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254 lines
6.9 KiB
C
254 lines
6.9 KiB
C
///////////////////////////////////////
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// Implements:
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// double sqrt(double x);
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///////////////////////////////////////
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// Notes:
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// This is taken from newlib.
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// Only included because pow() needs it.
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// @nolint
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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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#include <stdint.h>
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#include <pblibc_private.h>
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typedef union
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{
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double value;
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struct
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{
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uint32_t lsw;
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uint32_t msw;
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} parts;
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} ieee_double_shape_type;
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#define EXTRACT_WORDS(ix0,ix1,d) \
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do { \
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ieee_double_shape_type ew_u; \
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ew_u.value = (d); \
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(ix0) = ew_u.parts.msw; \
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(ix1) = ew_u.parts.lsw; \
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} while (0)
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/* Get the more significant 32 bit int from a double. */
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#define GET_HIGH_WORD(i,d) \
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do { \
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ieee_double_shape_type gh_u; \
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gh_u.value = (d); \
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(i) = gh_u.parts.msw; \
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} while (0)
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/* Get the less significant 32 bit int from a double. */
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#define GET_LOW_WORD(i,d) \
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do { \
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ieee_double_shape_type gl_u; \
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gl_u.value = (d); \
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(i) = gl_u.parts.lsw; \
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} while (0)
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#define INSERT_WORDS(d,ix0,ix1) \
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do { \
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ieee_double_shape_type iw_u; \
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iw_u.parts.msw = (ix0); \
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iw_u.parts.lsw = (ix1); \
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(d) = iw_u.value; \
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} while (0)
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/* Set the more significant 32 bits of a double from an int. */
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#define SET_HIGH_WORD(d,v) \
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do { \
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ieee_double_shape_type sh_u; \
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sh_u.value = (d); \
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sh_u.parts.msw = (v); \
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(d) = sh_u.value; \
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} while (0)
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/* Set the less significant 32 bits of a double from an int. */
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#define SET_LOW_WORD(d,v) \
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do { \
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ieee_double_shape_type sl_u; \
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sl_u.value = (d); \
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sl_u.parts.lsw = (v); \
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(d) = sl_u.value; \
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} while (0)
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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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static const double one = 1.0, tiny=1.0e-300;
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double sqrt(double x)
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{
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double z;
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int32_t sign = 0x80000000;
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uint32_t r,t1,s1,ix1,q1;
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int32_t ix0,s0,q,m,t,i;
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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) 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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/* 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(((t1&sign)==(uint32_t)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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/* 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==(uint32_t)0xffffffff) { q1=0; q += 1;}
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else if (z>one) {
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if (q1==(uint32_t)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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INSERT_WORDS(z,ix0,ix1);
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return z;
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}
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