libstdc++-v3/include/std/complex

   1 // The template and inlines for the -*- C++ -*- complex number classes.
   2
   3 // Copyright (C) 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005,
   4 // 2006, 2007, 2008, 2009, 2010
   5 // Free Software Foundation, Inc.
   6 //
   7 // This file is part of the GNU ISO C++ Library.  This library is free
   8 // software; you can redistribute it and/or modify it under the
   9 // terms of the GNU General Public License as published by the
  10 // Free Software Foundation; either version 3, or (at your option)
  11 // any later version.
  12
  13 // This library is distributed in the hope that it will be useful,
  14 // but WITHOUT ANY WARRANTY; without even the implied warranty of
  15 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16 // GNU General Public License for more details.
  17
  18 // Under Section 7 of GPL version 3, you are granted additional
  19 // permissions described in the GCC Runtime Library Exception, version
  20 // 3.1, as published by the Free Software Foundation.
  21
  22 // You should have received a copy of the GNU General Public License and
  23 // a copy of the GCC Runtime Library Exception along with this program;
  24 // see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
  25 // <http://www.gnu.org/licenses/>.
  26
  27 /** @file include/complex
  28  *  This is a Standard C++ Library header.
  29  */
  30
  31 //
  32 // ISO C++ 14882: 26.2  Complex Numbers
  33 // Note: this is not a conforming implementation.
  34 // Initially implemented by Ulrich Drepper <drepper@cygnus.com>
  35 // Improved by Gabriel Dos Reis <dosreis@cmla.ens-cachan.fr>
  36 //
  37
  38 #ifndef _GLIBCXX_COMPLEX
  39 #define _GLIBCXX_COMPLEX 1
  40
  41 #pragma GCC system_header
  42
  43 #include <bits/c++config.h>
  44 #include <bits/cpp_type_traits.h>
  45 #include <ext/type_traits.h>
  46 #include <cmath>
  47 #include <sstream>
  48
  49 namespace std _GLIBCXX_VISIBILITY(default)
  50 {
  51 _GLIBCXX_BEGIN_NAMESPACE_VERSION
  52
  53   /**
  54    * @defgroup complex_numbers Complex Numbers
  55    * @ingroup numerics
  56    *
  57    * Classes and functions for complex numbers.
  58    * @{
  59    */
  60
  61   // Forward declarations.
  62   template<typename _Tp> class complex;
  63   template<> class complex<float>;
  64   template<> class complex<double>;
  65   template<> class complex<long double>;
  66
  67   ///  Return magnitude of @a z.
  68   template<typename _Tp> _Tp abs(const complex<_Tp>&);
  69   ///  Return phase angle of @a z.
  70   template<typename _Tp> _Tp arg(const complex<_Tp>&);
  71   ///  Return @a z magnitude squared.
  72   template<typename _Tp> _Tp norm(const complex<_Tp>&);
  73
  74   ///  Return complex conjugate of @a z.
  75   template<typename _Tp> complex<_Tp> conj(const complex<_Tp>&);
  76   ///  Return complex with magnitude @a rho and angle @a theta.
  77   template<typename _Tp> complex<_Tp> polar(const _Tp&, const _Tp& = 0);
  78
  79   // Transcendentals:
  80   /// Return complex cosine of @a z.
  81   template<typename _Tp> complex<_Tp> cos(const complex<_Tp>&);
  82   /// Return complex hyperbolic cosine of @a z.
  83   template<typename _Tp> complex<_Tp> cosh(const complex<_Tp>&);
  84   /// Return complex base e exponential of @a z.
  85   template<typename _Tp> complex<_Tp> exp(const complex<_Tp>&);
  86   /// Return complex natural logarithm of @a z.
  87   template<typename _Tp> complex<_Tp> log(const complex<_Tp>&);
  88   /// Return complex base 10 logarithm of @a z.
  89   template<typename _Tp> complex<_Tp> log10(const complex<_Tp>&);
  90 #ifndef __GXX_EXPERIMENTAL_CXX0X__
  91   // DR 844.
  92   /// Return @a x to the @a y'th power.
  93   template<typename _Tp> complex<_Tp> pow(const complex<_Tp>&, int);
  94 #endif
  95   /// Return @a x to the @a y'th power.
  96   template<typename _Tp> complex<_Tp> pow(const complex<_Tp>&, const _Tp&);
  97   /// Return @a x to the @a y'th power.
  98   template<typename _Tp> complex<_Tp> pow(const complex<_Tp>&,
  99                                           const complex<_Tp>&);
 100   /// Return @a x to the @a y'th power.
 101   template<typename _Tp> complex<_Tp> pow(const _Tp&, const complex<_Tp>&);
 102   /// Return complex sine of @a z.
 103   template<typename _Tp> complex<_Tp> sin(const complex<_Tp>&);
 104   /// Return complex hyperbolic sine of @a z.
 105   template<typename _Tp> complex<_Tp> sinh(const complex<_Tp>&);
 106   /// Return complex square root of @a z.
 107   template<typename _Tp> complex<_Tp> sqrt(const complex<_Tp>&);
 108   /// Return complex tangent of @a z.
 109   template<typename _Tp> complex<_Tp> tan(const complex<_Tp>&);
 110   /// Return complex hyperbolic tangent of @a z.
 111   template<typename _Tp> complex<_Tp> tanh(const complex<_Tp>&);
 112
 113
 114   // 26.2.2  Primary template class complex
 115   /**
 116    *  Template to represent complex numbers.
 117    *
 118    *  Specializations for float, double, and long double are part of the
 119    *  library.  Results with any other type are not guaranteed.
 120    *
 121    *  @param  Tp  Type of real and imaginary values.
 122   */
 123   template<typename _Tp>
 124     struct complex
 125     {
 126       /// Value typedef.
 127       typedef _Tp value_type;
 128
 129       ///  Default constructor.  First parameter is x, second parameter is y.
 130       ///  Unspecified parameters default to 0.
 131       _GLIBCXX_CONSTEXPR complex(const _Tp& __r = _Tp(), const _Tp& __i = _Tp())
 132       : _M_real(__r), _M_imag(__i) { }
 133
 134       // Lets the compiler synthesize the copy constructor
 135       // complex (const complex<_Tp>&);
 136       ///  Copy constructor.
 137       template<typename _Up>
 138         _GLIBCXX_CONSTEXPR complex(const complex<_Up>& __z)
 139         : _M_real(__z.real()), _M_imag(__z.imag()) { }
 140
 141 #ifdef __GXX_EXPERIMENTAL_CXX0X__
 142       // _GLIBCXX_RESOLVE_LIB_DEFECTS
 143       // DR 387. std::complex over-encapsulated.
 144       constexpr _Tp
 145       real() const { return _M_real; }
 146
 147       constexpr _Tp
 148       imag() const { return _M_imag; }
 149 #else
 150       ///  Return real part of complex number.
 151       _Tp&
 152       real() { return _M_real; }
 153
 154       ///  Return real part of complex number.
 155       const _Tp&
 156       real() const { return _M_real; }
 157
 158       ///  Return imaginary part of complex number.
 159       _Tp&
 160       imag() { return _M_imag; }
 161
 162       ///  Return imaginary part of complex number.
 163       const _Tp&
 164       imag() const { return _M_imag; }
 165 #endif
 166
 167       // _GLIBCXX_RESOLVE_LIB_DEFECTS
 168       // DR 387. std::complex over-encapsulated.
 169       void
 170       real(_Tp __val) { _M_real = __val; }
 171
 172       void
 173       imag(_Tp __val) { _M_imag = __val; }
 174
 175       /// Assign this complex number to scalar @a t.
 176       complex<_Tp>& operator=(const _Tp&);
 177
 178       /// Add @a t to this complex number.
 179       // 26.2.5/1
 180       complex<_Tp>&
 181       operator+=(const _Tp& __t)
 182       {
 183         _M_real += __t;
 184         return *this;
 185       }
 186
 187       /// Subtract @a t from this complex number.
 188       // 26.2.5/3
 189       complex<_Tp>&
 190       operator-=(const _Tp& __t)
 191       {
 192         _M_real -= __t;
 193         return *this;
 194       }
 195
 196       /// Multiply this complex number by @a t.
 197       complex<_Tp>& operator*=(const _Tp&);
 198       /// Divide this complex number by @a t.
 199       complex<_Tp>& operator/=(const _Tp&);
 200
 201       // Lets the compiler synthesize the
 202       // copy and assignment operator
 203       // complex<_Tp>& operator= (const complex<_Tp>&);
 204       /// Assign this complex number to complex @a z.
 205       template<typename _Up>
 206         complex<_Tp>& operator=(const complex<_Up>&);
 207       /// Add @a z to this complex number.
 208       template<typename _Up>
 209         complex<_Tp>& operator+=(const complex<_Up>&);
 210       /// Subtract @a z from this complex number.
 211       template<typename _Up>
 212         complex<_Tp>& operator-=(const complex<_Up>&);
 213       /// Multiply this complex number by @a z.
 214       template<typename _Up>
 215         complex<_Tp>& operator*=(const complex<_Up>&);
 216       /// Divide this complex number by @a z.
 217       template<typename _Up>
 218         complex<_Tp>& operator/=(const complex<_Up>&);
 219
 220       _GLIBCXX_USE_CONSTEXPR complex __rep() const
 221       { return *this; }
 222
 223     private:
 224       _Tp _M_real;
 225       _Tp _M_imag;
 226     };
 227
 228   template<typename _Tp>
 229     complex<_Tp>&
 230     complex<_Tp>::operator=(const _Tp& __t)
 231     {
 232      _M_real = __t;
 233      _M_imag = _Tp();
 234      return *this;
 235     }
 236
 237   // 26.2.5/5
 238   template<typename _Tp>
 239     complex<_Tp>&
 240     complex<_Tp>::operator*=(const _Tp& __t)
 241     {
 242       _M_real *= __t;
 243       _M_imag *= __t;
 244       return *this;
 245     }
 246
 247   // 26.2.5/7
 248   template<typename _Tp>
 249     complex<_Tp>&
 250     complex<_Tp>::operator/=(const _Tp& __t)
 251     {
 252       _M_real /= __t;
 253       _M_imag /= __t;
 254       return *this;
 255     }
 256
 257   template<typename _Tp>
 258     template<typename _Up>
 259     complex<_Tp>&
 260     complex<_Tp>::operator=(const complex<_Up>& __z)
 261     {
 262       _M_real = __z.real();
 263       _M_imag = __z.imag();
 264       return *this;
 265     }
 266
 267   // 26.2.5/9
 268   template<typename _Tp>
 269     template<typename _Up>
 270     complex<_Tp>&
 271     complex<_Tp>::operator+=(const complex<_Up>& __z)
 272     {
 273       _M_real += __z.real();
 274       _M_imag += __z.imag();
 275       return *this;
 276     }
 277
 278   // 26.2.5/11
 279   template<typename _Tp>
 280     template<typename _Up>
 281     complex<_Tp>&
 282     complex<_Tp>::operator-=(const complex<_Up>& __z)
 283     {
 284       _M_real -= __z.real();
 285       _M_imag -= __z.imag();
 286       return *this;
 287     }
 288
 289   // 26.2.5/13
 290   // XXX: This is a grammar school implementation.
 291   template<typename _Tp>
 292     template<typename _Up>
 293     complex<_Tp>&
 294     complex<_Tp>::operator*=(const complex<_Up>& __z)
 295     {
 296       const _Tp __r = _M_real * __z.real() - _M_imag * __z.imag();
 297       _M_imag = _M_real * __z.imag() + _M_imag * __z.real();
 298       _M_real = __r;
 299       return *this;
 300     }
 301
 302   // 26.2.5/15
 303   // XXX: This is a grammar school implementation.
 304   template<typename _Tp>
 305     template<typename _Up>
 306     complex<_Tp>&
 307     complex<_Tp>::operator/=(const complex<_Up>& __z)
 308     {
 309       const _Tp __r =  _M_real * __z.real() + _M_imag * __z.imag();
 310       const _Tp __n = std::norm(__z);
 311       _M_imag = (_M_imag * __z.real() - _M_real * __z.imag()) / __n;
 312       _M_real = __r / __n;
 313       return *this;
 314     }
 315
 316   // Operators:
 317   //@{
 318   ///  Return new complex value @a x plus @a y.
 319   template<typename _Tp>
 320     inline complex<_Tp>
 321     operator+(const complex<_Tp>& __x, const complex<_Tp>& __y)
 322     {
 323       complex<_Tp> __r = __x;
 324       __r += __y;
 325       return __r;
 326     }
 327
 328   template<typename _Tp>
 329     inline complex<_Tp>
 330     operator+(const complex<_Tp>& __x, const _Tp& __y)
 331     {
 332       complex<_Tp> __r = __x;
 333       __r += __y;
 334       return __r;
 335     }
 336
 337   template<typename _Tp>
 338     inline complex<_Tp>
 339     operator+(const _Tp& __x, const complex<_Tp>& __y)
 340     {
 341       complex<_Tp> __r = __y;
 342       __r += __x;
 343       return __r;
 344     }
 345   //@}
 346
 347   //@{
 348   ///  Return new complex value @a x minus @a y.
 349   template<typename _Tp>
 350     inline complex<_Tp>
 351     operator-(const complex<_Tp>& __x, const complex<_Tp>& __y)
 352     {
 353       complex<_Tp> __r = __x;
 354       __r -= __y;
 355       return __r;
 356     }
 357
 358   template<typename _Tp>
 359     inline complex<_Tp>
 360     operator-(const complex<_Tp>& __x, const _Tp& __y)
 361     {
 362       complex<_Tp> __r = __x;
 363       __r -= __y;
 364       return __r;
 365     }
 366
 367   template<typename _Tp>
 368     inline complex<_Tp>
 369     operator-(const _Tp& __x, const complex<_Tp>& __y)
 370     {
 371       complex<_Tp> __r(__x, -__y.imag());
 372       __r -= __y.real();
 373       return __r;
 374     }
 375   //@}
 376
 377   //@{
 378   ///  Return new complex value @a x times @a y.
 379   template<typename _Tp>
 380     inline complex<_Tp>
 381     operator*(const complex<_Tp>& __x, const complex<_Tp>& __y)
 382     {
 383       complex<_Tp> __r = __x;
 384       __r *= __y;
 385       return __r;
 386     }
 387
 388   template<typename _Tp>
 389     inline complex<_Tp>
 390     operator*(const complex<_Tp>& __x, const _Tp& __y)
 391     {
 392       complex<_Tp> __r = __x;
 393       __r *= __y;
 394       return __r;
 395     }
 396
 397   template<typename _Tp>
 398     inline complex<_Tp>
 399     operator*(const _Tp& __x, const complex<_Tp>& __y)
 400     {
 401       complex<_Tp> __r = __y;
 402       __r *= __x;
 403       return __r;
 404     }
 405   //@}
 406
 407   //@{
 408   ///  Return new complex value @a x divided by @a y.
 409   template<typename _Tp>
 410     inline complex<_Tp>
 411     operator/(const complex<_Tp>& __x, const complex<_Tp>& __y)
 412     {
 413       complex<_Tp> __r = __x;
 414       __r /= __y;
 415       return __r;
 416     }
 417
 418   template<typename _Tp>
 419     inline complex<_Tp>
 420     operator/(const complex<_Tp>& __x, const _Tp& __y)
 421     {
 422       complex<_Tp> __r = __x;
 423       __r /= __y;
 424       return __r;
 425     }
 426
 427   template<typename _Tp>
 428     inline complex<_Tp>
 429     operator/(const _Tp& __x, const complex<_Tp>& __y)
 430     {
 431       complex<_Tp> __r = __x;
 432       __r /= __y;
 433       return __r;
 434     }
 435   //@}
 436
 437   ///  Return @a x.
 438   template<typename _Tp>
 439     inline complex<_Tp>
 440     operator+(const complex<_Tp>& __x)
 441     { return __x; }
 442
 443   ///  Return complex negation of @a x.
 444   template<typename _Tp>
 445     inline complex<_Tp>
 446     operator-(const complex<_Tp>& __x)
 447     {  return complex<_Tp>(-__x.real(), -__x.imag()); }
 448
 449   //@{
 450   ///  Return true if @a x is equal to @a y.
 451   template<typename _Tp>
 452     inline _GLIBCXX_CONSTEXPR bool
 453     operator==(const complex<_Tp>& __x, const complex<_Tp>& __y)
 454     { return __x.real() == __y.real() && __x.imag() == __y.imag(); }
 455
 456   template<typename _Tp>
 457     inline _GLIBCXX_CONSTEXPR bool
 458     operator==(const complex<_Tp>& __x, const _Tp& __y)
 459     { return __x.real() == __y && __x.imag() == _Tp(); }
 460
 461   template<typename _Tp>
 462     inline _GLIBCXX_CONSTEXPR bool
 463     operator==(const _Tp& __x, const complex<_Tp>& __y)
 464     { return __x == __y.real() && _Tp() == __y.imag(); }
 465   //@}
 466
 467   //@{
 468   ///  Return false if @a x is equal to @a y.
 469   template<typename _Tp>
 470     inline _GLIBCXX_CONSTEXPR bool
 471     operator!=(const complex<_Tp>& __x, const complex<_Tp>& __y)
 472     { return __x.real() != __y.real() || __x.imag() != __y.imag(); }
 473
 474   template<typename _Tp>
 475     inline _GLIBCXX_CONSTEXPR bool
 476     operator!=(const complex<_Tp>& __x, const _Tp& __y)
 477     { return __x.real() != __y || __x.imag() != _Tp(); }
 478
 479   template<typename _Tp>
 480     inline _GLIBCXX_CONSTEXPR bool
 481     operator!=(const _Tp& __x, const complex<_Tp>& __y)
 482     { return __x != __y.real() || _Tp() != __y.imag(); }
 483   //@}
 484
 485   ///  Extraction operator for complex values.
 486   template<typename _Tp, typename _CharT, class _Traits>
 487     basic_istream<_CharT, _Traits>&
 488     operator>>(basic_istream<_CharT, _Traits>& __is, complex<_Tp>& __x)
 489     {
 490       _Tp __re_x, __im_x;
 491       _CharT __ch;
 492       __is >> __ch;
 493       if (__ch == '(')
 494         {
 495           __is >> __re_x >> __ch;
 496           if (__ch == ',')
 497             {
 498               __is >> __im_x >> __ch;
 499               if (__ch == ')')
 500                 __x = complex<_Tp>(__re_x, __im_x);
 501               else
 502                 __is.setstate(ios_base::failbit);
 503             }
 504           else if (__ch == ')')
 505             __x = __re_x;
 506           else
 507             __is.setstate(ios_base::failbit);
 508         }
 509       else
 510         {
 511           __is.putback(__ch);
 512           __is >> __re_x;
 513           __x = __re_x;
 514         }
 515       return __is;
 516     }
 517
 518   ///  Insertion operator for complex values.
 519   template<typename _Tp, typename _CharT, class _Traits>
 520     basic_ostream<_CharT, _Traits>&
 521     operator<<(basic_ostream<_CharT, _Traits>& __os, const complex<_Tp>& __x)
 522     {
 523       basic_ostringstream<_CharT, _Traits> __s;
 524       __s.flags(__os.flags());
 525       __s.imbue(__os.getloc());
 526       __s.precision(__os.precision());
 527       __s << '(' << __x.real() << ',' << __x.imag() << ')';
 528       return __os << __s.str();
 529     }
 530
 531   // Values
 532 #ifdef __GXX_EXPERIMENTAL_CXX0X__
 533   template<typename _Tp>
 534     inline constexpr _Tp
 535     real(const complex<_Tp>& __z)
 536     { return __z.real(); }
 537
 538   template<typename _Tp>
 539     inline constexpr _Tp
 540     imag(const complex<_Tp>& __z)
 541     { return __z.imag(); }
 542 #else
 543   template<typename _Tp>
 544     inline _Tp&
 545     real(complex<_Tp>& __z)
 546     { return __z.real(); }
 547
 548   template<typename _Tp>
 549     inline const _Tp&
 550     real(const complex<_Tp>& __z)
 551     { return __z.real(); }
 552
 553   template<typename _Tp>
 554     inline _Tp&
 555     imag(complex<_Tp>& __z)
 556     { return __z.imag(); }
 557
 558   template<typename _Tp>
 559     inline const _Tp&
 560     imag(const complex<_Tp>& __z)
 561     { return __z.imag(); }
 562 #endif
 563
 564   // 26.2.7/3 abs(__z):  Returns the magnitude of __z.
 565   template<typename _Tp>
 566     inline _Tp
 567     __complex_abs(const complex<_Tp>& __z)
 568     {
 569       _Tp __x = __z.real();
 570       _Tp __y = __z.imag();
 571       const _Tp __s = std::max(abs(__x), abs(__y));
 572       if (__s == _Tp())  // well ...
 573         return __s;
 574       __x /= __s;
 575       __y /= __s;
 576       return __s * sqrt(__x * __x + __y * __y);
 577     }
 578
 579 #if _GLIBCXX_USE_C99_COMPLEX
 580   inline float
 581   __complex_abs(__complex__ float __z) { return __builtin_cabsf(__z); }
 582
 583   inline double
 584   __complex_abs(__complex__ double __z) { return __builtin_cabs(__z); }
 585
 586   inline long double
 587   __complex_abs(const __complex__ long double& __z)
 588   { return __builtin_cabsl(__z); }
 589
 590   template<typename _Tp>
 591     inline _Tp
 592     abs(const complex<_Tp>& __z) { return __complex_abs(__z.__rep()); }
 593 #else
 594   template<typename _Tp>
 595     inline _Tp
 596     abs(const complex<_Tp>& __z) { return __complex_abs(__z); }
 597 #endif
 598
 599
 600   // 26.2.7/4: arg(__z): Returns the phase angle of __z.
 601   template<typename _Tp>
 602     inline _Tp
 603     __complex_arg(const complex<_Tp>& __z)
 604     { return  atan2(__z.imag(), __z.real()); }
 605
 606 #if _GLIBCXX_USE_C99_COMPLEX
 607   inline float
 608   __complex_arg(__complex__ float __z) { return __builtin_cargf(__z); }
 609
 610   inline double
 611   __complex_arg(__complex__ double __z) { return __builtin_carg(__z); }
 612
 613   inline long double
 614   __complex_arg(const __complex__ long double& __z)
 615   { return __builtin_cargl(__z); }
 616
 617   template<typename _Tp>
 618     inline _Tp
 619     arg(const complex<_Tp>& __z) { return __complex_arg(__z.__rep()); }
 620 #else
 621   template<typename _Tp>
 622     inline _Tp
 623     arg(const complex<_Tp>& __z) { return __complex_arg(__z); }
 624 #endif
 625
 626   // 26.2.7/5: norm(__z) returns the squared magnitude of __z.
 627   //     As defined, norm() is -not- a norm is the common mathematical
 628   //     sens used in numerics.  The helper class _Norm_helper<> tries to
 629   //     distinguish between builtin floating point and the rest, so as
 630   //     to deliver an answer as close as possible to the real value.
 631   template<bool>
 632     struct _Norm_helper
 633     {
 634       template<typename _Tp>
 635         static inline _Tp _S_do_it(const complex<_Tp>& __z)
 636         {
 637           const _Tp __x = __z.real();
 638           const _Tp __y = __z.imag();
 639           return __x * __x + __y * __y;
 640         }
 641     };
 642
 643   template<>
 644     struct _Norm_helper<true>
 645     {
 646       template<typename _Tp>
 647         static inline _Tp _S_do_it(const complex<_Tp>& __z)
 648         {
 649           _Tp __res = std::abs(__z);
 650           return __res * __res;
 651         }
 652     };
 653
 654   template<typename _Tp>
 655     inline _Tp
 656     norm(const complex<_Tp>& __z)
 657     {
 658       return _Norm_helper<__is_floating<_Tp>::__value
 659         && !_GLIBCXX_FAST_MATH>::_S_do_it(__z);
 660     }
 661
 662   template<typename _Tp>
 663     inline complex<_Tp>
 664     polar(const _Tp& __rho, const _Tp& __theta)
 665     { return complex<_Tp>(__rho * cos(__theta), __rho * sin(__theta)); }
 666
 667   template<typename _Tp>
 668     inline complex<_Tp>
 669     conj(const complex<_Tp>& __z)
 670     { return complex<_Tp>(__z.real(), -__z.imag()); }
 671
 672   // Transcendentals
 673
 674   // 26.2.8/1 cos(__z):  Returns the cosine of __z.
 675   template<typename _Tp>
 676     inline complex<_Tp>
 677     __complex_cos(const complex<_Tp>& __z)
 678     {
 679       const _Tp __x = __z.real();
 680       const _Tp __y = __z.imag();
 681       return complex<_Tp>(cos(__x) * cosh(__y), -sin(__x) * sinh(__y));
 682     }
 683
 684 #if _GLIBCXX_USE_C99_COMPLEX
 685   inline __complex__ float
 686   __complex_cos(__complex__ float __z) { return __builtin_ccosf(__z); }
 687
 688   inline __complex__ double
 689   __complex_cos(__complex__ double __z) { return __builtin_ccos(__z); }
 690
 691   inline __complex__ long double
 692   __complex_cos(const __complex__ long double& __z)
 693   { return __builtin_ccosl(__z); }
 694
 695   template<typename _Tp>
 696     inline complex<_Tp>
 697     cos(const complex<_Tp>& __z) { return __complex_cos(__z.__rep()); }
 698 #else
 699   template<typename _Tp>
 700     inline complex<_Tp>
 701     cos(const complex<_Tp>& __z) { return __complex_cos(__z); }
 702 #endif
 703
 704   // 26.2.8/2 cosh(__z): Returns the hyperbolic cosine of __z.
 705   template<typename _Tp>
 706     inline complex<_Tp>
 707     __complex_cosh(const complex<_Tp>& __z)
 708     {
 709       const _Tp __x = __z.real();
 710       const _Tp __y = __z.imag();
 711       return complex<_Tp>(cosh(__x) * cos(__y), sinh(__x) * sin(__y));
 712     }
 713
 714 #if _GLIBCXX_USE_C99_COMPLEX
 715   inline __complex__ float
 716   __complex_cosh(__complex__ float __z) { return __builtin_ccoshf(__z); }
 717
 718   inline __complex__ double
 719   __complex_cosh(__complex__ double __z) { return __builtin_ccosh(__z); }
 720
 721   inline __complex__ long double
 722   __complex_cosh(const __complex__ long double& __z)
 723   { return __builtin_ccoshl(__z); }
 724
 725   template<typename _Tp>
 726     inline complex<_Tp>
 727     cosh(const complex<_Tp>& __z) { return __complex_cosh(__z.__rep()); }
 728 #else
 729   template<typename _Tp>
 730     inline complex<_Tp>
 731     cosh(const complex<_Tp>& __z) { return __complex_cosh(__z); }
 732 #endif
 733
 734   // 26.2.8/3 exp(__z): Returns the complex base e exponential of x
 735   template<typename _Tp>
 736     inline complex<_Tp>
 737     __complex_exp(const complex<_Tp>& __z)
 738     { return std::polar(exp(__z.real()), __z.imag()); }
 739
 740 #if _GLIBCXX_USE_C99_COMPLEX
 741   inline __complex__ float
 742   __complex_exp(__complex__ float __z) { return __builtin_cexpf(__z); }
 743
 744   inline __complex__ double
 745   __complex_exp(__complex__ double __z) { return __builtin_cexp(__z); }
 746
 747   inline __complex__ long double
 748   __complex_exp(const __complex__ long double& __z)
 749   { return __builtin_cexpl(__z); }
 750
 751   template<typename _Tp>
 752     inline complex<_Tp>
 753     exp(const complex<_Tp>& __z) { return __complex_exp(__z.__rep()); }
 754 #else
 755   template<typename _Tp>
 756     inline complex<_Tp>
 757     exp(const complex<_Tp>& __z) { return __complex_exp(__z); }
 758 #endif
 759
 760   // 26.2.8/5 log(__z): Returns the natural complex logarithm of __z.
 761   //                    The branch cut is along the negative axis.
 762   template<typename _Tp>
 763     inline complex<_Tp>
 764     __complex_log(const complex<_Tp>& __z)
 765     { return complex<_Tp>(log(std::abs(__z)), std::arg(__z)); }
 766
 767 #if _GLIBCXX_USE_C99_COMPLEX
 768   inline __complex__ float
 769   __complex_log(__complex__ float __z) { return __builtin_clogf(__z); }
 770
 771   inline __complex__ double
 772   __complex_log(__complex__ double __z) { return __builtin_clog(__z); }
 773
 774   inline __complex__ long double
 775   __complex_log(const __complex__ long double& __z)
 776   { return __builtin_clogl(__z); }
 777
 778   template<typename _Tp>
 779     inline complex<_Tp>
 780     log(const complex<_Tp>& __z) { return __complex_log(__z.__rep()); }
 781 #else
 782   template<typename _Tp>
 783     inline complex<_Tp>
 784     log(const complex<_Tp>& __z) { return __complex_log(__z); }
 785 #endif
 786
 787   template<typename _Tp>
 788     inline complex<_Tp>
 789     log10(const complex<_Tp>& __z)
 790     { return std::log(__z) / log(_Tp(10.0)); }
 791
 792   // 26.2.8/10 sin(__z): Returns the sine of __z.
 793   template<typename _Tp>
 794     inline complex<_Tp>
 795     __complex_sin(const complex<_Tp>& __z)
 796     {
 797       const _Tp __x = __z.real();
 798       const _Tp __y = __z.imag();
 799       return complex<_Tp>(sin(__x) * cosh(__y), cos(__x) * sinh(__y));
 800     }
 801
 802 #if _GLIBCXX_USE_C99_COMPLEX
 803   inline __complex__ float
 804   __complex_sin(__complex__ float __z) { return __builtin_csinf(__z); }
 805
 806   inline __complex__ double
 807   __complex_sin(__complex__ double __z) { return __builtin_csin(__z); }
 808
 809   inline __complex__ long double
 810   __complex_sin(const __complex__ long double& __z)
 811   { return __builtin_csinl(__z); }
 812
 813   template<typename _Tp>
 814     inline complex<_Tp>
 815     sin(const complex<_Tp>& __z) { return __complex_sin(__z.__rep()); }
 816 #else
 817   template<typename _Tp>
 818     inline complex<_Tp>
 819     sin(const complex<_Tp>& __z) { return __complex_sin(__z); }
 820 #endif
 821
 822   // 26.2.8/11 sinh(__z): Returns the hyperbolic sine of __z.
 823   template<typename _Tp>
 824     inline complex<_Tp>
 825     __complex_sinh(const complex<_Tp>& __z)
 826     {
 827       const _Tp __x = __z.real();
 828       const _Tp  __y = __z.imag();
 829       return complex<_Tp>(sinh(__x) * cos(__y), cosh(__x) * sin(__y));
 830     }
 831
 832 #if _GLIBCXX_USE_C99_COMPLEX
 833   inline __complex__ float
 834   __complex_sinh(__complex__ float __z) { return __builtin_csinhf(__z); }
 835
 836   inline __complex__ double
 837   __complex_sinh(__complex__ double __z) { return __builtin_csinh(__z); }
 838
 839   inline __complex__ long double
 840   __complex_sinh(const __complex__ long double& __z)
 841   { return __builtin_csinhl(__z); }
 842
 843   template<typename _Tp>
 844     inline complex<_Tp>
 845     sinh(const complex<_Tp>& __z) { return __complex_sinh(__z.__rep()); }
 846 #else
 847   template<typename _Tp>
 848     inline complex<_Tp>
 849     sinh(const complex<_Tp>& __z) { return __complex_sinh(__z); }
 850 #endif
 851
 852   // 26.2.8/13 sqrt(__z): Returns the complex square root of __z.
 853   //                     The branch cut is on the negative axis.
 854   template<typename _Tp>
 855     complex<_Tp>
 856     __complex_sqrt(const complex<_Tp>& __z)
 857     {
 858       _Tp __x = __z.real();
 859       _Tp __y = __z.imag();
 860
 861       if (__x == _Tp())
 862         {
 863           _Tp __t = sqrt(abs(__y) / 2);
 864           return complex<_Tp>(__t, __y < _Tp() ? -__t : __t);
 865         }
 866       else
 867         {
 868           _Tp __t = sqrt(2 * (std::abs(__z) + abs(__x)));
 869           _Tp __u = __t / 2;
 870           return __x > _Tp()
 871             ? complex<_Tp>(__u, __y / __t)
 872             : complex<_Tp>(abs(__y) / __t, __y < _Tp() ? -__u : __u);
 873         }
 874     }
 875
 876 #if _GLIBCXX_USE_C99_COMPLEX
 877   inline __complex__ float
 878   __complex_sqrt(__complex__ float __z) { return __builtin_csqrtf(__z); }
 879
 880   inline __complex__ double
 881   __complex_sqrt(__complex__ double __z) { return __builtin_csqrt(__z); }
 882
 883   inline __complex__ long double
 884   __complex_sqrt(const __complex__ long double& __z)
 885   { return __builtin_csqrtl(__z); }
 886
 887   template<typename _Tp>
 888     inline complex<_Tp>
 889     sqrt(const complex<_Tp>& __z) { return __complex_sqrt(__z.__rep()); }
 890 #else
 891   template<typename _Tp>
 892     inline complex<_Tp>
 893     sqrt(const complex<_Tp>& __z) { return __complex_sqrt(__z); }
 894 #endif
 895
 896   // 26.2.8/14 tan(__z):  Return the complex tangent of __z.
 897
 898   template<typename _Tp>
 899     inline complex<_Tp>
 900     __complex_tan(const complex<_Tp>& __z)
 901     { return std::sin(__z) / std::cos(__z); }
 902
 903 #if _GLIBCXX_USE_C99_COMPLEX
 904   inline __complex__ float
 905   __complex_tan(__complex__ float __z) { return __builtin_ctanf(__z); }
 906
 907   inline __complex__ double
 908   __complex_tan(__complex__ double __z) { return __builtin_ctan(__z); }
 909
 910   inline __complex__ long double
 911   __complex_tan(const __complex__ long double& __z)
 912   { return __builtin_ctanl(__z); }
 913
 914   template<typename _Tp>
 915     inline complex<_Tp>
 916     tan(const complex<_Tp>& __z) { return __complex_tan(__z.__rep()); }
 917 #else
 918   template<typename _Tp>
 919     inline complex<_Tp>
 920     tan(const complex<_Tp>& __z) { return __complex_tan(__z); }
 921 #endif
 922
 923
 924   // 26.2.8/15 tanh(__z):  Returns the hyperbolic tangent of __z.
 925
 926   template<typename _Tp>
 927     inline complex<_Tp>
 928     __complex_tanh(const complex<_Tp>& __z)
 929     { return std::sinh(__z) / std::cosh(__z); }
 930
 931 #if _GLIBCXX_USE_C99_COMPLEX
 932   inline __complex__ float
 933   __complex_tanh(__complex__ float __z) { return __builtin_ctanhf(__z); }
 934
 935   inline __complex__ double
 936   __complex_tanh(__complex__ double __z) { return __builtin_ctanh(__z); }
 937
 938   inline __complex__ long double
 939   __complex_tanh(const __complex__ long double& __z)
 940   { return __builtin_ctanhl(__z); }
 941
 942   template<typename _Tp>
 943     inline complex<_Tp>
 944     tanh(const complex<_Tp>& __z) { return __complex_tanh(__z.__rep()); }
 945 #else
 946   template<typename _Tp>
 947     inline complex<_Tp>
 948     tanh(const complex<_Tp>& __z) { return __complex_tanh(__z); }
 949 #endif
 950
 951
 952   // 26.2.8/9  pow(__x, __y): Returns the complex power base of __x
 953   //                          raised to the __y-th power.  The branch
 954   //                          cut is on the negative axis.
 955 #ifndef __GXX_EXPERIMENTAL_CXX0X__
 956   template<typename _Tp>
 957     complex<_Tp>
 958     __complex_pow_unsigned(complex<_Tp> __x, unsigned __n)
 959     {
 960       complex<_Tp> __y = __n % 2 ? __x : complex<_Tp>(1);
 961
 962       while (__n >>= 1)
 963         {
 964           __x *= __x;
 965           if (__n % 2)
 966             __y *= __x;
 967         }
 968
 969       return __y;
 970     }
 971
 972   // _GLIBCXX_RESOLVE_LIB_DEFECTS
 973   // DR 844. complex pow return type is ambiguous.
 974   template<typename _Tp>
 975     inline complex<_Tp>
 976     pow(const complex<_Tp>& __z, int __n)
 977     {
 978       return __n < 0
 979         ? complex<_Tp>(1) / std::__complex_pow_unsigned(__z, -__n)
 980         : std::__complex_pow_unsigned(__z, __n);
 981     }
 982 #endif
 983
 984   template<typename _Tp>
 985     complex<_Tp>
 986     pow(const complex<_Tp>& __x, const _Tp& __y)
 987     {
 988 #ifndef _GLIBCXX_USE_C99_COMPLEX
 989       if (__x == _Tp())
 990         return _Tp();
 991 #endif
 992       if (__x.imag() == _Tp() && __x.real() > _Tp())
 993         return pow(__x.real(), __y);
 994
 995       complex<_Tp> __t = std::log(__x);
 996       return std::polar(exp(__y * __t.real()), __y * __t.imag());
 997     }
 998
 999   template<typename _Tp>
1000     inline complex<_Tp>
1001     __complex_pow(const complex<_Tp>& __x, const complex<_Tp>& __y)
1002     { return __x == _Tp() ? _Tp() : std::exp(__y * std::log(__x)); }
1003
1004 #if _GLIBCXX_USE_C99_COMPLEX
1005   inline __complex__ float
1006   __complex_pow(__complex__ float __x, __complex__ float __y)
1007   { return __builtin_cpowf(__x, __y); }
1008
1009   inline __complex__ double
1010   __complex_pow(__complex__ double __x, __complex__ double __y)
1011   { return __builtin_cpow(__x, __y); }
1012
1013   inline __complex__ long double
1014   __complex_pow(const __complex__ long double& __x,
1015                 const __complex__ long double& __y)
1016   { return __builtin_cpowl(__x, __y); }
1017
1018   template<typename _Tp>
1019     inline complex<_Tp>
1020     pow(const complex<_Tp>& __x, const complex<_Tp>& __y)
1021     { return __complex_pow(__x.__rep(), __y.__rep()); }
1022 #else
1023   template<typename _Tp>
1024     inline complex<_Tp>
1025     pow(const complex<_Tp>& __x, const complex<_Tp>& __y)
1026     { return __complex_pow(__x, __y); }
1027 #endif
1028
1029   template<typename _Tp>
1030     inline complex<_Tp>
1031     pow(const _Tp& __x, const complex<_Tp>& __y)
1032     {
1033       return __x > _Tp() ? std::polar(pow(__x, __y.real()),
1034                                       __y.imag() * log(__x))
1035                          : std::pow(complex<_Tp>(__x), __y);
1036     }
1037
1038   // 26.2.3  complex specializations
1039   // complex<float> specialization
1040   template<>
1041     struct complex<float>
1042     {
1043       typedef float value_type;
1044       typedef __complex__ float _ComplexT;
1045
1046       _GLIBCXX_CONSTEXPR complex(_ComplexT __z) : _M_value(__z) { }
1047
1048       _GLIBCXX_CONSTEXPR complex(float __r = 0.0f, float __i = 0.0f)
1049       : _M_value(__r + __i * 1.0fi) { }
1050
1051       explicit _GLIBCXX_CONSTEXPR complex(const complex<double>&);
1052       explicit _GLIBCXX_CONSTEXPR complex(const complex<long double>&);
1053
1054 #ifdef __GXX_EXPERIMENTAL_CXX0X__
1055       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1056       // DR 387. std::complex over-encapsulated.
1057       constexpr float
1058       real() const { return __real__ _M_value; }
1059
1060       constexpr float
1061       imag() const { return __imag__ _M_value; }
1062 #else
1063       float&
1064       real() { return __real__ _M_value; }
1065
1066       const float&
1067       real() const { return __real__ _M_value; }
1068
1069       float&
1070       imag() { return __imag__ _M_value; }
1071
1072       const float&
1073       imag() const { return __imag__ _M_value; }
1074 #endif
1075
1076       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1077       // DR 387. std::complex over-encapsulated.
1078       void
1079       real(float __val) { __real__ _M_value = __val; }
1080
1081       void
1082       imag(float __val) { __imag__ _M_value = __val; }
1083
1084       complex&
1085       operator=(float __f)
1086       {
1087         _M_value = __f;
1088         return *this;
1089       }
1090
1091       complex&
1092       operator+=(float __f)
1093       {
1094         _M_value += __f;
1095         return *this;
1096       }
1097
1098       complex&
1099       operator-=(float __f)
1100       {
1101         _M_value -= __f;
1102         return *this;
1103       }
1104
1105       complex&
1106       operator*=(float __f)
1107       {
1108         _M_value *= __f;
1109         return *this;
1110       }
1111
1112       complex&
1113       operator/=(float __f)
1114       {
1115         _M_value /= __f;
1116         return *this;
1117       }
1118
1119       // Let the compiler synthesize the copy and assignment
1120       // operator.  It always does a pretty good job.
1121       // complex& operator=(const complex&);
1122
1123       template<typename _Tp>
1124         complex&
1125         operator=(const complex<_Tp>&  __z)
1126         {
1127           __real__ _M_value = __z.real();
1128           __imag__ _M_value = __z.imag();
1129           return *this;
1130         }
1131
1132       template<typename _Tp>
1133         complex&
1134         operator+=(const complex<_Tp>& __z)
1135         {
1136           __real__ _M_value += __z.real();
1137           __imag__ _M_value += __z.imag();
1138           return *this;
1139         }
1140
1141       template<class _Tp>
1142         complex&
1143         operator-=(const complex<_Tp>& __z)
1144         {
1145           __real__ _M_value -= __z.real();
1146           __imag__ _M_value -= __z.imag();
1147           return *this;
1148         }
1149
1150       template<class _Tp>
1151         complex&
1152         operator*=(const complex<_Tp>& __z)
1153         {
1154           _ComplexT __t;
1155           __real__ __t = __z.real();
1156           __imag__ __t = __z.imag();
1157           _M_value *= __t;
1158           return *this;
1159         }
1160
1161       template<class _Tp>
1162         complex&
1163         operator/=(const complex<_Tp>& __z)
1164         {
1165           _ComplexT __t;
1166           __real__ __t = __z.real();
1167           __imag__ __t = __z.imag();
1168           _M_value /= __t;
1169           return *this;
1170         }
1171
1172       _GLIBCXX_USE_CONSTEXPR _ComplexT __rep() const { return _M_value; }
1173
1174     private:
1175       _ComplexT _M_value;
1176     };
1177
1178   // 26.2.3  complex specializations
1179   // complex<double> specialization
1180   template<>
1181     struct complex<double>
1182     {
1183       typedef double value_type;
1184       typedef __complex__ double _ComplexT;
1185
1186       _GLIBCXX_CONSTEXPR complex(_ComplexT __z) : _M_value(__z) { }
1187
1188       _GLIBCXX_CONSTEXPR complex(double __r = 0.0, double __i = 0.0)
1189       : _M_value(__r + __i * 1.0i) { }
1190
1191       _GLIBCXX_CONSTEXPR complex(const complex<float>& __z)
1192       : _M_value(__z.__rep()) { }
1193
1194       explicit _GLIBCXX_CONSTEXPR complex(const complex<long double>&);
1195
1196 #ifdef __GXX_EXPERIMENTAL_CXX0X__
1197       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1198       // DR 387. std::complex over-encapsulated.
1199       constexpr double
1200       real() const { return __real__ _M_value; }
1201
1202       constexpr double
1203       imag() const { return __imag__ _M_value; }
1204 #else
1205       double&
1206       real() { return __real__ _M_value; }
1207
1208       const double&
1209       real() const { return __real__ _M_value; }
1210
1211       double&
1212       imag() { return __imag__ _M_value; }
1213
1214       const double&
1215       imag() const { return __imag__ _M_value; }
1216 #endif
1217
1218       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1219       // DR 387. std::complex over-encapsulated.
1220       void
1221       real(double __val) { __real__ _M_value = __val; }
1222
1223       void
1224       imag(double __val) { __imag__ _M_value = __val; }
1225
1226       complex&
1227       operator=(double __d)
1228       {
1229         _M_value = __d;
1230         return *this;
1231       }
1232
1233       complex&
1234       operator+=(double __d)
1235       {
1236         _M_value += __d;
1237         return *this;
1238       }
1239
1240       complex&
1241       operator-=(double __d)
1242       {
1243         _M_value -= __d;
1244         return *this;
1245       }
1246
1247       complex&
1248       operator*=(double __d)
1249       {
1250         _M_value *= __d;
1251         return *this;
1252       }
1253
1254       complex&
1255       operator/=(double __d)
1256       {
1257         _M_value /= __d;
1258         return *this;
1259       }
1260
1261       // The compiler will synthesize this, efficiently.
1262       // complex& operator=(const complex&);
1263
1264       template<typename _Tp>
1265         complex&
1266         operator=(const complex<_Tp>& __z)
1267         {
1268           __real__ _M_value = __z.real();
1269           __imag__ _M_value = __z.imag();
1270           return *this;
1271         }
1272
1273       template<typename _Tp>
1274         complex&
1275         operator+=(const complex<_Tp>& __z)
1276         {
1277           __real__ _M_value += __z.real();
1278           __imag__ _M_value += __z.imag();
1279           return *this;
1280         }
1281
1282       template<typename _Tp>
1283         complex&
1284         operator-=(const complex<_Tp>& __z)
1285         {
1286           __real__ _M_value -= __z.real();
1287           __imag__ _M_value -= __z.imag();
1288           return *this;
1289         }
1290
1291       template<typename _Tp>
1292         complex&
1293         operator*=(const complex<_Tp>& __z)
1294         {
1295           _ComplexT __t;
1296           __real__ __t = __z.real();
1297           __imag__ __t = __z.imag();
1298           _M_value *= __t;
1299           return *this;
1300         }
1301
1302       template<typename _Tp>
1303         complex&
1304         operator/=(const complex<_Tp>& __z)
1305         {
1306           _ComplexT __t;
1307           __real__ __t = __z.real();
1308           __imag__ __t = __z.imag();
1309           _M_value /= __t;
1310           return *this;
1311         }
1312
1313       _GLIBCXX_USE_CONSTEXPR _ComplexT __rep() const { return _M_value; }
1314
1315     private:
1316       _ComplexT _M_value;
1317     };
1318
1319   // 26.2.3  complex specializations
1320   // complex<long double> specialization
1321   template<>
1322     struct complex<long double>
1323     {
1324       typedef long double value_type;
1325       typedef __complex__ long double _ComplexT;
1326
1327       _GLIBCXX_CONSTEXPR complex(_ComplexT __z) : _M_value(__z) { }
1328
1329       _GLIBCXX_CONSTEXPR complex(long double __r = 0.0L,
1330                                  long double __i = 0.0L)
1331       : _M_value(__r + __i * 1.0Li) { }
1332
1333       _GLIBCXX_CONSTEXPR complex(const complex<float>& __z)
1334       : _M_value(__z.__rep()) { }
1335
1336       _GLIBCXX_CONSTEXPR complex(const complex<double>& __z)
1337       : _M_value(__z.__rep()) { }
1338
1339 #ifdef __GXX_EXPERIMENTAL_CXX0X__
1340       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1341       // DR 387. std::complex over-encapsulated.
1342       constexpr long double
1343       real() const { return __real__ _M_value; }
1344
1345       constexpr long double
1346       imag() const { return __imag__ _M_value; }
1347 #else
1348       long double&
1349       real() { return __real__ _M_value; }
1350
1351       const long double&
1352       real() const { return __real__ _M_value; }
1353
1354       long double&
1355       imag() { return __imag__ _M_value; }
1356
1357       const long double&
1358       imag() const { return __imag__ _M_value; }
1359 #endif
1360
1361       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1362       // DR 387. std::complex over-encapsulated.
1363       void
1364       real(long double __val) { __real__ _M_value = __val; }
1365
1366       void
1367       imag(long double __val) { __imag__ _M_value = __val; }
1368
1369       complex&
1370       operator=(long double __r)
1371       {
1372         _M_value = __r;
1373         return *this;
1374       }
1375
1376       complex&
1377       operator+=(long double __r)
1378       {
1379         _M_value += __r;
1380         return *this;
1381       }
1382
1383       complex&
1384       operator-=(long double __r)
1385       {
1386         _M_value -= __r;
1387         return *this;
1388       }
1389
1390       complex&
1391       operator*=(long double __r)
1392       {
1393         _M_value *= __r;
1394         return *this;
1395       }
1396
1397       complex&
1398       operator/=(long double __r)
1399       {
1400         _M_value /= __r;
1401         return *this;
1402       }
1403
1404       // The compiler knows how to do this efficiently
1405       // complex& operator=(const complex&);
1406
1407       template<typename _Tp>
1408         complex&
1409         operator=(const complex<_Tp>& __z)
1410         {
1411           __real__ _M_value = __z.real();
1412           __imag__ _M_value = __z.imag();
1413           return *this;
1414         }
1415
1416       template<typename _Tp>
1417         complex&
1418         operator+=(const complex<_Tp>& __z)
1419         {
1420           __real__ _M_value += __z.real();
1421           __imag__ _M_value += __z.imag();
1422           return *this;
1423         }
1424
1425       template<typename _Tp>
1426         complex&
1427         operator-=(const complex<_Tp>& __z)
1428         {
1429           __real__ _M_value -= __z.real();
1430           __imag__ _M_value -= __z.imag();
1431           return *this;
1432         }
1433
1434       template<typename _Tp>
1435         complex&
1436         operator*=(const complex<_Tp>& __z)
1437         {
1438           _ComplexT __t;
1439           __real__ __t = __z.real();
1440           __imag__ __t = __z.imag();
1441           _M_value *= __t;
1442           return *this;
1443         }
1444
1445       template<typename _Tp>
1446         complex&
1447         operator/=(const complex<_Tp>& __z)
1448         {
1449           _ComplexT __t;
1450           __real__ __t = __z.real();
1451           __imag__ __t = __z.imag();
1452           _M_value /= __t;
1453           return *this;
1454         }
1455
1456       _GLIBCXX_USE_CONSTEXPR _ComplexT __rep() const { return _M_value; }
1457
1458     private:
1459       _ComplexT _M_value;
1460     };
1461
1462   // These bits have to be at the end of this file, so that the
1463   // specializations have all been defined.
1464   inline _GLIBCXX_CONSTEXPR
1465   complex<float>::complex(const complex<double>& __z)
1466   : _M_value(__z.__rep()) { }
1467
1468   inline _GLIBCXX_CONSTEXPR
1469   complex<float>::complex(const complex<long double>& __z)
1470   : _M_value(__z.__rep()) { }
1471
1472   inline _GLIBCXX_CONSTEXPR
1473   complex<double>::complex(const complex<long double>& __z)
1474   : _M_value(__z.__rep()) { }
1475
1476   // Inhibit implicit instantiations for required instantiations,
1477   // which are defined via explicit instantiations elsewhere.
1478   // NB:  This syntax is a GNU extension.
1479 #if _GLIBCXX_EXTERN_TEMPLATE
1480   extern template istream& operator>>(istream&, complex<float>&);
1481   extern template ostream& operator<<(ostream&, const complex<float>&);
1482   extern template istream& operator>>(istream&, complex<double>&);
1483   extern template ostream& operator<<(ostream&, const complex<double>&);
1484   extern template istream& operator>>(istream&, complex<long double>&);
1485   extern template ostream& operator<<(ostream&, const complex<long double>&);
1486
1487 #ifdef _GLIBCXX_USE_WCHAR_T
1488   extern template wistream& operator>>(wistream&, complex<float>&);
1489   extern template wostream& operator<<(wostream&, const complex<float>&);
1490   extern template wistream& operator>>(wistream&, complex<double>&);
1491   extern template wostream& operator<<(wostream&, const complex<double>&);
1492   extern template wistream& operator>>(wistream&, complex<long double>&);
1493   extern template wostream& operator<<(wostream&, const complex<long double>&);
1494 #endif
1495 #endif
1496
1497   // @} group complex_numbers
1498
1499 _GLIBCXX_END_NAMESPACE_VERSION
1500 } // namespace
1501
1502 namespace __gnu_cxx _GLIBCXX_VISIBILITY(default)
1503 {
1504 _GLIBCXX_BEGIN_NAMESPACE_VERSION
1505
1506   // See ext/type_traits.h for the primary template.
1507   template<typename _Tp, typename _Up>
1508     struct __promote_2<std::complex<_Tp>, _Up>
1509     {
1510     public:
1511       typedef std::complex<typename __promote_2<_Tp, _Up>::__type> __type;
1512     };
1513
1514   template<typename _Tp, typename _Up>
1515     struct __promote_2<_Tp, std::complex<_Up> >
1516     {
1517     public:
1518       typedef std::complex<typename __promote_2<_Tp, _Up>::__type> __type;
1519     };
1520
1521   template<typename _Tp, typename _Up>
1522     struct __promote_2<std::complex<_Tp>, std::complex<_Up> >
1523     {
1524     public:
1525       typedef std::complex<typename __promote_2<_Tp, _Up>::__type> __type;
1526     };
1527
1528 _GLIBCXX_END_NAMESPACE_VERSION
1529 } // namespace
1530
1531 #ifdef __GXX_EXPERIMENTAL_CXX0X__
1532
1533 namespace std _GLIBCXX_VISIBILITY(default)
1534 {
1535 _GLIBCXX_BEGIN_NAMESPACE_VERSION
1536
1537   // Forward declarations.
1538   template<typename _Tp> std::complex<_Tp> acos(const std::complex<_Tp>&);
1539   template<typename _Tp> std::complex<_Tp> asin(const std::complex<_Tp>&);
1540   template<typename _Tp> std::complex<_Tp> atan(const std::complex<_Tp>&);
1541
1542   template<typename _Tp> std::complex<_Tp> acosh(const std::complex<_Tp>&);
1543   template<typename _Tp> std::complex<_Tp> asinh(const std::complex<_Tp>&);
1544   template<typename _Tp> std::complex<_Tp> atanh(const std::complex<_Tp>&);
1545   // DR 595.
1546   template<typename _Tp> _Tp               fabs(const std::complex<_Tp>&);
1547
1548   template<typename _Tp>
1549     inline std::complex<_Tp>
1550     __complex_acos(const std::complex<_Tp>& __z)
1551     {
1552       const std::complex<_Tp> __t = std::asin(__z);
1553       const _Tp __pi_2 = 1.5707963267948966192313216916397514L;
1554       return std::complex<_Tp>(__pi_2 - __t.real(), -__t.imag());
1555     }
1556
1557 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1558   inline __complex__ float
1559   __complex_acos(__complex__ float __z)
1560   { return __builtin_cacosf(__z); }
1561
1562   inline __complex__ double
1563   __complex_acos(__complex__ double __z)
1564   { return __builtin_cacos(__z); }
1565
1566   inline __complex__ long double
1567   __complex_acos(const __complex__ long double& __z)
1568   { return __builtin_cacosl(__z); }
1569
1570   template<typename _Tp>
1571     inline std::complex<_Tp>
1572     acos(const std::complex<_Tp>& __z)
1573     { return __complex_acos(__z.__rep()); }
1574 #else
1575   /// acos(__z) [8.1.2].
1576   //  Effects:  Behaves the same as C99 function cacos, defined
1577   //            in subclause 7.3.5.1.
1578   template<typename _Tp>
1579     inline std::complex<_Tp>
1580     acos(const std::complex<_Tp>& __z)
1581     { return __complex_acos(__z); }
1582 #endif
1583
1584   template<typename _Tp>
1585     inline std::complex<_Tp>
1586     __complex_asin(const std::complex<_Tp>& __z)
1587     {
1588       std::complex<_Tp> __t(-__z.imag(), __z.real());
1589       __t = std::asinh(__t);
1590       return std::complex<_Tp>(__t.imag(), -__t.real());
1591     }
1592
1593 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1594   inline __complex__ float
1595   __complex_asin(__complex__ float __z)
1596   { return __builtin_casinf(__z); }
1597
1598   inline __complex__ double
1599   __complex_asin(__complex__ double __z)
1600   { return __builtin_casin(__z); }
1601
1602   inline __complex__ long double
1603   __complex_asin(const __complex__ long double& __z)
1604   { return __builtin_casinl(__z); }
1605
1606   template<typename _Tp>
1607     inline std::complex<_Tp>
1608     asin(const std::complex<_Tp>& __z)
1609     { return __complex_asin(__z.__rep()); }
1610 #else
1611   /// asin(__z) [8.1.3].
1612   //  Effects:  Behaves the same as C99 function casin, defined
1613   //            in subclause 7.3.5.2.
1614   template<typename _Tp>
1615     inline std::complex<_Tp>
1616     asin(const std::complex<_Tp>& __z)
1617     { return __complex_asin(__z); }
1618 #endif
1619
1620   template<typename _Tp>
1621     std::complex<_Tp>
1622     __complex_atan(const std::complex<_Tp>& __z)
1623     {
1624       const _Tp __r2 = __z.real() * __z.real();
1625       const _Tp __x = _Tp(1.0) - __r2 - __z.imag() * __z.imag();
1626
1627       _Tp __num = __z.imag() + _Tp(1.0);
1628       _Tp __den = __z.imag() - _Tp(1.0);
1629
1630       __num = __r2 + __num * __num;
1631       __den = __r2 + __den * __den;
1632
1633       return std::complex<_Tp>(_Tp(0.5) * atan2(_Tp(2.0) * __z.real(), __x),
1634                                _Tp(0.25) * log(__num / __den));
1635     }
1636
1637 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1638   inline __complex__ float
1639   __complex_atan(__complex__ float __z)
1640   { return __builtin_catanf(__z); }
1641
1642   inline __complex__ double
1643   __complex_atan(__complex__ double __z)
1644   { return __builtin_catan(__z); }
1645
1646   inline __complex__ long double
1647   __complex_atan(const __complex__ long double& __z)
1648   { return __builtin_catanl(__z); }
1649
1650   template<typename _Tp>
1651     inline std::complex<_Tp>
1652     atan(const std::complex<_Tp>& __z)
1653     { return __complex_atan(__z.__rep()); }
1654 #else
1655   /// atan(__z) [8.1.4].
1656   //  Effects:  Behaves the same as C99 function catan, defined
1657   //            in subclause 7.3.5.3.
1658   template<typename _Tp>
1659     inline std::complex<_Tp>
1660     atan(const std::complex<_Tp>& __z)
1661     { return __complex_atan(__z); }
1662 #endif
1663
1664   template<typename _Tp>
1665     std::complex<_Tp>
1666     __complex_acosh(const std::complex<_Tp>& __z)
1667     {
1668       std::complex<_Tp> __t((__z.real() - __z.imag())
1669                             * (__z.real() + __z.imag()) - _Tp(1.0),
1670                             _Tp(2.0) * __z.real() * __z.imag());
1671       __t = std::sqrt(__t);
1672
1673       return std::log(__t + __z);
1674     }
1675
1676 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1677   inline __complex__ float
1678   __complex_acosh(__complex__ float __z)
1679   { return __builtin_cacoshf(__z); }
1680
1681   inline __complex__ double
1682   __complex_acosh(__complex__ double __z)
1683   { return __builtin_cacosh(__z); }
1684
1685   inline __complex__ long double
1686   __complex_acosh(const __complex__ long double& __z)
1687   { return __builtin_cacoshl(__z); }
1688
1689   template<typename _Tp>
1690     inline std::complex<_Tp>
1691     acosh(const std::complex<_Tp>& __z)
1692     { return __complex_acosh(__z.__rep()); }
1693 #else
1694   /// acosh(__z) [8.1.5].
1695   //  Effects:  Behaves the same as C99 function cacosh, defined
1696   //            in subclause 7.3.6.1.
1697   template<typename _Tp>
1698     inline std::complex<_Tp>
1699     acosh(const std::complex<_Tp>& __z)
1700     { return __complex_acosh(__z); }
1701 #endif
1702
1703   template<typename _Tp>
1704     std::complex<_Tp>
1705     __complex_asinh(const std::complex<_Tp>& __z)
1706     {
1707       std::complex<_Tp> __t((__z.real() - __z.imag())
1708                             * (__z.real() + __z.imag()) + _Tp(1.0),
1709                             _Tp(2.0) * __z.real() * __z.imag());
1710       __t = std::sqrt(__t);
1711
1712       return std::log(__t + __z);
1713     }
1714
1715 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1716   inline __complex__ float
1717   __complex_asinh(__complex__ float __z)
1718   { return __builtin_casinhf(__z); }
1719
1720   inline __complex__ double
1721   __complex_asinh(__complex__ double __z)
1722   { return __builtin_casinh(__z); }
1723
1724   inline __complex__ long double
1725   __complex_asinh(const __complex__ long double& __z)
1726   { return __builtin_casinhl(__z); }
1727
1728   template<typename _Tp>
1729     inline std::complex<_Tp>
1730     asinh(const std::complex<_Tp>& __z)
1731     { return __complex_asinh(__z.__rep()); }
1732 #else
1733   /// asinh(__z) [8.1.6].
1734   //  Effects:  Behaves the same as C99 function casin, defined
1735   //            in subclause 7.3.6.2.
1736   template<typename _Tp>
1737     inline std::complex<_Tp>
1738     asinh(const std::complex<_Tp>& __z)
1739     { return __complex_asinh(__z); }
1740 #endif
1741
1742   template<typename _Tp>
1743     std::complex<_Tp>
1744     __complex_atanh(const std::complex<_Tp>& __z)
1745     {
1746       const _Tp __i2 = __z.imag() * __z.imag();
1747       const _Tp __x = _Tp(1.0) - __i2 - __z.real() * __z.real();
1748
1749       _Tp __num = _Tp(1.0) + __z.real();
1750       _Tp __den = _Tp(1.0) - __z.real();
1751
1752       __num = __i2 + __num * __num;
1753       __den = __i2 + __den * __den;
1754
1755       return std::complex<_Tp>(_Tp(0.25) * (log(__num) - log(__den)),
1756                                _Tp(0.5) * atan2(_Tp(2.0) * __z.imag(), __x));
1757     }
1758
1759 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1760   inline __complex__ float
1761   __complex_atanh(__complex__ float __z)
1762   { return __builtin_catanhf(__z); }
1763
1764   inline __complex__ double
1765   __complex_atanh(__complex__ double __z)
1766   { return __builtin_catanh(__z); }
1767
1768   inline __complex__ long double
1769   __complex_atanh(const __complex__ long double& __z)
1770   { return __builtin_catanhl(__z); }
1771
1772   template<typename _Tp>
1773     inline std::complex<_Tp>
1774     atanh(const std::complex<_Tp>& __z)
1775     { return __complex_atanh(__z.__rep()); }
1776 #else
1777   /// atanh(__z) [8.1.7].
1778   //  Effects:  Behaves the same as C99 function catanh, defined
1779   //            in subclause 7.3.6.3.
1780   template<typename _Tp>
1781     inline std::complex<_Tp>
1782     atanh(const std::complex<_Tp>& __z)
1783     { return __complex_atanh(__z); }
1784 #endif
1785
1786   template<typename _Tp>
1787     inline _Tp
1788     /// fabs(__z) [8.1.8].
1789     //  Effects:  Behaves the same as C99 function cabs, defined
1790     //            in subclause 7.3.8.1.
1791     fabs(const std::complex<_Tp>& __z)
1792     { return std::abs(__z); }
1793
1794   /// Additional overloads [8.1.9].
1795   template<typename _Tp>
1796     inline typename __gnu_cxx::__promote<_Tp>::__type
1797     arg(_Tp __x)
1798     {
1799       typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
1800 #if (_GLIBCXX_USE_C99_MATH && !_GLIBCXX_USE_C99_FP_MACROS_DYNAMIC)
1801       return std::signbit(__x) ? __type(3.1415926535897932384626433832795029L)
1802                                : __type();
1803 #else
1804       return std::arg(std::complex<__type>(__x));
1805 #endif
1806     }
1807
1808   template<typename _Tp>
1809     inline typename __gnu_cxx::__promote<_Tp>::__type
1810     imag(_Tp)
1811     { return _Tp(); }
1812
1813   template<typename _Tp>
1814     inline typename __gnu_cxx::__promote<_Tp>::__type
1815     norm(_Tp __x)
1816     {
1817       typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
1818       return __type(__x) * __type(__x);
1819     }
1820
1821   template<typename _Tp>
1822     inline typename __gnu_cxx::__promote<_Tp>::__type
1823     real(_Tp __x)
1824     { return __x; }
1825
1826   template<typename _Tp, typename _Up>
1827     inline std::complex<typename __gnu_cxx::__promote_2<_Tp, _Up>::__type>
1828     pow(const std::complex<_Tp>& __x, const _Up& __y)
1829     {
1830       typedef typename __gnu_cxx::__promote_2<_Tp, _Up>::__type __type;
1831       return std::pow(std::complex<__type>(__x), __type(__y));
1832     }
1833
1834   template<typename _Tp, typename _Up>
1835     inline std::complex<typename __gnu_cxx::__promote_2<_Tp, _Up>::__type>
1836     pow(const _Tp& __x, const std::complex<_Up>& __y)
1837     {
1838       typedef typename __gnu_cxx::__promote_2<_Tp, _Up>::__type __type;
1839       return std::pow(__type(__x), std::complex<__type>(__y));
1840     }
1841
1842   template<typename _Tp, typename _Up>
1843     inline std::complex<typename __gnu_cxx::__promote_2<_Tp, _Up>::__type>
1844     pow(const std::complex<_Tp>& __x, const std::complex<_Up>& __y)
1845     {
1846       typedef typename __gnu_cxx::__promote_2<_Tp, _Up>::__type __type;
1847       return std::pow(std::complex<__type>(__x),
1848                       std::complex<__type>(__y));
1849     }
1850
1851   // Forward declarations.
1852   // DR 781.
1853   template<typename _Tp> std::complex<_Tp> proj(const std::complex<_Tp>&);
1854
1855   template<typename _Tp>
1856     std::complex<_Tp>
1857     __complex_proj(const std::complex<_Tp>& __z)
1858     {
1859       const _Tp __den = (__z.real() * __z.real()
1860                          + __z.imag() * __z.imag() + _Tp(1.0));
1861
1862       return std::complex<_Tp>((_Tp(2.0) * __z.real()) / __den,
1863                                (_Tp(2.0) * __z.imag()) / __den);
1864     }
1865
1866 #if _GLIBCXX_USE_C99_COMPLEX
1867   inline __complex__ float
1868   __complex_proj(__complex__ float __z)
1869   { return __builtin_cprojf(__z); }
1870
1871   inline __complex__ double
1872   __complex_proj(__complex__ double __z)
1873   { return __builtin_cproj(__z); }
1874
1875   inline __complex__ long double
1876   __complex_proj(const __complex__ long double& __z)
1877   { return __builtin_cprojl(__z); }
1878
1879   template<typename _Tp>
1880     inline std::complex<_Tp>
1881     proj(const std::complex<_Tp>& __z)
1882     { return __complex_proj(__z.__rep()); }
1883 #else
1884   template<typename _Tp>
1885     inline std::complex<_Tp>
1886     proj(const std::complex<_Tp>& __z)
1887     { return __complex_proj(__z); }
1888 #endif
1889
1890   // DR 1137.
1891   template<typename _Tp>
1892     inline typename __gnu_cxx::__promote<_Tp>::__type
1893     proj(_Tp __x)
1894     { return __x; }
1895
1896   template<typename _Tp>
1897     inline typename __gnu_cxx::__promote<_Tp>::__type
1898     conj(_Tp __x)
1899     { return __x; }
1900
1901 _GLIBCXX_END_NAMESPACE_VERSION
1902 } // namespace
1903
1904 #endif  // __GXX_EXPERIMENTAL_CXX0X__
1905
1906 #endif  /* _GLIBCXX_COMPLEX */