src/video/e_sqrt.h
author Edgar Simo <bobbens@gmail.com>
Sun, 06 Jul 2008 17:06:37 +0000
branchgsoc2008_force_feedback
changeset 2498 ab567bd667bf
parent 1895 c121d94672cb
permissions -rw-r--r--
Fixed various mistakes in the doxygen.
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/* @(#)e_sqrt.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_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $";
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#endif
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/* __ieee754_sqrt(x)
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 * Return correctly rounded sqrt.
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 *           ------------------------------------------
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 *	     |  Use the hardware sqrt if you have one |
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 *           ------------------------------------------
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 * Method:
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 *   Bit by bit method using integer arithmetic. (Slow, but portable)
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 *   1. Normalization
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 *	Scale x to y in [1,4) with even powers of 2:
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 *	find an integer k such that  1 <= (y=x*2^(2k)) < 4, then
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 *		sqrt(x) = 2^k * sqrt(y)
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 *   2. Bit by bit computation
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 *	Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
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 *	     i							 0
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 *                                     i+1         2
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 *	    s  = 2*q , and	y  =  2   * ( y - q  ).		(1)
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 *	     i      i            i                 i
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 *
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 *	To compute q    from q , one checks whether
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 *		    i+1       i
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 *
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 *			      -(i+1) 2
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 *			(q + 2      ) <= y.			(2)
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 *     			  i
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 *							      -(i+1)
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 *	If (2) is false, then q   = q ; otherwise q   = q  + 2      .
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 *		 	       i+1   i             i+1   i
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 *
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 *	With some algebric manipulation, it is not difficult to see
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 *	that (2) is equivalent to
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 *                             -(i+1)
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 *			s  +  2       <= y			(3)
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 *			 i                i
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 *
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 *	The advantage of (3) is that s  and y  can be computed by
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 *				      i      i
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 *	the following recurrence formula:
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 *	    if (3) is false
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 *
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 *	    s     =  s  ,	y    = y   ;			(4)
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 *	     i+1      i		 i+1    i
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 *
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 *	    otherwise,
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 *                         -i                     -(i+1)
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 *	    s	  =  s  + 2  ,  y    = y  -  s  - 2  		(5)
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 *           i+1      i          i+1    i     i
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 *
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 *	One may easily use induction to prove (4) and (5).
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 *	Note. Since the left hand side of (3) contain only i+2 bits,
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 *	      it does not necessary to do a full (53-bit) comparison
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 *	      in (3).
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 *   3. Final rounding
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 *	After generating the 53 bits result, we compute one more bit.
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 *	Together with the remainder, we can decide whether the
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 *	result is exact, bigger than 1/2ulp, or less than 1/2ulp
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 *	(it will never equal to 1/2ulp).
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 *	The rounding mode can be detected by checking whether
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 *	huge + tiny is equal to huge, and whether huge - tiny is
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 *	equal to huge for some floating point number "huge" and "tiny".
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 *
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 * Special cases:
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 *	sqrt(+-0) = +-0 	... exact
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 *	sqrt(inf) = inf
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 *	sqrt(-ve) = NaN		... with invalid signal
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 *	sqrt(NaN) = NaN		... with invalid signal for signaling NaN
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 *
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 * Other methods : see the appended file at the end of the program below.
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 *---------------
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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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double SDL_NAME(copysign) (double x, double y)
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#else
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double SDL_NAME(copysign) (x, y)
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     double
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         x,
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         y;
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#endif
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{
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    u_int32_t hx, hy;
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    GET_HIGH_WORD(hx, x);
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    GET_HIGH_WORD(hy, y);
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    SET_HIGH_WORD(x, (hx & 0x7fffffff) | (hy & 0x80000000));
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    return x;
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}
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#ifdef __STDC__
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double SDL_NAME(scalbn) (double x, int n)
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#else
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double SDL_NAME(scalbn) (x, n)
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     double
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         x;
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     int
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         n;
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#endif
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{
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    int32_t k, hx, lx;
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    EXTRACT_WORDS(hx, lx, x);
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    k = (hx & 0x7ff00000) >> 20;        /* extract exponent */
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    if (k == 0) {               /* 0 or subnormal x */
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        if ((lx | (hx & 0x7fffffff)) == 0)
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            return x;           /* +-0 */
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        x *= two54;
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        GET_HIGH_WORD(hx, x);
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        k = ((hx & 0x7ff00000) >> 20) - 54;
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        if (n < -50000)
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            return tiny * x;    /*underflow */
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    }
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    if (k == 0x7ff)
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        return x + x;           /* NaN or Inf */
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    k = k + n;
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    if (k > 0x7fe)
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        return huge * SDL_NAME(copysign) (huge, x);     /* overflow  */
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    if (k > 0) {                /* normal result */
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        SET_HIGH_WORD(x, (hx & 0x800fffff) | (k << 20));
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        return x;
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    }
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    if (k <= -54) {
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        if (n > 50000)          /* in case integer overflow in n+k */
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            return huge * SDL_NAME(copysign) (huge, x); /*overflow */
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        else
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            return tiny * SDL_NAME(copysign) (tiny, x); /*underflow */
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    }
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    k += 54;                    /* subnormal result */
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    SET_HIGH_WORD(x, (hx & 0x800fffff) | (k << 20));
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    return x * twom54;
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}
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#ifdef __STDC__
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double
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__ieee754_sqrt(double x)
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#else
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double
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__ieee754_sqrt(x)
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     double x;
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#endif
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{
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    double z;
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    int32_t sign = (int) 0x80000000;
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    int32_t ix0, s0, q, m, t, i;
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    u_int32_t r, t1, s1, ix1, q1;
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    EXTRACT_WORDS(ix0, ix1, x);
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    /* take care of Inf and NaN */
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    if ((ix0 & 0x7ff00000) == 0x7ff00000) {
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        return x * x + x;       /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
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                                   sqrt(-inf)=sNaN */
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    }
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    /* take care of zero */
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    if (ix0 <= 0) {
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        if (((ix0 & (~sign)) | ix1) == 0)
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            return x;           /* sqrt(+-0) = +-0 */
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        else if (ix0 < 0)
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            return (x - x) / (x - x);   /* sqrt(-ve) = sNaN */
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    }
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    /* normalize x */
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    m = (ix0 >> 20);
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    if (m == 0) {               /* subnormal x */
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        while (ix0 == 0) {
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            m -= 21;
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            ix0 |= (ix1 >> 11);
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            ix1 <<= 21;
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        }
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        for (i = 0; (ix0 & 0x00100000) == 0; i++)
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            ix0 <<= 1;
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        m -= i - 1;
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        ix0 |= (ix1 >> (32 - i));
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        ix1 <<= i;
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    }
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    m -= 1023;                  /* unbias exponent */
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    ix0 = (ix0 & 0x000fffff) | 0x00100000;
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    if (m & 1) {                /* odd m, double x to make it even */
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        ix0 += ix0 + ((ix1 & sign) >> 31);
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        ix1 += ix1;
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    }
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    m >>= 1;                    /* m = [m/2] */
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    /* generate sqrt(x) bit by bit */
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    ix0 += ix0 + ((ix1 & sign) >> 31);
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    ix1 += ix1;
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    q = q1 = s0 = s1 = 0;       /* [q,q1] = sqrt(x) */
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    r = 0x00200000;             /* r = moving bit from right to left */
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    while (r != 0) {
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        t = s0 + r;
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        if (t <= ix0) {
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            s0 = t + r;
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            ix0 -= t;
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            q += r;
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        }
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        ix0 += ix0 + ((ix1 & sign) >> 31);
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        ix1 += ix1;
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        r >>= 1;
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    }
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    r = sign;
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    while (r != 0) {
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        t1 = s1 + r;
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        t = s0;
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        if ((t < ix0) || ((t == ix0) && (t1 <= ix1))) {
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            s1 = t1 + r;
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            if (((int32_t) (t1 & sign) == sign) && (s1 & sign) == 0)
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                s0 += 1;
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            ix0 -= t;
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            if (ix1 < t1)
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                ix0 -= 1;
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            ix1 -= t1;
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            q1 += r;
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        }
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        ix0 += ix0 + ((ix1 & sign) >> 31);
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        ix1 += ix1;
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        r >>= 1;
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    }
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    /* use floating add to find out rounding direction */
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    if ((ix0 | ix1) != 0) {
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        z = one - tiny;         /* trigger inexact flag */
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        if (z >= one) {
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            z = one + tiny;
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            if (q1 == (u_int32_t) 0xffffffff) {
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                q1 = 0;
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                q += 1;
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            } else if (z > one) {
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                if (q1 == (u_int32_t) 0xfffffffe)
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                    q += 1;
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                q1 += 2;
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            } else
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                q1 += (q1 & 1);
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        }
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    }
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    ix0 = (q >> 1) + 0x3fe00000;
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    ix1 = q1 >> 1;
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    if ((q & 1) == 1)
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        ix1 |= sign;
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    ix0 += (m << 20);
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    INSERT_WORDS(z, ix0, ix1);
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    return z;
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}
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/*
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Other methods  (use floating-point arithmetic)
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-------------
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(This is a copy of a drafted paper by Prof W. Kahan
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and K.C. Ng, written in May, 1986)
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	Two algorithms are given here to implement sqrt(x)
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	(IEEE double precision arithmetic) in software.
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	Both supply sqrt(x) correctly rounded. The first algorithm (in
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	Section A) uses newton iterations and involves four divisions.
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	The second one uses reciproot iterations to avoid division, but
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	requires more multiplications. Both algorithms need the ability
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	to chop results of arithmetic operations instead of round them,
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	and the INEXACT flag to indicate when an arithmetic operation
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	is executed exactly with no roundoff error, all part of the
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	standard (IEEE 754-1985). The ability to perform shift, add,
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	subtract and logical AND operations upon 32-bit words is needed
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	too, though not part of the standard.
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A.  sqrt(x) by Newton Iteration
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   (1)	Initial approximation
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	Let x0 and x1 be the leading and the trailing 32-bit words of
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	a floating point number x (in IEEE double format) respectively
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	    1    11		     52				  ...widths
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	   ------------------------------------------------------
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	x: |s|	  e     |	      f				|
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	   ------------------------------------------------------
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	      msb    lsb  msb				      lsb ...order
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	     ------------------------  	     ------------------------
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	x0:  |s|   e    |    f1     |	 x1: |          f2           |
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	     ------------------------  	     ------------------------
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	By performing shifts and subtracts on x0 and x1 (both regarded
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	as integers), we obtain an 8-bit approximation of sqrt(x) as
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	follows.
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		k  := (x0>>1) + 0x1ff80000;
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		y0 := k - T1[31&(k>>15)].	... y ~ sqrt(x) to 8 bits
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	Here k is a 32-bit integer and T1[] is an integer array containing
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	correction terms. Now magically the floating value of y (y's
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	leading 32-bit word is y0, the value of its trailing word is 0)
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	approximates sqrt(x) to almost 8-bit.
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	Value of T1:
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	static int T1[32]= {
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	0,	1024,	3062,	5746,	9193,	13348,	18162,	23592,
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	29598,	36145,	43202,	50740,	58733,	67158,	75992,	85215,
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	83599,	71378,	60428,	50647,	41945,	34246,	27478,	21581,
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	16499,	12183,	8588,	5674,	3403,	1742,	661,	130,};
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    (2)	Iterative refinement
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	Apply Heron's rule three times to y, we have y approximates
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	sqrt(x) to within 1 ulp (Unit in the Last Place):
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		y := (y+x/y)/2		... almost 17 sig. bits
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		y := (y+x/y)/2		... almost 35 sig. bits
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		y := y-(y-x/y)/2	... within 1 ulp
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	Remark 1.
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	    Another way to improve y to within 1 ulp is:
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		y := (y+x/y)		... almost 17 sig. bits to 2*sqrt(x)
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		y := y - 0x00100006	... almost 18 sig. bits to sqrt(x)
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				2
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			    (x-y )*y
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		y := y + 2* ----------	...within 1 ulp
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			       2
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			     3y  + x
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	This formula has one division fewer than the one above; however,
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	it requires more multiplications and additions. Also x must be
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	scaled in advance to avoid spurious overflow in evaluating the
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	expression 3y*y+x. Hence it is not recommended uless division
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	is slow. If division is very slow, then one should use the
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   343
	reciproot algorithm given in section B.
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   344
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   345
    (3) Final adjustment
slouken@1330
   346
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   347
	By twiddling y's last bit it is possible to force y to be
slouken@1330
   348
	correctly rounded according to the prevailing rounding mode
slouken@1330
   349
	as follows. Let r and i be copies of the rounding mode and
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   350
	inexact flag before entering the square root program. Also we
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   351
	use the expression y+-ulp for the next representable floating
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   352
	numbers (up and down) of y. Note that y+-ulp = either fixed
slouken@1330
   353
	point y+-1, or multiply y by nextafter(1,+-inf) in chopped
slouken@1330
   354
	mode.
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   355
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   356
		I := FALSE;	... reset INEXACT flag I
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   357
		R := RZ;	... set rounding mode to round-toward-zero
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   358
		z := x/y;	... chopped quotient, possibly inexact
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   359
		If(not I) then {	... if the quotient is exact
slouken@1330
   360
		    if(z=y) {
slouken@1330
   361
		        I := i;	 ... restore inexact flag
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   362
		        R := r;  ... restore rounded mode
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   363
		        return sqrt(x):=y.
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   364
		    } else {
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   365
			z := z - ulp;	... special rounding
slouken@1330
   366
		    }
slouken@1330
   367
		}
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   368
		i := TRUE;		... sqrt(x) is inexact
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   369
		If (r=RN) then z=z+ulp	... rounded-to-nearest
slouken@1330
   370
		If (r=RP) then {	... round-toward-+inf
slouken@1330
   371
		    y = y+ulp; z=z+ulp;
slouken@1330
   372
		}
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   373
		y := y+z;		... chopped sum
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   374
		y0:=y0-0x00100000;	... y := y/2 is correctly rounded.
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   375
	        I := i;	 		... restore inexact flag
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   376
	        R := r;  		... restore rounded mode
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   377
	        return sqrt(x):=y.
slouken@1330
   378
slouken@1330
   379
    (4)	Special cases
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   380
slouken@1330
   381
	Square root of +inf, +-0, or NaN is itself;
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   382
	Square root of a negative number is NaN with invalid signal.
slouken@1330
   383
slouken@1330
   384
slouken@1330
   385
B.  sqrt(x) by Reciproot Iteration
slouken@1330
   386
slouken@1330
   387
   (1)	Initial approximation
slouken@1330
   388
slouken@1330
   389
	Let x0 and x1 be the leading and the trailing 32-bit words of
slouken@1330
   390
	a floating point number x (in IEEE double format) respectively
slouken@1330
   391
	(see section A). By performing shifs and subtracts on x0 and y0,
slouken@1330
   392
	we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
slouken@1330
   393
slouken@1330
   394
	    k := 0x5fe80000 - (x0>>1);
slouken@1330
   395
	    y0:= k - T2[63&(k>>14)].	... y ~ 1/sqrt(x) to 7.8 bits
slouken@1330
   396
slouken@1330
   397
	Here k is a 32-bit integer and T2[] is an integer array
slouken@1330
   398
	containing correction terms. Now magically the floating
slouken@1330
   399
	value of y (y's leading 32-bit word is y0, the value of
slouken@1330
   400
	its trailing word y1 is set to zero) approximates 1/sqrt(x)
slouken@1330
   401
	to almost 7.8-bit.
slouken@1330
   402
slouken@1330
   403
	Value of T2:
slouken@1330
   404
	static int T2[64]= {
slouken@1330
   405
	0x1500,	0x2ef8,	0x4d67,	0x6b02,	0x87be,	0xa395,	0xbe7a,	0xd866,
slouken@1330
   406
	0xf14a,	0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
slouken@1330
   407
	0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
slouken@1330
   408
	0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
slouken@1330
   409
	0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
slouken@1330
   410
	0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
slouken@1330
   411
	0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
slouken@1330
   412
	0x1527f,0x1334a,0x11051,0xe951,	0xbe01,	0x8e0d,	0x5924,	0x1edd,};
slouken@1330
   413
slouken@1330
   414
    (2)	Iterative refinement
slouken@1330
   415
slouken@1330
   416
	Apply Reciproot iteration three times to y and multiply the
slouken@1330
   417
	result by x to get an approximation z that matches sqrt(x)
slouken@1330
   418
	to about 1 ulp. To be exact, we will have
slouken@1330
   419
		-1ulp < sqrt(x)-z<1.0625ulp.
slouken@1330
   420
slouken@1330
   421
	... set rounding mode to Round-to-nearest
slouken@1330
   422
	   y := y*(1.5-0.5*x*y*y)	... almost 15 sig. bits to 1/sqrt(x)
slouken@1330
   423
	   y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
slouken@1330
   424
	... special arrangement for better accuracy
slouken@1330
   425
	   z := x*y			... 29 bits to sqrt(x), with z*y<1
slouken@1330
   426
	   z := z + 0.5*z*(1-z*y)	... about 1 ulp to sqrt(x)
slouken@1330
   427
slouken@1330
   428
	Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
slouken@1330
   429
	(a) the term z*y in the final iteration is always less than 1;
slouken@1330
   430
	(b) the error in the final result is biased upward so that
slouken@1330
   431
		-1 ulp < sqrt(x) - z < 1.0625 ulp
slouken@1330
   432
	    instead of |sqrt(x)-z|<1.03125ulp.
slouken@1330
   433
slouken@1330
   434
    (3)	Final adjustment
slouken@1330
   435
slouken@1330
   436
	By twiddling y's last bit it is possible to force y to be
slouken@1330
   437
	correctly rounded according to the prevailing rounding mode
slouken@1330
   438
	as follows. Let r and i be copies of the rounding mode and
slouken@1330
   439
	inexact flag before entering the square root program. Also we
slouken@1330
   440
	use the expression y+-ulp for the next representable floating
slouken@1330
   441
	numbers (up and down) of y. Note that y+-ulp = either fixed
slouken@1330
   442
	point y+-1, or multiply y by nextafter(1,+-inf) in chopped
slouken@1330
   443
	mode.
slouken@1330
   444
slouken@1330
   445
	R := RZ;		... set rounding mode to round-toward-zero
slouken@1330
   446
	switch(r) {
slouken@1330
   447
	    case RN:		... round-to-nearest
slouken@1330
   448
	       if(x<= z*(z-ulp)...chopped) z = z - ulp; else
slouken@1330
   449
	       if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
slouken@1330
   450
	       break;
slouken@1330
   451
	    case RZ:case RM:	... round-to-zero or round-to--inf
slouken@1330
   452
	       R:=RP;		... reset rounding mod to round-to-+inf
slouken@1330
   453
	       if(x<z*z ... rounded up) z = z - ulp; else
slouken@1330
   454
	       if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
slouken@1330
   455
	       break;
slouken@1330
   456
	    case RP:		... round-to-+inf
slouken@1330
   457
	       if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
slouken@1330
   458
	       if(x>z*z ...chopped) z = z+ulp;
slouken@1330
   459
	       break;
slouken@1330
   460
	}
slouken@1330
   461
slouken@1330
   462
	Remark 3. The above comparisons can be done in fixed point. For
slouken@1330
   463
	example, to compare x and w=z*z chopped, it suffices to compare
slouken@1330
   464
	x1 and w1 (the trailing parts of x and w), regarding them as
slouken@1330
   465
	two's complement integers.
slouken@1330
   466
slouken@1330
   467
	...Is z an exact square root?
slouken@1330
   468
	To determine whether z is an exact square root of x, let z1 be the
slouken@1330
   469
	trailing part of z, and also let x0 and x1 be the leading and
slouken@1330
   470
	trailing parts of x.
slouken@1330
   471
slouken@1330
   472
	If ((z1&0x03ffffff)!=0)	... not exact if trailing 26 bits of z!=0
slouken@1330
   473
	    I := 1;		... Raise Inexact flag: z is not exact
slouken@1330
   474
	else {
slouken@1330
   475
	    j := 1 - [(x0>>20)&1]	... j = logb(x) mod 2
slouken@1330
   476
	    k := z1 >> 26;		... get z's 25-th and 26-th
slouken@1330
   477
					    fraction bits
slouken@1330
   478
	    I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
slouken@1330
   479
	}
slouken@1330
   480
	R:= r		... restore rounded mode
slouken@1330
   481
	return sqrt(x):=z.
slouken@1330
   482
slouken@1330
   483
	If multiplication is cheaper then the foregoing red tape, the
slouken@1330
   484
	Inexact flag can be evaluated by
slouken@1330
   485
slouken@1330
   486
	    I := i;
slouken@1330
   487
	    I := (z*z!=x) or I.
slouken@1330
   488
slouken@1330
   489
	Note that z*z can overwrite I; this value must be sensed if it is
slouken@1330
   490
	True.
slouken@1330
   491
slouken@1330
   492
	Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
slouken@1330
   493
	zero.
slouken@1330
   494
slouken@1330
   495
		    --------------------
slouken@1330
   496
		z1: |        f2        |
slouken@1330
   497
		    --------------------
slouken@1330
   498
		bit 31		   bit 0
slouken@1330
   499
slouken@1330
   500
	Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
slouken@1330
   501
	or even of logb(x) have the following relations:
slouken@1330
   502
slouken@1330
   503
	-------------------------------------------------
slouken@1330
   504
	bit 27,26 of z1		bit 1,0 of x1	logb(x)
slouken@1330
   505
	-------------------------------------------------
slouken@1330
   506
	00			00		odd and even
slouken@1330
   507
	01			01		even
slouken@1330
   508
	10			10		odd
slouken@1330
   509
	10			00		even
slouken@1330
   510
	11			01		even
slouken@1330
   511
	-------------------------------------------------
slouken@1330
   512
slouken@1330
   513
    (4)	Special cases (see (4) of Section A).
slouken@1330
   514
slouken@1330
   515
 */
slouken@1895
   516
/* vi: set ts=4 sw=4 expandtab: */