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

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.

1 files changed, 8 insertions, 5 deletions

diff --git a/lib/msun/src/e_rem_pio2f.c b/lib/msun/src/e_rem_pio2f.c
index 5e37eaa..53d1c8e 100644
--- a/lib/msun/src/e_rem_pio2f.c
+++ b/lib/msun/src/e_rem_pio2f.c
@@ -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;
 	}