Tiny Vector Matrix library using Expression Templates

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---

include/tvmet/NumericTraits.h

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/*
 * Tiny Vector Matrix Library
 * Dense Vector Matrix Libary of Tiny size using Expression Templates
 *
 * Copyright (C) 2001 - 2007 Olaf Petzold <opetzold@users.sourceforge.net>
 *
 * This library is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public
 * License as published by the Free Software Foundation; either
 * version 2.1 of the License, 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
 * Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public
 * License along with this library; if not, write to the Free Software
 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 *
 * $Id: NumericTraits.h,v 1.16 2007-06-23 15:58:58 opetzold Exp $
 */

#ifndef TVMET_NUMERIC_TRAITS_H
#define TVMET_NUMERIC_TRAITS_H

#if defined(TVMET_HAVE_COMPLEX)
#  include <complex>
#endif
#include <cmath>
#include <limits>

#include <tvmet/CompileTimeError.h>


namespace tvmet {


template<class T>
struct NumericTraits {
  typedef T         base_type;
  typedef T         value_type;
  typedef value_type        sum_type;
  typedef value_type        diff_type;
  typedef value_type        float_type;
  typedef value_type        signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef const value_type&     argument_type;

  static inline
  base_type real(argument_type x);

  static inline
  base_type imag(argument_type x);

  static inline
  value_type conj(argument_type x);

  static inline
  base_type abs(argument_type x);

  static inline
  value_type sqrt(argument_type x);

  static inline
  base_type norm_1(argument_type x) {
    return NumericTraits<base_type>::abs(traits_type::real(x))
         + NumericTraits<base_type>::abs(traits_type::imag(x));
  }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(x)),
        NumericTraits<base_type>::abs(traits_type::imag(x)));
   }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
      traits_type::norm_inf(rhs)),
         std::numeric_limits<base_type>::min());
  }
};


/*
 * numeric traits for standard types
 */


template<>
struct NumericTraits<char> {
  typedef char          value_type;
  typedef value_type        base_type;
  typedef long          sum_type;
  typedef int         diff_type;
  typedef float         float_type;
  typedef char          signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};


template<>
struct NumericTraits<unsigned char> {
  typedef unsigned char       value_type;
  typedef value_type        base_type;
  typedef unsigned long       sum_type;
  typedef int         diff_type;
  typedef float         float_type;
  typedef int         signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};


template<>
struct NumericTraits<short int> {
  typedef short int         value_type;
  typedef value_type        base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef long long       sum_type;
#else
  typedef long          sum_type;
#endif
  typedef int         diff_type;
  typedef float         float_type;
  typedef short int       signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};


template<>
struct NumericTraits<short unsigned int> {
  typedef short unsigned int      value_type;
  typedef value_type        base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef unsigned long long      sum_type;
#else
  typedef unsigned long       sum_type;
#endif
  typedef int           diff_type;
  typedef float         float_type;
  typedef int         signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};


template<>
struct NumericTraits<int> {
  typedef int           value_type;
  typedef value_type        base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef long long       sum_type;
#else
  typedef long          sum_type;
#endif
  typedef int         diff_type;
  typedef double        float_type;
  typedef int         signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};


template<>
struct NumericTraits<unsigned int> {
  typedef unsigned int        value_type;
  typedef value_type        base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef unsigned long long      sum_type;
#else
  typedef unsigned long       sum_type;
#endif
  typedef int           diff_type;
  typedef double        float_type;
  typedef long          signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  base_type abs(argument_type x) { return x; }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};


template<>
struct NumericTraits<long> {
  typedef long          value_type;
  typedef value_type        base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef long long       sum_type;
#else
  typedef long            sum_type;
#endif
  typedef long          diff_type;
  typedef double        float_type;
  typedef long          signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};


template<>
struct NumericTraits<unsigned long> {
  typedef unsigned long       value_type;
  typedef value_type        base_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef unsigned long long      sum_type;
#else
  typedef unsigned long       sum_type;
#endif
  typedef unsigned long       diff_type;
  typedef double        float_type;
  typedef long          signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  base_type abs(argument_type x) { return x; }

  static inline
  value_type sqrt(argument_type x) {
    return static_cast<value_type>(std::sqrt(static_cast<float_type>(x)));
  }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) { return lhs == rhs; }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};


template<>
struct NumericTraits<float> {
  typedef float         value_type;
  typedef value_type        base_type;
  typedef double        sum_type;
  typedef float         diff_type;
  typedef float         float_type;
  typedef float         signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) { return std::sqrt(x); }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
      traits_type::norm_inf(rhs)),
         std::numeric_limits<base_type>::min());
  }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};


template<>
struct NumericTraits<double> {
  typedef double        value_type;
  typedef value_type        base_type;
#if defined(TVMET_HAVE_LONG_DOUBLE)
  typedef long double       sum_type;
#else
  typedef double        sum_type;
#endif
  typedef double        diff_type;
  typedef double        float_type;
  typedef double        signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) { return std::sqrt(x); }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
      traits_type::norm_inf(rhs)),
         std::numeric_limits<base_type>::min());
  }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};


#if defined(TVMET_HAVE_LONG_DOUBLE)

template<>
struct NumericTraits<long double> {
  typedef long double         value_type;
  typedef value_type        base_type;
  typedef long double       sum_type;
  typedef long double       diff_type;
  typedef long double         float_type;
  typedef long double       signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef value_type        argument_type;

  static inline
  base_type real(argument_type x) { return x; }

  static inline
  base_type imag(argument_type) { return 0; }

  static inline
  value_type conj(argument_type x) { return x; }

  static inline
  base_type abs(argument_type x) { return std::abs(x); }

  static inline
  value_type sqrt(argument_type x) { return std::sqrt(x); }

  static inline
  base_type norm_1(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_2(argument_type x) { return traits_type::abs(x); }

  static inline
  base_type norm_inf(argument_type x) { return traits_type::abs(x); }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
      traits_type::norm_inf(rhs)),
         std::numeric_limits<base_type>::min());
  }

  enum { is_complex = false };

  enum {
    ops_plus = 1, 
    ops_muls = 1  
  };
};
#endif // TVMET_HAVE_LONG_DOUBLE


/*
 * numeric traits for complex types
 */
#if defined(TVMET_HAVE_COMPLEX)

template<>
struct NumericTraits< std::complex<int> > {
  typedef int         base_type;
  typedef std::complex<int>     value_type;
  typedef std::complex<long>      sum_type;
  typedef std::complex<int>     diff_type;
  typedef std::complex<float>     float_type;
  typedef std::complex<int>     signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef const value_type&     argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  value_type conj(argument_type z) { return std::conj(z); }

  static inline
  base_type abs(argument_type z) {
    base_type x = z.real();
    base_type y = z.imag();

    // XXX probably case of overrun; header complex uses scaling
    return static_cast<base_type>(NumericTraits<base_type>::sqrt(x * x + y * y));
  }

  static /* inline */
  value_type sqrt(argument_type z) {
    // borrowed and adapted from header complex
    base_type x = z.real();
    base_type y = z.imag();

    if(x == base_type()) {
  base_type t = NumericTraits<base_type>::sqrt(
                        NumericTraits<base_type>::abs(y) / 2);
  return value_type(t, y < base_type() ? -t : t);
    }
    else {
      base_type t = NumericTraits<base_type>::sqrt(
          2 * (traits_type::abs(z)
                + NumericTraits<base_type>::abs(x)));
      base_type u = t / 2;
      return x > base_type()
  ? value_type(u, y / t)
  : value_type(NumericTraits<base_type>::abs(y) / t, y < base_type() ? -u : u);
    }
  }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
        NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    return (traits_type::real(lhs) == traits_type::real(rhs))
        && (traits_type::imag(lhs) == traits_type::imag(rhs));
  }

  enum { is_complex = true };

  enum {
    ops_plus = 2, 
    ops_muls = 6  
  };
};


template<>
struct NumericTraits< std::complex<unsigned int> > {
  typedef unsigned int        base_type;
  typedef std::complex<unsigned int>    value_type;
  typedef std::complex<unsigned long>     sum_type;
  typedef std::complex<int>     diff_type;
  typedef std::complex<float>     float_type;
  typedef std::complex<int>     signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef const value_type&     argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  base_type abs(argument_type z) {
    base_type x = z.real();
    base_type y = z.imag();

    // XXX probably case of overrun; header complex uses scaling
    return static_cast<base_type>(NumericTraits<base_type>::sqrt(x * x + y * y));
  }

  static /* inline */
  value_type sqrt(argument_type z) {
    // borrowed and adapted from header complex
    base_type x = z.real();
    base_type y = z.imag();

    if(x == base_type()) {
  base_type t = NumericTraits<base_type>::sqrt(
                        NumericTraits<base_type>::abs(y) / 2);
  return value_type(t, t);
    }
    else {
      base_type t = NumericTraits<base_type>::sqrt(
          2 * (traits_type::abs(z)
                + NumericTraits<base_type>::abs(x)));
      return value_type(t / 2, y / t);
    }
  }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
        NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    return (traits_type::real(lhs) == traits_type::real(rhs))
        && (traits_type::imag(lhs) == traits_type::imag(rhs));
  }

  enum { is_complex = true };

  enum {
    ops_plus = 2, 
    ops_muls = 6  
  };
};


template<>
struct NumericTraits< std::complex<long> > {
  typedef long          base_type;
  typedef std::complex<long>      value_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef std::complex<long long>   sum_type;
#else
  typedef std::complex<long>      sum_type;
#endif
  typedef std::complex<int>     diff_type;
  typedef std::complex<float>     float_type;
  typedef std::complex<int>     signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef const value_type&     argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  value_type conj(argument_type z) { return std::conj(z); }

  static inline
  base_type abs(argument_type z) {
    base_type x = z.real();
    base_type y = z.imag();

    // XXX probably case of overrun; header complex uses scaling
    return static_cast<base_type>(NumericTraits<base_type>::sqrt(x * x + y * y));
  }

  static /* inline */
  value_type sqrt(argument_type z) {
    // borrowed and adapted from header complex
    base_type x = z.real();
    base_type y = z.imag();

    if(x == base_type()) {
  base_type t = NumericTraits<base_type>::sqrt(
                        NumericTraits<base_type>::abs(y) / 2);
  return value_type(t, y < base_type() ? -t : t);
    }
    else {
      base_type t = NumericTraits<base_type>::sqrt(
          2 * (traits_type::abs(z)
                + NumericTraits<base_type>::abs(x)));
      base_type u = t / 2;
      return x > base_type()
  ? value_type(u, y / t)
  : value_type(NumericTraits<base_type>::abs(y) / t, y < base_type() ? -u : u);
    }
  }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
        NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    return (traits_type::real(lhs) == traits_type::real(rhs))
        && (traits_type::imag(lhs) == traits_type::imag(rhs));
  }

  enum { is_complex = true };

  enum {
    ops_plus = 2, 
    ops_muls = 6  
  };
};


template<>
struct NumericTraits< std::complex<unsigned long> > {
  typedef unsigned long       base_type;
  typedef std::complex<unsigned long>   value_type;
#if defined(TVMET_HAVE_LONG_LONG)
  typedef std::complex<unsigned long long>  sum_type;
#else
  typedef std::complex<unsigned long>   sum_type;
#endif
  typedef std::complex<long>      diff_type;
  typedef std::complex<float>     float_type;
  typedef std::complex<long>      signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef const value_type&     argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  base_type abs(argument_type z) {
    base_type x = z.real();
    base_type y = z.imag();

    // XXX probably case of overrun; header complex uses scaling
    return static_cast<base_type>(NumericTraits<base_type>::sqrt(x * x + y * y));
  }

  static /* inline */
  value_type sqrt(argument_type z) {
    // borrowed and adapted from header complex
    base_type x = z.real();
    base_type y = z.imag();

    if(x == base_type()) {
  base_type t = NumericTraits<base_type>::sqrt(
                        NumericTraits<base_type>::abs(y) / 2);
  return value_type(t, t);
    }
    else {
      base_type t = NumericTraits<base_type>::sqrt(
          2 * (traits_type::abs(z)
                + NumericTraits<base_type>::abs(x)));
      return value_type(t / 2, y / t);
    }
  }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
        NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    return (traits_type::real(lhs) == traits_type::real(rhs))
        && (traits_type::imag(lhs) == traits_type::imag(rhs));
  }

  enum { is_complex = true };

  enum {
    ops_plus = 2, 
    ops_muls = 6  
  };
};


template<>
struct NumericTraits< std::complex<float> > {
  typedef float         base_type;
  typedef std::complex<float>     value_type;
  typedef std::complex<double>      sum_type;
  typedef std::complex<float>     diff_type;
  typedef std::complex<float>     float_type;
  typedef std::complex<float>     signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef const value_type&     argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  value_type conj(argument_type z) { return std::conj(z); }

  static inline
  base_type abs(argument_type z) { return std::abs(z); }

  static inline
  value_type sqrt(argument_type z) { return std::sqrt(z); }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
              NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

 static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
      traits_type::norm_inf(rhs)),
         std::numeric_limits<base_type>::min());
  }

  enum { is_complex = true };

  enum {
    ops_plus = 2, 
    ops_muls = 6  
  };
};


template<>
struct NumericTraits< std::complex<double> > {
  typedef double        base_type;
  typedef std::complex<double>      value_type;
#if defined(TVMET_HAVE_LONG_DOUBLE)
  typedef std::complex<long double>     sum_type;
#else
  typedef std::complex<double>      sum_type;
#endif
  typedef std::complex<double>      diff_type;
  typedef std::complex<double>      float_type;
  typedef std::complex<double>      signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef const value_type&     argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  value_type conj(argument_type z) { return std::conj(z); }

  static inline
  base_type abs(argument_type z) { return std::abs(z); }

  static inline
  value_type sqrt(argument_type z) { return std::sqrt(z); }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
              NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

 static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
      traits_type::norm_inf(rhs)),
         std::numeric_limits<base_type>::min());
  }

  enum { is_complex = true };

  enum {
    ops_plus = 2, 
    ops_muls = 6  
  };
};


#if defined(TVMET_HAVE_LONG_DOUBLE)

template<>
struct NumericTraits< std::complex<long double> > {
  typedef long double       base_type;
  typedef std::complex<long double>   value_type;
  typedef std::complex<long double>     sum_type;
  typedef std::complex<long double>   diff_type;
  typedef std::complex<long double>   float_type;
  typedef std::complex<long double>   signed_type;

  typedef NumericTraits<value_type>   traits_type;
  typedef const value_type&     argument_type;

  static inline
  base_type real(argument_type z) { return std::real(z); }

  static inline
  base_type imag(argument_type z) { return std::imag(z); }

  static inline
  value_type conj(argument_type z) { return std::conj(z); }

  static inline
  base_type abs(argument_type z) { return std::abs(z); }

  static inline
  value_type sqrt(argument_type z) { return std::sqrt(z); }

  static inline
  base_type norm_1(argument_type z) {
    return NumericTraits<base_type>::abs((traits_type::real(z)))
         + NumericTraits<base_type>::abs((traits_type::imag(z)));
  }

  static inline
  base_type norm_2(argument_type z) { return traits_type::abs(z); }

  static inline
  base_type norm_inf(argument_type z) {
    return std::max(NumericTraits<base_type>::abs(traits_type::real(z)),
              NumericTraits<base_type>::abs(traits_type::imag(z)));
  }

  static inline
  bool equals(argument_type lhs, argument_type rhs) {
    static base_type sqrt_epsilon(
      NumericTraits<base_type>::sqrt(
        std::numeric_limits<base_type>::epsilon()));

    return traits_type::norm_inf(lhs - rhs) < sqrt_epsilon *
      std::max(std::max(traits_type::norm_inf(lhs),
      traits_type::norm_inf(rhs)),
         std::numeric_limits<base_type>::min());
  }

  enum { is_complex = true };

  enum {
    ops_plus = 2, 
    ops_muls = 6  
  };
};
#endif // defined(TVMET_HAVE_LONG_DOUBLE)


#endif // defined(TVMET_HAVE_COMPLEX)


} // namespace tvmet


#endif //  TVMET_NUMERIC_TRAITS_H


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