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, 2011
   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 #if __cplusplus < 201103L
  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 #if __cplusplus >= 201103L
 142       // _GLIBCXX_RESOLVE_LIB_DEFECTS
 143       // DR 387. std::complex over-encapsulated.
 144       __attribute ((__abi_tag__ ("cxx11")))
 145       constexpr _Tp
 146       real() { return _M_real; }
 147
 148       __attribute ((__abi_tag__ ("cxx11")))
 149       constexpr _Tp
 150       imag() { return _M_imag; }
 151 #else
 152       ///  Return real part of complex number.
 153       _Tp&
 154       real() { return _M_real; }
 155
 156       ///  Return real part of complex number.
 157       const _Tp&
 158       real() const { return _M_real; }
 159
 160       ///  Return imaginary part of complex number.
 161       _Tp&
 162       imag() { return _M_imag; }
 163
 164       ///  Return imaginary part of complex number.
 165       const _Tp&
 166       imag() const { return _M_imag; }
 167 #endif
 168
 169       // _GLIBCXX_RESOLVE_LIB_DEFECTS
 170       // DR 387. std::complex over-encapsulated.
 171       void
 172       real(_Tp __val) { _M_real = __val; }
 173
 174       void
 175       imag(_Tp __val) { _M_imag = __val; }
 176
 177       /// Assign this complex number to scalar @a t.
 178       complex<_Tp>& operator=(const _Tp&);
 179
 180       /// Add @a t to this complex number.
 181       // 26.2.5/1
 182       complex<_Tp>&
 183       operator+=(const _Tp& __t)
 184       {
 185         _M_real += __t;
 186         return *this;
 187       }
 188
 189       /// Subtract @a t from this complex number.
 190       // 26.2.5/3
 191       complex<_Tp>&
 192       operator-=(const _Tp& __t)
 193       {
 194         _M_real -= __t;
 195         return *this;
 196       }
 197
 198       /// Multiply this complex number by @a t.
 199       complex<_Tp>& operator*=(const _Tp&);
 200       /// Divide this complex number by @a t.
 201       complex<_Tp>& operator/=(const _Tp&);
 202
 203       // Lets the compiler synthesize the
 204       // copy and assignment operator
 205       // complex<_Tp>& operator= (const complex<_Tp>&);
 206       /// Assign this complex number to complex @a z.
 207       template<typename _Up>
 208         complex<_Tp>& operator=(const complex<_Up>&);
 209       /// Add @a z to this complex number.
 210       template<typename _Up>
 211         complex<_Tp>& operator+=(const complex<_Up>&);
 212       /// Subtract @a z from this complex number.
 213       template<typename _Up>
 214         complex<_Tp>& operator-=(const complex<_Up>&);
 215       /// Multiply this complex number by @a z.
 216       template<typename _Up>
 217         complex<_Tp>& operator*=(const complex<_Up>&);
 218       /// Divide this complex number by @a z.
 219       template<typename _Up>
 220         complex<_Tp>& operator/=(const complex<_Up>&);
 221
 222       _GLIBCXX_USE_CONSTEXPR complex __rep() const
 223       { return *this; }
 224
 225     private:
 226       _Tp _M_real;
 227       _Tp _M_imag;
 228     };
 229
 230   template<typename _Tp>
 231     complex<_Tp>&
 232     complex<_Tp>::operator=(const _Tp& __t)
 233     {
 234      _M_real = __t;
 235      _M_imag = _Tp();
 236      return *this;
 237     }
 238
 239   // 26.2.5/5
 240   template<typename _Tp>
 241     complex<_Tp>&
 242     complex<_Tp>::operator*=(const _Tp& __t)
 243     {
 244       _M_real *= __t;
 245       _M_imag *= __t;
 246       return *this;
 247     }
 248
 249   // 26.2.5/7
 250   template<typename _Tp>
 251     complex<_Tp>&
 252     complex<_Tp>::operator/=(const _Tp& __t)
 253     {
 254       _M_real /= __t;
 255       _M_imag /= __t;
 256       return *this;
 257     }
 258
 259   template<typename _Tp>
 260     template<typename _Up>
 261     complex<_Tp>&
 262     complex<_Tp>::operator=(const complex<_Up>& __z)
 263     {
 264       _M_real = __z.real();
 265       _M_imag = __z.imag();
 266       return *this;
 267     }
 268
 269   // 26.2.5/9
 270   template<typename _Tp>
 271     template<typename _Up>
 272     complex<_Tp>&
 273     complex<_Tp>::operator+=(const complex<_Up>& __z)
 274     {
 275       _M_real += __z.real();
 276       _M_imag += __z.imag();
 277       return *this;
 278     }
 279
 280   // 26.2.5/11
 281   template<typename _Tp>
 282     template<typename _Up>
 283     complex<_Tp>&
 284     complex<_Tp>::operator-=(const complex<_Up>& __z)
 285     {
 286       _M_real -= __z.real();
 287       _M_imag -= __z.imag();
 288       return *this;
 289     }
 290
 291   // 26.2.5/13
 292   // XXX: This is a grammar school implementation.
 293   template<typename _Tp>
 294     template<typename _Up>
 295     complex<_Tp>&
 296     complex<_Tp>::operator*=(const complex<_Up>& __z)
 297     {
 298       const _Tp __r = _M_real * __z.real() - _M_imag * __z.imag();
 299       _M_imag = _M_real * __z.imag() + _M_imag * __z.real();
 300       _M_real = __r;
 301       return *this;
 302     }
 303
 304   // 26.2.5/15
 305   // XXX: This is a grammar school implementation.
 306   template<typename _Tp>
 307     template<typename _Up>
 308     complex<_Tp>&
 309     complex<_Tp>::operator/=(const complex<_Up>& __z)
 310     {
 311       const _Tp __r =  _M_real * __z.real() + _M_imag * __z.imag();
 312       const _Tp __n = std::norm(__z);
 313       _M_imag = (_M_imag * __z.real() - _M_real * __z.imag()) / __n;
 314       _M_real = __r / __n;
 315       return *this;
 316     }
 317
 318   // Operators:
 319   //@{
 320   ///  Return new complex value @a x plus @a y.
 321   template<typename _Tp>
 322     inline complex<_Tp>
 323     operator+(const complex<_Tp>& __x, const complex<_Tp>& __y)
 324     {
 325       complex<_Tp> __r = __x;
 326       __r += __y;
 327       return __r;
 328     }
 329
 330   template<typename _Tp>
 331     inline complex<_Tp>
 332     operator+(const complex<_Tp>& __x, const _Tp& __y)
 333     {
 334       complex<_Tp> __r = __x;
 335       __r += __y;
 336       return __r;
 337     }
 338
 339   template<typename _Tp>
 340     inline complex<_Tp>
 341     operator+(const _Tp& __x, const complex<_Tp>& __y)
 342     {
 343       complex<_Tp> __r = __y;
 344       __r += __x;
 345       return __r;
 346     }
 347   //@}
 348
 349   //@{
 350   ///  Return new complex value @a x minus @a y.
 351   template<typename _Tp>
 352     inline complex<_Tp>
 353     operator-(const complex<_Tp>& __x, const complex<_Tp>& __y)
 354     {
 355       complex<_Tp> __r = __x;
 356       __r -= __y;
 357       return __r;
 358     }
 359
 360   template<typename _Tp>
 361     inline complex<_Tp>
 362     operator-(const complex<_Tp>& __x, const _Tp& __y)
 363     {
 364       complex<_Tp> __r = __x;
 365       __r -= __y;
 366       return __r;
 367     }
 368
 369   template<typename _Tp>
 370     inline complex<_Tp>
 371     operator-(const _Tp& __x, const complex<_Tp>& __y)
 372     {
 373       complex<_Tp> __r(__x, -__y.imag());
 374       __r -= __y.real();
 375       return __r;
 376     }
 377   //@}
 378
 379   //@{
 380   ///  Return new complex value @a x times @a y.
 381   template<typename _Tp>
 382     inline complex<_Tp>
 383     operator*(const complex<_Tp>& __x, const complex<_Tp>& __y)
 384     {
 385       complex<_Tp> __r = __x;
 386       __r *= __y;
 387       return __r;
 388     }
 389
 390   template<typename _Tp>
 391     inline complex<_Tp>
 392     operator*(const complex<_Tp>& __x, const _Tp& __y)
 393     {
 394       complex<_Tp> __r = __x;
 395       __r *= __y;
 396       return __r;
 397     }
 398
 399   template<typename _Tp>
 400     inline complex<_Tp>
 401     operator*(const _Tp& __x, const complex<_Tp>& __y)
 402     {
 403       complex<_Tp> __r = __y;
 404       __r *= __x;
 405       return __r;
 406     }
 407   //@}
 408
 409   //@{
 410   ///  Return new complex value @a x divided by @a y.
 411   template<typename _Tp>
 412     inline complex<_Tp>
 413     operator/(const complex<_Tp>& __x, const complex<_Tp>& __y)
 414     {
 415       complex<_Tp> __r = __x;
 416       __r /= __y;
 417       return __r;
 418     }
 419
 420   template<typename _Tp>
 421     inline complex<_Tp>
 422     operator/(const complex<_Tp>& __x, const _Tp& __y)
 423     {
 424       complex<_Tp> __r = __x;
 425       __r /= __y;
 426       return __r;
 427     }
 428
 429   template<typename _Tp>
 430     inline complex<_Tp>
 431     operator/(const _Tp& __x, const complex<_Tp>& __y)
 432     {
 433       complex<_Tp> __r = __x;
 434       __r /= __y;
 435       return __r;
 436     }
 437   //@}
 438
 439   ///  Return @a x.
 440   template<typename _Tp>
 441     inline complex<_Tp>
 442     operator+(const complex<_Tp>& __x)
 443     { return __x; }
 444
 445   ///  Return complex negation of @a x.
 446   template<typename _Tp>
 447     inline complex<_Tp>
 448     operator-(const complex<_Tp>& __x)
 449     {  return complex<_Tp>(-__x.real(), -__x.imag()); }
 450
 451   //@{
 452   ///  Return true if @a x is equal to @a y.
 453   template<typename _Tp>
 454     inline _GLIBCXX_CONSTEXPR bool
 455     operator==(const complex<_Tp>& __x, const complex<_Tp>& __y)
 456     { return __x.real() == __y.real() && __x.imag() == __y.imag(); }
 457
 458   template<typename _Tp>
 459     inline _GLIBCXX_CONSTEXPR bool
 460     operator==(const complex<_Tp>& __x, const _Tp& __y)
 461     { return __x.real() == __y && __x.imag() == _Tp(); }
 462
 463   template<typename _Tp>
 464     inline _GLIBCXX_CONSTEXPR bool
 465     operator==(const _Tp& __x, const complex<_Tp>& __y)
 466     { return __x == __y.real() && _Tp() == __y.imag(); }
 467   //@}
 468
 469   //@{
 470   ///  Return false if @a x is equal to @a y.
 471   template<typename _Tp>
 472     inline _GLIBCXX_CONSTEXPR bool
 473     operator!=(const complex<_Tp>& __x, const complex<_Tp>& __y)
 474     { return __x.real() != __y.real() || __x.imag() != __y.imag(); }
 475
 476   template<typename _Tp>
 477     inline _GLIBCXX_CONSTEXPR bool
 478     operator!=(const complex<_Tp>& __x, const _Tp& __y)
 479     { return __x.real() != __y || __x.imag() != _Tp(); }
 480
 481   template<typename _Tp>
 482     inline _GLIBCXX_CONSTEXPR bool
 483     operator!=(const _Tp& __x, const complex<_Tp>& __y)
 484     { return __x != __y.real() || _Tp() != __y.imag(); }
 485   //@}
 486
 487   ///  Extraction operator for complex values.
 488   template<typename _Tp, typename _CharT, class _Traits>
 489     basic_istream<_CharT, _Traits>&
 490     operator>>(basic_istream<_CharT, _Traits>& __is, complex<_Tp>& __x)
 491     {
 492       _Tp __re_x, __im_x;
 493       _CharT __ch;
 494       __is >> __ch;
 495       if (__ch == '(')
 496         {
 497           __is >> __re_x >> __ch;
 498           if (__ch == ',')
 499             {
 500               __is >> __im_x >> __ch;
 501               if (__ch == ')')
 502                 __x = complex<_Tp>(__re_x, __im_x);
 503               else
 504                 __is.setstate(ios_base::failbit);
 505             }
 506           else if (__ch == ')')
 507             __x = __re_x;
 508           else
 509             __is.setstate(ios_base::failbit);
 510         }
 511       else
 512         {
 513           __is.putback(__ch);
 514           __is >> __re_x;
 515           __x = __re_x;
 516         }
 517       return __is;
 518     }
 519
 520   ///  Insertion operator for complex values.
 521   template<typename _Tp, typename _CharT, class _Traits>
 522     basic_ostream<_CharT, _Traits>&
 523     operator<<(basic_ostream<_CharT, _Traits>& __os, const complex<_Tp>& __x)
 524     {
 525       basic_ostringstream<_CharT, _Traits> __s;
 526       __s.flags(__os.flags());
 527       __s.imbue(__os.getloc());
 528       __s.precision(__os.precision());
 529       __s << '(' << __x.real() << ',' << __x.imag() << ')';
 530       return __os << __s.str();
 531     }
 532
 533   // Values
 534 #if __cplusplus >= 201103L
 535   template<typename _Tp>
 536     constexpr _Tp
 537     real(const complex<_Tp>& __z)
 538     { return __z.real(); }
 539
 540   template<typename _Tp>
 541     constexpr _Tp
 542     imag(const complex<_Tp>& __z)
 543     { return __z.imag(); }
 544 #else
 545   template<typename _Tp>
 546     inline _Tp&
 547     real(complex<_Tp>& __z)
 548     { return __z.real(); }
 549
 550   template<typename _Tp>
 551     inline const _Tp&
 552     real(const complex<_Tp>& __z)
 553     { return __z.real(); }
 554
 555   template<typename _Tp>
 556     inline _Tp&
 557     imag(complex<_Tp>& __z)
 558     { return __z.imag(); }
 559
 560   template<typename _Tp>
 561     inline const _Tp&
 562     imag(const complex<_Tp>& __z)
 563     { return __z.imag(); }
 564 #endif
 565
 566   // 26.2.7/3 abs(__z):  Returns the magnitude of __z.
 567   template<typename _Tp>
 568     inline _Tp
 569     __complex_abs(const complex<_Tp>& __z)
 570     {
 571       _Tp __x = __z.real();
 572       _Tp __y = __z.imag();
 573       const _Tp __s = std::max(abs(__x), abs(__y));
 574       if (__s == _Tp())  // well ...
 575         return __s;
 576       __x /= __s;
 577       __y /= __s;
 578       return __s * sqrt(__x * __x + __y * __y);
 579     }
 580
 581 #if _GLIBCXX_USE_C99_COMPLEX
 582   inline float
 583   __complex_abs(__complex__ float __z) { return __builtin_cabsf(__z); }
 584
 585   inline double
 586   __complex_abs(__complex__ double __z) { return __builtin_cabs(__z); }
 587
 588   inline long double
 589   __complex_abs(const __complex__ long double& __z)
 590   { return __builtin_cabsl(__z); }
 591
 592   template<typename _Tp>
 593     inline _Tp
 594     abs(const complex<_Tp>& __z) { return __complex_abs(__z.__rep()); }
 595 #else
 596   template<typename _Tp>
 597     inline _Tp
 598     abs(const complex<_Tp>& __z) { return __complex_abs(__z); }
 599 #endif
 600
 601
 602   // 26.2.7/4: arg(__z): Returns the phase angle of __z.
 603   template<typename _Tp>
 604     inline _Tp
 605     __complex_arg(const complex<_Tp>& __z)
 606     { return  atan2(__z.imag(), __z.real()); }
 607
 608 #if _GLIBCXX_USE_C99_COMPLEX
 609   inline float
 610   __complex_arg(__complex__ float __z) { return __builtin_cargf(__z); }
 611
 612   inline double
 613   __complex_arg(__complex__ double __z) { return __builtin_carg(__z); }
 614
 615   inline long double
 616   __complex_arg(const __complex__ long double& __z)
 617   { return __builtin_cargl(__z); }
 618
 619   template<typename _Tp>
 620     inline _Tp
 621     arg(const complex<_Tp>& __z) { return __complex_arg(__z.__rep()); }
 622 #else
 623   template<typename _Tp>
 624     inline _Tp
 625     arg(const complex<_Tp>& __z) { return __complex_arg(__z); }
 626 #endif
 627
 628   // 26.2.7/5: norm(__z) returns the squared magnitude of __z.
 629   //     As defined, norm() is -not- a norm is the common mathematical
 630   //     sens used in numerics.  The helper class _Norm_helper<> tries to
 631   //     distinguish between builtin floating point and the rest, so as
 632   //     to deliver an answer as close as possible to the real value.
 633   template<bool>
 634     struct _Norm_helper
 635     {
 636       template<typename _Tp>
 637         static inline _Tp _S_do_it(const complex<_Tp>& __z)
 638         {
 639           const _Tp __x = __z.real();
 640           const _Tp __y = __z.imag();
 641           return __x * __x + __y * __y;
 642         }
 643     };
 644
 645   template<>
 646     struct _Norm_helper<true>
 647     {
 648       template<typename _Tp>
 649         static inline _Tp _S_do_it(const complex<_Tp>& __z)
 650         {
 651           _Tp __res = std::abs(__z);
 652           return __res * __res;
 653         }
 654     };
 655
 656   template<typename _Tp>
 657     inline _Tp
 658     norm(const complex<_Tp>& __z)
 659     {
 660       return _Norm_helper<__is_floating<_Tp>::__value
 661         && !_GLIBCXX_FAST_MATH>::_S_do_it(__z);
 662     }
 663
 664   template<typename _Tp>
 665     inline complex<_Tp>
 666     polar(const _Tp& __rho, const _Tp& __theta)
 667     { return complex<_Tp>(__rho * cos(__theta), __rho * sin(__theta)); }
 668
 669   template<typename _Tp>
 670     inline complex<_Tp>
 671     conj(const complex<_Tp>& __z)
 672     { return complex<_Tp>(__z.real(), -__z.imag()); }
 673
 674   // Transcendentals
 675
 676   // 26.2.8/1 cos(__z):  Returns the cosine of __z.
 677   template<typename _Tp>
 678     inline complex<_Tp>
 679     __complex_cos(const complex<_Tp>& __z)
 680     {
 681       const _Tp __x = __z.real();
 682       const _Tp __y = __z.imag();
 683       return complex<_Tp>(cos(__x) * cosh(__y), -sin(__x) * sinh(__y));
 684     }
 685
 686 #if _GLIBCXX_USE_C99_COMPLEX
 687   inline __complex__ float
 688   __complex_cos(__complex__ float __z) { return __builtin_ccosf(__z); }
 689
 690   inline __complex__ double
 691   __complex_cos(__complex__ double __z) { return __builtin_ccos(__z); }
 692
 693   inline __complex__ long double
 694   __complex_cos(const __complex__ long double& __z)
 695   { return __builtin_ccosl(__z); }
 696
 697   template<typename _Tp>
 698     inline complex<_Tp>
 699     cos(const complex<_Tp>& __z) { return __complex_cos(__z.__rep()); }
 700 #else
 701   template<typename _Tp>
 702     inline complex<_Tp>
 703     cos(const complex<_Tp>& __z) { return __complex_cos(__z); }
 704 #endif
 705
 706   // 26.2.8/2 cosh(__z): Returns the hyperbolic cosine of __z.
 707   template<typename _Tp>
 708     inline complex<_Tp>
 709     __complex_cosh(const complex<_Tp>& __z)
 710     {
 711       const _Tp __x = __z.real();
 712       const _Tp __y = __z.imag();
 713       return complex<_Tp>(cosh(__x) * cos(__y), sinh(__x) * sin(__y));
 714     }
 715
 716 #if _GLIBCXX_USE_C99_COMPLEX
 717   inline __complex__ float
 718   __complex_cosh(__complex__ float __z) { return __builtin_ccoshf(__z); }
 719
 720   inline __complex__ double
 721   __complex_cosh(__complex__ double __z) { return __builtin_ccosh(__z); }
 722
 723   inline __complex__ long double
 724   __complex_cosh(const __complex__ long double& __z)
 725   { return __builtin_ccoshl(__z); }
 726
 727   template<typename _Tp>
 728     inline complex<_Tp>
 729     cosh(const complex<_Tp>& __z) { return __complex_cosh(__z.__rep()); }
 730 #else
 731   template<typename _Tp>
 732     inline complex<_Tp>
 733     cosh(const complex<_Tp>& __z) { return __complex_cosh(__z); }
 734 #endif
 735
 736   // 26.2.8/3 exp(__z): Returns the complex base e exponential of x
 737   template<typename _Tp>
 738     inline complex<_Tp>
 739     __complex_exp(const complex<_Tp>& __z)
 740     { return std::polar(exp(__z.real()), __z.imag()); }
 741
 742 #if _GLIBCXX_USE_C99_COMPLEX
 743   inline __complex__ float
 744   __complex_exp(__complex__ float __z) { return __builtin_cexpf(__z); }
 745
 746   inline __complex__ double
 747   __complex_exp(__complex__ double __z) { return __builtin_cexp(__z); }
 748
 749   inline __complex__ long double
 750   __complex_exp(const __complex__ long double& __z)
 751   { return __builtin_cexpl(__z); }
 752
 753   template<typename _Tp>
 754     inline complex<_Tp>
 755     exp(const complex<_Tp>& __z) { return __complex_exp(__z.__rep()); }
 756 #else
 757   template<typename _Tp>
 758     inline complex<_Tp>
 759     exp(const complex<_Tp>& __z) { return __complex_exp(__z); }
 760 #endif
 761
 762   // 26.2.8/5 log(__z): Returns the natural complex logarithm of __z.
 763   //                    The branch cut is along the negative axis.
 764   template<typename _Tp>
 765     inline complex<_Tp>
 766     __complex_log(const complex<_Tp>& __z)
 767     { return complex<_Tp>(log(std::abs(__z)), std::arg(__z)); }
 768
 769 #if _GLIBCXX_USE_C99_COMPLEX
 770   inline __complex__ float
 771   __complex_log(__complex__ float __z) { return __builtin_clogf(__z); }
 772
 773   inline __complex__ double
 774   __complex_log(__complex__ double __z) { return __builtin_clog(__z); }
 775
 776   inline __complex__ long double
 777   __complex_log(const __complex__ long double& __z)
 778   { return __builtin_clogl(__z); }
 779
 780   template<typename _Tp>
 781     inline complex<_Tp>
 782     log(const complex<_Tp>& __z) { return __complex_log(__z.__rep()); }
 783 #else
 784   template<typename _Tp>
 785     inline complex<_Tp>
 786     log(const complex<_Tp>& __z) { return __complex_log(__z); }
 787 #endif
 788
 789   template<typename _Tp>
 790     inline complex<_Tp>
 791     log10(const complex<_Tp>& __z)
 792     { return std::log(__z) / log(_Tp(10.0)); }
 793
 794   // 26.2.8/10 sin(__z): Returns the sine of __z.
 795   template<typename _Tp>
 796     inline complex<_Tp>
 797     __complex_sin(const complex<_Tp>& __z)
 798     {
 799       const _Tp __x = __z.real();
 800       const _Tp __y = __z.imag();
 801       return complex<_Tp>(sin(__x) * cosh(__y), cos(__x) * sinh(__y));
 802     }
 803
 804 #if _GLIBCXX_USE_C99_COMPLEX
 805   inline __complex__ float
 806   __complex_sin(__complex__ float __z) { return __builtin_csinf(__z); }
 807
 808   inline __complex__ double
 809   __complex_sin(__complex__ double __z) { return __builtin_csin(__z); }
 810
 811   inline __complex__ long double
 812   __complex_sin(const __complex__ long double& __z)
 813   { return __builtin_csinl(__z); }
 814
 815   template<typename _Tp>
 816     inline complex<_Tp>
 817     sin(const complex<_Tp>& __z) { return __complex_sin(__z.__rep()); }
 818 #else
 819   template<typename _Tp>
 820     inline complex<_Tp>
 821     sin(const complex<_Tp>& __z) { return __complex_sin(__z); }
 822 #endif
 823
 824   // 26.2.8/11 sinh(__z): Returns the hyperbolic sine of __z.
 825   template<typename _Tp>
 826     inline complex<_Tp>
 827     __complex_sinh(const complex<_Tp>& __z)
 828     {
 829       const _Tp __x = __z.real();
 830       const _Tp  __y = __z.imag();
 831       return complex<_Tp>(sinh(__x) * cos(__y), cosh(__x) * sin(__y));
 832     }
 833
 834 #if _GLIBCXX_USE_C99_COMPLEX
 835   inline __complex__ float
 836   __complex_sinh(__complex__ float __z) { return __builtin_csinhf(__z); }
 837
 838   inline __complex__ double
 839   __complex_sinh(__complex__ double __z) { return __builtin_csinh(__z); }
 840
 841   inline __complex__ long double
 842   __complex_sinh(const __complex__ long double& __z)
 843   { return __builtin_csinhl(__z); }
 844
 845   template<typename _Tp>
 846     inline complex<_Tp>
 847     sinh(const complex<_Tp>& __z) { return __complex_sinh(__z.__rep()); }
 848 #else
 849   template<typename _Tp>
 850     inline complex<_Tp>
 851     sinh(const complex<_Tp>& __z) { return __complex_sinh(__z); }
 852 #endif
 853
 854   // 26.2.8/13 sqrt(__z): Returns the complex square root of __z.
 855   //                     The branch cut is on the negative axis.
 856   template<typename _Tp>
 857     complex<_Tp>
 858     __complex_sqrt(const complex<_Tp>& __z)
 859     {
 860       _Tp __x = __z.real();
 861       _Tp __y = __z.imag();
 862
 863       if (__x == _Tp())
 864         {
 865           _Tp __t = sqrt(abs(__y) / 2);
 866           return complex<_Tp>(__t, __y < _Tp() ? -__t : __t);
 867         }
 868       else
 869         {
 870           _Tp __t = sqrt(2 * (std::abs(__z) + abs(__x)));
 871           _Tp __u = __t / 2;
 872           return __x > _Tp()
 873             ? complex<_Tp>(__u, __y / __t)
 874             : complex<_Tp>(abs(__y) / __t, __y < _Tp() ? -__u : __u);
 875         }
 876     }
 877
 878 #if _GLIBCXX_USE_C99_COMPLEX
 879   inline __complex__ float
 880   __complex_sqrt(__complex__ float __z) { return __builtin_csqrtf(__z); }
 881
 882   inline __complex__ double
 883   __complex_sqrt(__complex__ double __z) { return __builtin_csqrt(__z); }
 884
 885   inline __complex__ long double
 886   __complex_sqrt(const __complex__ long double& __z)
 887   { return __builtin_csqrtl(__z); }
 888
 889   template<typename _Tp>
 890     inline complex<_Tp>
 891     sqrt(const complex<_Tp>& __z) { return __complex_sqrt(__z.__rep()); }
 892 #else
 893   template<typename _Tp>
 894     inline complex<_Tp>
 895     sqrt(const complex<_Tp>& __z) { return __complex_sqrt(__z); }
 896 #endif
 897
 898   // 26.2.8/14 tan(__z):  Return the complex tangent of __z.
 899
 900   template<typename _Tp>
 901     inline complex<_Tp>
 902     __complex_tan(const complex<_Tp>& __z)
 903     { return std::sin(__z) / std::cos(__z); }
 904
 905 #if _GLIBCXX_USE_C99_COMPLEX
 906   inline __complex__ float
 907   __complex_tan(__complex__ float __z) { return __builtin_ctanf(__z); }
 908
 909   inline __complex__ double
 910   __complex_tan(__complex__ double __z) { return __builtin_ctan(__z); }
 911
 912   inline __complex__ long double
 913   __complex_tan(const __complex__ long double& __z)
 914   { return __builtin_ctanl(__z); }
 915
 916   template<typename _Tp>
 917     inline complex<_Tp>
 918     tan(const complex<_Tp>& __z) { return __complex_tan(__z.__rep()); }
 919 #else
 920   template<typename _Tp>
 921     inline complex<_Tp>
 922     tan(const complex<_Tp>& __z) { return __complex_tan(__z); }
 923 #endif
 924
 925
 926   // 26.2.8/15 tanh(__z):  Returns the hyperbolic tangent of __z.
 927
 928   template<typename _Tp>
 929     inline complex<_Tp>
 930     __complex_tanh(const complex<_Tp>& __z)
 931     { return std::sinh(__z) / std::cosh(__z); }
 932
 933 #if _GLIBCXX_USE_C99_COMPLEX
 934   inline __complex__ float
 935   __complex_tanh(__complex__ float __z) { return __builtin_ctanhf(__z); }
 936
 937   inline __complex__ double
 938   __complex_tanh(__complex__ double __z) { return __builtin_ctanh(__z); }
 939
 940   inline __complex__ long double
 941   __complex_tanh(const __complex__ long double& __z)
 942   { return __builtin_ctanhl(__z); }
 943
 944   template<typename _Tp>
 945     inline complex<_Tp>
 946     tanh(const complex<_Tp>& __z) { return __complex_tanh(__z.__rep()); }
 947 #else
 948   template<typename _Tp>
 949     inline complex<_Tp>
 950     tanh(const complex<_Tp>& __z) { return __complex_tanh(__z); }
 951 #endif
 952
 953
 954   // 26.2.8/9  pow(__x, __y): Returns the complex power base of __x
 955   //                          raised to the __y-th power.  The branch
 956   //                          cut is on the negative axis.
 957 #if __cplusplus < 201103L
 958   template<typename _Tp>
 959     complex<_Tp>
 960     __complex_pow_unsigned(complex<_Tp> __x, unsigned __n)
 961     {
 962       complex<_Tp> __y = __n % 2 ? __x : complex<_Tp>(1);
 963
 964       while (__n >>= 1)
 965         {
 966           __x *= __x;
 967           if (__n % 2)
 968             __y *= __x;
 969         }
 970
 971       return __y;
 972     }
 973
 974   // _GLIBCXX_RESOLVE_LIB_DEFECTS
 975   // DR 844. complex pow return type is ambiguous.
 976   template<typename _Tp>
 977     inline complex<_Tp>
 978     pow(const complex<_Tp>& __z, int __n)
 979     {
 980       return __n < 0
 981         ? complex<_Tp>(1) / std::__complex_pow_unsigned(__z, -__n)
 982         : std::__complex_pow_unsigned(__z, __n);
 983     }
 984 #endif
 985
 986   template<typename _Tp>
 987     complex<_Tp>
 988     pow(const complex<_Tp>& __x, const _Tp& __y)
 989     {
 990 #ifndef _GLIBCXX_USE_C99_COMPLEX
 991       if (__x == _Tp())
 992         return _Tp();
 993 #endif
 994       if (__x.imag() == _Tp() && __x.real() > _Tp())
 995         return pow(__x.real(), __y);
 996
 997       complex<_Tp> __t = std::log(__x);
 998       return std::polar(exp(__y * __t.real()), __y * __t.imag());
 999     }
1000
1001   template<typename _Tp>
1002     inline complex<_Tp>
1003     __complex_pow(const complex<_Tp>& __x, const complex<_Tp>& __y)
1004     { return __x == _Tp() ? _Tp() : std::exp(__y * std::log(__x)); }
1005
1006 #if _GLIBCXX_USE_C99_COMPLEX
1007   inline __complex__ float
1008   __complex_pow(__complex__ float __x, __complex__ float __y)
1009   { return __builtin_cpowf(__x, __y); }
1010
1011   inline __complex__ double
1012   __complex_pow(__complex__ double __x, __complex__ double __y)
1013   { return __builtin_cpow(__x, __y); }
1014
1015   inline __complex__ long double
1016   __complex_pow(const __complex__ long double& __x,
1017                 const __complex__ long double& __y)
1018   { return __builtin_cpowl(__x, __y); }
1019
1020   template<typename _Tp>
1021     inline complex<_Tp>
1022     pow(const complex<_Tp>& __x, const complex<_Tp>& __y)
1023     { return __complex_pow(__x.__rep(), __y.__rep()); }
1024 #else
1025   template<typename _Tp>
1026     inline complex<_Tp>
1027     pow(const complex<_Tp>& __x, const complex<_Tp>& __y)
1028     { return __complex_pow(__x, __y); }
1029 #endif
1030
1031   template<typename _Tp>
1032     inline complex<_Tp>
1033     pow(const _Tp& __x, const complex<_Tp>& __y)
1034     {
1035       return __x > _Tp() ? std::polar(pow(__x, __y.real()),
1036                                       __y.imag() * log(__x))
1037                          : std::pow(complex<_Tp>(__x), __y);
1038     }
1039
1040   /// 26.2.3  complex specializations
1041   /// complex<float> specialization
1042   template<>
1043     struct complex<float>
1044     {
1045       typedef float value_type;
1046       typedef __complex__ float _ComplexT;
1047
1048       _GLIBCXX_CONSTEXPR complex(_ComplexT __z) : _M_value(__z) { }
1049
1050       _GLIBCXX_CONSTEXPR complex(float __r = 0.0f, float __i = 0.0f)
1051 #if __cplusplus >= 201103L
1052       : _M_value{ __r, __i } { }
1053 #else
1054       {
1055         __real__ _M_value = __r;
1056         __imag__ _M_value = __i;
1057       }
1058 #endif
1059
1060       explicit _GLIBCXX_CONSTEXPR complex(const complex<double>&);
1061       explicit _GLIBCXX_CONSTEXPR complex(const complex<long double>&);
1062
1063 #if __cplusplus >= 201103L
1064       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1065       // DR 387. std::complex over-encapsulated.
1066       __attribute ((__abi_tag__ ("cxx11")))
1067       constexpr float
1068       real() { return __real__ _M_value; }
1069
1070       __attribute ((__abi_tag__ ("cxx11")))
1071       constexpr float
1072       imag() { return __imag__ _M_value; }
1073 #else
1074       float&
1075       real() { return __real__ _M_value; }
1076
1077       const float&
1078       real() const { return __real__ _M_value; }
1079
1080       float&
1081       imag() { return __imag__ _M_value; }
1082
1083       const float&
1084       imag() const { return __imag__ _M_value; }
1085 #endif
1086
1087       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1088       // DR 387. std::complex over-encapsulated.
1089       void
1090       real(float __val) { __real__ _M_value = __val; }
1091
1092       void
1093       imag(float __val) { __imag__ _M_value = __val; }
1094
1095       complex&
1096       operator=(float __f)
1097       {
1098         _M_value = __f;
1099         return *this;
1100       }
1101
1102       complex&
1103       operator+=(float __f)
1104       {
1105         _M_value += __f;
1106         return *this;
1107       }
1108
1109       complex&
1110       operator-=(float __f)
1111       {
1112         _M_value -= __f;
1113         return *this;
1114       }
1115
1116       complex&
1117       operator*=(float __f)
1118       {
1119         _M_value *= __f;
1120         return *this;
1121       }
1122
1123       complex&
1124       operator/=(float __f)
1125       {
1126         _M_value /= __f;
1127         return *this;
1128       }
1129
1130       // Let the compiler synthesize the copy and assignment
1131       // operator.  It always does a pretty good job.
1132       // complex& operator=(const complex&);
1133
1134       template<typename _Tp>
1135         complex&
1136         operator=(const complex<_Tp>&  __z)
1137         {
1138           __real__ _M_value = __z.real();
1139           __imag__ _M_value = __z.imag();
1140           return *this;
1141         }
1142
1143       template<typename _Tp>
1144         complex&
1145         operator+=(const complex<_Tp>& __z)
1146         {
1147           __real__ _M_value += __z.real();
1148           __imag__ _M_value += __z.imag();
1149           return *this;
1150         }
1151
1152       template<class _Tp>
1153         complex&
1154         operator-=(const complex<_Tp>& __z)
1155         {
1156           __real__ _M_value -= __z.real();
1157           __imag__ _M_value -= __z.imag();
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       template<class _Tp>
1173         complex&
1174         operator/=(const complex<_Tp>& __z)
1175         {
1176           _ComplexT __t;
1177           __real__ __t = __z.real();
1178           __imag__ __t = __z.imag();
1179           _M_value /= __t;
1180           return *this;
1181         }
1182
1183       _GLIBCXX_USE_CONSTEXPR _ComplexT __rep() const { return _M_value; }
1184
1185     private:
1186       _ComplexT _M_value;
1187     };
1188
1189   /// 26.2.3  complex specializations
1190   /// complex<double> specialization
1191   template<>
1192     struct complex<double>
1193     {
1194       typedef double value_type;
1195       typedef __complex__ double _ComplexT;
1196
1197       _GLIBCXX_CONSTEXPR complex(_ComplexT __z) : _M_value(__z) { }
1198
1199       _GLIBCXX_CONSTEXPR complex(double __r = 0.0, double __i = 0.0)
1200 #if __cplusplus >= 201103L
1201       : _M_value{ __r, __i } { }
1202 #else
1203       {
1204         __real__ _M_value = __r;
1205         __imag__ _M_value = __i;
1206       }
1207 #endif
1208
1209       _GLIBCXX_CONSTEXPR complex(const complex<float>& __z)
1210       : _M_value(__z.__rep()) { }
1211
1212       explicit _GLIBCXX_CONSTEXPR complex(const complex<long double>&);
1213
1214 #if __cplusplus >= 201103L
1215       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1216       // DR 387. std::complex over-encapsulated.
1217       __attribute ((__abi_tag__ ("cxx11")))
1218       constexpr double
1219       real() { return __real__ _M_value; }
1220
1221       __attribute ((__abi_tag__ ("cxx11")))
1222       constexpr double
1223       imag() { return __imag__ _M_value; }
1224 #else
1225       double&
1226       real() { return __real__ _M_value; }
1227
1228       const double&
1229       real() const { return __real__ _M_value; }
1230
1231       double&
1232       imag() { return __imag__ _M_value; }
1233
1234       const double&
1235       imag() const { return __imag__ _M_value; }
1236 #endif
1237
1238       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1239       // DR 387. std::complex over-encapsulated.
1240       void
1241       real(double __val) { __real__ _M_value = __val; }
1242
1243       void
1244       imag(double __val) { __imag__ _M_value = __val; }
1245
1246       complex&
1247       operator=(double __d)
1248       {
1249         _M_value = __d;
1250         return *this;
1251       }
1252
1253       complex&
1254       operator+=(double __d)
1255       {
1256         _M_value += __d;
1257         return *this;
1258       }
1259
1260       complex&
1261       operator-=(double __d)
1262       {
1263         _M_value -= __d;
1264         return *this;
1265       }
1266
1267       complex&
1268       operator*=(double __d)
1269       {
1270         _M_value *= __d;
1271         return *this;
1272       }
1273
1274       complex&
1275       operator/=(double __d)
1276       {
1277         _M_value /= __d;
1278         return *this;
1279       }
1280
1281       // The compiler will synthesize this, efficiently.
1282       // complex& operator=(const complex&);
1283
1284       template<typename _Tp>
1285         complex&
1286         operator=(const complex<_Tp>& __z)
1287         {
1288           __real__ _M_value = __z.real();
1289           __imag__ _M_value = __z.imag();
1290           return *this;
1291         }
1292
1293       template<typename _Tp>
1294         complex&
1295         operator+=(const complex<_Tp>& __z)
1296         {
1297           __real__ _M_value += __z.real();
1298           __imag__ _M_value += __z.imag();
1299           return *this;
1300         }
1301
1302       template<typename _Tp>
1303         complex&
1304         operator-=(const complex<_Tp>& __z)
1305         {
1306           __real__ _M_value -= __z.real();
1307           __imag__ _M_value -= __z.imag();
1308           return *this;
1309         }
1310
1311       template<typename _Tp>
1312         complex&
1313         operator*=(const complex<_Tp>& __z)
1314         {
1315           _ComplexT __t;
1316           __real__ __t = __z.real();
1317           __imag__ __t = __z.imag();
1318           _M_value *= __t;
1319           return *this;
1320         }
1321
1322       template<typename _Tp>
1323         complex&
1324         operator/=(const complex<_Tp>& __z)
1325         {
1326           _ComplexT __t;
1327           __real__ __t = __z.real();
1328           __imag__ __t = __z.imag();
1329           _M_value /= __t;
1330           return *this;
1331         }
1332
1333       _GLIBCXX_USE_CONSTEXPR _ComplexT __rep() const { return _M_value; }
1334
1335     private:
1336       _ComplexT _M_value;
1337     };
1338
1339   /// 26.2.3  complex specializations
1340   /// complex<long double> specialization
1341   template<>
1342     struct complex<long double>
1343     {
1344       typedef long double value_type;
1345       typedef __complex__ long double _ComplexT;
1346
1347       _GLIBCXX_CONSTEXPR complex(_ComplexT __z) : _M_value(__z) { }
1348
1349       _GLIBCXX_CONSTEXPR complex(long double __r = 0.0L,
1350                                  long double __i = 0.0L)
1351 #if __cplusplus >= 201103L
1352       : _M_value{ __r, __i } { }
1353 #else
1354       {
1355         __real__ _M_value = __r;
1356         __imag__ _M_value = __i;
1357       }
1358 #endif
1359
1360       _GLIBCXX_CONSTEXPR complex(const complex<float>& __z)
1361       : _M_value(__z.__rep()) { }
1362
1363       _GLIBCXX_CONSTEXPR complex(const complex<double>& __z)
1364       : _M_value(__z.__rep()) { }
1365
1366 #if __cplusplus >= 201103L
1367       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1368       // DR 387. std::complex over-encapsulated.
1369       __attribute ((__abi_tag__ ("cxx11")))
1370       constexpr long double
1371       real() { return __real__ _M_value; }
1372
1373       __attribute ((__abi_tag__ ("cxx11")))
1374       constexpr long double
1375       imag() { return __imag__ _M_value; }
1376 #else
1377       long double&
1378       real() { return __real__ _M_value; }
1379
1380       const long double&
1381       real() const { return __real__ _M_value; }
1382
1383       long double&
1384       imag() { return __imag__ _M_value; }
1385
1386       const long double&
1387       imag() const { return __imag__ _M_value; }
1388 #endif
1389
1390       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1391       // DR 387. std::complex over-encapsulated.
1392       void
1393       real(long double __val) { __real__ _M_value = __val; }
1394
1395       void
1396       imag(long double __val) { __imag__ _M_value = __val; }
1397
1398       complex&
1399       operator=(long double __r)
1400       {
1401         _M_value = __r;
1402         return *this;
1403       }
1404
1405       complex&
1406       operator+=(long double __r)
1407       {
1408         _M_value += __r;
1409         return *this;
1410       }
1411
1412       complex&
1413       operator-=(long double __r)
1414       {
1415         _M_value -= __r;
1416         return *this;
1417       }
1418
1419       complex&
1420       operator*=(long double __r)
1421       {
1422         _M_value *= __r;
1423         return *this;
1424       }
1425
1426       complex&
1427       operator/=(long double __r)
1428       {
1429         _M_value /= __r;
1430         return *this;
1431       }
1432
1433       // The compiler knows how to do this efficiently
1434       // complex& operator=(const complex&);
1435
1436       template<typename _Tp>
1437         complex&
1438         operator=(const complex<_Tp>& __z)
1439         {
1440           __real__ _M_value = __z.real();
1441           __imag__ _M_value = __z.imag();
1442           return *this;
1443         }
1444
1445       template<typename _Tp>
1446         complex&
1447         operator+=(const complex<_Tp>& __z)
1448         {
1449           __real__ _M_value += __z.real();
1450           __imag__ _M_value += __z.imag();
1451           return *this;
1452         }
1453
1454       template<typename _Tp>
1455         complex&
1456         operator-=(const complex<_Tp>& __z)
1457         {
1458           __real__ _M_value -= __z.real();
1459           __imag__ _M_value -= __z.imag();
1460           return *this;
1461         }
1462
1463       template<typename _Tp>
1464         complex&
1465         operator*=(const complex<_Tp>& __z)
1466         {
1467           _ComplexT __t;
1468           __real__ __t = __z.real();
1469           __imag__ __t = __z.imag();
1470           _M_value *= __t;
1471           return *this;
1472         }
1473
1474       template<typename _Tp>
1475         complex&
1476         operator/=(const complex<_Tp>& __z)
1477         {
1478           _ComplexT __t;
1479           __real__ __t = __z.real();
1480           __imag__ __t = __z.imag();
1481           _M_value /= __t;
1482           return *this;
1483         }
1484
1485       _GLIBCXX_USE_CONSTEXPR _ComplexT __rep() const { return _M_value; }
1486
1487     private:
1488       _ComplexT _M_value;
1489     };
1490
1491   // These bits have to be at the end of this file, so that the
1492   // specializations have all been defined.
1493   inline _GLIBCXX_CONSTEXPR
1494   complex<float>::complex(const complex<double>& __z)
1495   : _M_value(__z.__rep()) { }
1496
1497   inline _GLIBCXX_CONSTEXPR
1498   complex<float>::complex(const complex<long double>& __z)
1499   : _M_value(__z.__rep()) { }
1500
1501   inline _GLIBCXX_CONSTEXPR
1502   complex<double>::complex(const complex<long double>& __z)
1503   : _M_value(__z.__rep()) { }
1504
1505   // Inhibit implicit instantiations for required instantiations,
1506   // which are defined via explicit instantiations elsewhere.
1507   // NB:  This syntax is a GNU extension.
1508 #if _GLIBCXX_EXTERN_TEMPLATE
1509   extern template istream& operator>>(istream&, complex<float>&);
1510   extern template ostream& operator<<(ostream&, const complex<float>&);
1511   extern template istream& operator>>(istream&, complex<double>&);
1512   extern template ostream& operator<<(ostream&, const complex<double>&);
1513   extern template istream& operator>>(istream&, complex<long double>&);
1514   extern template ostream& operator<<(ostream&, const complex<long double>&);
1515
1516 #ifdef _GLIBCXX_USE_WCHAR_T
1517   extern template wistream& operator>>(wistream&, complex<float>&);
1518   extern template wostream& operator<<(wostream&, const complex<float>&);
1519   extern template wistream& operator>>(wistream&, complex<double>&);
1520   extern template wostream& operator<<(wostream&, const complex<double>&);
1521   extern template wistream& operator>>(wistream&, complex<long double>&);
1522   extern template wostream& operator<<(wostream&, const complex<long double>&);
1523 #endif
1524 #endif
1525
1526   // @} group complex_numbers
1527
1528 _GLIBCXX_END_NAMESPACE_VERSION
1529 } // namespace
1530
1531 namespace __gnu_cxx _GLIBCXX_VISIBILITY(default)
1532 {
1533 _GLIBCXX_BEGIN_NAMESPACE_VERSION
1534
1535   // See ext/type_traits.h for the primary template.
1536   template<typename _Tp, typename _Up>
1537     struct __promote_2<std::complex<_Tp>, _Up>
1538     {
1539     public:
1540       typedef std::complex<typename __promote_2<_Tp, _Up>::__type> __type;
1541     };
1542
1543   template<typename _Tp, typename _Up>
1544     struct __promote_2<_Tp, std::complex<_Up> >
1545     {
1546     public:
1547       typedef std::complex<typename __promote_2<_Tp, _Up>::__type> __type;
1548     };
1549
1550   template<typename _Tp, typename _Up>
1551     struct __promote_2<std::complex<_Tp>, std::complex<_Up> >
1552     {
1553     public:
1554       typedef std::complex<typename __promote_2<_Tp, _Up>::__type> __type;
1555     };
1556
1557 _GLIBCXX_END_NAMESPACE_VERSION
1558 } // namespace
1559
1560 #if __cplusplus >= 201103L
1561
1562 namespace std _GLIBCXX_VISIBILITY(default)
1563 {
1564 _GLIBCXX_BEGIN_NAMESPACE_VERSION
1565
1566   // Forward declarations.
1567   template<typename _Tp> std::complex<_Tp> acos(const std::complex<_Tp>&);
1568   template<typename _Tp> std::complex<_Tp> asin(const std::complex<_Tp>&);
1569   template<typename _Tp> std::complex<_Tp> atan(const std::complex<_Tp>&);
1570
1571   template<typename _Tp> std::complex<_Tp> acosh(const std::complex<_Tp>&);
1572   template<typename _Tp> std::complex<_Tp> asinh(const std::complex<_Tp>&);
1573   template<typename _Tp> std::complex<_Tp> atanh(const std::complex<_Tp>&);
1574   // DR 595.
1575   template<typename _Tp> _Tp               fabs(const std::complex<_Tp>&);
1576
1577   template<typename _Tp>
1578     inline std::complex<_Tp>
1579     __complex_acos(const std::complex<_Tp>& __z)
1580     {
1581       const std::complex<_Tp> __t = std::asin(__z);
1582       const _Tp __pi_2 = 1.5707963267948966192313216916397514L;
1583       return std::complex<_Tp>(__pi_2 - __t.real(), -__t.imag());
1584     }
1585
1586 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1587   inline __complex__ float
1588   __complex_acos(__complex__ float __z)
1589   { return __builtin_cacosf(__z); }
1590
1591   inline __complex__ double
1592   __complex_acos(__complex__ double __z)
1593   { return __builtin_cacos(__z); }
1594
1595   inline __complex__ long double
1596   __complex_acos(const __complex__ long double& __z)
1597   { return __builtin_cacosl(__z); }
1598
1599   template<typename _Tp>
1600     inline std::complex<_Tp>
1601     acos(const std::complex<_Tp>& __z)
1602     { return __complex_acos(__z.__rep()); }
1603 #else
1604   /// acos(__z) [8.1.2].
1605   //  Effects:  Behaves the same as C99 function cacos, defined
1606   //            in subclause 7.3.5.1.
1607   template<typename _Tp>
1608     inline std::complex<_Tp>
1609     acos(const std::complex<_Tp>& __z)
1610     { return __complex_acos(__z); }
1611 #endif
1612
1613   template<typename _Tp>
1614     inline std::complex<_Tp>
1615     __complex_asin(const std::complex<_Tp>& __z)
1616     {
1617       std::complex<_Tp> __t(-__z.imag(), __z.real());
1618       __t = std::asinh(__t);
1619       return std::complex<_Tp>(__t.imag(), -__t.real());
1620     }
1621
1622 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1623   inline __complex__ float
1624   __complex_asin(__complex__ float __z)
1625   { return __builtin_casinf(__z); }
1626
1627   inline __complex__ double
1628   __complex_asin(__complex__ double __z)
1629   { return __builtin_casin(__z); }
1630
1631   inline __complex__ long double
1632   __complex_asin(const __complex__ long double& __z)
1633   { return __builtin_casinl(__z); }
1634
1635   template<typename _Tp>
1636     inline std::complex<_Tp>
1637     asin(const std::complex<_Tp>& __z)
1638     { return __complex_asin(__z.__rep()); }
1639 #else
1640   /// asin(__z) [8.1.3].
1641   //  Effects:  Behaves the same as C99 function casin, defined
1642   //            in subclause 7.3.5.2.
1643   template<typename _Tp>
1644     inline std::complex<_Tp>
1645     asin(const std::complex<_Tp>& __z)
1646     { return __complex_asin(__z); }
1647 #endif
1648
1649   template<typename _Tp>
1650     std::complex<_Tp>
1651     __complex_atan(const std::complex<_Tp>& __z)
1652     {
1653       const _Tp __r2 = __z.real() * __z.real();
1654       const _Tp __x = _Tp(1.0) - __r2 - __z.imag() * __z.imag();
1655
1656       _Tp __num = __z.imag() + _Tp(1.0);
1657       _Tp __den = __z.imag() - _Tp(1.0);
1658
1659       __num = __r2 + __num * __num;
1660       __den = __r2 + __den * __den;
1661
1662       return std::complex<_Tp>(_Tp(0.5) * atan2(_Tp(2.0) * __z.real(), __x),
1663                                _Tp(0.25) * log(__num / __den));
1664     }
1665
1666 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1667   inline __complex__ float
1668   __complex_atan(__complex__ float __z)
1669   { return __builtin_catanf(__z); }
1670
1671   inline __complex__ double
1672   __complex_atan(__complex__ double __z)
1673   { return __builtin_catan(__z); }
1674
1675   inline __complex__ long double
1676   __complex_atan(const __complex__ long double& __z)
1677   { return __builtin_catanl(__z); }
1678
1679   template<typename _Tp>
1680     inline std::complex<_Tp>
1681     atan(const std::complex<_Tp>& __z)
1682     { return __complex_atan(__z.__rep()); }
1683 #else
1684   /// atan(__z) [8.1.4].
1685   //  Effects:  Behaves the same as C99 function catan, defined
1686   //            in subclause 7.3.5.3.
1687   template<typename _Tp>
1688     inline std::complex<_Tp>
1689     atan(const std::complex<_Tp>& __z)
1690     { return __complex_atan(__z); }
1691 #endif
1692
1693   template<typename _Tp>
1694     std::complex<_Tp>
1695     __complex_acosh(const std::complex<_Tp>& __z)
1696     {
1697       // Kahan's formula.
1698       return _Tp(2.0) * std::log(std::sqrt(_Tp(0.5) * (__z + _Tp(1.0)))
1699                                  + std::sqrt(_Tp(0.5) * (__z - _Tp(1.0))));
1700     }
1701
1702 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1703   inline __complex__ float
1704   __complex_acosh(__complex__ float __z)
1705   { return __builtin_cacoshf(__z); }
1706
1707   inline __complex__ double
1708   __complex_acosh(__complex__ double __z)
1709   { return __builtin_cacosh(__z); }
1710
1711   inline __complex__ long double
1712   __complex_acosh(const __complex__ long double& __z)
1713   { return __builtin_cacoshl(__z); }
1714
1715   template<typename _Tp>
1716     inline std::complex<_Tp>
1717     acosh(const std::complex<_Tp>& __z)
1718     { return __complex_acosh(__z.__rep()); }
1719 #else
1720   /// acosh(__z) [8.1.5].
1721   //  Effects:  Behaves the same as C99 function cacosh, defined
1722   //            in subclause 7.3.6.1.
1723   template<typename _Tp>
1724     inline std::complex<_Tp>
1725     acosh(const std::complex<_Tp>& __z)
1726     { return __complex_acosh(__z); }
1727 #endif
1728
1729   template<typename _Tp>
1730     std::complex<_Tp>
1731     __complex_asinh(const std::complex<_Tp>& __z)
1732     {
1733       std::complex<_Tp> __t((__z.real() - __z.imag())
1734                             * (__z.real() + __z.imag()) + _Tp(1.0),
1735                             _Tp(2.0) * __z.real() * __z.imag());
1736       __t = std::sqrt(__t);
1737
1738       return std::log(__t + __z);
1739     }
1740
1741 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1742   inline __complex__ float
1743   __complex_asinh(__complex__ float __z)
1744   { return __builtin_casinhf(__z); }
1745
1746   inline __complex__ double
1747   __complex_asinh(__complex__ double __z)
1748   { return __builtin_casinh(__z); }
1749
1750   inline __complex__ long double
1751   __complex_asinh(const __complex__ long double& __z)
1752   { return __builtin_casinhl(__z); }
1753
1754   template<typename _Tp>
1755     inline std::complex<_Tp>
1756     asinh(const std::complex<_Tp>& __z)
1757     { return __complex_asinh(__z.__rep()); }
1758 #else
1759   /// asinh(__z) [8.1.6].
1760   //  Effects:  Behaves the same as C99 function casin, defined
1761   //            in subclause 7.3.6.2.
1762   template<typename _Tp>
1763     inline std::complex<_Tp>
1764     asinh(const std::complex<_Tp>& __z)
1765     { return __complex_asinh(__z); }
1766 #endif
1767
1768   template<typename _Tp>
1769     std::complex<_Tp>
1770     __complex_atanh(const std::complex<_Tp>& __z)
1771     {
1772       const _Tp __i2 = __z.imag() * __z.imag();
1773       const _Tp __x = _Tp(1.0) - __i2 - __z.real() * __z.real();
1774
1775       _Tp __num = _Tp(1.0) + __z.real();
1776       _Tp __den = _Tp(1.0) - __z.real();
1777
1778       __num = __i2 + __num * __num;
1779       __den = __i2 + __den * __den;
1780
1781       return std::complex<_Tp>(_Tp(0.25) * (log(__num) - log(__den)),
1782                                _Tp(0.5) * atan2(_Tp(2.0) * __z.imag(), __x));
1783     }
1784
1785 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1786   inline __complex__ float
1787   __complex_atanh(__complex__ float __z)
1788   { return __builtin_catanhf(__z); }
1789
1790   inline __complex__ double
1791   __complex_atanh(__complex__ double __z)
1792   { return __builtin_catanh(__z); }
1793
1794   inline __complex__ long double
1795   __complex_atanh(const __complex__ long double& __z)
1796   { return __builtin_catanhl(__z); }
1797
1798   template<typename _Tp>
1799     inline std::complex<_Tp>
1800     atanh(const std::complex<_Tp>& __z)
1801     { return __complex_atanh(__z.__rep()); }
1802 #else
1803   /// atanh(__z) [8.1.7].
1804   //  Effects:  Behaves the same as C99 function catanh, defined
1805   //            in subclause 7.3.6.3.
1806   template<typename _Tp>
1807     inline std::complex<_Tp>
1808     atanh(const std::complex<_Tp>& __z)
1809     { return __complex_atanh(__z); }
1810 #endif
1811
1812   template<typename _Tp>
1813     inline _Tp
1814     /// fabs(__z) [8.1.8].
1815     //  Effects:  Behaves the same as C99 function cabs, defined
1816     //            in subclause 7.3.8.1.
1817     fabs(const std::complex<_Tp>& __z)
1818     { return std::abs(__z); }
1819
1820   /// Additional overloads [8.1.9].
1821   template<typename _Tp>
1822     inline typename __gnu_cxx::__promote<_Tp>::__type
1823     arg(_Tp __x)
1824     {
1825       typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
1826 #if (_GLIBCXX_USE_C99_MATH && !_GLIBCXX_USE_C99_FP_MACROS_DYNAMIC)
1827       return std::signbit(__x) ? __type(3.1415926535897932384626433832795029L)
1828                                : __type();
1829 #else
1830       return std::arg(std::complex<__type>(__x));
1831 #endif
1832     }
1833
1834   template<typename _Tp>
1835     inline typename __gnu_cxx::__promote<_Tp>::__type
1836     imag(_Tp)
1837     { return _Tp(); }
1838
1839   template<typename _Tp>
1840     inline typename __gnu_cxx::__promote<_Tp>::__type
1841     norm(_Tp __x)
1842     {
1843       typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
1844       return __type(__x) * __type(__x);
1845     }
1846
1847   template<typename _Tp>
1848     inline typename __gnu_cxx::__promote<_Tp>::__type
1849     real(_Tp __x)
1850     { return __x; }
1851
1852   template<typename _Tp, typename _Up>
1853     inline std::complex<typename __gnu_cxx::__promote_2<_Tp, _Up>::__type>
1854     pow(const std::complex<_Tp>& __x, const _Up& __y)
1855     {
1856       typedef typename __gnu_cxx::__promote_2<_Tp, _Up>::__type __type;
1857       return std::pow(std::complex<__type>(__x), __type(__y));
1858     }
1859
1860   template<typename _Tp, typename _Up>
1861     inline std::complex<typename __gnu_cxx::__promote_2<_Tp, _Up>::__type>
1862     pow(const _Tp& __x, const std::complex<_Up>& __y)
1863     {
1864       typedef typename __gnu_cxx::__promote_2<_Tp, _Up>::__type __type;
1865       return std::pow(__type(__x), std::complex<__type>(__y));
1866     }
1867
1868   template<typename _Tp, typename _Up>
1869     inline std::complex<typename __gnu_cxx::__promote_2<_Tp, _Up>::__type>
1870     pow(const std::complex<_Tp>& __x, const std::complex<_Up>& __y)
1871     {
1872       typedef typename __gnu_cxx::__promote_2<_Tp, _Up>::__type __type;
1873       return std::pow(std::complex<__type>(__x),
1874                       std::complex<__type>(__y));
1875     }
1876
1877   // Forward declarations.
1878   // DR 781.
1879   template<typename _Tp> std::complex<_Tp> proj(const std::complex<_Tp>&);
1880
1881   template<typename _Tp>
1882     std::complex<_Tp>
1883     __complex_proj(const std::complex<_Tp>& __z)
1884     {
1885       const _Tp __den = (__z.real() * __z.real()
1886                          + __z.imag() * __z.imag() + _Tp(1.0));
1887
1888       return std::complex<_Tp>((_Tp(2.0) * __z.real()) / __den,
1889                                (_Tp(2.0) * __z.imag()) / __den);
1890     }
1891
1892 #if _GLIBCXX_USE_C99_COMPLEX
1893   inline __complex__ float
1894   __complex_proj(__complex__ float __z)
1895   { return __builtin_cprojf(__z); }
1896
1897   inline __complex__ double
1898   __complex_proj(__complex__ double __z)
1899   { return __builtin_cproj(__z); }
1900
1901   inline __complex__ long double
1902   __complex_proj(const __complex__ long double& __z)
1903   { return __builtin_cprojl(__z); }
1904
1905   template<typename _Tp>
1906     inline std::complex<_Tp>
1907     proj(const std::complex<_Tp>& __z)
1908     { return __complex_proj(__z.__rep()); }
1909 #else
1910   template<typename _Tp>
1911     inline std::complex<_Tp>
1912     proj(const std::complex<_Tp>& __z)
1913     { return __complex_proj(__z); }
1914 #endif
1915
1916   // DR 1137.
1917   template<typename _Tp>
1918     inline typename __gnu_cxx::__promote<_Tp>::__type
1919     proj(_Tp __x)
1920     { return __x; }
1921
1922   template<typename _Tp>
1923     inline typename __gnu_cxx::__promote<_Tp>::__type
1924     conj(_Tp __x)
1925     { return __x; }
1926
1927 _GLIBCXX_END_NAMESPACE_VERSION
1928 } // namespace
1929
1930 #endif  // C++11
1931
1932 #endif  /* _GLIBCXX_COMPLEX */