d/freebsd-skq
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

More fixes for arg reduction near pi/2 on systems with broken assignment

Browse Source 
to floats (mainly i386's).  All errors of more than 1 ulp for float
precision trig functions were supposed to have been fixed; however,
compiling with gcc -O2 uncovered 18250 more such errors for cosf(),
with a maximum error of 1.409 ulps.

Use essentially the same fix as in rev.1.8 of k_rem_pio2f.c (access a
non-volatile variable as a volatile).  Here the -O1 case apparently
worked because the variable is in a 2-element array and it takes -O2
to mess up such a variable by putting it in a register.

The maximum error for cosf() on i386 with gcc -O2 is now 0.5467 (it
is still 0.5650 with gcc -O1).  This shows that -O2 still causes some
extra precision, but the extra precision is now good.

Extra precision is harmful mainly for implementing extra precision in
software.  We want to represent x+y as w+r where both "+" operations
are in infinite precision and r is tiny compared with w.  There is a
standard algorithm for this (Knuth (1981) 4.2.2 Theorem C), and fdlibm
uses this routinely, but the algorithm requires w and r to have the
same precision as x and y.  w is just x+y (calculated in the same
finite precision as x and y), and r is a tiny correction term.  The
i386 gcc bugs tend to give extra precision in w, and then using this
extra precision in the calculation of r results in the correction
mostly staying in w and being missing from r.  There still tends to
be no problem if the result is a simple expression involving w and r
-- modulo spills, w keeps its extra precision and r remains the right
correction for this wrong w.  However, here we want to pass w and r
to extern functions.  Extra precision is not retained in function args,
so w gets fixed up, but the change to the tiny r is tinier, so r almost
remains as a wrong correction for the right w.
This commit is contained in:
Bruce Evans 2005-10-25 12:13:37 +00:00
parent ebeecabae3
commit a92cb60b4e
1 changed files with 8 additions and 5 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

13
lib/msun/src/e_rem_pio2f.c
View File

@ -26,6 +26,9 @@ static char rcsid[] = "$FreeBSD$";
#include "math.h"
#include "math_private.h"
/* Clip any extra precision in the float variable v. */
#define cliptofloat(v) (*(volatile float *)&(v))
/*
* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
*/
@ -117,22 +120,22 @@ pio2_3t = 6.1232342629e-17; /* 0x248d3132 */
z = x - pio2_1;
if((ix&0xfffe0000)!=0x3fc80000) { /* 17+24 bit pi OK */
y[0] = z - pio2_1t;
y[1] = (z-y[0])-pio2_1t;
y[1] = (z-cliptofloat(y[0]))-pio2_1t;
} else { /* near pi/2, use 17+17+24 bit pi */
z -= pio2_2;
y[0] = z - pio2_2t;
y[1] = (z-y[0])-pio2_2t;
y[1] = (z-cliptofloat(y[0]))-pio2_2t;
}
return 1;
} else { /* negative x */
z = x + pio2_1;
if((ix&0xfffe0000)!=0x3fc80000) { /* 17+24 bit pi OK */
y[0] = z + pio2_1t;
y[1] = (z-y[0])+pio2_1t;
y[1] = (z-cliptofloat(y[0]))+pio2_1t;
} else { /* near pi/2, use 17+17+24 bit pi */
z += pio2_2;
y[0] = z + pio2_2t;
y[1] = (z-y[0])+pio2_2t;
y[1] = (z-cliptofloat(y[0]))+pio2_2t;
}
return -1;
}
@ -168,7 +171,7 @@ pio2_3t = 6.1232342629e-17; /* 0x248d3132 */
}
}
}
y[1] = (r-y[0])-w;
y[1] = (r-cliptofloat(y[0]))-w;
if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
else return n;
}