Optimize the 9pi/2 < |x| <= 2**19pi/2 case some more by avoiding an

fabs(), a conditional branch, and sign adjustments of 3 variables for
x < 0 when the branch is taken.  In double precision, even when the
branch is perfectly predicted, this saves about 10 cycles or 10% on
amd64 (A64) and i386 (A64) for the negative half of the range, but
makes little difference for the positive half of the range.  In float
precision, it also saves about 4 cycles for the positive half of the
range on i386, and many more cycles in both halves on amd64 (28 in the
negative half and 11 in the positive half for tanf), but the amd64
times for float precision are anomalously slow so the larger
improvement is only a side effect.

Previous commits arranged for the x < 0 case to be handled simply:
- one part of the rounding method uses the magic number 0x1.8p52
  instead of the usual 0x1.0p52.  The latter is required for large |x|,
  but it doesn't work for negative x and we don't need it for large |x|.
- another part of the rounding method no longer needs to add `half'.
  It would have needed to add -half for negative x.
- removing the "quick check no cancellation" in the double precision
  case removed the need to take the absolute value of the quadrant
  number.

Add my noncopyright in e_rem_pio2.c

lib/msun/src/e_rem_pio2.c

@@ -10,6 +10,7 @@
  * is preserved.
  * ====================================================
  *
+ * Optimized by Bruce D. Evans.
  */
 
 #include <sys/cdefs.h>
@@ -127,16 +128,15 @@ __ieee754_rem_pio2(double x, double *y)
 	}
 	if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
 medium:
-	    t  = fabs(x);
 	    /* Use a specialized rint() to get fn.  Assume round-to-nearest. */
-	    STRICT_ASSIGN(double,fn,t*invpio2+0x1.8p52);
+	    STRICT_ASSIGN(double,fn,x*invpio2+0x1.8p52);
 	    fn = fn-0x1.8p52;
 #ifdef HAVE_EFFICIENT_IRINT
 	    n  = irint(fn);
 #else
 	    n  = (int32_t)fn;
 #endif
-	    r  = t-fn*pio2_1;
+	    r  = x-fn*pio2_1;
 	    w  = fn*pio2_1t;	/* 1st round good to 85 bit */
 	    {
 	        u_int32_t high;
@@ -162,8 +162,7 @@ __ieee754_rem_pio2(double x, double *y)
 		}
 	    }
 	    y[1] = (r-y[0])-w;
-	    if(hx<0) 	{y[0] = -y[0]; y[1] = -y[1]; return -n;}
-	    else	 return n;
+	    return n;
 	}
     /* 
      * all other (large) arguments

lib/msun/src/e_rem_pio2f.c

@@ -44,7 +44,7 @@ pio2_1t =  6.07710050650619224932e-11; /* 0x3DD0B461, 0x1A626331 */
 int
 __ieee754_rem_pio2f(float x, float *y)
 {
-	double w,t,r,fn;
+	double w,r,fn;
 	double tx[1],ty[2];
 	float z;
 	int32_t e0,n,ix,hx;
@@ -53,21 +53,19 @@ __ieee754_rem_pio2f(float x, float *y)
 	ix = hx&0x7fffffff;
     /* 33+53 bit pi is good enough for medium size */
 	if(ix<=0x49490f80) {		/* |x| ~<= 2^19*(pi/2), medium size */
-	    t  = fabsf(x);
 	    /* Use a specialized rint() to get fn.  Assume round-to-nearest. */
-	    STRICT_ASSIGN(double,fn,t*invpio2+0x1.8p52);
+	    STRICT_ASSIGN(double,fn,x*invpio2+0x1.8p52);
 	    fn = fn-0x1.8p52;
 #ifdef HAVE_EFFICIENT_IRINT
 	    n  = irint(fn);
 #else
 	    n  = (int32_t)fn;
 #endif
-	    r  = t-fn*pio2_1;
+	    r  = x-fn*pio2_1;
 	    w  = fn*pio2_1t;
 	    y[0] = r-w;
 	    y[1] = (r-y[0])-w;
-	    if(hx<0) 	{y[0] = -y[0]; y[1] = -y[1]; return -n;}
-	    else	 return n;
+	    return n;
 	}
     /*
      * all other (large) arguments