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+// The template and inlines for the numeric_limits classes. -*- C++ -*-
+
+// Copyright (C) 1999-2013 Free Software Foundation, Inc.
+//
+// This file is part of the GNU ISO C++ Library.  This library is free
+// software; you can redistribute it and/or modify it under the
+// terms of the GNU General Public License as published by the
+// Free Software Foundation; either version 3, or (at your option)
+// any later version.
+
+// This library is distributed in the hope that it will be useful,
+// but WITHOUT ANY WARRANTY; without even the implied warranty of
+// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+// GNU General Public License for more details.
+
+// Under Section 7 of GPL version 3, you are granted additional
+// permissions described in the GCC Runtime Library Exception, version
+// 3.1, as published by the Free Software Foundation.
+
+// You should have received a copy of the GNU General Public License and
+// a copy of the GCC Runtime Library Exception along with this program;
+// see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
+// <http://www.gnu.org/licenses/>.
+
+/** @file include/limits
+ *  This is a Standard C++ Library header.
+ */
+
+// Note: this is not a conforming implementation.
+// Written by Gabriel Dos Reis <gdr@codesourcery.com>
+
+//
+// ISO 14882:1998
+// 18.2.1
+//
+
+#ifndef _GLIBCXX_NUMERIC_LIMITS
+#define _GLIBCXX_NUMERIC_LIMITS 1
+
+#pragma GCC system_header
+
+#include <float.h>
+#include <limits.h>
+#include <math.h>
+
+
+//
+// The numeric_limits<> traits document implementation-defined aspects
+// of fundamental arithmetic data types (integers and floating points).
+// From Standard C++ point of view, there are 14 such types:
+//   * integers
+//         bool							(1)
+//         char, signed char, unsigned char, wchar_t            (4)
+//         short, unsigned short				(2)
+//         int, unsigned					(2)
+//         long, unsigned long					(2)
+//
+//   * floating points
+//         float						(1)
+//         double						(1)
+//         long double						(1)
+//
+// GNU C++ understands (where supported by the host C-library)
+//   * integer
+//         long long, unsigned long long			(2)
+//
+// which brings us to 16 fundamental arithmetic data types in GNU C++.
+//
+//
+// Since a numeric_limits<> is a bit tricky to get right, we rely on
+// an interface composed of macros which should be defined in config/os
+// or config/cpu when they differ from the generic (read arbitrary)
+// definitions given here.
+//
+
+// These values can be overridden in the target configuration file.
+// The default values are appropriate for many 32-bit targets.
+
+// GCC only intrinsically supports modulo integral types.  The only remaining
+// integral exceptional values is division by zero.  Only targets that do not
+// signal division by zero in some "hard to ignore" way should use false.
+#ifndef __glibcxx_integral_traps
+# define __glibcxx_integral_traps true
+#endif
+
+// float
+//
+
+// Default values.  Should be overridden in configuration files if necessary.
+
+#ifndef __glibcxx_float_has_denorm_loss
+#  define __glibcxx_float_has_denorm_loss false
+#endif
+#ifndef __glibcxx_float_traps
+#  define __glibcxx_float_traps false
+#endif
+#ifndef __glibcxx_float_tinyness_before
+#  define __glibcxx_float_tinyness_before false
+#endif
+
+// double
+
+// Default values.  Should be overridden in configuration files if necessary.
+
+#ifndef __glibcxx_double_has_denorm_loss
+#  define __glibcxx_double_has_denorm_loss false
+#endif
+#ifndef __glibcxx_double_traps
+#  define __glibcxx_double_traps false
+#endif
+#ifndef __glibcxx_double_tinyness_before
+#  define __glibcxx_double_tinyness_before false
+#endif
+
+// long double
+
+// Default values.  Should be overridden in configuration files if necessary.
+
+#ifndef __glibcxx_long_double_has_denorm_loss
+#  define __glibcxx_long_double_has_denorm_loss false
+#endif
+#ifndef __glibcxx_long_double_traps
+#  define __glibcxx_long_double_traps false
+#endif
+#ifndef __glibcxx_long_double_tinyness_before
+#  define __glibcxx_long_double_tinyness_before false
+#endif
+
+// You should not need to define any macros below this point.
+
+#define __glibcxx_signed(T)	((T)(-1) < 0)
+
+#define __glibcxx_min(T) \
+  (__glibcxx_signed (T) ? -__glibcxx_max (T) - 1 : (T)0)
+
+#define __glibcxx_max(T) \
+  (__glibcxx_signed (T) ? \
+   (((((T)1 << (__glibcxx_digits (T) - 1)) - 1) << 1) + 1) : ~(T)0)
+
+#define __glibcxx_digits(T) \
+  (sizeof(T) * CHAR_BIT - __glibcxx_signed (T))
+
+// The fraction 643/2136 approximates log10(2) to 7 significant digits.
+#define __glibcxx_digits10(T) \
+  (__glibcxx_digits (T) * 643L / 2136)
+
+#define __glibcxx_max_digits10(T) \
+  (2 + (T) * 643L / 2136)
+
+namespace std
+{
+
+  /**
+   *  @brief Describes the rounding style for floating-point types.
+   *
+   *  This is used in the std::numeric_limits class.
+  */
+  enum float_round_style
+  {
+    round_indeterminate       = -1,    /// Intermediate.
+    round_toward_zero         = 0,     /// To zero.
+    round_to_nearest          = 1,     /// To the nearest representable value.
+    round_toward_infinity     = 2,     /// To infinity.
+    round_toward_neg_infinity = 3      /// To negative infinity.
+  };
+
+  /**
+   *  @brief Describes the denormalization for floating-point types.
+   *
+   *  These values represent the presence or absence of a variable number
+   *  of exponent bits.  This type is used in the std::numeric_limits class.
+  */
+  enum float_denorm_style
+  {
+    /// Indeterminate at compile time whether denormalized values are allowed.
+    denorm_indeterminate = -1,
+    /// The type does not allow denormalized values.
+    denorm_absent        = 0,
+    /// The type allows denormalized values.
+    denorm_present       = 1
+  };
+
+  /**
+   *  @brief Part of std::numeric_limits.
+   *
+   *  The @c static @c const members are usable as integral constant
+   *  expressions.
+   *
+   *  @note This is a separate class for purposes of efficiency; you
+   *        should only access these members as part of an instantiation
+   *        of the std::numeric_limits class.
+  */
+  struct __numeric_limits_base
+  {
+    /** This will be true for all fundamental types (which have
+	specializations), and false for everything else.  */
+    static const bool is_specialized = false;
+
+    /** The number of @c radix digits that be represented without change:  for
+	integer types, the number of non-sign bits in the mantissa; for
+	floating types, the number of @c radix digits in the mantissa.  */
+    static const int digits = 0;
+
+    /** The number of base 10 digits that can be represented without change. */
+    static const int digits10 = 0;
+
+#if __cplusplus >= 201103L
+    /** The number of base 10 digits required to ensure that values which
+	differ are always differentiated.  */
+    static constexpr int max_digits10 = 0;
+#endif
+
+    /** True if the type is signed.  */
+    static const bool is_signed = false;
+
+    /** True if the type is integer.  */
+    static const bool is_integer = false;
+
+    /** True if the type uses an exact representation. All integer types are
+	exact, but not all exact types are integer.  For example, rational and
+	fixed-exponent representations are exact but not integer. */
+    static const bool is_exact = false;
+
+    /** For integer types, specifies the base of the representation.  For
+	floating types, specifies the base of the exponent representation.  */
+    static const int radix = 0;
+
+    /** The minimum negative integer such that @c radix raised to the power of
+	(one less than that integer) is a normalized floating point number.  */
+    static const int min_exponent = 0;
+
+    /** The minimum negative integer such that 10 raised to that power is in
+	the range of normalized floating point numbers.  */
+    static const int min_exponent10 = 0;
+
+    /** The maximum positive integer such that @c radix raised to the power of
+	(one less than that integer) is a representable finite floating point
+	number.  */
+    static const int max_exponent = 0;
+
+    /** The maximum positive integer such that 10 raised to that power is in
+	the range of representable finite floating point numbers.  */
+    static const int max_exponent10 = 0;
+
+    /** True if the type has a representation for positive infinity.  */
+    static const bool has_infinity = false;
+
+    /** True if the type has a representation for a quiet (non-signaling)
+	Not a Number.  */
+    static const bool has_quiet_NaN = false;
+
+    /** True if the type has a representation for a signaling
+	Not a Number.  */
+    static const bool has_signaling_NaN = false;
+
+    /** See std::float_denorm_style for more information.  */
+    static const float_denorm_style has_denorm = denorm_absent;
+
+    /** True if loss of accuracy is detected as a denormalization loss,
+	rather than as an inexact result. */
+    static const bool has_denorm_loss = false;
+
+    /** True if-and-only-if the type adheres to the IEC 559 standard, also
+	known as IEEE 754.  (Only makes sense for floating point types.)  */
+    static const bool is_iec559 = false;
+
+    /** True if the set of values representable by the type is
+	finite.  All built-in types are bounded, this member would be
+	false for arbitrary precision types. */
+    static const bool is_bounded = false;
+
+    /** True if the type is @e modulo. A type is modulo if, for any
+	operation involving +, -, or * on values of that type whose
+	result would fall outside the range [min(),max()], the value
+	returned differs from the true value by an integer multiple of
+	max() - min() + 1. On most machines, this is false for floating
+	types, true for unsigned integers, and true for signed integers.
+	See PR22200 about signed integers.  */
+    static const bool is_modulo = false;
+
+    /** True if trapping is implemented for this type.  */
+    static const bool traps = false;
+
+    /** True if tininess is detected before rounding.  (see IEC 559)  */
+    static const bool tinyness_before = false;
+
+    /** See std::float_round_style for more information.  This is only
+	meaningful for floating types; integer types will all be
+	round_toward_zero.  */
+    static const float_round_style round_style = 
+						    round_toward_zero;
+  };
+
+  /**
+   *  @brief Properties of fundamental types.
+   *
+   *  This class allows a program to obtain information about the
+   *  representation of a fundamental type on a given platform.  For
+   *  non-fundamental types, the functions will return 0 and the data
+   *  members will all be @c false.
+   *
+   *  _GLIBCXX_RESOLVE_LIB_DEFECTS:  DRs 201 and 184 (hi Gaby!) are
+   *  noted, but not incorporated in this documented (yet).
+  */
+  template<typename _Tp>
+    struct numeric_limits : public __numeric_limits_base
+    {
+      /** The minimum finite value, or for floating types with
+	  denormalization, the minimum positive normalized value.  */
+      static const _Tp
+      min() { return _Tp(); }
+
+      /** The maximum finite value.  */
+      static const _Tp
+      max() { return _Tp(); }
+
+#if __cplusplus >= 201103L
+      /** A finite value x such that there is no other finite value y
+       *  where y < x.  */
+      static constexpr _Tp
+      lowest() noexcept { return _Tp(); }
+#endif
+
+      /** The @e machine @e epsilon:  the difference between 1 and the least
+	  value greater than 1 that is representable.  */
+      static const _Tp
+      epsilon() { return _Tp(); }
+
+      /** The maximum rounding error measurement (see LIA-1).  */
+      static const _Tp
+      round_error() { return _Tp(); }
+
+      /** The representation of positive infinity, if @c has_infinity.  */
+      static const _Tp
+      infinity() { return _Tp(); }
+
+      /** The representation of a quiet Not a Number,
+	  if @c has_quiet_NaN. */
+      static const _Tp
+      quiet_NaN() { return _Tp(); }
+
+      /** The representation of a signaling Not a Number, if
+	  @c has_signaling_NaN. */
+      static const _Tp
+      signaling_NaN() { return _Tp(); }
+
+      /** The minimum positive denormalized value.  For types where
+	  @c has_denorm is false, this is the minimum positive normalized
+	  value.  */
+      static const _Tp
+      denorm_min() { return _Tp(); }
+    };
+
+#if __cplusplus >= 201103L
+  template<typename _Tp>
+    struct numeric_limits<const _Tp>
+    : public numeric_limits<_Tp> { };
+
+  template<typename _Tp>
+    struct numeric_limits<volatile _Tp>
+    : public numeric_limits<_Tp> { };
+
+  template<typename _Tp>
+    struct numeric_limits<const volatile _Tp>
+    : public numeric_limits<_Tp> { };
+#endif
+
+  // Now there follow 16 explicit specializations.  Yes, 16.  Make sure
+  // you get the count right. (18 in c++0x mode)
+
+  /// numeric_limits<bool> specialization.
+  template<>
+    struct numeric_limits<bool>
+    {
+      static const bool is_specialized = true;
+
+      static const bool 
+      min() { return false; }
+
+      static const bool 
+      max() { return true; }
+
+#if __cplusplus >= 201103L
+      static constexpr bool
+      lowest() noexcept { return min(); }
+#endif
+      static const int digits = 1;
+      static const int digits10 = 0;
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = false;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const bool 
+      epsilon() { return false; }
+
+      static const bool 
+      round_error() { return false; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const bool 
+      infinity() { return false; }
+
+      static const bool 
+      quiet_NaN() { return false; }
+
+      static const bool 
+      signaling_NaN() { return false; }
+
+      static const bool 
+      denorm_min() { return false; }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = false;
+
+      // It is not clear what it means for a boolean type to trap.
+      // This is a DR on the LWG issue list.  Here, I use integer
+      // promotion semantics.
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<char> specialization.
+  template<>
+    struct numeric_limits<char>
+    {
+      static const bool is_specialized = true;
+
+      static const char 
+      min() { return __glibcxx_min(char); }
+
+      static const char 
+      max() { return __glibcxx_max(char); }
+
+#if __cplusplus >= 201103L
+      static constexpr char 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits = __glibcxx_digits (char);
+      static const int digits10 = __glibcxx_digits10 (char);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = __glibcxx_signed (char);
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const char 
+      epsilon() { return 0; }
+
+      static const char 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const 
+      char infinity() { return char(); }
+
+      static const char 
+      quiet_NaN() { return char(); }
+
+      static const char 
+      signaling_NaN() { return char(); }
+
+      static const char 
+      denorm_min() { return static_cast<char>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = !is_signed;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<signed char> specialization.
+  template<>
+    struct numeric_limits<signed char>
+    {
+      static const bool is_specialized = true;
+
+      static const signed char 
+      min() { return -SCHAR_MAX - 1; }
+
+      static const signed char 
+      max() { return SCHAR_MAX; }
+
+#if __cplusplus >= 201103L
+      static constexpr signed char 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits = __glibcxx_digits (signed char);
+      static const int digits10 
+       = __glibcxx_digits10 (signed char);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = true;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const signed char 
+      epsilon() { return 0; }
+
+      static const signed char 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const signed char 
+      infinity() { return static_cast<signed char>(0); }
+
+      static const signed char 
+      quiet_NaN() { return static_cast<signed char>(0); }
+
+      static const signed char 
+      signaling_NaN() 
+      { return static_cast<signed char>(0); }
+
+      static const signed char 
+      denorm_min() 
+      { return static_cast<signed char>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = false;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<unsigned char> specialization.
+  template<>
+    struct numeric_limits<unsigned char>
+    {
+      static const bool is_specialized = true;
+
+      static const unsigned char 
+      min() { return 0; }
+
+      static const unsigned char 
+      max() { return SCHAR_MAX * 2U + 1; }
+
+#if __cplusplus >= 201103L
+      static constexpr unsigned char 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits 
+       = __glibcxx_digits (unsigned char);
+      static const int digits10 
+       = __glibcxx_digits10 (unsigned char);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = false;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const unsigned char 
+      epsilon() { return 0; }
+
+      static const unsigned char 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const unsigned char 
+      infinity() 
+      { return static_cast<unsigned char>(0); }
+
+      static const unsigned char 
+      quiet_NaN() 
+      { return static_cast<unsigned char>(0); }
+
+      static const unsigned char 
+      signaling_NaN() 
+      { return static_cast<unsigned char>(0); }
+
+      static const unsigned char 
+      denorm_min() 
+      { return static_cast<unsigned char>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = true;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<wchar_t> specialization.
+  template<>
+    struct numeric_limits<wchar_t>
+    {
+      static const bool is_specialized = true;
+
+      static const wchar_t 
+      min() { return __glibcxx_min (wchar_t); }
+
+      static const wchar_t 
+      max() { return __glibcxx_max (wchar_t); }
+
+#if __cplusplus >= 201103L
+      static constexpr wchar_t
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits = __glibcxx_digits (wchar_t);
+      static const int digits10 
+       = __glibcxx_digits10 (wchar_t);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = __glibcxx_signed (wchar_t);
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const wchar_t 
+      epsilon() { return 0; }
+
+      static const wchar_t 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const wchar_t 
+      infinity() { return wchar_t(); }
+
+      static const wchar_t 
+      quiet_NaN() { return wchar_t(); }
+
+      static const wchar_t 
+      signaling_NaN() { return wchar_t(); }
+
+      static const wchar_t 
+      denorm_min() { return wchar_t(); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = !is_signed;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+#if __cplusplus >= 201103L
+  /// numeric_limits<char16_t> specialization.
+  template<>
+    struct numeric_limits<char16_t>
+    {
+      static constexpr bool is_specialized = true;
+
+      static constexpr char16_t 
+      min() noexcept { return __glibcxx_min (char16_t); }
+
+      static constexpr char16_t 
+      max() noexcept { return __glibcxx_max (char16_t); }
+
+      static constexpr char16_t 
+      lowest() noexcept { return min(); }
+
+      static constexpr int digits = __glibcxx_digits (char16_t);
+      static constexpr int digits10 = __glibcxx_digits10 (char16_t);
+      static constexpr int max_digits10 = 0;
+      static constexpr bool is_signed = __glibcxx_signed (char16_t);
+      static constexpr bool is_integer = true;
+      static constexpr bool is_exact = true;
+      static constexpr int radix = 2;
+
+      static constexpr char16_t 
+      epsilon() noexcept { return 0; }
+
+      static constexpr char16_t 
+      round_error() noexcept { return 0; }
+
+      static constexpr int min_exponent = 0;
+      static constexpr int min_exponent10 = 0;
+      static constexpr int max_exponent = 0;
+      static constexpr int max_exponent10 = 0;
+
+      static constexpr bool has_infinity = false;
+      static constexpr bool has_quiet_NaN = false;
+      static constexpr bool has_signaling_NaN = false;
+      static constexpr float_denorm_style has_denorm = denorm_absent;
+      static constexpr bool has_denorm_loss = false;
+
+      static constexpr char16_t 
+      infinity() noexcept { return char16_t(); }
+
+      static constexpr char16_t 
+      quiet_NaN() noexcept { return char16_t(); }
+
+      static constexpr char16_t 
+      signaling_NaN() noexcept { return char16_t(); }
+
+      static constexpr char16_t 
+      denorm_min() noexcept { return char16_t(); }
+
+      static constexpr bool is_iec559 = false;
+      static constexpr bool is_bounded = true;
+      static constexpr bool is_modulo = !is_signed;
+
+      static constexpr bool traps = __glibcxx_integral_traps;
+      static constexpr bool tinyness_before = false;
+      static constexpr float_round_style round_style = round_toward_zero;
+    };
+
+  /// numeric_limits<char32_t> specialization.
+  template<>
+    struct numeric_limits<char32_t>
+    {
+      static constexpr bool is_specialized = true;
+
+      static constexpr char32_t 
+      min() noexcept { return __glibcxx_min (char32_t); }
+
+      static constexpr char32_t 
+      max() noexcept { return __glibcxx_max (char32_t); }
+
+      static constexpr char32_t 
+      lowest() noexcept { return min(); }
+
+      static constexpr int digits = __glibcxx_digits (char32_t);
+      static constexpr int digits10 = __glibcxx_digits10 (char32_t);
+      static constexpr int max_digits10 = 0;
+      static constexpr bool is_signed = __glibcxx_signed (char32_t);
+      static constexpr bool is_integer = true;
+      static constexpr bool is_exact = true;
+      static constexpr int radix = 2;
+
+      static constexpr char32_t 
+      epsilon() noexcept { return 0; }
+
+      static constexpr char32_t 
+      round_error() noexcept { return 0; }
+
+      static constexpr int min_exponent = 0;
+      static constexpr int min_exponent10 = 0;
+      static constexpr int max_exponent = 0;
+      static constexpr int max_exponent10 = 0;
+
+      static constexpr bool has_infinity = false;
+      static constexpr bool has_quiet_NaN = false;
+      static constexpr bool has_signaling_NaN = false;
+      static constexpr float_denorm_style has_denorm = denorm_absent;
+      static constexpr bool has_denorm_loss = false;
+
+      static constexpr char32_t 
+      infinity() noexcept { return char32_t(); }
+
+      static constexpr char32_t 
+      quiet_NaN() noexcept { return char32_t(); }
+
+      static constexpr char32_t 
+      signaling_NaN() noexcept { return char32_t(); }
+
+      static constexpr char32_t 
+      denorm_min() noexcept { return char32_t(); }
+
+      static constexpr bool is_iec559 = false;
+      static constexpr bool is_bounded = true;
+      static constexpr bool is_modulo = !is_signed;
+
+      static constexpr bool traps = __glibcxx_integral_traps;
+      static constexpr bool tinyness_before = false;
+      static constexpr float_round_style round_style = round_toward_zero;
+    };
+#endif
+
+  /// numeric_limits<short> specialization.
+  template<>
+    struct numeric_limits<short>
+    {
+      static const bool is_specialized = true;
+
+      static const short 
+      min() { return -SHRT_MAX - 1; }
+
+      static const short 
+      max() { return SHRT_MAX; }
+
+#if __cplusplus >= 201103L
+      static constexpr short 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits = __glibcxx_digits (short);
+      static const int digits10 = __glibcxx_digits10 (short);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = true;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const short 
+      epsilon() { return 0; }
+
+      static const short 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const short 
+      infinity() { return short(); }
+
+      static const short 
+      quiet_NaN() { return short(); }
+
+      static const short 
+      signaling_NaN() { return short(); }
+
+      static const short 
+      denorm_min() { return short(); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = false;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<unsigned short> specialization.
+  template<>
+    struct numeric_limits<unsigned short>
+    {
+      static const bool is_specialized = true;
+
+      static const unsigned short 
+      min() { return 0; }
+
+      static const unsigned short 
+      max() { return SHRT_MAX * 2U + 1; }
+
+#if __cplusplus >= 201103L
+      static constexpr unsigned short 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits 
+       = __glibcxx_digits (unsigned short);
+      static const int digits10 
+       = __glibcxx_digits10 (unsigned short);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = false;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const unsigned short 
+      epsilon() { return 0; }
+
+      static const unsigned short 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const unsigned short 
+      infinity() 
+      { return static_cast<unsigned short>(0); }
+
+      static const unsigned short 
+      quiet_NaN() 
+      { return static_cast<unsigned short>(0); }
+
+      static const unsigned short 
+      signaling_NaN() 
+      { return static_cast<unsigned short>(0); }
+
+      static const unsigned short 
+      denorm_min() 
+      { return static_cast<unsigned short>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = true;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<int> specialization.
+  template<>
+    struct numeric_limits<int>
+    {
+      static const bool is_specialized = true;
+
+      static const int 
+      min() { return -__INT_MAX__ - 1; }
+
+      static const int 
+      max() { return __INT_MAX__; }
+
+#if __cplusplus >= 201103L
+      static constexpr int 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits = __glibcxx_digits (int);
+      static const int digits10 = __glibcxx_digits10 (int);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = true;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const int 
+      epsilon() { return 0; }
+
+      static const int 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const int 
+      infinity() { return static_cast<int>(0); }
+
+      static const int 
+      quiet_NaN() { return static_cast<int>(0); }
+
+      static const int 
+      signaling_NaN() { return static_cast<int>(0); }
+
+      static const int 
+      denorm_min() { return static_cast<int>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = false;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<unsigned int> specialization.
+  template<>
+    struct numeric_limits<unsigned int>
+    {
+      static const bool is_specialized = true;
+
+      static const unsigned int 
+      min() { return 0; }
+
+      static const unsigned int 
+      max() { return __INT_MAX__ * 2U + 1; }
+
+#if __cplusplus >= 201103L
+      static constexpr unsigned int 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits 
+       = __glibcxx_digits (unsigned int);
+      static const int digits10 
+       = __glibcxx_digits10 (unsigned int);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = false;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const unsigned int 
+      epsilon() { return 0; }
+
+      static const unsigned int 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const unsigned int 
+      infinity() { return static_cast<unsigned int>(0); }
+
+      static const unsigned int 
+      quiet_NaN() 
+      { return static_cast<unsigned int>(0); }
+
+      static const unsigned int 
+      signaling_NaN() 
+      { return static_cast<unsigned int>(0); }
+
+      static const unsigned int 
+      denorm_min() 
+      { return static_cast<unsigned int>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = true;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<long> specialization.
+  template<>
+    struct numeric_limits<long>
+    {
+      static const bool is_specialized = true;
+
+      static const long
+      min() { return -__LONG_MAX__ - 1; }
+
+      static const long 
+      max() { return __LONG_MAX__; }
+
+#if __cplusplus >= 201103L
+      static constexpr long 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits = __glibcxx_digits (long);
+      static const int digits10 = __glibcxx_digits10 (long);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = true;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const long 
+      epsilon() { return 0; }
+
+      static const long 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const long 
+      infinity() { return static_cast<long>(0); }
+
+      static const long 
+      quiet_NaN() { return static_cast<long>(0); }
+
+      static const long 
+      signaling_NaN() { return static_cast<long>(0); }
+
+      static const long 
+      denorm_min() { return static_cast<long>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = false;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<unsigned long> specialization.
+  template<>
+    struct numeric_limits<unsigned long>
+    {
+      static const bool is_specialized = true;
+
+      static const unsigned long 
+      min() { return 0; }
+
+      static const unsigned long 
+      max() { return __LONG_MAX__ * 2UL + 1; }
+
+#if __cplusplus >= 201103L
+      static constexpr unsigned long 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits 
+       = __glibcxx_digits (unsigned long);
+      static const int digits10 
+       = __glibcxx_digits10 (unsigned long);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = false;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const unsigned long 
+      epsilon() { return 0; }
+
+      static const unsigned long 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const unsigned long 
+      infinity() 
+      { return static_cast<unsigned long>(0); }
+
+      static const unsigned long 
+      quiet_NaN() 
+      { return static_cast<unsigned long>(0); }
+
+      static const unsigned long 
+      signaling_NaN() 
+      { return static_cast<unsigned long>(0); }
+
+      static const unsigned long 
+      denorm_min() 
+      { return static_cast<unsigned long>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = true;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<long long> specialization.
+  template<>
+    struct numeric_limits<long long>
+    {
+      static const bool is_specialized = true;
+
+      static const long long 
+      min() { return -__LONG_LONG_MAX__ - 1; }
+
+      static const long long 
+      max() { return __LONG_LONG_MAX__; }
+
+#if __cplusplus >= 201103L
+      static constexpr long long 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits 
+       = __glibcxx_digits (long long);
+      static const int digits10 
+       = __glibcxx_digits10 (long long);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = true;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const long long 
+      epsilon() { return 0; }
+
+      static const long long 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const long long 
+      infinity() { return static_cast<long long>(0); }
+
+      static const long long 
+      quiet_NaN() { return static_cast<long long>(0); }
+
+      static const long long 
+      signaling_NaN() 
+      { return static_cast<long long>(0); }
+
+      static const long long 
+      denorm_min() { return static_cast<long long>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = false;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<unsigned long long> specialization.
+  template<>
+    struct numeric_limits<unsigned long long>
+    {
+      static const bool is_specialized = true;
+
+      static const unsigned long long 
+      min() { return 0; }
+
+      static const unsigned long long 
+      max() { return __LONG_LONG_MAX__ * 2ULL + 1; }
+
+#if __cplusplus >= 201103L
+      static constexpr unsigned long long 
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits 
+       = __glibcxx_digits (unsigned long long);
+      static const int digits10 
+       = __glibcxx_digits10 (unsigned long long);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = false;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const unsigned long long 
+      epsilon() { return 0; }
+
+      static const unsigned long long 
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm 
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const unsigned long long 
+      infinity() 
+      { return static_cast<unsigned long long>(0); }
+
+      static const unsigned long long 
+      quiet_NaN() 
+      { return static_cast<unsigned long long>(0); }
+
+      static const unsigned long long 
+      signaling_NaN() 
+      { return static_cast<unsigned long long>(0); }
+
+      static const unsigned long long 
+      denorm_min() 
+      { return static_cast<unsigned long long>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = true;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style 
+       = round_toward_zero;
+    };
+
+#if !defined(__STRICT_ANSI__) && defined(_GLIBCXX_USE_INT128)
+  /// numeric_limits<__int128> specialization.
+  template<>
+    struct numeric_limits<__int128>
+    {
+      static const bool is_specialized = true;
+
+      static const __int128
+      min() { return __glibcxx_min (__int128); }
+
+      static const __int128
+      max() { return __glibcxx_max (__int128); }
+
+#if __cplusplus >= 201103L
+      static constexpr __int128
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits
+       = __glibcxx_digits (__int128);
+      static const int digits10
+       = __glibcxx_digits10 (__int128);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = true;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const __int128
+      epsilon() { return 0; }
+
+      static const __int128
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const __int128
+      infinity() 
+      { return static_cast<__int128>(0); }
+
+      static const __int128
+      quiet_NaN() 
+      { return static_cast<__int128>(0); }
+      
+      static const __int128
+      signaling_NaN() 
+      { return static_cast<__int128>(0); }
+      
+      static const __int128
+      denorm_min() 
+      { return static_cast<__int128>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = false;
+
+      static const bool traps
+       = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style
+       = round_toward_zero;
+    };
+
+  /// numeric_limits<unsigned __int128> specialization.
+  template<>
+    struct numeric_limits<unsigned __int128>
+    {
+      static const bool is_specialized = true;
+
+      static const unsigned __int128
+      min() { return 0; }
+
+      static const unsigned __int128
+      max() { return __glibcxx_max (unsigned __int128); }
+
+#if __cplusplus >= 201103L
+      static constexpr unsigned __int128
+      lowest() noexcept { return min(); }
+#endif
+
+      static const int digits
+       = __glibcxx_digits (unsigned __int128);
+      static const int digits10
+       = __glibcxx_digits10 (unsigned __int128);
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10 = 0;
+#endif
+      static const bool is_signed = false;
+      static const bool is_integer = true;
+      static const bool is_exact = true;
+      static const int radix = 2;
+
+      static const unsigned __int128
+      epsilon() { return 0; }
+
+      static const unsigned __int128
+      round_error() { return 0; }
+
+      static const int min_exponent = 0;
+      static const int min_exponent10 = 0;
+      static const int max_exponent = 0;
+      static const int max_exponent10 = 0;
+
+      static const bool has_infinity = false;
+      static const bool has_quiet_NaN = false;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm
+       = denorm_absent;
+      static const bool has_denorm_loss = false;
+
+      static const unsigned __int128
+      infinity() 
+      { return static_cast<unsigned __int128>(0); }
+
+      static const unsigned __int128
+      quiet_NaN() 
+      { return static_cast<unsigned __int128>(0); }
+
+      static const unsigned __int128
+      signaling_NaN() 
+      { return static_cast<unsigned __int128>(0); }
+
+      static const unsigned __int128
+      denorm_min() 
+      { return static_cast<unsigned __int128>(0); }
+
+      static const bool is_iec559 = false;
+      static const bool is_bounded = true;
+      static const bool is_modulo = true;
+
+      static const bool traps = __glibcxx_integral_traps;
+      static const bool tinyness_before = false;
+      static const float_round_style round_style
+       = round_toward_zero;
+    };
+#endif
+
+  /// numeric_limits<float> specialization.
+  template<>
+    struct numeric_limits<float>
+    {
+      static const bool is_specialized = true;
+
+      static const float 
+      min() { return FLT_MIN; }
+
+      static const float 
+      max() { return FLT_MAX; }
+
+#if __cplusplus >= 201103L
+      static constexpr float 
+      lowest() noexcept { return -FLT_MAX; }
+#endif
+
+      static const int digits = FLT_MANT_DIG;
+      static const int digits10 = FLT_DIG;
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10
+	 = __glibcxx_max_digits10 (FLT_MANT_DIG);
+#endif
+      static const bool is_signed = true;
+      static const bool is_integer = false;
+      static const bool is_exact = false;
+      static const int radix = FLT_RADIX;
+
+      static const float 
+      epsilon() { return FLT_EPSILON; }
+
+      static const float 
+      round_error() { return 0.5F; }
+
+      static const int min_exponent = FLT_MIN_EXP;
+      static const int min_exponent10 = FLT_MIN_10_EXP;
+      static const int max_exponent = FLT_MAX_EXP;
+      static const int max_exponent10 = FLT_MAX_10_EXP;
+
+      static const bool has_infinity = true;
+      static const bool has_quiet_NaN = true;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm
+	= denorm_present;
+      static const bool has_denorm_loss 
+       = __glibcxx_float_has_denorm_loss;
+
+      static const float 
+      infinity() { return HUGE_VALF; }
+
+      static const float 
+      quiet_NaN() { return nanf(""); }
+
+      static const float 
+      signaling_NaN() { return nanf(""); }
+
+      static const float 
+      denorm_min() { return FLT_MIN; }
+
+      static const bool is_iec559
+	= has_infinity && has_quiet_NaN && has_denorm == denorm_present;
+      static const bool is_bounded = true;
+      static const bool is_modulo = false;
+
+      static const bool traps = __glibcxx_float_traps;
+      static const bool tinyness_before 
+       = __glibcxx_float_tinyness_before;
+      static const float_round_style round_style 
+       = round_to_nearest;
+    };
+
+#undef __glibcxx_float_has_denorm_loss
+#undef __glibcxx_float_traps
+#undef __glibcxx_float_tinyness_before
+
+  /// numeric_limits<double> specialization.
+  template<>
+    struct numeric_limits<double>
+    {
+      static const bool is_specialized = true;
+
+      static const double 
+      min() { return DBL_MIN; }
+
+      static const double 
+      max() { return DBL_MAX; }
+
+#if __cplusplus >= 201103L
+      static constexpr double 
+      lowest() noexcept { return -DBL_MAX; }
+#endif
+
+      static const int digits = DBL_MANT_DIG;
+      static const int digits10 = DBL_DIG;
+#if __cplusplus >= 201103L
+      static constexpr int max_digits10
+	 = __glibcxx_max_digits10 (DBL_MANT_DIG);
+#endif
+      static const bool is_signed = true;
+      static const bool is_integer = false;
+      static const bool is_exact = false;
+      static const int radix = FLT_RADIX;
+
+      static const double 
+      epsilon() { return DBL_EPSILON; }
+
+      static const double 
+      round_error() { return 0.5; }
+
+      static const int min_exponent = DBL_MIN_EXP;
+      static const int min_exponent10 = DBL_MIN_10_EXP;
+      static const int max_exponent = DBL_MAX_EXP;
+      static const int max_exponent10 = DBL_MAX_10_EXP;
+
+      static const bool has_infinity = true;
+      static const bool has_quiet_NaN = true;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm
+	= denorm_present;
+      static const bool has_denorm_loss 
+        = __glibcxx_double_has_denorm_loss;
+
+      static const double 
+      infinity() { return HUGE_VAL; }
+
+      static const double 
+      quiet_NaN() { return nan(""); }
+
+      static const double 
+      signaling_NaN() { return nan(""); }
+
+      static const double 
+      denorm_min() { return DBL_MIN; }
+
+      static const bool is_iec559
+	= has_infinity && has_quiet_NaN && has_denorm == denorm_present;
+      static const bool is_bounded = true;
+      static const bool is_modulo = false;
+
+      static const bool traps = __glibcxx_double_traps;
+      static const bool tinyness_before 
+       = __glibcxx_double_tinyness_before;
+      static const float_round_style round_style 
+       = round_to_nearest;
+    };
+
+#undef __glibcxx_double_has_denorm_loss
+#undef __glibcxx_double_traps
+#undef __glibcxx_double_tinyness_before
+
+  /// numeric_limits<long double> specialization.
+  template<>
+    struct numeric_limits<long double>
+    {
+      static const bool is_specialized = true;
+
+      static const long double 
+      min() { return LDBL_MIN; }
+
+      static const long double 
+      max() { return LDBL_MAX; }
+
+#if __cplusplus >= 201103L
+      static constexpr long double 
+      lowest() noexcept { return -LDBL_MAX; }
+#endif
+
+      static const int digits = LDBL_MANT_DIG;
+      static const int digits10 = LDBL_DIG;
+#if __cplusplus >= 201103L
+      static const int max_digits10
+	 = __glibcxx_max_digits10 (LDBL_MANT_DIG);
+#endif
+      static const bool is_signed = true;
+      static const bool is_integer = false;
+      static const bool is_exact = false;
+      static const int radix = FLT_RADIX;
+
+      static const long double 
+      epsilon() { return LDBL_EPSILON; }
+
+      static const long double 
+      round_error() { return 0.5L; }
+
+      static const int min_exponent = LDBL_MIN_EXP;
+      static const int min_exponent10 = LDBL_MIN_10_EXP;
+      static const int max_exponent = LDBL_MAX_EXP;
+      static const int max_exponent10 = LDBL_MAX_10_EXP;
+
+      static const bool has_infinity = true;
+      static const bool has_quiet_NaN = true;
+      static const bool has_signaling_NaN = false;
+      static const float_denorm_style has_denorm
+	= denorm_present;
+      static const bool has_denorm_loss
+	= __glibcxx_long_double_has_denorm_loss;
+
+      static const long double 
+      infinity() { return HUGE_VAL; }
+
+      static const long double 
+      quiet_NaN() { return nanl(""); }
+
+      static const long double 
+      signaling_NaN() { return nanl(""); }
+
+      static const long double 
+      denorm_min() { return LDBL_MIN; }
+
+      static const bool is_iec559
+	= has_infinity && has_quiet_NaN && has_denorm == denorm_present;
+      static const bool is_bounded = true;
+      static const bool is_modulo = false;
+
+      static const bool traps = __glibcxx_long_double_traps;
+      static const bool tinyness_before = 
+					 __glibcxx_long_double_tinyness_before;
+      static const float_round_style round_style = 
+						      round_to_nearest;
+    };
+
+#undef __glibcxx_long_double_has_denorm_loss
+#undef __glibcxx_long_double_traps
+#undef __glibcxx_long_double_tinyness_before
+
+} // namespace
+
+#undef __glibcxx_signed
+#undef __glibcxx_min
+#undef __glibcxx_max
+#undef __glibcxx_digits
+#undef __glibcxx_digits10
+#undef __glibcxx_max_digits10
+
+#endif // _GLIBCXX_NUMERIC_LIMITS