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/* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*- |
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* |
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* ***** BEGIN LICENSE BLOCK ***** |
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* Version: MPL 1.1/GPL 2.0/LGPL 2.1 |
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* |
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* The contents of this file are subject to the Mozilla Public License Version |
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* 1.1 (the "License"); you may not use this file except in compliance with |
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* the License. You may obtain a copy of the License at |
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* http://www.mozilla.org/MPL/ |
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* |
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* Software distributed under the License is distributed on an "AS IS" basis, |
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* WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License |
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* for the specific language governing rights and limitations under the |
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* License. |
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* |
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* The Original Code is Mozilla Communicator client code, released |
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* March 31, 1998. |
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* |
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* The Initial Developer of the Original Code is |
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* Sun Microsystems, Inc. |
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* Portions created by the Initial Developer are Copyright (C) 1998 |
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* the Initial Developer. All Rights Reserved. |
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* |
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* Contributor(s): |
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* |
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* Alternatively, the contents of this file may be used under the terms of |
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* either of the GNU General Public License Version 2 or later (the "GPL"), |
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* or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"), |
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* in which case the provisions of the GPL or the LGPL are applicable instead |
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* of those above. If you wish to allow use of your version of this file only |
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* under the terms of either the GPL or the LGPL, and not to allow others to |
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* use your version of this file under the terms of the MPL, indicate your |
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* decision by deleting the provisions above and replace them with the notice |
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* and other provisions required by the GPL or the LGPL. If you do not delete |
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* the provisions above, a recipient may use your version of this file under |
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* the terms of any one of the MPL, the GPL or the LGPL. |
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* |
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* ***** END LICENSE BLOCK ***** */ |
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/* @(#)e_sqrt.c 1.3 95/01/18 */ |
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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 SunSoft, 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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|
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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(y) = 2^k * sqrt(x) |
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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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|
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#include "fdlibm.h" |
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|
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#if defined(_MSC_VER) |
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/* Microsoft Compiler */ |
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#pragma warning( disable : 4723 ) /* disables potential divide by 0 warning */ |
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#endif |
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|
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#ifdef __STDC__ |
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static const double one = 1.0, tiny=1.0e-300; |
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#else |
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static double one = 1.0, tiny=1.0e-300; |
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#endif |
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|
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#ifdef __STDC__ |
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double __ieee754_sqrt(double x) |
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#else |
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double __ieee754_sqrt(x) |
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double x; |
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#endif |
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{ |
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fd_twoints u; |
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double z; |
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int sign = (int)0x80000000; |
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unsigned r,t1,s1,ix1,q1; |
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int ix0,s0,q,m,t,i; |
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|
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u.d = x; |
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ix0 = __HI(u); /* high word of x */ |
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ix1 = __LO(u); /* low word of x */ |
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|
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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) 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); ix1 <<= 21; |
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} |
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for(i=0;(ix0&0x00100000)==0;i++) 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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|
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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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|
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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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|
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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(((int)(t1&sign)==sign)&&(s1&sign)==0) s0 += 1; |
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ix0 -= t; |
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if (ix1 < t1) 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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|
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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==(unsigned)0xffffffff) { q1=0; q += 1;} |
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else if (z>one) { |
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if (q1==(unsigned)0xfffffffe) 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) ix1 |= sign; |
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ix0 += (m <<20); |
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u.d = z; |
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__HI(u) = ix0; |
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__LO(u) = ix1; |
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z = u.d; |
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return z; |
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} |
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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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|
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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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|
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A. sqrt(x) by Newton Iteration |
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|
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(1) Initial approximation |
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|
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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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|
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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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|
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------------------------ ------------------------ |
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x0: |s| e | f1 | x1: | f2 | |
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------------------------ ------------------------ |
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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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|
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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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|
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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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|
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(2) Iterative refinement |
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|
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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): |
| 300 |
|
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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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|
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|
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Remark 1. |
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Another way to improve y to within 1 ulp is: |
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|
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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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|
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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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|
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|
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This formula has one division fewer than the one above; however, |
| 320 |
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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reciproot algorithm given in section B. |
| 325 |
|
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(3) Final adjustment |
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|
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By twiddling y's last bit it is possible to force y to be |
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correctly rounded according to the prevailing rounding mode |
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as follows. Let r and i be copies of the rounding mode and |
| 331 |
inexact flag before entering the square root program. Also we |
| 332 |
use the expression y+-ulp for the next representable floating |
| 333 |
numbers (up and down) of y. Note that y+-ulp = either fixed |
| 334 |
point y+-1, or multiply y by nextafter(1,+-inf) in chopped |
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mode. |
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|
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I := FALSE; ... reset INEXACT flag I |
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R := RZ; ... set rounding mode to round-toward-zero |
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z := x/y; ... chopped quotient, possibly inexact |
| 340 |
If(not I) then { ... if the quotient is exact |
| 341 |
if(z=y) { |
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I := i; ... restore inexact flag |
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R := r; ... restore rounded mode |
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return sqrt(x):=y. |
| 345 |
} else { |
| 346 |
z := z - ulp; ... special rounding |
| 347 |
} |
| 348 |
} |
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i := TRUE; ... sqrt(x) is inexact |
| 350 |
If (r=RN) then z=z+ulp ... rounded-to-nearest |
| 351 |
If (r=RP) then { ... round-toward-+inf |
| 352 |
y = y+ulp; z=z+ulp; |
| 353 |
} |
| 354 |
y := y+z; ... chopped sum |
| 355 |
y0:=y0-0x00100000; ... y := y/2 is correctly rounded. |
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I := i; ... restore inexact flag |
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R := r; ... restore rounded mode |
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return sqrt(x):=y. |
| 359 |
|
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(4) Special cases |
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|
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Square root of +inf, +-0, or NaN is itself; |
| 363 |
Square root of a negative number is NaN with invalid signal. |
| 364 |
|
| 365 |
|
| 366 |
B. sqrt(x) by Reciproot Iteration |
| 367 |
|
| 368 |
(1) Initial approximation |
| 369 |
|
| 370 |
Let x0 and x1 be the leading and the trailing 32-bit words of |
| 371 |
a floating point number x (in IEEE double format) respectively |
| 372 |
(see section A). By performing shifs and subtracts on x0 and y0, |
| 373 |
we obtain a 7.8-bit approximation of 1/sqrt(x) as follows. |
| 374 |
|
| 375 |
k := 0x5fe80000 - (x0>>1); |
| 376 |
y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits |
| 377 |
|
| 378 |
Here k is a 32-bit integer and T2[] is an integer array |
| 379 |
containing correction terms. Now magically the floating |
| 380 |
value of y (y's leading 32-bit word is y0, the value of |
| 381 |
its trailing word y1 is set to zero) approximates 1/sqrt(x) |
| 382 |
to almost 7.8-bit. |
| 383 |
|
| 384 |
Value of T2: |
| 385 |
static int T2[64]= { |
| 386 |
0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866, |
| 387 |
0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f, |
| 388 |
0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d, |
| 389 |
0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0, |
| 390 |
0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989, |
| 391 |
0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd, |
| 392 |
0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e, |
| 393 |
0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,}; |
| 394 |
|
| 395 |
(2) Iterative refinement |
| 396 |
|
| 397 |
Apply Reciproot iteration three times to y and multiply the |
| 398 |
result by x to get an approximation z that matches sqrt(x) |
| 399 |
to about 1 ulp. To be exact, we will have |
| 400 |
-1ulp < sqrt(x)-z<1.0625ulp. |
| 401 |
|
| 402 |
... set rounding mode to Round-to-nearest |
| 403 |
y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x) |
| 404 |
y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x) |
| 405 |
... special arrangement for better accuracy |
| 406 |
z := x*y ... 29 bits to sqrt(x), with z*y<1 |
| 407 |
z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x) |
| 408 |
|
| 409 |
Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that |
| 410 |
(a) the term z*y in the final iteration is always less than 1; |
| 411 |
(b) the error in the final result is biased upward so that |
| 412 |
-1 ulp < sqrt(x) - z < 1.0625 ulp |
| 413 |
instead of |sqrt(x)-z|<1.03125ulp. |
| 414 |
|
| 415 |
(3) Final adjustment |
| 416 |
|
| 417 |
By twiddling y's last bit it is possible to force y to be |
| 418 |
correctly rounded according to the prevailing rounding mode |
| 419 |
as follows. Let r and i be copies of the rounding mode and |
| 420 |
inexact flag before entering the square root program. Also we |
| 421 |
use the expression y+-ulp for the next representable floating |
| 422 |
numbers (up and down) of y. Note that y+-ulp = either fixed |
| 423 |
point y+-1, or multiply y by nextafter(1,+-inf) in chopped |
| 424 |
mode. |
| 425 |
|
| 426 |
R := RZ; ... set rounding mode to round-toward-zero |
| 427 |
switch(r) { |
| 428 |
case RN: ... round-to-nearest |
| 429 |
if(x<= z*(z-ulp)...chopped) z = z - ulp; else |
| 430 |
if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp; |
| 431 |
break; |
| 432 |
case RZ:case RM: ... round-to-zero or round-to--inf |
| 433 |
R:=RP; ... reset rounding mod to round-to-+inf |
| 434 |
if(x<z*z ... rounded up) z = z - ulp; else |
| 435 |
if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp; |
| 436 |
break; |
| 437 |
case RP: ... round-to-+inf |
| 438 |
if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else |
| 439 |
if(x>z*z ...chopped) z = z+ulp; |
| 440 |
break; |
| 441 |
} |
| 442 |
|
| 443 |
Remark 3. The above comparisons can be done in fixed point. For |
| 444 |
example, to compare x and w=z*z chopped, it suffices to compare |
| 445 |
x1 and w1 (the trailing parts of x and w), regarding them as |
| 446 |
two's complement integers. |
| 447 |
|
| 448 |
...Is z an exact square root? |
| 449 |
To determine whether z is an exact square root of x, let z1 be the |
| 450 |
trailing part of z, and also let x0 and x1 be the leading and |
| 451 |
trailing parts of x. |
| 452 |
|
| 453 |
If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0 |
| 454 |
I := 1; ... Raise Inexact flag: z is not exact |
| 455 |
else { |
| 456 |
j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2 |
| 457 |
k := z1 >> 26; ... get z's 25-th and 26-th |
| 458 |
fraction bits |
| 459 |
I := i or (k&j) or ((k&(j+j+1))!=(x1&3)); |
| 460 |
} |
| 461 |
R:= r ... restore rounded mode |
| 462 |
return sqrt(x):=z. |
| 463 |
|
| 464 |
If multiplication is cheaper then the foregoing red tape, the |
| 465 |
Inexact flag can be evaluated by |
| 466 |
|
| 467 |
I := i; |
| 468 |
I := (z*z!=x) or I. |
| 469 |
|
| 470 |
Note that z*z can overwrite I; this value must be sensed if it is |
| 471 |
True. |
| 472 |
|
| 473 |
Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be |
| 474 |
zero. |
| 475 |
|
| 476 |
-------------------- |
| 477 |
z1: | f2 | |
| 478 |
-------------------- |
| 479 |
bit 31 bit 0 |
| 480 |
|
| 481 |
Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd |
| 482 |
or even of logb(x) have the following relations: |
| 483 |
|
| 484 |
------------------------------------------------- |
| 485 |
bit 27,26 of z1 bit 1,0 of x1 logb(x) |
| 486 |
------------------------------------------------- |
| 487 |
00 00 odd and even |
| 488 |
01 01 even |
| 489 |
10 10 odd |
| 490 |
10 00 even |
| 491 |
11 01 even |
| 492 |
------------------------------------------------- |
| 493 |
|
| 494 |
(4) Special cases (see (4) of Section A). |
| 495 |
|
| 496 |
*/ |
| 497 |
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