 * Accuracy:
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 *	according to an error analysis, the error is always less than
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 *	1 ulp (unit in the last place).
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 *
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 * Constants:
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 * The hexadecimal values are the intended ones for the following
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 * constants. The decimal values may be used, provided that the
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 * compiler will convert from decimal to binary accurately enough
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 * to produce the hexadecimal values shown.
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 */
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#include "math.h"
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#include "math_private.h"
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#ifdef __STDC__
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static const double
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#else
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static double
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#endif
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  ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
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    ln2_lo = 1.90821492927058770002e-10,        /* 3dea39ef 35793c76 */
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    two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
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    Lg1 = 6.666666666666735130e-01,     /* 3FE55555 55555593 */
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    Lg2 = 3.999999999940941908e-01,     /* 3FD99999 9997FA04 */
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    Lg3 = 2.857142874366239149e-01,     /* 3FD24924 94229359 */
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    Lg4 = 2.222219843214978396e-01,     /* 3FCC71C5 1D8E78AF */
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    Lg5 = 1.818357216161805012e-01,     /* 3FC74664 96CB03DE */
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    Lg6 = 1.531383769920937332e-01,     /* 3FC39A09 D078C69F */
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    Lg7 = 1.479819860511658591e-01;     /* 3FC2F112 DF3E5244 */
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#ifdef __STDC__
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static const double zero = 0.0;
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#else
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static double zero = 0.0;
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#endif
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#ifdef __STDC__
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double attribute_hidden
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__ieee754_log(double x)
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#else
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double attribute_hidden
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__ieee754_log(x)
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     double x;
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#endif
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{
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    double hfsq, f, s, z, R, w, t1, t2, dk;
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    int32_t k, hx, i, j;
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    u_int32_t lx;
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    EXTRACT_WORDS(hx, lx, x);
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    k = 0;
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    if (hx < 0x00100000) {      /* x < 2**-1022  */
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        if (((hx & 0x7fffffff) | lx) == 0)
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            return -two54 / zero;       /* log(+-0)=-inf */
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        if (hx < 0)
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            return (x - x) / zero;      /* log(-#) = NaN */
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        k -= 54;
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        x *= two54;             /* subnormal number, scale up x */
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        GET_HIGH_WORD(hx, x);
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    }
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    if (hx >= 0x7ff00000)
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        return x + x;
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    k += (hx >> 20) - 1023;
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    hx &= 0x000fffff;
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    i = (hx + 0x95f64) & 0x100000;
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    SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000));    /* normalize x or x/2 */
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    k += (i >> 20);
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    f = x - 1.0;
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    if ((0x000fffff & (2 + hx)) < 3) {  /* |f| < 2**-20 */
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        if (f == zero) {
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            if (k == 0)
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                return zero;
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            else {
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                dk = (double) k;
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                return dk * ln2_hi + dk * ln2_lo;
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            }
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        }
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        R = f * f * (0.5 - 0.33333333333333333 * f);
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        if (k == 0)
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            return f - R;
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        else {
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            dk = (double) k;
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            return dk * ln2_hi - ((R - dk * ln2_lo) - f);
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        }
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    }
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    s = f / (2.0 + f);
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    dk = (double) k;
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    z = s * s;
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    i = hx - 0x6147a;
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    w = z * z;
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    j = 0x6b851 - hx;
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    t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
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    t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
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    i |= j;
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    R = t2 + t1;
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    if (i > 0) {
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        hfsq = 0.5 * f * f;
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        if (k == 0)
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            return f - (hfsq - s * (hfsq + R));
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        else
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            return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -
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                                  f);
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    } else {
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        if (k == 0)
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            return f - s * (f - R);
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        else
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            return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
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    }
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}