libstdc++-v3/include/std/complex

   1 // The template and inlines for the -*- C++ -*- complex number classes.
   2
   3 // Copyright (C) 1997-2014 Free Software Foundation, Inc.
   4 //
   5 // This file is part of the GNU ISO C++ Library.  This library is free
   6 // software; you can redistribute it and/or modify it under the
   7 // terms of the GNU General Public License as published by the
   8 // Free Software Foundation; either version 3, or (at your option)
   9 // any later version.
  10
  11 // This library is distributed in the hope that it will be useful,
  12 // but WITHOUT ANY WARRANTY; without even the implied warranty of
  13 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  14 // GNU General Public License for more details.
  15
  16 // Under Section 7 of GPL version 3, you are granted additional
  17 // permissions described in the GCC Runtime Library Exception, version
  18 // 3.1, as published by the Free Software Foundation.
  19
  20 // You should have received a copy of the GNU General Public License and
  21 // a copy of the GCC Runtime Library Exception along with this program;
  22 // see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
  23 // <http://www.gnu.org/licenses/>.
  24
  25 /** @file include/complex
  26  *  This is a Standard C++ Library header.
  27  */
  28
  29 //
  30 // ISO C++ 14882: 26.2  Complex Numbers
  31 // Note: this is not a conforming implementation.
  32 // Initially implemented by Ulrich Drepper <drepper@cygnus.com>
  33 // Improved by Gabriel Dos Reis <dosreis@cmla.ens-cachan.fr>
  34 //
  35
  36 #ifndef _GLIBCXX_COMPLEX
  37 #define _GLIBCXX_COMPLEX 1
  38
  39 #pragma GCC system_header
  40
  41 #include <bits/c++config.h>
  42 #include <bits/cpp_type_traits.h>
  43 #include <ext/type_traits.h>
  44 #include <cmath>
  45 #include <sstream>
  46
  47 // Get rid of a macro possibly defined in <complex.h>
  48 #undef complex
  49
  50 namespace std _GLIBCXX_VISIBILITY(default)
  51 {
  52 _GLIBCXX_BEGIN_NAMESPACE_VERSION
  53
  54   /**
  55    * @defgroup complex_numbers Complex Numbers
  56    * @ingroup numerics
  57    *
  58    * Classes and functions for complex numbers.
  59    * @{
  60    */
  61
  62   // Forward declarations.
  63   template<typename _Tp> class complex;
  64   template<> class complex<float>;
  65   template<> class complex<double>;
  66   template<> class complex<long double>;
  67
  68   ///  Return magnitude of @a z.
  69   template<typename _Tp> _Tp abs(const complex<_Tp>&);
  70   ///  Return phase angle of @a z.
  71   template<typename _Tp> _Tp arg(const complex<_Tp>&);
  72   ///  Return @a z magnitude squared.
  73   template<typename _Tp> _Tp norm(const complex<_Tp>&);
  74
  75   ///  Return complex conjugate of @a z.
  76   template<typename _Tp> complex<_Tp> conj(const complex<_Tp>&);
  77   ///  Return complex with magnitude @a rho and angle @a theta.
  78   template<typename _Tp> complex<_Tp> polar(const _Tp&, const _Tp& = 0);
  79
  80   // Transcendentals:
  81   /// Return complex cosine of @a z.
  82   template<typename _Tp> complex<_Tp> cos(const complex<_Tp>&);
  83   /// Return complex hyperbolic cosine of @a z.
  84   template<typename _Tp> complex<_Tp> cosh(const complex<_Tp>&);
  85   /// Return complex base e exponential of @a z.
  86   template<typename _Tp> complex<_Tp> exp(const complex<_Tp>&);
  87   /// Return complex natural logarithm of @a z.
  88   template<typename _Tp> complex<_Tp> log(const complex<_Tp>&);
  89   /// Return complex base 10 logarithm of @a z.
  90   template<typename _Tp> complex<_Tp> log10(const complex<_Tp>&);
  91   /// Return @a x to the @a y'th power.
  92   template<typename _Tp> complex<_Tp> pow(const complex<_Tp>&, int);
  93   /// Return @a x to the @a y'th power.
  94   template<typename _Tp> complex<_Tp> pow(const complex<_Tp>&, const _Tp&);
  95   /// Return @a x to the @a y'th power.
  96   template<typename _Tp> complex<_Tp> pow(const complex<_Tp>&,
  97                                           const complex<_Tp>&);
  98   /// Return @a x to the @a y'th power.
  99   template<typename _Tp> complex<_Tp> pow(const _Tp&, const complex<_Tp>&);
 100   /// Return complex sine of @a z.
 101   template<typename _Tp> complex<_Tp> sin(const complex<_Tp>&);
 102   /// Return complex hyperbolic sine of @a z.
 103   template<typename _Tp> complex<_Tp> sinh(const complex<_Tp>&);
 104   /// Return complex square root of @a z.
 105   template<typename _Tp> complex<_Tp> sqrt(const complex<_Tp>&);
 106   /// Return complex tangent of @a z.
 107   template<typename _Tp> complex<_Tp> tan(const complex<_Tp>&);
 108   /// Return complex hyperbolic tangent of @a z.
 109   template<typename _Tp> complex<_Tp> tanh(const complex<_Tp>&);
 110
 111
 112   // 26.2.2  Primary template class complex
 113   /**
 114    *  Template to represent complex numbers.
 115    *
 116    *  Specializations for float, double, and long double are part of the
 117    *  library.  Results with any other type are not guaranteed.
 118    *
 119    *  @param  Tp  Type of real and imaginary values.
 120   */
 121   template<typename _Tp>
 122     struct complex
 123     {
 124       /// Value typedef.
 125       typedef _Tp value_type;
 126
 127       ///  Default constructor.  First parameter is x, second parameter is y.
 128       ///  Unspecified parameters default to 0.
 129       _GLIBCXX_CONSTEXPR complex(const _Tp& __r = _Tp(), const _Tp& __i = _Tp())
 130       : _M_real(__r), _M_imag(__i) { }
 131
 132       // Lets the compiler synthesize the copy constructor
 133       // complex (const complex<_Tp>&);
 134       ///  Copy constructor.
 135       template<typename _Up>
 136         _GLIBCXX_CONSTEXPR complex(const complex<_Up>& __z)
 137         : _M_real(__z.real()), _M_imag(__z.imag()) { }
 138
 139 #if __cplusplus >= 201103L
 140       // _GLIBCXX_RESOLVE_LIB_DEFECTS
 141       // DR 387. std::complex over-encapsulated.
 142       _GLIBCXX_ABI_TAG_CXX11
 143       constexpr _Tp
 144       real() { return _M_real; }
 145
 146       _GLIBCXX_ABI_TAG_CXX11
 147       constexpr _Tp
 148       imag() { 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 #if __cplusplus >= 201103L
 533   template<typename _Tp>
 534     constexpr _Tp
 535     real(const complex<_Tp>& __z)
 536     { return __z.real(); }
 537
 538   template<typename _Tp>
 539     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   template<typename _Tp>
 956     complex<_Tp>
 957     __complex_pow_unsigned(complex<_Tp> __x, unsigned __n)
 958     {
 959       complex<_Tp> __y = __n % 2 ? __x : complex<_Tp>(1);
 960
 961       while (__n >>= 1)
 962         {
 963           __x *= __x;
 964           if (__n % 2)
 965             __y *= __x;
 966         }
 967
 968       return __y;
 969     }
 970
 971   // In C++11 mode we used to implement the resolution of
 972   // DR 844. complex pow return type is ambiguous.
 973   // thus the following overload was disabled in that mode.  However, doing
 974   // that causes all sorts of issues, see, for example:
 975   //   http://gcc.gnu.org/ml/libstdc++/2013-01/msg00058.html
 976   // and also PR57974.
 977   template<typename _Tp>
 978     inline complex<_Tp>
 979     pow(const complex<_Tp>& __z, int __n)
 980     {
 981       return __n < 0
 982         ? complex<_Tp>(1) / std::__complex_pow_unsigned(__z, -(unsigned)__n)
 983         : std::__complex_pow_unsigned(__z, __n);
 984     }
 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() const { return __real__ _M_value; }
1069
1070       __attribute ((__abi_tag__ ("cxx11")))
1071       constexpr float
1072       imag() const { 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() const { return __real__ _M_value; }
1220
1221       __attribute ((__abi_tag__ ("cxx11")))
1222       constexpr double
1223       imag() const { 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() const { return __real__ _M_value; }
1372
1373       __attribute ((__abi_tag__ ("cxx11")))
1374       constexpr long double
1375       imag() const { 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 #if __cplusplus > 201103L
1928
1929 inline namespace literals {
1930 inline namespace complex_literals {
1931
1932   constexpr std::complex<float>
1933   operator""if(long double __num)
1934   { return std::complex<float>{0.0F, static_cast<float>(__num)}; }
1935
1936   constexpr std::complex<float>
1937   operator""if(unsigned long long __num)
1938   { return std::complex<float>{0.0F, static_cast<float>(__num)}; }
1939
1940   constexpr std::complex<double>
1941   operator""i(long double __num)
1942   { return std::complex<double>{0.0, static_cast<double>(__num)}; }
1943
1944   constexpr std::complex<double>
1945   operator""i(unsigned long long __num)
1946   { return std::complex<double>{0.0, static_cast<double>(__num)}; }
1947
1948   constexpr std::complex<long double>
1949   operator""il(long double __num)
1950   { return std::complex<long double>{0.0L, __num}; }
1951
1952   constexpr std::complex<long double>
1953   operator""il(unsigned long long __num)
1954   { return std::complex<long double>{0.0L, static_cast<long double>(__num)}; }
1955
1956 } // inline namespace complex_literals
1957 } // inline namespace literals
1958
1959 #endif // C++14
1960
1961 _GLIBCXX_END_NAMESPACE_VERSION
1962 } // namespace
1963
1964 #endif  // C++11
1965
1966 #endif  /* _GLIBCXX_COMPLEX */