[multiple changes]
Fri May 28 22:20:03 1999 Anthony Green <green@cygnus.com> * java/lang/fdlibm.h: Don't use __uint32_t. Include mprec.h. * java/lang/e_log.c: Don't use __uint32_t. 1999-05-27 Eric Christopher <echristo@cygnus.com> * configure: Rebuilt * configure.in: Fixed ISO C9X and namespace collision with __uint32_t * acconfig.h: Rebuilt * include/config.h.in: Rebuilt * java/lang/mprec.h, java/lang/e_acos.c, java/lang/e_asin.c, java/lang/e_atan2.c, java/lang/e_exp.c, java/lang/e_fmod.c, e_log.c, java/lang/e_pow.c, java/lang/e_rem_pio2.c, java/lang/e_remainder.c, java/lang/e_sqrt.c, java/lang/fdlibm.h, k_tan.c, java/lang/mprec.h, java/lang/s_atan.c, java/lang/s_ceil.c, java/lang/s_copysign.c, java/lang/s_fabs.c, s_floor.c, java/lang/s_rint.c, java/lang/sf_rint.c: Fixed ISO C9X and namespace collision with __uint32_t From-SVN: r27729
This commit is contained in:
Tom Tromey
parent
fe574d5d92
commit
0d16618c58
31 changed files with 614 additions and 526 deletions
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@ -6,7 +6,7 @@
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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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* 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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@ -16,10 +16,10 @@
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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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* 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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* 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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@ -28,9 +28,9 @@
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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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* 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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@ -40,12 +40,12 @@
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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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* 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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* 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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@ -57,10 +57,10 @@
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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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*
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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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* 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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@ -70,7 +70,7 @@
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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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*
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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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@ -99,17 +99,17 @@ static double one = 1.0, tiny=1.0e-300;
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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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__uint32_t r,t1,s1,ix1,q1;
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__int32_t ix0,s0,q,m,t,i;
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int32_t sign = (int)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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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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}
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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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@ -143,12 +143,12 @@ static double one = 1.0, tiny=1.0e-300;
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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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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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@ -156,11 +156,11 @@ static double one = 1.0, tiny=1.0e-300;
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r = sign;
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while(r!=0) {
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t1 = s1+r;
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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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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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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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@ -176,10 +176,10 @@ static double one = 1.0, tiny=1.0e-300;
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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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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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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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@ -191,24 +191,24 @@ static double one = 1.0, tiny=1.0e-300;
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INSERT_WORDS(z,ix0,ix1);
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return z;
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}
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#endif /* defined(_DOUBLE_IS_32BITS) */
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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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(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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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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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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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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@ -218,7 +218,7 @@ 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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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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@ -226,7 +226,7 @@ A. sqrt(x) by Newton Iteration
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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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@ -251,7 +251,7 @@ A. sqrt(x) by Newton Iteration
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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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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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@ -276,12 +276,12 @@ A. sqrt(x) by Newton Iteration
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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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is slow. If division is very slow, then one should use the
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reciproot algorithm given in section B.
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(3) Final adjustment
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By twiddling y's last bit it is possible to force y to be
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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
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inexact flag before entering the square root program. Also we
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@ -312,7 +312,7 @@ A. sqrt(x) by Newton Iteration
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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.
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(4) Special cases
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Square root of +inf, +-0, or NaN is itself;
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@ -331,7 +331,7 @@ B. sqrt(x) by Reciproot Iteration
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k := 0x5fe80000 - (x0>>1);
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y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
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Here k is a 32-bit integer and T2[] is an integer array
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Here k is a 32-bit integer and T2[] is an integer array
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containing correction terms. Now magically the floating
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value of y (y's leading 32-bit word is y0, the value of
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its trailing word y1 is set to zero) approximates 1/sqrt(x)
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@ -352,9 +352,9 @@ B. sqrt(x) by Reciproot Iteration
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Apply Reciproot iteration three times to y and multiply the
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result by x to get an approximation z that matches sqrt(x)
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to about 1 ulp. To be exact, we will have
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to about 1 ulp. To be exact, we will have
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-1ulp < sqrt(x)-z<1.0625ulp.
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... set rounding mode to Round-to-nearest
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y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
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y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
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@ -363,14 +363,14 @@ B. sqrt(x) by Reciproot Iteration
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z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
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Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
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(a) the term z*y in the final iteration is always less than 1;
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(a) the term z*y in the final iteration is always less than 1;
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(b) the error in the final result is biased upward so that
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-1 ulp < sqrt(x) - z < 1.0625 ulp
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instead of |sqrt(x)-z|<1.03125ulp.
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(3) Final adjustment
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By twiddling y's last bit it is possible to force y to be
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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
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inexact flag before entering the square root program. Also we
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@ -410,27 +410,27 @@ B. sqrt(x) by Reciproot Iteration
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I := 1; ... Raise Inexact flag: z is not exact
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else {
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j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
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k := z1 >> 26; ... get z's 25-th and 26-th
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k := z1 >> 26; ... get z's 25-th and 26-th
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fraction bits
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I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
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}
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R:= r ... restore rounded mode
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return sqrt(x):=z.
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If multiplication is cheaper then the foregoing red tape, the
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If multiplication is cheaper then the foregoing red tape, the
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Inexact flag can be evaluated by
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I := i;
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I := (z*z!=x) or I.
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Note that z*z can overwrite I; this value must be sensed if it is
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Note that z*z can overwrite I; this value must be sensed if it is
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True.
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Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
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zero.
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--------------------
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z1: | f2 |
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z1: | f2 |
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--------------------
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bit 31 bit 0
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@ -447,6 +447,6 @@ B. sqrt(x) by Reciproot Iteration
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11 01 even
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-------------------------------------------------
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(4) Special cases (see (4) of Section A).
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(4) Special cases (see (4) of Section A).
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*/
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