[src/root.c] Fixed the case |x| = 1 and k > 100.

[tests/troot.c] Added non-regression tests.
(merged changesets r11978-11981 from the trunk)

[From SVN r11982 (branches/3.1)]

src/root.c

@@ -107,6 +107,11 @@ mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
       MPFR_RET_NAN;
     }
 
+  /* Special case |x| = 1. Note that if x = -1, then k is odd
+     (NaN results have already been filtered), so that y = -1. */
+  if (mpfr_cmpabs (x, __gmpfr_one) == 0)
+    return mpfr_set (y, x, rnd_mode);
+
   /* General case */
 
   /* For large k, use exp(log(x)/k). The threshold of 100 seems to be quite
@@ -188,6 +193,14 @@ mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
    Assume all special cases have been eliminated before.
    In the extended exponent range, overflows/underflows are not possible.
    Assume x > 0, or x < 0 and k odd.
+   Also assume |x| <> 1 because log(1) = 0, which does not have an exponent
+   and would yield a failure in the error bound computation. A priori, this
+   constraint is quite artificial because if |x| is close enough to 1, then
+   the exponent of log|x| does not need to be used (in the code, err would
+   be 1 in such a domain). So this constraint |x| <> 1 could be avoided in
+   the code. However, this is an exact case easy to detect, so that such a
+   change would be useless. Values very close to 1 are not an issue, since
+   an underflow is not possible before the MPFR_GET_EXP.
 */
 static int
 mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
@@ -219,7 +232,8 @@ mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
       mpfr_log (t, absx, MPFR_RNDN);
       /* t = log|x| * (1 + theta) with |theta| <= 2^(-w) */
       mpfr_div_ui (t, t, k, MPFR_RNDN);
-      expt = MPFR_GET_EXP (t);
+      /* No possible underflow in mpfr_log and mpfr_div_ui. */
+      expt = MPFR_GET_EXP (t);  /* assumes t <> 0 */
       /* t = log|x|/k * (1 + theta) + eps with |theta| <= 2^(-w)
          and |eps| <= 1/2 ulp(t), thus the total error is bounded
          by 1.5 * 2^(expt - w) */

tests/troot.c

@@ -405,6 +405,26 @@ cmp_pow (void)
   mpfr_clears (x, y1, y2, (mpfr_ptr) 0);
 }
 
+static void
+bug20171214 (void)
+{
+  mpfr_t x, y;
+  int inex;
+
+  mpfr_init2 (x, 805);
+  mpfr_init2 (y, 837);
+  mpfr_set_ui (x, 1, MPFR_RNDN);
+  inex = mpfr_root (y, x, 120, MPFR_RNDN);
+  MPFR_ASSERTN (inex == 0);
+  MPFR_ASSERTN (mpfr_cmp_ui (y, 1) == 0);
+  mpfr_set_si (x, -1, MPFR_RNDN);
+  inex = mpfr_root (y, x, 121, MPFR_RNDN);
+  MPFR_ASSERTN (inex == 0);
+  MPFR_ASSERTN (mpfr_cmp_si (y, -1) == 0);
+  mpfr_clear (x);
+  mpfr_clear (y);
+}
+
 int
 main (void)
 {
@@ -415,6 +435,7 @@ main (void)
 
   tests_start_mpfr ();
 
+  bug20171214 ();
   exact_powers (3, 1000);
   special ();
   bigint ();