[doc/algorithms.tex,src/tanh.c] Fixed the bound, in particular from

the recent improvements in the error analysis.

git-svn-id: svn://scm.gforge.inria.fr/svn/mpfr/trunk@12007 280ebfd0-de03-0410-8827-d642c229c3f4

2 files changed, 7 insertions, 3 deletions

diff --git a/doc/algorithms.tex b/doc/algorithms.tex
index c2bb689ad..3010a2738 100644
--- a/doc/algorithms.tex
+++ b/doc/algorithms.tex
@@ -1892,7 +1892,9 @@ $|\theta_5| \leq 2^{-p}$.
 Since $2^k+4 \leq 2^{{\rm max}(3,k+1)}$,
 the relative error on $s$ is thus bounded by $2^{{\rm max}(4,k+2)-p}$.
 
-The condition $2^k+4 \leq 2^{p/2}$ is checked in the code.
+The condition $2^k+4 \leq 2^{p/2}$ is checked in the code with
+${\rm max}(3,k+1) \leq \lfloor p/2 \rfloor$, so that:
+\[2^k+4 \leq 2^{{\rm max}(3,k+1)} \leq 2^{\lfloor p/2 \rfloor} \leq 2^{p/2}.\]
 
 \subsection{The inverse hyperbolic tangent function}
 
diff --git a/src/tanh.c b/src/tanh.c
index e731f2623..7182d0e79 100644
--- a/src/tanh.c
+++ b/src/tanh.c
@@ -131,10 +131,12 @@ mpfr_tanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
       d = d - MPFR_GET_EXP (te);
       mpfr_div (t, te, t, MPFR_RNDN);      /* (exp(2x)-1)/(exp(2x)+1) */
 
-      /* Calculation of the error, see algorithms.tex */
-      d = MAX(4, d + 2);
+      /* Calculation of the error, see algorithms.tex; the current value
+         of d is k in algorithms.tex. */
+      d = MAX(3, d + 1);  /* d = exponent in 2^(max(3,k+1)) */
       err = Nt - (d + 1);
 
+      /* The inequality is the condition max(3,k+1) <= floor(p/2). */
       if (MPFR_LIKELY ((d <= Nt / 2) && MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
         {
           inexact = mpfr_set4 (y, t, rnd_mode, sign);