e_sqrt.c

ecos实时嵌入式操作系统

//===========================================================================
//
//      e_sqrt.c
//
//      Part of the standard mathematical function library
//
//===========================================================================
//####ECOSGPLCOPYRIGHTBEGIN####
// -------------------------------------------
// This file is part of eCos, the Embedded Configurable Operating System.
// Copyright (C) 1998, 1999, 2000, 2001, 2002 Red Hat, Inc.
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// or inline functions from this file, or you compile this file and link it
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//####ECOSGPLCOPYRIGHTEND####
//===========================================================================
//#####DESCRIPTIONBEGIN####
//
// Author(s):   jlarmour
// Contributors:  jlarmour
// Date:        1998-02-13
// Purpose:     
// Description: 
// Usage:       
//
//####DESCRIPTIONEND####
//
//===========================================================================

// CONFIGURATION

#include <pkgconf/libm.h>   // Configuration header

// Include the Math library?
#ifdef CYGPKG_LIBM     

// Derived from code with the following copyright

/* @(#)e_sqrt.c 1.3 95/01/18 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice 
 * is preserved.
 * ====================================================
 */

/* __ieee754_sqrt(x)
 * Return correctly rounded sqrt.
 *           ------------------------------------------
 *           |  Use the hardware sqrt if you have one |
 *           ------------------------------------------
 * Method: 
 *   Bit by bit method using integer arithmetic. (Slow, but portable) 
 *   1. Normalization
 *      Scale x to y in [1,4) with even powers of 2: 
 *      find an integer k such that  1 <= (y=x*2^(2k)) < 4, then
 *              sqrt(x) = 2^k * sqrt(y)
 *   2. Bit by bit computation
 *      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
 *           i                                                   0
 *                                     i+1         2
 *          s  = 2*q , and      y  =  2   * ( y - q  ).         (1)
 *           i      i            i                 i
 *                                                        
 *      To compute q    from q , one checks whether 
 *                  i+1       i                       
 *
 *                            -(i+1) 2
 *                      (q + 2      ) <= y.                     (2)
 *                        i
 *                                                            -(i+1)
 *      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
 *                             i+1   i             i+1   i
 *
 *      With some algebric manipulation, it is not difficult to see
 *      that (2) is equivalent to 
 *                             -(i+1)
 *                      s  +  2       <= y                      (3)
 *                       i                i
 *
 *      The advantage of (3) is that s  and y  can be computed by 
 *                                    i      i
 *      the following recurrence formula:
 *          if (3) is false
 *
 *          s     =  s  ,       y    = y   ;                    (4)
 *           i+1      i          i+1    i
 *
 *          otherwise,
 *                         -i                     -(i+1)
 *          s     =  s  + 2  ,  y    = y  -  s  - 2             (5)
 *           i+1      i          i+1    i     i
 *                              
 *      One may easily use induction to prove (4) and (5). 
 *      Note. Since the left hand side of (3) contain only i+2 bits,
 *            it does not necessary to do a full (53-bit) comparison 
 *            in (3).
 *   3. Final rounding
 *      After generating the 53 bits result, we compute one more bit.
 *      Together with the remainder, we can decide whether the
 *      result is exact, bigger than 1/2ulp, or less than 1/2ulp
 *      (it will never equal to 1/2ulp).
 *      The rounding mode can be detected by checking whether
 *      huge + tiny is equal to huge, and whether huge - tiny is
 *      equal to huge for some floating point number "huge" and "tiny".
 *              
 * Special cases:
 *      sqrt(+-0) = +-0         ... exact
 *      sqrt(inf) = inf
 *      sqrt(-ve) = NaN         ... with invalid signal
 *      sqrt(NaN) = NaN         ... with invalid signal for signaling NaN
 *
 * Other methods : see the appended file at the end of the program below.
 *---------------
 */

#include "mathincl/fdlibm.h"

static  const double    one     = 1.0, tiny=1.0e-300;

        double __ieee754_sqrt(double x)
{
        double z;
        int     sign = (int)0x80000000; 
        unsigned r,t1,s1,ix1,q1;
        int ix0,s0,q,m,t,i;

        ix0 = CYG_LIBM_HI(x);                   /* high word of x */
        ix1 = CYG_LIBM_LO(x);           /* low word of x */

    /* take care of Inf and NaN */
        if((ix0&0x7ff00000)==0x7ff00000) {                      
            return x*x+x;               /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
                                           sqrt(-inf)=sNaN */
        } 
    /* take care of zero */
        if(ix0<=0) {
            if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
            else if(ix0<0)
                return (x-x)/(x-x);             /* sqrt(-ve) = sNaN */
        }
    /* normalize x */
        m = (ix0>>20);
        if(m==0) {                              /* subnormal x */
            while(ix0==0) {
                m -= 21;
                ix0 |= (ix1>>11); ix1 <<= 21;
            }
            for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
            m -= i-1;
            ix0 |= (ix1>>(32-i));
            ix1 <<= i;
        }
        m -= 1023;      /* unbias exponent */
        ix0 = (ix0&0x000fffff)|0x00100000;
        if(m&1){        /* odd m, double x to make it even */
            ix0 += ix0 + ((ix1&sign)>>31);
            ix1 += ix1;
        }
        m >>= 1;        /* m = [m/2] */

    /* generate sqrt(x) bit by bit */
        ix0 += ix0 + ((ix1&sign)>>31);
        ix1 += ix1;
        q = q1 = s0 = s1 = 0;   /* [q,q1] = sqrt(x) */
        r = 0x00200000;         /* r = moving bit from right to left */

        while(r!=0) {
            t = s0+r; 
            if(t<=ix0) { 
                s0   = t+r; 
                ix0 -= t; 
                q   += r; 
            } 
            ix0 += ix0 + ((ix1&sign)>>31);
            ix1 += ix1;
            r>>=1;
        }

        r = sign;
        while(r!=0) {
            t1 = s1+r; 
            t  = s0;
            if((t<ix0)||((t==ix0)&&(t1<=ix1))) { 
                s1  = t1+r;
                if(((t1&sign)==(unsigned)sign)&&(s1&sign)==0) s0 += 1;
                ix0 -= t;
                if (ix1 < t1) ix0 -= 1;
                ix1 -= t1;
                q1  += r;
            }
            ix0 += ix0 + ((ix1&sign)>>31);
            ix1 += ix1;
            r>>=1;
        }

    /* use floating add to find out rounding direction */
        if((ix0|ix1)!=0) {
            z = one-tiny; /* trigger inexact flag */
            if (z>=one) {
                z = one+tiny;
                if (q1==(unsigned)0xffffffff) { q1=0; q += 1;}
                else if (z>one) {
                    if (q1==(unsigned)0xfffffffe) q+=1;
                    q1+=2; 
                } else
                    q1 += (q1&1);
            }
        }
        ix0 = (q>>1)+0x3fe00000;
        ix1 =  q1>>1;
        if ((q&1)==1) ix1 |= sign;
        ix0 += (m <<20);
        CYG_LIBM_HI(z) = ix0;
        CYG_LIBM_LO(z) = ix1;
        return z;
}

/*
Other methods  (use floating-point arithmetic)
-------------
(This is a copy of a drafted paper by Prof W. Kahan 
and K.C. Ng, written in May, 1986)

        Two algorithms are given here to implement sqrt(x) 
        (IEEE double precision arithmetic) in software.