128-bit long double implementation of e_log2.

diff --git a/sysdeps/ieee754/ldbl-128/e_log2l.c b/sysdeps/ieee754/ldbl-128/e_log2l.c
new file mode 100644 (file)
index 0000000..5f887a5
--- /dev/null
+++ b/sysdeps/ieee754/ldbl-128/e_log2l.c
@@ -0,0 +1,237 @@
+/*                                                      log2l.c
+ *      Base 2 logarithm, 128-bit long double precision
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * long double x, y, log2l();
+ *
+ * y = log2l( x );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * Returns the base 2 logarithm of x.
+ *
+ * The argument is separated into its exponent and fractional
+ * parts.  If the exponent is between -1 and +1, the (natural)
+ * logarithm of the fraction is approximated by
+ *
+ *     log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
+ *
+ * Otherwise, setting  z = 2(x-1)/x+1),
+ *
+ *     log(x) = z + z^3 P(z)/Q(z).
+ *
+ *
+ *
+ * ACCURACY:
+ *
+ *                      Relative error:
+ * arithmetic   domain     # trials      peak         rms
+ *    IEEE      0.5, 2.0     100,000    2.6e-34     4.9e-35
+ *    IEEE     exp(+-10000)  100,000    9.6e-35     4.0e-35
+ *
+ * In the tests over the interval exp(+-10000), the logarithms
+ * of the random arguments were uniformly distributed over
+ * [-10000, +10000].
+ *
+ */
+
+/*
+   Cephes Math Library Release 2.2:  January, 1991
+   Copyright 1984, 1991 by Stephen L. Moshier
+   Adapted for glibc November, 2001
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+/* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
+ * 1/sqrt(2) <= x < sqrt(2)
+ * Theoretical peak relative error = 5.3e-37,
+ * relative peak error spread = 2.3e-14
+ */
+static const long double P[13] =
+{
+  1.313572404063446165910279910527789794488E4L,
+  7.771154681358524243729929227226708890930E4L,
+  2.014652742082537582487669938141683759923E5L,
+  3.007007295140399532324943111654767187848E5L,
+  2.854829159639697837788887080758954924001E5L,
+  1.797628303815655343403735250238293741397E5L,
+  7.594356839258970405033155585486712125861E4L,
+  2.128857716871515081352991964243375186031E4L,
+  3.824952356185897735160588078446136783779E3L,
+  4.114517881637811823002128927449878962058E2L,
+  2.321125933898420063925789532045674660756E1L,
+  4.998469661968096229986658302195402690910E-1L,
+  1.538612243596254322971797716843006400388E-6L
+};
+static const long double Q[12] =
+{
+  3.940717212190338497730839731583397586124E4L,
+  2.626900195321832660448791748036714883242E5L,
+  7.777690340007566932935753241556479363645E5L,
+  1.347518538384329112529391120390701166528E6L,
+  1.514882452993549494932585972882995548426E6L,
+  1.158019977462989115839826904108208787040E6L,
+  6.132189329546557743179177159925690841200E5L,
+  2.248234257620569139969141618556349415120E5L,
+  5.605842085972455027590989944010492125825E4L,
+  9.147150349299596453976674231612674085381E3L,
+  9.104928120962988414618126155557301584078E2L,
+  4.839208193348159620282142911143429644326E1L
+/* 1.000000000000000000000000000000000000000E0L, */
+};
+
+/* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
+ * where z = 2(x-1)/(x+1)
+ * 1/sqrt(2) <= x < sqrt(2)
+ * Theoretical peak relative error = 1.1e-35,
+ * relative peak error spread 1.1e-9
+ */
+static const long double R[6] =
+{
+  1.418134209872192732479751274970992665513E5L,
+ -8.977257995689735303686582344659576526998E4L,
+  2.048819892795278657810231591630928516206E4L,
+ -2.024301798136027039250415126250455056397E3L,
+  8.057002716646055371965756206836056074715E1L,
+ -8.828896441624934385266096344596648080902E-1L
+};
+static const long double S[6] =
+{
+  1.701761051846631278975701529965589676574E6L,
+ -1.332535117259762928288745111081235577029E6L,
+  4.001557694070773974936904547424676279307E5L,
+ -5.748542087379434595104154610899551484314E4L,
+  3.998526750980007367835804959888064681098E3L,
+ -1.186359407982897997337150403816839480438E2L
+/* 1.000000000000000000000000000000000000000E0L, */
+};
+
+static const long double
+/* log2(e) - 1 */
+LOG2EA = 4.4269504088896340735992468100189213742664595E-1L,
+/* sqrt(2)/2 */
+SQRTH = 7.071067811865475244008443621048490392848359E-1L;
+
+
+/* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
+
+static long double
+neval (long double x, const long double *p, int n)
+{
+  long double y;
+
+  p += n;
+  y = *p--;
+  do
+    {
+      y = y * x + *p--;
+    }
+  while (--n > 0);
+  return y;
+}
+
+
+/* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
+
+static long double
+deval (long double x, const long double *p, int n)
+{
+  long double y;
+
+  p += n;
+  y = x + *p--;
+  do
+    {
+      y = y * x + *p--;
+    }
+  while (--n > 0);
+  return y;
+}
+
+
+
+long double
+__ieee754_log2l (x)
+     long double x;
+{
+  long double z;
+  long double y;
+  int e;
+
+/* Test for domain */
+  if (x <= 0.0L)
+    {
+      if (x == 0.0L)
+       return (-1.0L / (x - x));
+      else
+       return (x - x) / (x - x);
+    }
+  if (!__finitel (x))
+    return (x + x);
+
+/* separate mantissa from exponent */
+
+/* Note, frexp is used so that denormal numbers
+ * will be handled properly.
+ */
+  x = __frexpl (x, &e);
+
+
+/* logarithm using log(x) = z + z**3 P(z)/Q(z),
+ * where z = 2(x-1)/x+1)
+ */
+  if ((e > 2) || (e < -2))
+    {
+      if (x < SQRTH)
+       {                       /* 2( 2x-1 )/( 2x+1 ) */
+         e -= 1;
+         z = x - 0.5L;
+         y = 0.5L * z + 0.5L;
+       }
+      else
+       {                       /*  2 (x-1)/(x+1)   */
+         z = x - 0.5L;
+         z -= 0.5L;
+         y = 0.5L * x + 0.5L;
+       }
+      x = z / y;
+      z = x * x;
+      y = x * (z * neval (z, R, 5) / deval (z, S, 5));
+      goto done;
+    }
+
+
+/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
+
+  if (x < SQRTH)
+    {
+      e -= 1;
+      x = 2.0 * x - 1.0L;      /*  2x - 1  */
+    }
+  else
+    {
+      x = x - 1.0L;
+    }
+  z = x * x;
+  y = x * (z * neval (x, P, 12) / deval (x, Q, 11));
+  y = y - 0.5 * z;
+
+done:
+
+/* Multiply log of fraction by log2(e)
+ * and base 2 exponent by 1
+ */
+  z = y * LOG2EA;
+  z += x * LOG2EA;
+  z += y;
+  z += x;
+  z += e;
+  return (z);
+}