mpfr-4.0.0

git-svn-id: svn://tug.org/texlive/trunk@46140 c570f23f-e606-0410-a88d-b1316a301751

1 files changed, 101 insertions, 16 deletions

diff --git a/Build/source/libs/mpfr/mpfr-src/src/atanh.c b/Build/source/libs/mpfr/mpfr-src/src/atanh.c
index ec5a6cf6f1f..5f9953c4e7c 100644
--- a/Build/source/libs/mpfr/mpfr-src/src/atanh.c
+++ b/Build/source/libs/mpfr/mpfr-src/src/atanh.c
@@ -23,11 +23,84 @@ http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
 #define MPFR_NEED_LONGLONG_H
 #include "mpfr-impl.h"
 
- /* The computation of atanh is done by
-       atanh= 1/2*ln(x+1)-1/2*ln(1-x)   */
+/* Put in y an approximation of atanh(x) for x small.
+   We assume x <= 1/2, in which case:
+   x <= y ~ atanh(x) = x + x^3/3 + x^5/5 + x^7/7 + ... <= 2*x.
+   Return k such that the error is bounded by 2^k*ulp(y).
+*/
+static int
+mpfr_atanh_small (mpfr_ptr y, mpfr_srcptr x)
+{
+  mpfr_prec_t p = MPFR_PREC(y), err;
+  mpfr_t x2, t, u;
+  unsigned long i;
+  int k;
+
+  MPFR_ASSERTD(MPFR_GET_EXP (x) <= -1);
+
+  /* in the following, theta represents a value with |theta| <= 2^(1-p)
+     (might be a different value each time) */
+
+  mpfr_init2 (t, p);
+  mpfr_init2 (u, p);
+  mpfr_init2 (x2, p);
+  mpfr_set (t, x, MPFR_RNDF); /* t = x * (1 + theta) */
+  mpfr_set (y, t, MPFR_RNDF); /* exact */
+  mpfr_mul (x2, x, x, MPFR_RNDF); /* x2 = x^2 * (1 + theta) */
+  for (i = 3; ; i += 2)
+    {
+      mpfr_mul (t, t, x2, MPFR_RNDF);    /* t = x^i * (1 + theta)^i */
+      mpfr_div_ui (u, t, i, MPFR_RNDF); /* u = x^i/i * (1 + theta)^(i+1) */
+      if (MPFR_GET_EXP (u) <= MPFR_GET_EXP (y) - p) /* |u| < ulp(y) */
+        break;
+      mpfr_add (y, y, u, MPFR_RNDF); /* error <= ulp(y) */
+    }
+  /* We assume |(1 + theta)^(i+1)| <= 2.
+     The neglected part is at most |u| + |u|/4 + |u|/16 + ... <= 4/3*|u|,
+     which has to be multiplied by |(1 + theta)^(i+1)| <= 2, thus at most
+     3 ulp(y).
+     The rounding error on y is bounded by:
+     * for the (i-3)/2 add/sub, each error is bounded by ulp(y_i),
+       where y_i is the current value of y, which is bounded by ulp(y)
+       for y the final value (since it increases in absolute value),
+       this yields (i-3)/2*ulp(y)
+     * from Lemma 3.1 from [Higham02] (see algorithms.tex),
+       the relative error on u at step i is bounded by:
+       (i+1)*epsilon/(1-(i+1)*epsilon) where epsilon = 2^(1-p).
+       If (i+1)*epsilon <= 1/2, then the relative error on u at
+       step i is bounded by 2*(i+1)*epsilon, and since |u| <= 1/2^(i+1)
+       at step i, this gives an absolute error bound of;
+       2*epsilon*x*(4/2^4 + 6/2^6 + 8/2^8 + ...) = 2*2^(1-p)*x*(7/18) =
+       14/9*2^(-p)*x <= 2*ulp(x).
+
+     If (i+1)*epsilon <= 1/2, then the relative error on u at step i
+     is bounded by (i+1)*epsilon/(1-(i+1)*epsilon) <= 1, thus it follows
+     |(1 + theta)^(i+1)| <= 2.
+
+     Finally the total error is bounded by 3*ulp(y) + (i-3)/2*ulp(y) +2*ulp(x).
+     Since x <= 2*y, we have ulp(x) <= 2*ulp(y), thus the error is bounded by:
+     (i+7)/2*ulp(y).
+  */
+  err = (i + 8) / 2; /* ceil((i+7)/2) */
+  k = __gmpfr_int_ceil_log2 (err);
+  MPFR_ASSERTN(k + 2 < p);
+  /* if k + 2 < p, since k = ceil(log2(err)), we have err <= 2^k <= 2^(p-3),
+     thus i+7 <= 2*err <= 2^(p-2), thus (i+7)*epsilon <= 1/2, which implies
+     our assumption (i+1)*epsilon <= 1/2. */
+  mpfr_clear (t);
+  mpfr_clear (u);
+  mpfr_clear (x2);
+  return k;
+}
+
+/* The computation of atanh is done by:
+   atanh = ln((1+x)/(1-x)) / 2
+   except when x is very small, in which case atanh = x + tiny error,
+   and when x is small, where we use directly the Taylor expansion.
+*/
 
 int
-mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
+mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt, mpfr_rnd_t rnd_mode)
 {
   int inexact;
   mpfr_t x, t, te;
@@ -67,7 +140,7 @@ mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
         {
           MPFR_SET_INF (y);
           MPFR_SET_SAME_SIGN (y, xt);
-          mpfr_set_divby0 ();
+          MPFR_SET_DIVBY0 ();
           MPFR_RET (0);
         }
       MPFR_SET_NAN (y);
@@ -85,29 +158,42 @@ mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
   MPFR_TMP_INIT_ABS (x, xt);
   Ny = MPFR_PREC (y);
   Nt = MAX (Nx, Ny);
-  /* the optimal number of bits : see algorithms.ps */
   Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4;
 
-  /* initialise of intermediary variable */
+  /* initialize of intermediary variable */
   mpfr_init2 (t, Nt);
   mpfr_init2 (te, Nt);
 
-  /* First computation of cosh */
   MPFR_ZIV_INIT (loop, Nt);
   for (;;)
     {
+      int k;
+
+        /* small case: assuming the AGM algorithm used by mpfr_log uses
+           log2(p) steps for a precision of p bits, we try the special
+           variant whenever EXP(x) <= -p/log2(p). */
+        k = 1 + __gmpfr_int_ceil_log2 (Ny); /* the +1 avoids a division by 0
+                                               when Ny=1 */
+        if (MPFR_GET_EXP (x) <= - 1 - (mpfr_exp_t) (Ny / k))
+          /* this implies EXP(x) <= -1 thus x < 1/2 */
+          {
+            err = Nt - mpfr_atanh_small (t, x);
+            goto round;
+          }
+
       /* compute atanh */
-      mpfr_ui_sub (te, 1, x, MPFR_RNDU);   /* (1-xt)*/
-      mpfr_add_ui (t,  x, 1, MPFR_RNDD);   /* (xt+1)*/
-      mpfr_div (t, t, te, MPFR_RNDN);      /* (1+xt)/(1-xt)*/
-      mpfr_log (t, t, MPFR_RNDN);          /* ln((1+xt)/(1-xt))*/
-      mpfr_div_2ui (t, t, 1, MPFR_RNDN);   /* (1/2)*ln((1+xt)/(1-xt))*/
+      mpfr_ui_sub (te, 1, x, MPFR_RNDU);   /* (1-x) with x = |xt| */
+      mpfr_add_ui (t, x, 1, MPFR_RNDD);    /* (1+x) */
+      mpfr_div (t, t, te, MPFR_RNDN);      /* (1+x)/(1-x) */
+      mpfr_log (t, t, MPFR_RNDN);          /* ln((1+x)/(1-x)) */
+      mpfr_div_2ui (t, t, 1, MPFR_RNDN);   /* ln((1+x)/(1-x)) / 2 */
 
       /* error estimate: see algorithms.tex */
       /* FIXME: this does not correspond to the value in algorithms.tex!!! */
-      /* err=Nt-__gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t)));*/
+      /* err = Nt - __gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t))); */
       err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1);
 
+    round:
       if (MPFR_LIKELY (MPFR_IS_ZERO (t)
                        || MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
         break;
@@ -121,10 +207,9 @@ mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
 
   inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
 
-  mpfr_clear(t);
-  mpfr_clear(te);
+  mpfr_clear (t);
+  mpfr_clear (te);
 
   MPFR_SAVE_EXPO_FREE (expo);
   return mpfr_check_range (y, inexact, rnd_mode);
 }
-