Copy cos(3) software implementation from FreeBSD-13

The result passes more tests from msun suite. In particular,
	testacc(cos, -0.34061437849088045332L, 0.94254960031831729956L,
		ALL_STD_EXCEPT, FE_INEXACT);
matches instead of being 1e-16 off.

OK mbuhl@, tested by mbuhl@ and phessler@

diff --git a/lib/libm/src/k_cos.c b/lib/libm/src/k_cos.c
index 8f3882b..0839243 100644 (file)
--- a/lib/libm/src/k_cos.c
+++ b/lib/libm/src/k_cos.c
@@ -36,13 +36,17 @@
  *                       ~ cos(x) - x*y,
  *        a correction term is necessary in cos(x) and hence
  *             cos(x+y) = 1 - (x*x/2 - (r - x*y))
- *        For better accuracy when x > 0.3, let qx = |x|/4 with
- *        the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
- *        Then
- *             cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
- *        Note that 1-qx and (x*x/2-qx) is EXACT here, and the
- *        magnitude of the latter is at least a quarter of x*x/2,
- *        thus, reducing the rounding error in the subtraction.
+ *        For better accuracy, rearrange to
+ *             cos(x+y) ~ w + (tmp + (r-x*y))
+ *        where w = 1 - x*x/2 and tmp is a tiny correction term
+ *        (1 - x*x/2 == w + tmp exactly in infinite precision).
+ *        The exactness of w + tmp in infinite precision depends on w
+ *        and tmp having the same precision as x.  If they have extra
+ *        precision due to compiler bugs, then the extra precision is
+ *        only good provided it is retained in all terms of the final
+ *        expression for cos().  Retention happens in all cases tested
+ *        under FreeBSD, so don't pessimize things by forcibly clipping
+ *        any extra precision in w.
  */
 
 #include "math.h"
@@ -60,25 +64,12 @@ C6  = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
 double
 __kernel_cos(double x, double y)
 {
-       double a,hz,z,r,qx;
-       int32_t ix;
-       GET_HIGH_WORD(ix,x);
-       ix &= 0x7fffffff;                       /* ix = |x|'s high word*/
-       if(ix<0x3e400000) {                     /* if x < 2**27 */
-           if(((int)x)==0) return one;         /* generate inexact */
-       }
+       double hz,z,r,w;
+
        z  = x*x;
-       r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
-       if(ix < 0x3FD33333)                     /* if |x| < 0.3 */ 
-           return one - (0.5*z - (z*r - x*y));
-       else {
-           if(ix > 0x3fe90000) {               /* x > 0.78125 */
-               qx = 0.28125;
-           } else {
-               INSERT_WORDS(qx,ix-0x00200000,0);       /* x/4 */
-           }
-           hz = 0.5*z-qx;
-           a  = one-qx;
-           return a - (hz - (z*r-x*y));
-       }
+       w  = z*z;
+       r  = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
+       hz = 0.5*z;
+       w  = one-hz;
+       return w + (((one-w)-hz) + (z*r-x*y));
 }

diff --git a/lib/libm/src/s_cos.c b/lib/libm/src/s_cos.c
index 8b923d5..1406504 100644 (file)
--- a/lib/libm/src/s_cos.c
+++ b/lib/libm/src/s_cos.c
@@ -57,7 +57,11 @@ cos(double x)
 
     /* |x| ~< pi/4 */
        ix &= 0x7fffffff;
-       if(ix <= 0x3fe921fb) return __kernel_cos(x,z);
+       if(ix <= 0x3fe921fb) {
+           if(ix<0x3e46a09e)                   /* if x < 2**-27 * sqrt(2) */
+               if(((int)x)==0) return 1.0;     /* generate inexact */
+           return __kernel_cos(x,z);
+       }
 
     /* cos(Inf or NaN) is NaN */
        else if (ix>=0x7ff00000) return x-x;