[newlib-cygwin] cacosh: Use approximation for large input.

[Corinna Vinschen <corinna@sourceware.org>]

https://sourceware.org/git/gitweb.cgi?p=newlib-cygwin.git;h=ab31794e8a1da13224eca78b58df9a72d78fb6ae

commit ab31794e8a1da13224eca78b58df9a72d78fb6ae
Author: Markus Mützel <markus.muetzel@gmx.de>
Date:   Thu Aug 4 12:55:52 2022 +0200

    cacosh: Use approximation for large input.
    
    Signed-off-by: Martin Storsjö <martin@martin.st>
    Signed-off-by: Corinna Vinschen <corinna@vinschen.de>

Diff:
---
 winsup/cygwin/math/cacosh.def.h | 21 +++++++++++++++++++++
 1 file changed, 21 insertions(+)

diff --git a/winsup/cygwin/math/cacosh.def.h b/winsup/cygwin/math/cacosh.def.h
index 5105b426e..c2c2031eb 100644
--- a/winsup/cygwin/math/cacosh.def.h
+++ b/winsup/cygwin/math/cacosh.def.h
@@ -80,6 +80,27 @@ __FLT_ABI(cacosh) (__FLT_TYPE __complex__ z)
     return ret;
   }
 
+  /* cacosh(z) = log(z + sqrt(z*z - 1)) */
+
+  if (__FLT_ABI(fabs) (__real__ z) >= __FLT_CST(1.0)/__FLT_EPSILON
+      || __FLT_ABI(fabs) (__imag__ z) >= __FLT_CST(1.0)/__FLT_EPSILON)
+  {
+    /* For large z, z + sqrt(z*z - 1) is approximately 2*z.
+    Use that approximation to avoid overflow when squaring.
+    Additionally, use symmetries to perform the calculation in the positive
+    half plane. */
+    __real__ x = __real__ z;
+    __imag__ x = __FLT_ABI(fabs) (__imag__ z);
+    x = __FLT_ABI(clog) (x);
+    __real__ x += M_LN2;
+
+    /* adjust signs for input */
+    __real__ ret = __real__ x;
+    __imag__ ret = __FLT_ABI(copysign) (__imag__ x, __imag__ z);
+
+    return ret;
+  }
+
   __real__ x = (__real__ z - __imag__ z) * (__real__ z + __imag__ z) - __FLT_CST(1.0);
   __imag__ x = __FLT_CST(2.0) * __real__ z * __imag__ z;