[newlib-cygwin] Fix for k_tan.c specific inputs

[newlib-cygwin] Fix for k_tan.c specific inputs

[Corinna Vinschen]

https://sourceware.org/git/gitweb.cgi?p=newlib-cygwin.git;h=9e8da7bd2138aaefcb746be3bcce2787c75a5849

commit 9e8da7bd2138aaefcb746be3bcce2787c75a5849
Author: Fabian Schriever <fabian.schriever@gtd-gmbh.de>
Date:   Tue Mar 17 15:48:44 2020 +0100

    Fix for k_tan.c specific inputs
    
    This fix for k_tan.c is a copy from fdlibm version 5.3 (see also
    http://www.netlib.org/fdlibm/readme), adjusted to use the macros
    available in newlib (SET_LOW_WORD).
    
    This fix reduces the ULP error of the value shown in the fdlibm readme
    (tan(1.7765241907548024E+269)) to 0.45 (thereby reducing the error by
    1).
    
    This issue only happens for large numbers that get reduced by the range
    reduction to a value smaller in magnitude than 2^-28, that is also
    reduced an uneven number of times. This seems rather unlikely given that
    one ULP is (much) larger than 2^-28 for the values that may cause an
    issue.  Although given the sheer number of values a double can
    represent, it is still possible that there are more affected values,
    finding them however will be quite hard, if not impossible.
    
    We also took a look at how another library (libm in FreeBSD) handles the
    issue: In FreeBSD the complete if branch which checks for values smaller
    than 2^-28 (or rather 2^-27, another change done by FreeBSD) is moved
    out of the kernel function and into the external function. This means
    that the value that gets checked for this condition is the unreduced
    value. Therefore the input value which caused a problem in the
    fdlibm/newlib kernel tan will run through the full polynomial, including
    the careful calculation of -1/(x+r). So the difference is really whether
    r or y is used. r = y + p with p being the result of the polynomial with
    1/3*x^3 being the largest (and magnitude defining) value. With x being
    <2^-27 we therefore know that p is smaller than y (y has to be at least
    the size of the value of x last mantissa bit divided by 2, which is at
    least x*2^-51 for doubles) by enough to warrant saying that r ~ y.  So
    we can conclude that the general implementation of this special case is
    the same, FreeBSD simply has a different philosophy on when to handle
    especially small numbers.

Diff:
---
 newlib/libm/math/k_tan.c | 29 +++++++++++++++++++++--------
 1 file changed, 21 insertions(+), 8 deletions(-)

diff --git a/newlib/libm/math/k_tan.c b/newlib/libm/math/k_tan.c
index 9f5b30760..4be82d599 100644
--- a/newlib/libm/math/k_tan.c
+++ b/newlib/libm/math/k_tan.c
@@ -84,14 +84,27 @@ T[] =  {
 	__int32_t ix,hx;
 	GET_HIGH_WORD(hx,x);
 	ix = hx&0x7fffffff;	/* high word of |x| */
-	if(ix<0x3e300000)			/* x < 2**-28 */
-	    {if((int)x==0) {			/* generate inexact */
-	        __uint32_t low;
-		GET_LOW_WORD(low,x);
-		if(((ix|low)|(iy+1))==0) return one/fabs(x);
-		else return (iy==1)? x: -one/x;
-	    }
-	    }
+	if(ix<0x3e300000) {			/* x < 2**-28 */
+		if((int)x==0) {			/* generate inexact */
+			__uint32_t low;
+			GET_LOW_WORD(low,x);
+			if(((ix|low)|(iy+1))==0) return one/fabs(x);
+			else {
+				if(iy==1)
+					return x;
+				else {
+					double a, t;
+					z = w = x + y;
+					SET_LOW_WORD(z,0);
+					v = y - (z - x);
+					t = a = -one / w;
+					SET_LOW_WORD(t,0);
+					s = one + t * z;
+					return t + a * (s + t * v);
+				}
+			}
+		}
+	}
 	if(ix>=0x3FE59428) { 			/* |x|>=0.6744 */
 	    if(hx<0) {x = -x; y = -y;}
 	    z = pio4-x;