[tests/ttanh.c] added test for bug in mpfr_tanh

[doc/algorithms.tex] fixed error analysis for mpfr_tanh
[src/tanh.c] fixed error analysis
Note after r12016: Even though mpfr_tanh was incorrect, the failure
of the test added in ttanh.c was actually *only* due to a bug in the
mpfr_div code specific to the trunk (fixed in r12002), i.e. this was
not a non-regression test for the mpfr_tanh bug itself (in particular,
this test does not introduce a failure in the 3.1 branch, which still
has the same incorrect mpfr_tanh code but a correct mpfr_div).

git-svn-id: svn://scm.gforge.inria.fr/svn/mpfr/trunk@11993 280ebfd0-de03-0410-8827-d642c229c3f4

3 files changed, 47 insertions, 17 deletions

diff --git a/doc/algorithms.tex b/doc/algorithms.tex
index d5311d1c3..bc1a1eaeb 100644
--- a/doc/algorithms.tex
+++ b/doc/algorithms.tex
@@ -1854,7 +1854,8 @@ that $x \geq 0$.
 
 We use Higham's notation, with $\theta_i$ denoting variables such that
 $|\theta_i| \leq 2^{-p}$.
-Firstly, $u$ is exact. Then $v = e^{2x} (1+\theta_1)$ and
+Firstly, $u$ is exact, assuming $x$ is exact with precision $p$.
+Then $v = e^{2x} (1+\theta_1)$ and
 $w = (e^{2x}+1) (1+\theta_2)^2$.
 The error on $r$ is bounded by $\frac{1}{2} \ulp(v) + \frac{1}{2} \ulp(r)$.
 Assume $\ulp(v) = 2^e \ulp(r)$, with $e \geq 0$;
@@ -1864,27 +1865,29 @@ and then $s = \tanh(x) (1+\theta_4)^{2^e+4}$.
 
 \begin{lemma}
 For $|x| \leq 1/2$, and $|y| \leq |x|^{-1/2}$, we have:
-\[ |(1+x)^y-1| \leq 2 |y| x. \]
+\[ |(1+x)^y-1| \leq 2.5 |y| x. \]
 \end{lemma}
 \begin{proof}
 We have $(1+x)^y = e^{y \log (1+x)}$,
 with $|y \log (1+x)| \leq |x|^{-1/2} |\log (1+x)|$.
 The function $|x|^{-1/2} \log (1+x)$ is increasing on $[-1/2,1/2]$, and
-takes as values $\approx -0.490$ in $x=-1/2$ and $\approx 0.286$ in $x=1/2$,
-thus is bounded in absolute value by $1/2$.
-This yields $|y \log (1+x)| \leq 1/2$.
-Now it is easy to see that for $|t| \leq 1/2$, we have
-$|e^t-1| \leq 1.3 |t|$.
-Thus $|(1+x)^y-1| \leq 1.3 |y| |\log (1+x)|$.
+takes as values $\approx -0.980$ in $x=-1/2$ and $\approx 0.573$ in $x=1/2$,
+thus is bounded in absolute value by $1$.
+This yields $|y \log (1+x)| \leq 1$.
+Now it is easy to see that for $|t| \leq 1$, we have
+$|e^t-1| \leq 1.72 |t|$.
+Thus $|(1+x)^y-1| \leq 1.72 |y| |\log (1+x)|$.
 The result follows from $|\log (1+x)| \leq 1.4 |x|$ for $|x| \leq 1/2$,
-and $1.3 \times 1.4 \leq 2$.
+and $1.72 \times 1.4 \leq 2.5$.
 \end{proof}
 
 Applying the above lemma for $x=\theta_4$ and $y=2^e+4$,
 assuming $2^e+4 \leq 2^{p/2}$,
-we get $s = \tanh(x) [1 + 2(2^e+4)\theta_5]$.
+we get $|(1+\theta_4)^{2^e+4} - 1| \leq 2.5 (2^e+4) |\theta_4|$, and thus
+we can write $s = \tanh(x) [1 + 2.5 (2^e+4)\theta_5]$ with
+$|\theta_5| \leq 2^{-p}$.
 Since $2^e+4 \leq 2^{{\rm max}(3,e+1)}$,
-the relative error on $s$ is thus bounded by $2^{{\rm max}(4,e+2)-p}$.
+the relative error on $s$ is thus bounded by $2^{{\rm max}(5,e+3)-p}$.
 
 \subsection{The inverse hyperbolic tangent function}
 
diff --git a/src/tanh.c b/src/tanh.c
index 06adf7874..e731f2623 100644
--- a/src/tanh.c
+++ b/src/tanh.c
@@ -98,13 +98,20 @@ mpfr_tanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
     if (MPFR_GET_EXP (x) < 0)
       Nt += -MPFR_GET_EXP (x);
 
+    /* The error analysis in algorithms.tex assumes that x is exactly
+       representable with working precision Nt.
+       FIXME: adapt the error analysis for the case Nt < PREC(x). */
+    if (Nt < MPFR_PREC(x))
+      Nt = MPFR_PREC(x);
+
     /* initialize of intermediary variable */
     MPFR_GROUP_INIT_2 (group, Nt, t, te);
 
     MPFR_ZIV_INIT (loop, Nt);
     for (;;) {
       /* tanh = (exp(2x)-1)/(exp(2x)+1) */
-      mpfr_mul_2ui (te, x, 1, MPFR_RNDN);  /* 2x */
+      inexact = mpfr_mul_2ui (te, x, 1, MPFR_RNDN);  /* 2x */
+      MPFR_ASSERTD(inexact == 0); /* see FIXME above */
       /* since x > 0, we can only have an overflow */
       mpfr_exp (te, te, MPFR_RNDN);        /* exp(2x) */
       if (MPFR_UNLIKELY (MPFR_IS_INF (te))) {
@@ -119,13 +126,13 @@ mpfr_tanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
         break;
       }
       d = MPFR_GET_EXP (te);              /* For Error calculation */
-      mpfr_add_ui (t, te, 1, MPFR_RNDD);   /* exp(2x) + 1*/
-      mpfr_sub_ui (te, te, 1, MPFR_RNDU);  /* exp(2x) - 1*/
+      mpfr_add_ui (t, te, 1, MPFR_RNDD);   /* exp(2x) + 1 */
+      mpfr_sub_ui (te, te, 1, MPFR_RNDU);  /* exp(2x) - 1 */
       d = d - MPFR_GET_EXP (te);
-      mpfr_div (t, te, t, MPFR_RNDN);      /* (exp(2x)-1)/(exp(2x)+1)*/
+      mpfr_div (t, te, t, MPFR_RNDN);      /* (exp(2x)-1)/(exp(2x)+1) */
 
-      /* Calculation of the error */
-      d = MAX(3, d + 1);
+      /* Calculation of the error, see algorithms.tex */
+      d = MAX(4, d + 2);
       err = Nt - (d + 1);
 
       if (MPFR_LIKELY ((d <= Nt / 2) && MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
diff --git a/tests/ttanh.c b/tests/ttanh.c
index 90798d3ff..e2628a2e3 100644
--- a/tests/ttanh.c
+++ b/tests/ttanh.c
@@ -116,11 +116,31 @@ special_overflow (void)
   mpfr_clear (x);
 }
 
+/* This test was generated from bad_cases, with input y=-7.778@-1 = -3823/8192.
+   For the x value below, we have atanh(y) < x, thus since tanh() is increasing,
+   y < tanh(x), and thus tanh(x) rounded towards zero should give -3822/8192. */
+static void
+bug20171218 (void)
+{
+  mpfr_t x, y, z;
+  mpfr_init2 (x, 813);
+  mpfr_init2 (y, 12);
+  mpfr_init2 (z, 12);
+  mpfr_set_str (x, "-8.17cd20bfc17ae00935dc3abad8e17ab43d3ef7740c320798eefb93191f4a62dba9a2daa5efb6eace21130abd87e3ee2eadd2ad8ddae883d2f2db5dee1ac7ce3c59d16eca09e2ca3f21dc2a0386c037a0d3972e62d5b6e82446032020705553c566b1df24f40@-1", 16, MPFR_RNDN);
+  mpfr_tanh (y, x, MPFR_RNDZ);
+  mpfr_set_str (z, "-7.770@-1", 16, MPFR_RNDN);
+  MPFR_ASSERTN(mpfr_equal_p (y, z));
+  mpfr_clear (x);
+  mpfr_clear (y);
+  mpfr_clear (z);
+}
+
 int
 main (int argc, char *argv[])
 {
   tests_start_mpfr ();
 
+  bug20171218 ();
   special_overflow ();
   special ();