Optimize for 3pi/4 <= |x| <= 9pi/4 in much the same way as for

pi/4 <= |x| <= 3pi/4.  Use the same branch ladder as for float precision.
Remove the optimization for |x| near pi/2 and don't do it near the
multiples of pi/2 in the newly optimized range, since it requires
fairly large code to handle only relativley few cases.  Ifdef out
optimization for |x| <= pi/4 since this case can't occur because it
is done in callers.

On amd64 (A64), for cos() and sin() with uniformly distributed args,
no cache misses, some parallelism in the caller, and good but not great
CC and CFLAGS, etc., this saves about 40 cycles or 38% in the newly
optimized range, or about 27% on average across the range |x| <= 2pi
(~65 cycles for most args, while the A64 hardware fcos and fsin take
~75 cycles for half the args and 125 cycles for the other half).  The
speedup for tan() is much smaller, especially relatively.  The speedup
on i386 (A64) is slightly smaller, especially relatively.  i386 is
still much slower than amd64 here (unlike in the float case where it
is slightly faster).

1 files changed, 56 insertions, 18 deletions

diff --git a/lib/msun/src/e_rem_pio2.c b/lib/msun/src/e_rem_pio2.c
index 5467744..c7d2064 100644
--- a/lib/msun/src/e_rem_pio2.c
+++ b/lib/msun/src/e_rem_pio2.c
@@ -68,34 +68,72 @@ __ieee754_rem_pio2(double x, double *y)
 
 	GET_HIGH_WORD(hx,x);		/* high word of x */
 	ix = hx&0x7fffffff;
+#if 0 /* Must be handled in caller. */
 	if(ix<=0x3fe921fb)   /* |x| ~<= pi/4 , no need for reduction */
 	    {y[0] = x; y[1] = 0; return 0;}
-	if(ix<0x4002d97c) {  /* |x| < 3pi/4, special case with n=+-1 */
-	    if(hx>0) { 
-		z = x - pio2_1;
-		if(ix!=0x3ff921fb) { 	/* 33+53 bit pi is good enough */
+#endif
+	if (ix <= 0x400f6a7a) {		/* |x| ~<= 5pi/4 */
+	    if ((ix & 0xfffff) == 0x921fb)  /* |x| ~= pi/2 or 2pi/2 */
+		goto medium;		/* cancellation -- use medium case */
+	    if (ix <= 0x4002d97c) {	/* |x| ~<= 3pi/4 */
+		if (hx > 0) {
+		    z = x - pio2_1;	/* one round good to 85 bits */
 		    y[0] = z - pio2_1t;
 		    y[1] = (z-y[0])-pio2_1t;
-		} else {		/* near pi/2, use 33+33+53 bit pi */
-		    z -= pio2_2;
-		    y[0] = z - pio2_2t;
-		    y[1] = (z-y[0])-pio2_2t;
-		}
-		return 1;
-	    } else {	/* negative x */
-		z = x + pio2_1;
-		if(ix!=0x3ff921fb) { 	/* 33+53 bit pi is good enough */
+		    return 1;
+		} else {
+		    z = x + pio2_1;
 		    y[0] = z + pio2_1t;
 		    y[1] = (z-y[0])+pio2_1t;
-		} else {		/* near pi/2, use 33+33+53 bit pi */
-		    z += pio2_2;
-		    y[0] = z + pio2_2t;
-		    y[1] = (z-y[0])+pio2_2t;
+		    return -1;
+		}
+	    } else {
+		if (hx > 0) {
+		    z = x - 2*pio2_1;
+		    y[0] = z - 2*pio2_1t;
+		    y[1] = (z-y[0])-2*pio2_1t;
+		    return 2;
+		} else {
+		    z = x + 2*pio2_1;
+		    y[0] = z + 2*pio2_1t;
+		    y[1] = (z-y[0])+2*pio2_1t;
+		    return -2;
+		}
+	    }
+	}
+	if (ix <= 0x401c463b) {		/* |x| ~<= 9pi/4 */
+	    if (ix <= 0x4015fdbc) {	/* |x| ~<= 7pi/4 */
+		if (ix == 0x4012d97c)	/* |x| ~= 3pi/2 */
+		    goto medium;
+		if (hx > 0) {
+		    z = x - 3*pio2_1;
+		    y[0] = z - 3*pio2_1t;
+		    y[1] = (z-y[0])-3*pio2_1t;
+		    return 3;
+		} else {
+		    z = x + 3*pio2_1;
+		    y[0] = z + 3*pio2_1t;
+		    y[1] = (z-y[0])+3*pio2_1t;
+		    return -3;
+		}
+	    } else {
+		if (ix == 0x401921fb)	/* |x| ~= 4pi/2 */
+		    goto medium;
+		if (hx > 0) {
+		    z = x - 4*pio2_1;
+		    y[0] = z - 4*pio2_1t;
+		    y[1] = (z-y[0])-4*pio2_1t;
+		    return 4;
+		} else {
+		    z = x + 4*pio2_1;
+		    y[0] = z + 4*pio2_1t;
+		    y[1] = (z-y[0])+4*pio2_1t;
+		    return -4;
 		}
-		return -1;
 	    }
 	}
 	if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
+medium:
 	    t  = fabs(x);
 	    n  = (int32_t) (t*invpio2+half);
 	    fn = (double)n;