Fixed bug for zeta(s) with s near an even negative integer.

(merged changesets r9852-9854 from the trunk)

git-svn-id: svn://scm.gforge.inria.fr/svn/mpfr/branches/3.1@9855 280ebfd0-de03-0410-8827-d642c229c3f4

3 files changed, 47 insertions, 10 deletions

diff --git a/doc/algorithms.tex b/doc/algorithms.tex
index 3b987b0ad..208dec1e1 100644
--- a/doc/algorithms.tex
+++ b/doc/algorithms.tex
@@ -3097,7 +3097,9 @@ The algorithm for the Riemann Zeta function is due to Jean-Luc R\'emy
 and Sapphorain P\'etermann \cite{PeRe06,PeRe07}. For $s < 1/2$ we use the
 functional equation
 \[ \zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s)
-   \zeta(1-s). \]
+   \zeta(1-s) \]
+(in that case, one should take care of cancellation when $\sin(\pi s/2)$ is
+small, i.e., when $s$ is near an even negative integer).
 For $s \geq 1/2$ we use the Euler-MacLaurin summation formula, applied
 to the real function $f(x) = x^{-s}$ for $s > 1$:
 \[ \zeta(s) = \sum_{k=1}^{N-1} \frac{1}{k^s} + \frac{1}{2N^s}
diff --git a/src/zeta.c b/src/zeta.c
index f29e5d0f5..e7042d0a6 100644
--- a/src/zeta.c
+++ b/src/zeta.c
@@ -377,8 +377,8 @@ mpfr_zeta (mpfr_t z, mpfr_srcptr s, mpfr_rnd_t rnd_mode)
         }
     }
 
-  /* Check for case s= 1 before changing the exponent range */
-  if (mpfr_cmp (s, __gmpfr_one) ==0)
+  /* Check for case s=1 before changing the exponent range */
+  if (mpfr_cmp (s, __gmpfr_one) == 0)
     {
       MPFR_SET_INF (z);
       MPFR_SET_POS (z);
@@ -420,7 +420,7 @@ mpfr_zeta (mpfr_t z, mpfr_srcptr s, mpfr_rnd_t rnd_mode)
       MPFR_ZIV_INIT (loop, prec1);
       for (;;)
         {
-          mpfr_sub (s1, __gmpfr_one, s, MPFR_RNDN);/* s1 = 1-s */
+          mpfr_sub (s1, __gmpfr_one, s, MPFR_RNDN); /* s1 = 1-s */
           mpfr_zeta_pos (z_pre, s1, MPFR_RNDN);   /* zeta(1-s)  */
           mpfr_gamma (y, s1, MPFR_RNDN);          /* gamma(1-s) */
           if (MPFR_IS_INF (y)) /* Zeta(s) < 0 for -4k-2 < s < -4k,
@@ -432,17 +432,32 @@ mpfr_zeta (mpfr_t z, mpfr_srcptr s, mpfr_rnd_t rnd_mode)
               break;
             }
           mpfr_mul (z_pre, z_pre, y, MPFR_RNDN);  /* gamma(1-s)*zeta(1-s) */
-          mpfr_const_pi (p, MPFR_RNDD);
-          mpfr_mul (y, s, p, MPFR_RNDN);
-          mpfr_div_2ui (y, y, 1, MPFR_RNDN);      /* s*Pi/2 */
-          mpfr_sin (y, y, MPFR_RNDN);             /* sin(Pi*s/2) */
-          mpfr_mul (z_pre, z_pre, y, MPFR_RNDN);
+
+          mpfr_const_pi (p, MPFR_RNDD); /* p is Pi */
+
+          /* multiply z_pre by 2^s*Pi^(s-1) where p=Pi, s1=1-s */
           mpfr_mul_2ui (y, p, 1, MPFR_RNDN);      /* 2*Pi */
           mpfr_neg (s1, s1, MPFR_RNDN);           /* s-1 */
           mpfr_pow (y, y, s1, MPFR_RNDN);         /* (2*Pi)^(s-1) */
           mpfr_mul (z_pre, z_pre, y, MPFR_RNDN);
           mpfr_mul_2ui (z_pre, z_pre, 1, MPFR_RNDN);
 
+          /* multiply z_pre by sin(Pi*s/2) */
+          mpfr_mul (y, s, p, MPFR_RNDN);
+          mpfr_div_2ui (p, y, 1, MPFR_RNDN);      /* p = s*Pi/2 */
+          mpfr_sin (y, p, MPFR_RNDN);             /* y = sin(Pi*s/2) */
+          if (MPFR_GET_EXP(y) < 0) /* take account of cancellation in sin(p) */
+            {
+              mpfr_t t;
+              mpfr_init2 (t, prec1 - MPFR_GET_EXP(y));
+              mpfr_const_pi (t, MPFR_RNDD);
+              mpfr_mul (t, s, t, MPFR_RNDN);
+              mpfr_div_2ui (t, t, 1, MPFR_RNDN);
+              mpfr_sin (y, t, MPFR_RNDN);
+              mpfr_clear (t);
+            }
+          mpfr_mul (z_pre, z_pre, y, MPFR_RNDN);
+
           if (MPFR_LIKELY (MPFR_CAN_ROUND (z_pre, prec1 - add, precz,
                                            rnd_mode)))
             break;
diff --git a/tests/tzeta.c b/tests/tzeta.c
index 7bdf65cdd..ab0665983 100644
--- a/tests/tzeta.c
+++ b/tests/tzeta.c
@@ -243,7 +243,6 @@ main (int argc, char *argv[])
   mpfr_set_str_binary (s, "1.10010e4");
   mpfr_zeta (z, s, MPFR_RNDZ);
 
-
   mpfr_set_prec (s, 53);
   mpfr_set_prec (y, 53);
   mpfr_set_prec (z, 53);
@@ -394,6 +393,27 @@ main (int argc, char *argv[])
   mpfr_nextabove (s);
   MPFR_ASSERTN (mpfr_equal_p (z, s) && inex > 0);
 
+  /* bug reported by Fredrik Johansson on 19 Jan 2016 */
+  mpfr_set_prec (s, 536);
+  mpfr_set_ui_2exp (s, 1, -424, MPFR_RNDN);
+  mpfr_sub_ui (s, s, 128, MPFR_RNDN);  /* -128 + 2^(-424) */
+  for (prec = 6; prec <= 536; prec += 8) /* should go through 318 */
+    {
+      mpfr_set_prec (z, prec);
+      mpfr_zeta (z, s, MPFR_RNDD);
+      mpfr_set_prec (y, prec + 10);
+      mpfr_zeta (y, s, MPFR_RNDD);
+      mpfr_prec_round (y, prec, MPFR_RNDD);
+      if (! mpfr_equal_p (z, y))
+        {
+          printf ("mpfr_zeta fails near -128 for inprec=%lu outprec=%lu\n",
+                  (unsigned long) mpfr_get_prec (s), (unsigned long) prec);
+          printf ("expected "); mpfr_dump (y);
+          printf ("got      "); mpfr_dump (z);
+          exit (1);
+        }
+    }
+
   mpfr_clear (s);
   mpfr_clear (y);
   mpfr_clear (z);