libstdc++-v3/include/std/complex

   1 // The template and inlines for the -*- C++ -*- complex number classes.
   2
   3 // Copyright (C) 1997-2013 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 #if __cplusplus < 201103L
  92   // DR 844.
  93   /// Return @a x to the @a y'th power.
  94   template<typename _Tp> complex<_Tp> pow(const complex<_Tp>&, int);
  95 #endif
  96   /// Return @a x to the @a y'th power.
  97   template<typename _Tp> complex<_Tp> pow(const complex<_Tp>&, const _Tp&);
  98   /// Return @a x to the @a y'th power.
  99   template<typename _Tp> complex<_Tp> pow(const complex<_Tp>&,
 100                                           const complex<_Tp>&);
 101   /// Return @a x to the @a y'th power.
 102   template<typename _Tp> complex<_Tp> pow(const _Tp&, const complex<_Tp>&);
 103   /// Return complex sine of @a z.
 104   template<typename _Tp> complex<_Tp> sin(const complex<_Tp>&);
 105   /// Return complex hyperbolic sine of @a z.
 106   template<typename _Tp> complex<_Tp> sinh(const complex<_Tp>&);
 107   /// Return complex square root of @a z.
 108   template<typename _Tp> complex<_Tp> sqrt(const complex<_Tp>&);
 109   /// Return complex tangent of @a z.
 110   template<typename _Tp> complex<_Tp> tan(const complex<_Tp>&);
 111   /// Return complex hyperbolic tangent of @a z.
 112   template<typename _Tp> complex<_Tp> tanh(const complex<_Tp>&);
 113
 114
 115   // 26.2.2  Primary template class complex
 116   /**
 117    *  Template to represent complex numbers.
 118    *
 119    *  Specializations for float, double, and long double are part of the
 120    *  library.  Results with any other type are not guaranteed.
 121    *
 122    *  @param  Tp  Type of real and imaginary values.
 123   */
 124   template<typename _Tp>
 125     struct complex
 126     {
 127       /// Value typedef.
 128       typedef _Tp value_type;
 129
 130       ///  Default constructor.  First parameter is x, second parameter is y.
 131       ///  Unspecified parameters default to 0.
 132       _GLIBCXX_CONSTEXPR complex(const _Tp& __r = _Tp(), const _Tp& __i = _Tp())
 133       : _M_real(__r), _M_imag(__i) { }
 134
 135       // Lets the compiler synthesize the copy constructor
 136       // complex (const complex<_Tp>&);
 137       ///  Copy constructor.
 138       template<typename _Up>
 139         _GLIBCXX_CONSTEXPR complex(const complex<_Up>& __z)
 140         : _M_real(__z.real()), _M_imag(__z.imag()) { }
 141
 142 #if __cplusplus >= 201103L
 143       // _GLIBCXX_RESOLVE_LIB_DEFECTS
 144       // DR 387. std::complex over-encapsulated.
 145       __attribute ((__abi_tag__ ("cxx11")))
 146       constexpr _Tp
 147       real() { return _M_real; }
 148
 149       __attribute ((__abi_tag__ ("cxx11")))
 150       constexpr _Tp
 151       imag() { return _M_imag; }
 152 #else
 153       ///  Return real part of complex number.
 154       _Tp&
 155       real() { return _M_real; }
 156
 157       ///  Return real part of complex number.
 158       const _Tp&
 159       real() const { return _M_real; }
 160
 161       ///  Return imaginary part of complex number.
 162       _Tp&
 163       imag() { return _M_imag; }
 164
 165       ///  Return imaginary part of complex number.
 166       const _Tp&
 167       imag() const { return _M_imag; }
 168 #endif
 169
 170       // _GLIBCXX_RESOLVE_LIB_DEFECTS
 171       // DR 387. std::complex over-encapsulated.
 172       void
 173       real(_Tp __val) { _M_real = __val; }
 174
 175       void
 176       imag(_Tp __val) { _M_imag = __val; }
 177
 178       /// Assign this complex number to scalar @a t.
 179       complex<_Tp>& operator=(const _Tp&);
 180
 181       /// Add @a t to this complex number.
 182       // 26.2.5/1
 183       complex<_Tp>&
 184       operator+=(const _Tp& __t)
 185       {
 186         _M_real += __t;
 187         return *this;
 188       }
 189
 190       /// Subtract @a t from this complex number.
 191       // 26.2.5/3
 192       complex<_Tp>&
 193       operator-=(const _Tp& __t)
 194       {
 195         _M_real -= __t;
 196         return *this;
 197       }
 198
 199       /// Multiply this complex number by @a t.
 200       complex<_Tp>& operator*=(const _Tp&);
 201       /// Divide this complex number by @a t.
 202       complex<_Tp>& operator/=(const _Tp&);
 203
 204       // Lets the compiler synthesize the
 205       // copy and assignment operator
 206       // complex<_Tp>& operator= (const complex<_Tp>&);
 207       /// Assign this complex number to complex @a z.
 208       template<typename _Up>
 209         complex<_Tp>& operator=(const complex<_Up>&);
 210       /// Add @a z to this complex number.
 211       template<typename _Up>
 212         complex<_Tp>& operator+=(const complex<_Up>&);
 213       /// Subtract @a z from this complex number.
 214       template<typename _Up>
 215         complex<_Tp>& operator-=(const complex<_Up>&);
 216       /// Multiply this complex number by @a z.
 217       template<typename _Up>
 218         complex<_Tp>& operator*=(const complex<_Up>&);
 219       /// Divide this complex number by @a z.
 220       template<typename _Up>
 221         complex<_Tp>& operator/=(const complex<_Up>&);
 222
 223       _GLIBCXX_USE_CONSTEXPR complex __rep() const
 224       { return *this; }
 225
 226     private:
 227       _Tp _M_real;
 228       _Tp _M_imag;
 229     };
 230
 231   template<typename _Tp>
 232     complex<_Tp>&
 233     complex<_Tp>::operator=(const _Tp& __t)
 234     {
 235      _M_real = __t;
 236      _M_imag = _Tp();
 237      return *this;
 238     }
 239
 240   // 26.2.5/5
 241   template<typename _Tp>
 242     complex<_Tp>&
 243     complex<_Tp>::operator*=(const _Tp& __t)
 244     {
 245       _M_real *= __t;
 246       _M_imag *= __t;
 247       return *this;
 248     }
 249
 250   // 26.2.5/7
 251   template<typename _Tp>
 252     complex<_Tp>&
 253     complex<_Tp>::operator/=(const _Tp& __t)
 254     {
 255       _M_real /= __t;
 256       _M_imag /= __t;
 257       return *this;
 258     }
 259
 260   template<typename _Tp>
 261     template<typename _Up>
 262     complex<_Tp>&
 263     complex<_Tp>::operator=(const complex<_Up>& __z)
 264     {
 265       _M_real = __z.real();
 266       _M_imag = __z.imag();
 267       return *this;
 268     }
 269
 270   // 26.2.5/9
 271   template<typename _Tp>
 272     template<typename _Up>
 273     complex<_Tp>&
 274     complex<_Tp>::operator+=(const complex<_Up>& __z)
 275     {
 276       _M_real += __z.real();
 277       _M_imag += __z.imag();
 278       return *this;
 279     }
 280
 281   // 26.2.5/11
 282   template<typename _Tp>
 283     template<typename _Up>
 284     complex<_Tp>&
 285     complex<_Tp>::operator-=(const complex<_Up>& __z)
 286     {
 287       _M_real -= __z.real();
 288       _M_imag -= __z.imag();
 289       return *this;
 290     }
 291
 292   // 26.2.5/13
 293   // XXX: This is a grammar school implementation.
 294   template<typename _Tp>
 295     template<typename _Up>
 296     complex<_Tp>&
 297     complex<_Tp>::operator*=(const complex<_Up>& __z)
 298     {
 299       const _Tp __r = _M_real * __z.real() - _M_imag * __z.imag();
 300       _M_imag = _M_real * __z.imag() + _M_imag * __z.real();
 301       _M_real = __r;
 302       return *this;
 303     }
 304
 305   // 26.2.5/15
 306   // XXX: This is a grammar school implementation.
 307   template<typename _Tp>
 308     template<typename _Up>
 309     complex<_Tp>&
 310     complex<_Tp>::operator/=(const complex<_Up>& __z)
 311     {
 312       const _Tp __r =  _M_real * __z.real() + _M_imag * __z.imag();
 313       const _Tp __n = std::norm(__z);
 314       _M_imag = (_M_imag * __z.real() - _M_real * __z.imag()) / __n;
 315       _M_real = __r / __n;
 316       return *this;
 317     }
 318
 319   // Operators:
 320   //@{
 321   ///  Return new complex value @a x plus @a y.
 322   template<typename _Tp>
 323     inline complex<_Tp>
 324     operator+(const complex<_Tp>& __x, const complex<_Tp>& __y)
 325     {
 326       complex<_Tp> __r = __x;
 327       __r += __y;
 328       return __r;
 329     }
 330
 331   template<typename _Tp>
 332     inline complex<_Tp>
 333     operator+(const complex<_Tp>& __x, const _Tp& __y)
 334     {
 335       complex<_Tp> __r = __x;
 336       __r += __y;
 337       return __r;
 338     }
 339
 340   template<typename _Tp>
 341     inline complex<_Tp>
 342     operator+(const _Tp& __x, const complex<_Tp>& __y)
 343     {
 344       complex<_Tp> __r = __y;
 345       __r += __x;
 346       return __r;
 347     }
 348   //@}
 349
 350   //@{
 351   ///  Return new complex value @a x minus @a y.
 352   template<typename _Tp>
 353     inline complex<_Tp>
 354     operator-(const complex<_Tp>& __x, const complex<_Tp>& __y)
 355     {
 356       complex<_Tp> __r = __x;
 357       __r -= __y;
 358       return __r;
 359     }
 360
 361   template<typename _Tp>
 362     inline complex<_Tp>
 363     operator-(const complex<_Tp>& __x, const _Tp& __y)
 364     {
 365       complex<_Tp> __r = __x;
 366       __r -= __y;
 367       return __r;
 368     }
 369
 370   template<typename _Tp>
 371     inline complex<_Tp>
 372     operator-(const _Tp& __x, const complex<_Tp>& __y)
 373     {
 374       complex<_Tp> __r(__x, -__y.imag());
 375       __r -= __y.real();
 376       return __r;
 377     }
 378   //@}
 379
 380   //@{
 381   ///  Return new complex value @a x times @a y.
 382   template<typename _Tp>
 383     inline complex<_Tp>
 384     operator*(const complex<_Tp>& __x, const complex<_Tp>& __y)
 385     {
 386       complex<_Tp> __r = __x;
 387       __r *= __y;
 388       return __r;
 389     }
 390
 391   template<typename _Tp>
 392     inline complex<_Tp>
 393     operator*(const complex<_Tp>& __x, const _Tp& __y)
 394     {
 395       complex<_Tp> __r = __x;
 396       __r *= __y;
 397       return __r;
 398     }
 399
 400   template<typename _Tp>
 401     inline complex<_Tp>
 402     operator*(const _Tp& __x, const complex<_Tp>& __y)
 403     {
 404       complex<_Tp> __r = __y;
 405       __r *= __x;
 406       return __r;
 407     }
 408   //@}
 409
 410   //@{
 411   ///  Return new complex value @a x divided by @a y.
 412   template<typename _Tp>
 413     inline complex<_Tp>
 414     operator/(const complex<_Tp>& __x, const complex<_Tp>& __y)
 415     {
 416       complex<_Tp> __r = __x;
 417       __r /= __y;
 418       return __r;
 419     }
 420
 421   template<typename _Tp>
 422     inline complex<_Tp>
 423     operator/(const complex<_Tp>& __x, const _Tp& __y)
 424     {
 425       complex<_Tp> __r = __x;
 426       __r /= __y;
 427       return __r;
 428     }
 429
 430   template<typename _Tp>
 431     inline complex<_Tp>
 432     operator/(const _Tp& __x, const complex<_Tp>& __y)
 433     {
 434       complex<_Tp> __r = __x;
 435       __r /= __y;
 436       return __r;
 437     }
 438   //@}
 439
 440   ///  Return @a x.
 441   template<typename _Tp>
 442     inline complex<_Tp>
 443     operator+(const complex<_Tp>& __x)
 444     { return __x; }
 445
 446   ///  Return complex negation of @a x.
 447   template<typename _Tp>
 448     inline complex<_Tp>
 449     operator-(const complex<_Tp>& __x)
 450     {  return complex<_Tp>(-__x.real(), -__x.imag()); }
 451
 452   //@{
 453   ///  Return true if @a x is equal to @a y.
 454   template<typename _Tp>
 455     inline _GLIBCXX_CONSTEXPR bool
 456     operator==(const complex<_Tp>& __x, const complex<_Tp>& __y)
 457     { return __x.real() == __y.real() && __x.imag() == __y.imag(); }
 458
 459   template<typename _Tp>
 460     inline _GLIBCXX_CONSTEXPR bool
 461     operator==(const complex<_Tp>& __x, const _Tp& __y)
 462     { return __x.real() == __y && __x.imag() == _Tp(); }
 463
 464   template<typename _Tp>
 465     inline _GLIBCXX_CONSTEXPR bool
 466     operator==(const _Tp& __x, const complex<_Tp>& __y)
 467     { return __x == __y.real() && _Tp() == __y.imag(); }
 468   //@}
 469
 470   //@{
 471   ///  Return false if @a x is equal to @a y.
 472   template<typename _Tp>
 473     inline _GLIBCXX_CONSTEXPR bool
 474     operator!=(const complex<_Tp>& __x, const complex<_Tp>& __y)
 475     { return __x.real() != __y.real() || __x.imag() != __y.imag(); }
 476
 477   template<typename _Tp>
 478     inline _GLIBCXX_CONSTEXPR bool
 479     operator!=(const complex<_Tp>& __x, const _Tp& __y)
 480     { return __x.real() != __y || __x.imag() != _Tp(); }
 481
 482   template<typename _Tp>
 483     inline _GLIBCXX_CONSTEXPR bool
 484     operator!=(const _Tp& __x, const complex<_Tp>& __y)
 485     { return __x != __y.real() || _Tp() != __y.imag(); }
 486   //@}
 487
 488   ///  Extraction operator for complex values.
 489   template<typename _Tp, typename _CharT, class _Traits>
 490     basic_istream<_CharT, _Traits>&
 491     operator>>(basic_istream<_CharT, _Traits>& __is, complex<_Tp>& __x)
 492     {
 493       _Tp __re_x, __im_x;
 494       _CharT __ch;
 495       __is >> __ch;
 496       if (__ch == '(')
 497         {
 498           __is >> __re_x >> __ch;
 499           if (__ch == ',')
 500             {
 501               __is >> __im_x >> __ch;
 502               if (__ch == ')')
 503                 __x = complex<_Tp>(__re_x, __im_x);
 504               else
 505                 __is.setstate(ios_base::failbit);
 506             }
 507           else if (__ch == ')')
 508             __x = __re_x;
 509           else
 510             __is.setstate(ios_base::failbit);
 511         }
 512       else
 513         {
 514           __is.putback(__ch);
 515           __is >> __re_x;
 516           __x = __re_x;
 517         }
 518       return __is;
 519     }
 520
 521   ///  Insertion operator for complex values.
 522   template<typename _Tp, typename _CharT, class _Traits>
 523     basic_ostream<_CharT, _Traits>&
 524     operator<<(basic_ostream<_CharT, _Traits>& __os, const complex<_Tp>& __x)
 525     {
 526       basic_ostringstream<_CharT, _Traits> __s;
 527       __s.flags(__os.flags());
 528       __s.imbue(__os.getloc());
 529       __s.precision(__os.precision());
 530       __s << '(' << __x.real() << ',' << __x.imag() << ')';
 531       return __os << __s.str();
 532     }
 533
 534   // Values
 535 #if __cplusplus >= 201103L
 536   template<typename _Tp>
 537     constexpr _Tp
 538     real(const complex<_Tp>& __z)
 539     { return __z.real(); }
 540
 541   template<typename _Tp>
 542     constexpr _Tp
 543     imag(const complex<_Tp>& __z)
 544     { return __z.imag(); }
 545 #else
 546   template<typename _Tp>
 547     inline _Tp&
 548     real(complex<_Tp>& __z)
 549     { return __z.real(); }
 550
 551   template<typename _Tp>
 552     inline const _Tp&
 553     real(const complex<_Tp>& __z)
 554     { return __z.real(); }
 555
 556   template<typename _Tp>
 557     inline _Tp&
 558     imag(complex<_Tp>& __z)
 559     { return __z.imag(); }
 560
 561   template<typename _Tp>
 562     inline const _Tp&
 563     imag(const complex<_Tp>& __z)
 564     { return __z.imag(); }
 565 #endif
 566
 567   // 26.2.7/3 abs(__z):  Returns the magnitude of __z.
 568   template<typename _Tp>
 569     inline _Tp
 570     __complex_abs(const complex<_Tp>& __z)
 571     {
 572       _Tp __x = __z.real();
 573       _Tp __y = __z.imag();
 574       const _Tp __s = std::max(abs(__x), abs(__y));
 575       if (__s == _Tp())  // well ...
 576         return __s;
 577       __x /= __s;
 578       __y /= __s;
 579       return __s * sqrt(__x * __x + __y * __y);
 580     }
 581
 582 #if _GLIBCXX_USE_C99_COMPLEX
 583   inline float
 584   __complex_abs(__complex__ float __z) { return __builtin_cabsf(__z); }
 585
 586   inline double
 587   __complex_abs(__complex__ double __z) { return __builtin_cabs(__z); }
 588
 589   inline long double
 590   __complex_abs(const __complex__ long double& __z)
 591   { return __builtin_cabsl(__z); }
 592
 593   template<typename _Tp>
 594     inline _Tp
 595     abs(const complex<_Tp>& __z) { return __complex_abs(__z.__rep()); }
 596 #else
 597   template<typename _Tp>
 598     inline _Tp
 599     abs(const complex<_Tp>& __z) { return __complex_abs(__z); }
 600 #endif
 601
 602
 603   // 26.2.7/4: arg(__z): Returns the phase angle of __z.
 604   template<typename _Tp>
 605     inline _Tp
 606     __complex_arg(const complex<_Tp>& __z)
 607     { return  atan2(__z.imag(), __z.real()); }
 608
 609 #if _GLIBCXX_USE_C99_COMPLEX
 610   inline float
 611   __complex_arg(__complex__ float __z) { return __builtin_cargf(__z); }
 612
 613   inline double
 614   __complex_arg(__complex__ double __z) { return __builtin_carg(__z); }
 615
 616   inline long double
 617   __complex_arg(const __complex__ long double& __z)
 618   { return __builtin_cargl(__z); }
 619
 620   template<typename _Tp>
 621     inline _Tp
 622     arg(const complex<_Tp>& __z) { return __complex_arg(__z.__rep()); }
 623 #else
 624   template<typename _Tp>
 625     inline _Tp
 626     arg(const complex<_Tp>& __z) { return __complex_arg(__z); }
 627 #endif
 628
 629   // 26.2.7/5: norm(__z) returns the squared magnitude of __z.
 630   //     As defined, norm() is -not- a norm is the common mathematical
 631   //     sens used in numerics.  The helper class _Norm_helper<> tries to
 632   //     distinguish between builtin floating point and the rest, so as
 633   //     to deliver an answer as close as possible to the real value.
 634   template<bool>
 635     struct _Norm_helper
 636     {
 637       template<typename _Tp>
 638         static inline _Tp _S_do_it(const complex<_Tp>& __z)
 639         {
 640           const _Tp __x = __z.real();
 641           const _Tp __y = __z.imag();
 642           return __x * __x + __y * __y;
 643         }
 644     };
 645
 646   template<>
 647     struct _Norm_helper<true>
 648     {
 649       template<typename _Tp>
 650         static inline _Tp _S_do_it(const complex<_Tp>& __z)
 651         {
 652           _Tp __res = std::abs(__z);
 653           return __res * __res;
 654         }
 655     };
 656
 657   template<typename _Tp>
 658     inline _Tp
 659     norm(const complex<_Tp>& __z)
 660     {
 661       return _Norm_helper<__is_floating<_Tp>::__value
 662         && !_GLIBCXX_FAST_MATH>::_S_do_it(__z);
 663     }
 664
 665   template<typename _Tp>
 666     inline complex<_Tp>
 667     polar(const _Tp& __rho, const _Tp& __theta)
 668     { return complex<_Tp>(__rho * cos(__theta), __rho * sin(__theta)); }
 669
 670   template<typename _Tp>
 671     inline complex<_Tp>
 672     conj(const complex<_Tp>& __z)
 673     { return complex<_Tp>(__z.real(), -__z.imag()); }
 674
 675   // Transcendentals
 676
 677   // 26.2.8/1 cos(__z):  Returns the cosine of __z.
 678   template<typename _Tp>
 679     inline complex<_Tp>
 680     __complex_cos(const complex<_Tp>& __z)
 681     {
 682       const _Tp __x = __z.real();
 683       const _Tp __y = __z.imag();
 684       return complex<_Tp>(cos(__x) * cosh(__y), -sin(__x) * sinh(__y));
 685     }
 686
 687 #if _GLIBCXX_USE_C99_COMPLEX
 688   inline __complex__ float
 689   __complex_cos(__complex__ float __z) { return __builtin_ccosf(__z); }
 690
 691   inline __complex__ double
 692   __complex_cos(__complex__ double __z) { return __builtin_ccos(__z); }
 693
 694   inline __complex__ long double
 695   __complex_cos(const __complex__ long double& __z)
 696   { return __builtin_ccosl(__z); }
 697
 698   template<typename _Tp>
 699     inline complex<_Tp>
 700     cos(const complex<_Tp>& __z) { return __complex_cos(__z.__rep()); }
 701 #else
 702   template<typename _Tp>
 703     inline complex<_Tp>
 704     cos(const complex<_Tp>& __z) { return __complex_cos(__z); }
 705 #endif
 706
 707   // 26.2.8/2 cosh(__z): Returns the hyperbolic cosine of __z.
 708   template<typename _Tp>
 709     inline complex<_Tp>
 710     __complex_cosh(const complex<_Tp>& __z)
 711     {
 712       const _Tp __x = __z.real();
 713       const _Tp __y = __z.imag();
 714       return complex<_Tp>(cosh(__x) * cos(__y), sinh(__x) * sin(__y));
 715     }
 716
 717 #if _GLIBCXX_USE_C99_COMPLEX
 718   inline __complex__ float
 719   __complex_cosh(__complex__ float __z) { return __builtin_ccoshf(__z); }
 720
 721   inline __complex__ double
 722   __complex_cosh(__complex__ double __z) { return __builtin_ccosh(__z); }
 723
 724   inline __complex__ long double
 725   __complex_cosh(const __complex__ long double& __z)
 726   { return __builtin_ccoshl(__z); }
 727
 728   template<typename _Tp>
 729     inline complex<_Tp>
 730     cosh(const complex<_Tp>& __z) { return __complex_cosh(__z.__rep()); }
 731 #else
 732   template<typename _Tp>
 733     inline complex<_Tp>
 734     cosh(const complex<_Tp>& __z) { return __complex_cosh(__z); }
 735 #endif
 736
 737   // 26.2.8/3 exp(__z): Returns the complex base e exponential of x
 738   template<typename _Tp>
 739     inline complex<_Tp>
 740     __complex_exp(const complex<_Tp>& __z)
 741     { return std::polar(exp(__z.real()), __z.imag()); }
 742
 743 #if _GLIBCXX_USE_C99_COMPLEX
 744   inline __complex__ float
 745   __complex_exp(__complex__ float __z) { return __builtin_cexpf(__z); }
 746
 747   inline __complex__ double
 748   __complex_exp(__complex__ double __z) { return __builtin_cexp(__z); }
 749
 750   inline __complex__ long double
 751   __complex_exp(const __complex__ long double& __z)
 752   { return __builtin_cexpl(__z); }
 753
 754   template<typename _Tp>
 755     inline complex<_Tp>
 756     exp(const complex<_Tp>& __z) { return __complex_exp(__z.__rep()); }
 757 #else
 758   template<typename _Tp>
 759     inline complex<_Tp>
 760     exp(const complex<_Tp>& __z) { return __complex_exp(__z); }
 761 #endif
 762
 763   // 26.2.8/5 log(__z): Returns the natural complex logarithm of __z.
 764   //                    The branch cut is along the negative axis.
 765   template<typename _Tp>
 766     inline complex<_Tp>
 767     __complex_log(const complex<_Tp>& __z)
 768     { return complex<_Tp>(log(std::abs(__z)), std::arg(__z)); }
 769
 770 #if _GLIBCXX_USE_C99_COMPLEX
 771   inline __complex__ float
 772   __complex_log(__complex__ float __z) { return __builtin_clogf(__z); }
 773
 774   inline __complex__ double
 775   __complex_log(__complex__ double __z) { return __builtin_clog(__z); }
 776
 777   inline __complex__ long double
 778   __complex_log(const __complex__ long double& __z)
 779   { return __builtin_clogl(__z); }
 780
 781   template<typename _Tp>
 782     inline complex<_Tp>
 783     log(const complex<_Tp>& __z) { return __complex_log(__z.__rep()); }
 784 #else
 785   template<typename _Tp>
 786     inline complex<_Tp>
 787     log(const complex<_Tp>& __z) { return __complex_log(__z); }
 788 #endif
 789
 790   template<typename _Tp>
 791     inline complex<_Tp>
 792     log10(const complex<_Tp>& __z)
 793     { return std::log(__z) / log(_Tp(10.0)); }
 794
 795   // 26.2.8/10 sin(__z): Returns the sine of __z.
 796   template<typename _Tp>
 797     inline complex<_Tp>
 798     __complex_sin(const complex<_Tp>& __z)
 799     {
 800       const _Tp __x = __z.real();
 801       const _Tp __y = __z.imag();
 802       return complex<_Tp>(sin(__x) * cosh(__y), cos(__x) * sinh(__y));
 803     }
 804
 805 #if _GLIBCXX_USE_C99_COMPLEX
 806   inline __complex__ float
 807   __complex_sin(__complex__ float __z) { return __builtin_csinf(__z); }
 808
 809   inline __complex__ double
 810   __complex_sin(__complex__ double __z) { return __builtin_csin(__z); }
 811
 812   inline __complex__ long double
 813   __complex_sin(const __complex__ long double& __z)
 814   { return __builtin_csinl(__z); }
 815
 816   template<typename _Tp>
 817     inline complex<_Tp>
 818     sin(const complex<_Tp>& __z) { return __complex_sin(__z.__rep()); }
 819 #else
 820   template<typename _Tp>
 821     inline complex<_Tp>
 822     sin(const complex<_Tp>& __z) { return __complex_sin(__z); }
 823 #endif
 824
 825   // 26.2.8/11 sinh(__z): Returns the hyperbolic sine of __z.
 826   template<typename _Tp>
 827     inline complex<_Tp>
 828     __complex_sinh(const complex<_Tp>& __z)
 829     {
 830       const _Tp __x = __z.real();
 831       const _Tp  __y = __z.imag();
 832       return complex<_Tp>(sinh(__x) * cos(__y), cosh(__x) * sin(__y));
 833     }
 834
 835 #if _GLIBCXX_USE_C99_COMPLEX
 836   inline __complex__ float
 837   __complex_sinh(__complex__ float __z) { return __builtin_csinhf(__z); }
 838
 839   inline __complex__ double
 840   __complex_sinh(__complex__ double __z) { return __builtin_csinh(__z); }
 841
 842   inline __complex__ long double
 843   __complex_sinh(const __complex__ long double& __z)
 844   { return __builtin_csinhl(__z); }
 845
 846   template<typename _Tp>
 847     inline complex<_Tp>
 848     sinh(const complex<_Tp>& __z) { return __complex_sinh(__z.__rep()); }
 849 #else
 850   template<typename _Tp>
 851     inline complex<_Tp>
 852     sinh(const complex<_Tp>& __z) { return __complex_sinh(__z); }
 853 #endif
 854
 855   // 26.2.8/13 sqrt(__z): Returns the complex square root of __z.
 856   //                     The branch cut is on the negative axis.
 857   template<typename _Tp>
 858     complex<_Tp>
 859     __complex_sqrt(const complex<_Tp>& __z)
 860     {
 861       _Tp __x = __z.real();
 862       _Tp __y = __z.imag();
 863
 864       if (__x == _Tp())
 865         {
 866           _Tp __t = sqrt(abs(__y) / 2);
 867           return complex<_Tp>(__t, __y < _Tp() ? -__t : __t);
 868         }
 869       else
 870         {
 871           _Tp __t = sqrt(2 * (std::abs(__z) + abs(__x)));
 872           _Tp __u = __t / 2;
 873           return __x > _Tp()
 874             ? complex<_Tp>(__u, __y / __t)
 875             : complex<_Tp>(abs(__y) / __t, __y < _Tp() ? -__u : __u);
 876         }
 877     }
 878
 879 #if _GLIBCXX_USE_C99_COMPLEX
 880   inline __complex__ float
 881   __complex_sqrt(__complex__ float __z) { return __builtin_csqrtf(__z); }
 882
 883   inline __complex__ double
 884   __complex_sqrt(__complex__ double __z) { return __builtin_csqrt(__z); }
 885
 886   inline __complex__ long double
 887   __complex_sqrt(const __complex__ long double& __z)
 888   { return __builtin_csqrtl(__z); }
 889
 890   template<typename _Tp>
 891     inline complex<_Tp>
 892     sqrt(const complex<_Tp>& __z) { return __complex_sqrt(__z.__rep()); }
 893 #else
 894   template<typename _Tp>
 895     inline complex<_Tp>
 896     sqrt(const complex<_Tp>& __z) { return __complex_sqrt(__z); }
 897 #endif
 898
 899   // 26.2.8/14 tan(__z):  Return the complex tangent of __z.
 900
 901   template<typename _Tp>
 902     inline complex<_Tp>
 903     __complex_tan(const complex<_Tp>& __z)
 904     { return std::sin(__z) / std::cos(__z); }
 905
 906 #if _GLIBCXX_USE_C99_COMPLEX
 907   inline __complex__ float
 908   __complex_tan(__complex__ float __z) { return __builtin_ctanf(__z); }
 909
 910   inline __complex__ double
 911   __complex_tan(__complex__ double __z) { return __builtin_ctan(__z); }
 912
 913   inline __complex__ long double
 914   __complex_tan(const __complex__ long double& __z)
 915   { return __builtin_ctanl(__z); }
 916
 917   template<typename _Tp>
 918     inline complex<_Tp>
 919     tan(const complex<_Tp>& __z) { return __complex_tan(__z.__rep()); }
 920 #else
 921   template<typename _Tp>
 922     inline complex<_Tp>
 923     tan(const complex<_Tp>& __z) { return __complex_tan(__z); }
 924 #endif
 925
 926
 927   // 26.2.8/15 tanh(__z):  Returns the hyperbolic tangent of __z.
 928
 929   template<typename _Tp>
 930     inline complex<_Tp>
 931     __complex_tanh(const complex<_Tp>& __z)
 932     { return std::sinh(__z) / std::cosh(__z); }
 933
 934 #if _GLIBCXX_USE_C99_COMPLEX
 935   inline __complex__ float
 936   __complex_tanh(__complex__ float __z) { return __builtin_ctanhf(__z); }
 937
 938   inline __complex__ double
 939   __complex_tanh(__complex__ double __z) { return __builtin_ctanh(__z); }
 940
 941   inline __complex__ long double
 942   __complex_tanh(const __complex__ long double& __z)
 943   { return __builtin_ctanhl(__z); }
 944
 945   template<typename _Tp>
 946     inline complex<_Tp>
 947     tanh(const complex<_Tp>& __z) { return __complex_tanh(__z.__rep()); }
 948 #else
 949   template<typename _Tp>
 950     inline complex<_Tp>
 951     tanh(const complex<_Tp>& __z) { return __complex_tanh(__z); }
 952 #endif
 953
 954
 955   // 26.2.8/9  pow(__x, __y): Returns the complex power base of __x
 956   //                          raised to the __y-th power.  The branch
 957   //                          cut is on the negative axis.
 958 #if __cplusplus < 201103L
 959   template<typename _Tp>
 960     complex<_Tp>
 961     __complex_pow_unsigned(complex<_Tp> __x, unsigned __n)
 962     {
 963       complex<_Tp> __y = __n % 2 ? __x : complex<_Tp>(1);
 964
 965       while (__n >>= 1)
 966         {
 967           __x *= __x;
 968           if (__n % 2)
 969             __y *= __x;
 970         }
 971
 972       return __y;
 973     }
 974
 975   // _GLIBCXX_RESOLVE_LIB_DEFECTS
 976   // DR 844. complex pow return type is ambiguous.
 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 #endif
 986
 987   template<typename _Tp>
 988     complex<_Tp>
 989     pow(const complex<_Tp>& __x, const _Tp& __y)
 990     {
 991 #ifndef _GLIBCXX_USE_C99_COMPLEX
 992       if (__x == _Tp())
 993         return _Tp();
 994 #endif
 995       if (__x.imag() == _Tp() && __x.real() > _Tp())
 996         return pow(__x.real(), __y);
 997
 998       complex<_Tp> __t = std::log(__x);
 999       return std::polar(exp(__y * __t.real()), __y * __t.imag());
1000     }
1001
1002   template<typename _Tp>
1003     inline complex<_Tp>
1004     __complex_pow(const complex<_Tp>& __x, const complex<_Tp>& __y)
1005     { return __x == _Tp() ? _Tp() : std::exp(__y * std::log(__x)); }
1006
1007 #if _GLIBCXX_USE_C99_COMPLEX
1008   inline __complex__ float
1009   __complex_pow(__complex__ float __x, __complex__ float __y)
1010   { return __builtin_cpowf(__x, __y); }
1011
1012   inline __complex__ double
1013   __complex_pow(__complex__ double __x, __complex__ double __y)
1014   { return __builtin_cpow(__x, __y); }
1015
1016   inline __complex__ long double
1017   __complex_pow(const __complex__ long double& __x,
1018                 const __complex__ long double& __y)
1019   { return __builtin_cpowl(__x, __y); }
1020
1021   template<typename _Tp>
1022     inline complex<_Tp>
1023     pow(const complex<_Tp>& __x, const complex<_Tp>& __y)
1024     { return __complex_pow(__x.__rep(), __y.__rep()); }
1025 #else
1026   template<typename _Tp>
1027     inline complex<_Tp>
1028     pow(const complex<_Tp>& __x, const complex<_Tp>& __y)
1029     { return __complex_pow(__x, __y); }
1030 #endif
1031
1032   template<typename _Tp>
1033     inline complex<_Tp>
1034     pow(const _Tp& __x, const complex<_Tp>& __y)
1035     {
1036       return __x > _Tp() ? std::polar(pow(__x, __y.real()),
1037                                       __y.imag() * log(__x))
1038                          : std::pow(complex<_Tp>(__x), __y);
1039     }
1040
1041   /// 26.2.3  complex specializations
1042   /// complex<float> specialization
1043   template<>
1044     struct complex<float>
1045     {
1046       typedef float value_type;
1047       typedef __complex__ float _ComplexT;
1048
1049       _GLIBCXX_CONSTEXPR complex(_ComplexT __z) : _M_value(__z) { }
1050
1051       _GLIBCXX_CONSTEXPR complex(float __r = 0.0f, float __i = 0.0f)
1052 #if __cplusplus >= 201103L
1053       : _M_value{ __r, __i } { }
1054 #else
1055       {
1056         __real__ _M_value = __r;
1057         __imag__ _M_value = __i;
1058       }
1059 #endif
1060
1061       explicit _GLIBCXX_CONSTEXPR complex(const complex<double>&);
1062       explicit _GLIBCXX_CONSTEXPR complex(const complex<long double>&);
1063
1064 #if __cplusplus >= 201103L
1065       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1066       // DR 387. std::complex over-encapsulated.
1067       __attribute ((__abi_tag__ ("cxx11")))
1068       constexpr float
1069       real() { return __real__ _M_value; }
1070
1071       __attribute ((__abi_tag__ ("cxx11")))
1072       constexpr float
1073       imag() { return __imag__ _M_value; }
1074 #else
1075       float&
1076       real() { return __real__ _M_value; }
1077
1078       const float&
1079       real() const { return __real__ _M_value; }
1080
1081       float&
1082       imag() { return __imag__ _M_value; }
1083
1084       const float&
1085       imag() const { return __imag__ _M_value; }
1086 #endif
1087
1088       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1089       // DR 387. std::complex over-encapsulated.
1090       void
1091       real(float __val) { __real__ _M_value = __val; }
1092
1093       void
1094       imag(float __val) { __imag__ _M_value = __val; }
1095
1096       complex&
1097       operator=(float __f)
1098       {
1099         _M_value = __f;
1100         return *this;
1101       }
1102
1103       complex&
1104       operator+=(float __f)
1105       {
1106         _M_value += __f;
1107         return *this;
1108       }
1109
1110       complex&
1111       operator-=(float __f)
1112       {
1113         _M_value -= __f;
1114         return *this;
1115       }
1116
1117       complex&
1118       operator*=(float __f)
1119       {
1120         _M_value *= __f;
1121         return *this;
1122       }
1123
1124       complex&
1125       operator/=(float __f)
1126       {
1127         _M_value /= __f;
1128         return *this;
1129       }
1130
1131       // Let the compiler synthesize the copy and assignment
1132       // operator.  It always does a pretty good job.
1133       // complex& operator=(const complex&);
1134
1135       template<typename _Tp>
1136         complex&
1137         operator=(const complex<_Tp>&  __z)
1138         {
1139           __real__ _M_value = __z.real();
1140           __imag__ _M_value = __z.imag();
1141           return *this;
1142         }
1143
1144       template<typename _Tp>
1145         complex&
1146         operator+=(const complex<_Tp>& __z)
1147         {
1148           __real__ _M_value += __z.real();
1149           __imag__ _M_value += __z.imag();
1150           return *this;
1151         }
1152
1153       template<class _Tp>
1154         complex&
1155         operator-=(const complex<_Tp>& __z)
1156         {
1157           __real__ _M_value -= __z.real();
1158           __imag__ _M_value -= __z.imag();
1159           return *this;
1160         }
1161
1162       template<class _Tp>
1163         complex&
1164         operator*=(const complex<_Tp>& __z)
1165         {
1166           _ComplexT __t;
1167           __real__ __t = __z.real();
1168           __imag__ __t = __z.imag();
1169           _M_value *= __t;
1170           return *this;
1171         }
1172
1173       template<class _Tp>
1174         complex&
1175         operator/=(const complex<_Tp>& __z)
1176         {
1177           _ComplexT __t;
1178           __real__ __t = __z.real();
1179           __imag__ __t = __z.imag();
1180           _M_value /= __t;
1181           return *this;
1182         }
1183
1184       _GLIBCXX_USE_CONSTEXPR _ComplexT __rep() const { return _M_value; }
1185
1186     private:
1187       _ComplexT _M_value;
1188     };
1189
1190   /// 26.2.3  complex specializations
1191   /// complex<double> specialization
1192   template<>
1193     struct complex<double>
1194     {
1195       typedef double value_type;
1196       typedef __complex__ double _ComplexT;
1197
1198       _GLIBCXX_CONSTEXPR complex(_ComplexT __z) : _M_value(__z) { }
1199
1200       _GLIBCXX_CONSTEXPR complex(double __r = 0.0, double __i = 0.0)
1201 #if __cplusplus >= 201103L
1202       : _M_value{ __r, __i } { }
1203 #else
1204       {
1205         __real__ _M_value = __r;
1206         __imag__ _M_value = __i;
1207       }
1208 #endif
1209
1210       _GLIBCXX_CONSTEXPR complex(const complex<float>& __z)
1211       : _M_value(__z.__rep()) { }
1212
1213       explicit _GLIBCXX_CONSTEXPR complex(const complex<long double>&);
1214
1215 #if __cplusplus >= 201103L
1216       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1217       // DR 387. std::complex over-encapsulated.
1218       __attribute ((__abi_tag__ ("cxx11")))
1219       constexpr double
1220       real() { return __real__ _M_value; }
1221
1222       __attribute ((__abi_tag__ ("cxx11")))
1223       constexpr double
1224       imag() { return __imag__ _M_value; }
1225 #else
1226       double&
1227       real() { return __real__ _M_value; }
1228
1229       const double&
1230       real() const { return __real__ _M_value; }
1231
1232       double&
1233       imag() { return __imag__ _M_value; }
1234
1235       const double&
1236       imag() const { return __imag__ _M_value; }
1237 #endif
1238
1239       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1240       // DR 387. std::complex over-encapsulated.
1241       void
1242       real(double __val) { __real__ _M_value = __val; }
1243
1244       void
1245       imag(double __val) { __imag__ _M_value = __val; }
1246
1247       complex&
1248       operator=(double __d)
1249       {
1250         _M_value = __d;
1251         return *this;
1252       }
1253
1254       complex&
1255       operator+=(double __d)
1256       {
1257         _M_value += __d;
1258         return *this;
1259       }
1260
1261       complex&
1262       operator-=(double __d)
1263       {
1264         _M_value -= __d;
1265         return *this;
1266       }
1267
1268       complex&
1269       operator*=(double __d)
1270       {
1271         _M_value *= __d;
1272         return *this;
1273       }
1274
1275       complex&
1276       operator/=(double __d)
1277       {
1278         _M_value /= __d;
1279         return *this;
1280       }
1281
1282       // The compiler will synthesize this, efficiently.
1283       // complex& operator=(const complex&);
1284
1285       template<typename _Tp>
1286         complex&
1287         operator=(const complex<_Tp>& __z)
1288         {
1289           __real__ _M_value = __z.real();
1290           __imag__ _M_value = __z.imag();
1291           return *this;
1292         }
1293
1294       template<typename _Tp>
1295         complex&
1296         operator+=(const complex<_Tp>& __z)
1297         {
1298           __real__ _M_value += __z.real();
1299           __imag__ _M_value += __z.imag();
1300           return *this;
1301         }
1302
1303       template<typename _Tp>
1304         complex&
1305         operator-=(const complex<_Tp>& __z)
1306         {
1307           __real__ _M_value -= __z.real();
1308           __imag__ _M_value -= __z.imag();
1309           return *this;
1310         }
1311
1312       template<typename _Tp>
1313         complex&
1314         operator*=(const complex<_Tp>& __z)
1315         {
1316           _ComplexT __t;
1317           __real__ __t = __z.real();
1318           __imag__ __t = __z.imag();
1319           _M_value *= __t;
1320           return *this;
1321         }
1322
1323       template<typename _Tp>
1324         complex&
1325         operator/=(const complex<_Tp>& __z)
1326         {
1327           _ComplexT __t;
1328           __real__ __t = __z.real();
1329           __imag__ __t = __z.imag();
1330           _M_value /= __t;
1331           return *this;
1332         }
1333
1334       _GLIBCXX_USE_CONSTEXPR _ComplexT __rep() const { return _M_value; }
1335
1336     private:
1337       _ComplexT _M_value;
1338     };
1339
1340   /// 26.2.3  complex specializations
1341   /// complex<long double> specialization
1342   template<>
1343     struct complex<long double>
1344     {
1345       typedef long double value_type;
1346       typedef __complex__ long double _ComplexT;
1347
1348       _GLIBCXX_CONSTEXPR complex(_ComplexT __z) : _M_value(__z) { }
1349
1350       _GLIBCXX_CONSTEXPR complex(long double __r = 0.0L,
1351                                  long double __i = 0.0L)
1352 #if __cplusplus >= 201103L
1353       : _M_value{ __r, __i } { }
1354 #else
1355       {
1356         __real__ _M_value = __r;
1357         __imag__ _M_value = __i;
1358       }
1359 #endif
1360
1361       _GLIBCXX_CONSTEXPR complex(const complex<float>& __z)
1362       : _M_value(__z.__rep()) { }
1363
1364       _GLIBCXX_CONSTEXPR complex(const complex<double>& __z)
1365       : _M_value(__z.__rep()) { }
1366
1367 #if __cplusplus >= 201103L
1368       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1369       // DR 387. std::complex over-encapsulated.
1370       __attribute ((__abi_tag__ ("cxx11")))
1371       constexpr long double
1372       real() { return __real__ _M_value; }
1373
1374       __attribute ((__abi_tag__ ("cxx11")))
1375       constexpr long double
1376       imag() { return __imag__ _M_value; }
1377 #else
1378       long double&
1379       real() { return __real__ _M_value; }
1380
1381       const long double&
1382       real() const { return __real__ _M_value; }
1383
1384       long double&
1385       imag() { return __imag__ _M_value; }
1386
1387       const long double&
1388       imag() const { return __imag__ _M_value; }
1389 #endif
1390
1391       // _GLIBCXX_RESOLVE_LIB_DEFECTS
1392       // DR 387. std::complex over-encapsulated.
1393       void
1394       real(long double __val) { __real__ _M_value = __val; }
1395
1396       void
1397       imag(long double __val) { __imag__ _M_value = __val; }
1398
1399       complex&
1400       operator=(long double __r)
1401       {
1402         _M_value = __r;
1403         return *this;
1404       }
1405
1406       complex&
1407       operator+=(long double __r)
1408       {
1409         _M_value += __r;
1410         return *this;
1411       }
1412
1413       complex&
1414       operator-=(long double __r)
1415       {
1416         _M_value -= __r;
1417         return *this;
1418       }
1419
1420       complex&
1421       operator*=(long double __r)
1422       {
1423         _M_value *= __r;
1424         return *this;
1425       }
1426
1427       complex&
1428       operator/=(long double __r)
1429       {
1430         _M_value /= __r;
1431         return *this;
1432       }
1433
1434       // The compiler knows how to do this efficiently
1435       // complex& operator=(const complex&);
1436
1437       template<typename _Tp>
1438         complex&
1439         operator=(const complex<_Tp>& __z)
1440         {
1441           __real__ _M_value = __z.real();
1442           __imag__ _M_value = __z.imag();
1443           return *this;
1444         }
1445
1446       template<typename _Tp>
1447         complex&
1448         operator+=(const complex<_Tp>& __z)
1449         {
1450           __real__ _M_value += __z.real();
1451           __imag__ _M_value += __z.imag();
1452           return *this;
1453         }
1454
1455       template<typename _Tp>
1456         complex&
1457         operator-=(const complex<_Tp>& __z)
1458         {
1459           __real__ _M_value -= __z.real();
1460           __imag__ _M_value -= __z.imag();
1461           return *this;
1462         }
1463
1464       template<typename _Tp>
1465         complex&
1466         operator*=(const complex<_Tp>& __z)
1467         {
1468           _ComplexT __t;
1469           __real__ __t = __z.real();
1470           __imag__ __t = __z.imag();
1471           _M_value *= __t;
1472           return *this;
1473         }
1474
1475       template<typename _Tp>
1476         complex&
1477         operator/=(const complex<_Tp>& __z)
1478         {
1479           _ComplexT __t;
1480           __real__ __t = __z.real();
1481           __imag__ __t = __z.imag();
1482           _M_value /= __t;
1483           return *this;
1484         }
1485
1486       _GLIBCXX_USE_CONSTEXPR _ComplexT __rep() const { return _M_value; }
1487
1488     private:
1489       _ComplexT _M_value;
1490     };
1491
1492   // These bits have to be at the end of this file, so that the
1493   // specializations have all been defined.
1494   inline _GLIBCXX_CONSTEXPR
1495   complex<float>::complex(const complex<double>& __z)
1496   : _M_value(__z.__rep()) { }
1497
1498   inline _GLIBCXX_CONSTEXPR
1499   complex<float>::complex(const complex<long double>& __z)
1500   : _M_value(__z.__rep()) { }
1501
1502   inline _GLIBCXX_CONSTEXPR
1503   complex<double>::complex(const complex<long double>& __z)
1504   : _M_value(__z.__rep()) { }
1505
1506   // Inhibit implicit instantiations for required instantiations,
1507   // which are defined via explicit instantiations elsewhere.
1508   // NB:  This syntax is a GNU extension.
1509 #if _GLIBCXX_EXTERN_TEMPLATE
1510   extern template istream& operator>>(istream&, complex<float>&);
1511   extern template ostream& operator<<(ostream&, const complex<float>&);
1512   extern template istream& operator>>(istream&, complex<double>&);
1513   extern template ostream& operator<<(ostream&, const complex<double>&);
1514   extern template istream& operator>>(istream&, complex<long double>&);
1515   extern template ostream& operator<<(ostream&, const complex<long double>&);
1516
1517 #ifdef _GLIBCXX_USE_WCHAR_T
1518   extern template wistream& operator>>(wistream&, complex<float>&);
1519   extern template wostream& operator<<(wostream&, const complex<float>&);
1520   extern template wistream& operator>>(wistream&, complex<double>&);
1521   extern template wostream& operator<<(wostream&, const complex<double>&);
1522   extern template wistream& operator>>(wistream&, complex<long double>&);
1523   extern template wostream& operator<<(wostream&, const complex<long double>&);
1524 #endif
1525 #endif
1526
1527   // @} group complex_numbers
1528
1529 _GLIBCXX_END_NAMESPACE_VERSION
1530 } // namespace
1531
1532 namespace __gnu_cxx _GLIBCXX_VISIBILITY(default)
1533 {
1534 _GLIBCXX_BEGIN_NAMESPACE_VERSION
1535
1536   // See ext/type_traits.h for the primary template.
1537   template<typename _Tp, typename _Up>
1538     struct __promote_2<std::complex<_Tp>, _Up>
1539     {
1540     public:
1541       typedef std::complex<typename __promote_2<_Tp, _Up>::__type> __type;
1542     };
1543
1544   template<typename _Tp, typename _Up>
1545     struct __promote_2<_Tp, std::complex<_Up> >
1546     {
1547     public:
1548       typedef std::complex<typename __promote_2<_Tp, _Up>::__type> __type;
1549     };
1550
1551   template<typename _Tp, typename _Up>
1552     struct __promote_2<std::complex<_Tp>, std::complex<_Up> >
1553     {
1554     public:
1555       typedef std::complex<typename __promote_2<_Tp, _Up>::__type> __type;
1556     };
1557
1558 _GLIBCXX_END_NAMESPACE_VERSION
1559 } // namespace
1560
1561 #if __cplusplus >= 201103L
1562
1563 namespace std _GLIBCXX_VISIBILITY(default)
1564 {
1565 _GLIBCXX_BEGIN_NAMESPACE_VERSION
1566
1567   // Forward declarations.
1568   template<typename _Tp> std::complex<_Tp> acos(const std::complex<_Tp>&);
1569   template<typename _Tp> std::complex<_Tp> asin(const std::complex<_Tp>&);
1570   template<typename _Tp> std::complex<_Tp> atan(const std::complex<_Tp>&);
1571
1572   template<typename _Tp> std::complex<_Tp> acosh(const std::complex<_Tp>&);
1573   template<typename _Tp> std::complex<_Tp> asinh(const std::complex<_Tp>&);
1574   template<typename _Tp> std::complex<_Tp> atanh(const std::complex<_Tp>&);
1575   // DR 595.
1576   template<typename _Tp> _Tp               fabs(const std::complex<_Tp>&);
1577
1578   template<typename _Tp>
1579     inline std::complex<_Tp>
1580     __complex_acos(const std::complex<_Tp>& __z)
1581     {
1582       const std::complex<_Tp> __t = std::asin(__z);
1583       const _Tp __pi_2 = 1.5707963267948966192313216916397514L;
1584       return std::complex<_Tp>(__pi_2 - __t.real(), -__t.imag());
1585     }
1586
1587 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1588   inline __complex__ float
1589   __complex_acos(__complex__ float __z)
1590   { return __builtin_cacosf(__z); }
1591
1592   inline __complex__ double
1593   __complex_acos(__complex__ double __z)
1594   { return __builtin_cacos(__z); }
1595
1596   inline __complex__ long double
1597   __complex_acos(const __complex__ long double& __z)
1598   { return __builtin_cacosl(__z); }
1599
1600   template<typename _Tp>
1601     inline std::complex<_Tp>
1602     acos(const std::complex<_Tp>& __z)
1603     { return __complex_acos(__z.__rep()); }
1604 #else
1605   /// acos(__z) [8.1.2].
1606   //  Effects:  Behaves the same as C99 function cacos, defined
1607   //            in subclause 7.3.5.1.
1608   template<typename _Tp>
1609     inline std::complex<_Tp>
1610     acos(const std::complex<_Tp>& __z)
1611     { return __complex_acos(__z); }
1612 #endif
1613
1614   template<typename _Tp>
1615     inline std::complex<_Tp>
1616     __complex_asin(const std::complex<_Tp>& __z)
1617     {
1618       std::complex<_Tp> __t(-__z.imag(), __z.real());
1619       __t = std::asinh(__t);
1620       return std::complex<_Tp>(__t.imag(), -__t.real());
1621     }
1622
1623 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1624   inline __complex__ float
1625   __complex_asin(__complex__ float __z)
1626   { return __builtin_casinf(__z); }
1627
1628   inline __complex__ double
1629   __complex_asin(__complex__ double __z)
1630   { return __builtin_casin(__z); }
1631
1632   inline __complex__ long double
1633   __complex_asin(const __complex__ long double& __z)
1634   { return __builtin_casinl(__z); }
1635
1636   template<typename _Tp>
1637     inline std::complex<_Tp>
1638     asin(const std::complex<_Tp>& __z)
1639     { return __complex_asin(__z.__rep()); }
1640 #else
1641   /// asin(__z) [8.1.3].
1642   //  Effects:  Behaves the same as C99 function casin, defined
1643   //            in subclause 7.3.5.2.
1644   template<typename _Tp>
1645     inline std::complex<_Tp>
1646     asin(const std::complex<_Tp>& __z)
1647     { return __complex_asin(__z); }
1648 #endif
1649
1650   template<typename _Tp>
1651     std::complex<_Tp>
1652     __complex_atan(const std::complex<_Tp>& __z)
1653     {
1654       const _Tp __r2 = __z.real() * __z.real();
1655       const _Tp __x = _Tp(1.0) - __r2 - __z.imag() * __z.imag();
1656
1657       _Tp __num = __z.imag() + _Tp(1.0);
1658       _Tp __den = __z.imag() - _Tp(1.0);
1659
1660       __num = __r2 + __num * __num;
1661       __den = __r2 + __den * __den;
1662
1663       return std::complex<_Tp>(_Tp(0.5) * atan2(_Tp(2.0) * __z.real(), __x),
1664                                _Tp(0.25) * log(__num / __den));
1665     }
1666
1667 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1668   inline __complex__ float
1669   __complex_atan(__complex__ float __z)
1670   { return __builtin_catanf(__z); }
1671
1672   inline __complex__ double
1673   __complex_atan(__complex__ double __z)
1674   { return __builtin_catan(__z); }
1675
1676   inline __complex__ long double
1677   __complex_atan(const __complex__ long double& __z)
1678   { return __builtin_catanl(__z); }
1679
1680   template<typename _Tp>
1681     inline std::complex<_Tp>
1682     atan(const std::complex<_Tp>& __z)
1683     { return __complex_atan(__z.__rep()); }
1684 #else
1685   /// atan(__z) [8.1.4].
1686   //  Effects:  Behaves the same as C99 function catan, defined
1687   //            in subclause 7.3.5.3.
1688   template<typename _Tp>
1689     inline std::complex<_Tp>
1690     atan(const std::complex<_Tp>& __z)
1691     { return __complex_atan(__z); }
1692 #endif
1693
1694   template<typename _Tp>
1695     std::complex<_Tp>
1696     __complex_acosh(const std::complex<_Tp>& __z)
1697     {
1698       // Kahan's formula.
1699       return _Tp(2.0) * std::log(std::sqrt(_Tp(0.5) * (__z + _Tp(1.0)))
1700                                  + std::sqrt(_Tp(0.5) * (__z - _Tp(1.0))));
1701     }
1702
1703 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1704   inline __complex__ float
1705   __complex_acosh(__complex__ float __z)
1706   { return __builtin_cacoshf(__z); }
1707
1708   inline __complex__ double
1709   __complex_acosh(__complex__ double __z)
1710   { return __builtin_cacosh(__z); }
1711
1712   inline __complex__ long double
1713   __complex_acosh(const __complex__ long double& __z)
1714   { return __builtin_cacoshl(__z); }
1715
1716   template<typename _Tp>
1717     inline std::complex<_Tp>
1718     acosh(const std::complex<_Tp>& __z)
1719     { return __complex_acosh(__z.__rep()); }
1720 #else
1721   /// acosh(__z) [8.1.5].
1722   //  Effects:  Behaves the same as C99 function cacosh, defined
1723   //            in subclause 7.3.6.1.
1724   template<typename _Tp>
1725     inline std::complex<_Tp>
1726     acosh(const std::complex<_Tp>& __z)
1727     { return __complex_acosh(__z); }
1728 #endif
1729
1730   template<typename _Tp>
1731     std::complex<_Tp>
1732     __complex_asinh(const std::complex<_Tp>& __z)
1733     {
1734       std::complex<_Tp> __t((__z.real() - __z.imag())
1735                             * (__z.real() + __z.imag()) + _Tp(1.0),
1736                             _Tp(2.0) * __z.real() * __z.imag());
1737       __t = std::sqrt(__t);
1738
1739       return std::log(__t + __z);
1740     }
1741
1742 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1743   inline __complex__ float
1744   __complex_asinh(__complex__ float __z)
1745   { return __builtin_casinhf(__z); }
1746
1747   inline __complex__ double
1748   __complex_asinh(__complex__ double __z)
1749   { return __builtin_casinh(__z); }
1750
1751   inline __complex__ long double
1752   __complex_asinh(const __complex__ long double& __z)
1753   { return __builtin_casinhl(__z); }
1754
1755   template<typename _Tp>
1756     inline std::complex<_Tp>
1757     asinh(const std::complex<_Tp>& __z)
1758     { return __complex_asinh(__z.__rep()); }
1759 #else
1760   /// asinh(__z) [8.1.6].
1761   //  Effects:  Behaves the same as C99 function casin, defined
1762   //            in subclause 7.3.6.2.
1763   template<typename _Tp>
1764     inline std::complex<_Tp>
1765     asinh(const std::complex<_Tp>& __z)
1766     { return __complex_asinh(__z); }
1767 #endif
1768
1769   template<typename _Tp>
1770     std::complex<_Tp>
1771     __complex_atanh(const std::complex<_Tp>& __z)
1772     {
1773       const _Tp __i2 = __z.imag() * __z.imag();
1774       const _Tp __x = _Tp(1.0) - __i2 - __z.real() * __z.real();
1775
1776       _Tp __num = _Tp(1.0) + __z.real();
1777       _Tp __den = _Tp(1.0) - __z.real();
1778
1779       __num = __i2 + __num * __num;
1780       __den = __i2 + __den * __den;
1781
1782       return std::complex<_Tp>(_Tp(0.25) * (log(__num) - log(__den)),
1783                                _Tp(0.5) * atan2(_Tp(2.0) * __z.imag(), __x));
1784     }
1785
1786 #if _GLIBCXX_USE_C99_COMPLEX_TR1
1787   inline __complex__ float
1788   __complex_atanh(__complex__ float __z)
1789   { return __builtin_catanhf(__z); }
1790
1791   inline __complex__ double
1792   __complex_atanh(__complex__ double __z)
1793   { return __builtin_catanh(__z); }
1794
1795   inline __complex__ long double
1796   __complex_atanh(const __complex__ long double& __z)
1797   { return __builtin_catanhl(__z); }
1798
1799   template<typename _Tp>
1800     inline std::complex<_Tp>
1801     atanh(const std::complex<_Tp>& __z)
1802     { return __complex_atanh(__z.__rep()); }
1803 #else
1804   /// atanh(__z) [8.1.7].
1805   //  Effects:  Behaves the same as C99 function catanh, defined
1806   //            in subclause 7.3.6.3.
1807   template<typename _Tp>
1808     inline std::complex<_Tp>
1809     atanh(const std::complex<_Tp>& __z)
1810     { return __complex_atanh(__z); }
1811 #endif
1812
1813   template<typename _Tp>
1814     inline _Tp
1815     /// fabs(__z) [8.1.8].
1816     //  Effects:  Behaves the same as C99 function cabs, defined
1817     //            in subclause 7.3.8.1.
1818     fabs(const std::complex<_Tp>& __z)
1819     { return std::abs(__z); }
1820
1821   /// Additional overloads [8.1.9].
1822   template<typename _Tp>
1823     inline typename __gnu_cxx::__promote<_Tp>::__type
1824     arg(_Tp __x)
1825     {
1826       typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
1827 #if (_GLIBCXX_USE_C99_MATH && !_GLIBCXX_USE_C99_FP_MACROS_DYNAMIC)
1828       return std::signbit(__x) ? __type(3.1415926535897932384626433832795029L)
1829                                : __type();
1830 #else
1831       return std::arg(std::complex<__type>(__x));
1832 #endif
1833     }
1834
1835   template<typename _Tp>
1836     inline typename __gnu_cxx::__promote<_Tp>::__type
1837     imag(_Tp)
1838     { return _Tp(); }
1839
1840   template<typename _Tp>
1841     inline typename __gnu_cxx::__promote<_Tp>::__type
1842     norm(_Tp __x)
1843     {
1844       typedef typename __gnu_cxx::__promote<_Tp>::__type __type;
1845       return __type(__x) * __type(__x);
1846     }
1847
1848   template<typename _Tp>
1849     inline typename __gnu_cxx::__promote<_Tp>::__type
1850     real(_Tp __x)
1851     { return __x; }
1852
1853   template<typename _Tp, typename _Up>
1854     inline std::complex<typename __gnu_cxx::__promote_2<_Tp, _Up>::__type>
1855     pow(const std::complex<_Tp>& __x, const _Up& __y)
1856     {
1857       typedef typename __gnu_cxx::__promote_2<_Tp, _Up>::__type __type;
1858       return std::pow(std::complex<__type>(__x), __type(__y));
1859     }
1860
1861   template<typename _Tp, typename _Up>
1862     inline std::complex<typename __gnu_cxx::__promote_2<_Tp, _Up>::__type>
1863     pow(const _Tp& __x, const std::complex<_Up>& __y)
1864     {
1865       typedef typename __gnu_cxx::__promote_2<_Tp, _Up>::__type __type;
1866       return std::pow(__type(__x), std::complex<__type>(__y));
1867     }
1868
1869   template<typename _Tp, typename _Up>
1870     inline std::complex<typename __gnu_cxx::__promote_2<_Tp, _Up>::__type>
1871     pow(const std::complex<_Tp>& __x, const std::complex<_Up>& __y)
1872     {
1873       typedef typename __gnu_cxx::__promote_2<_Tp, _Up>::__type __type;
1874       return std::pow(std::complex<__type>(__x),
1875                       std::complex<__type>(__y));
1876     }
1877
1878   // Forward declarations.
1879   // DR 781.
1880   template<typename _Tp> std::complex<_Tp> proj(const std::complex<_Tp>&);
1881
1882   template<typename _Tp>
1883     std::complex<_Tp>
1884     __complex_proj(const std::complex<_Tp>& __z)
1885     {
1886       const _Tp __den = (__z.real() * __z.real()
1887                          + __z.imag() * __z.imag() + _Tp(1.0));
1888
1889       return std::complex<_Tp>((_Tp(2.0) * __z.real()) / __den,
1890                                (_Tp(2.0) * __z.imag()) / __den);
1891     }
1892
1893 #if _GLIBCXX_USE_C99_COMPLEX
1894   inline __complex__ float
1895   __complex_proj(__complex__ float __z)
1896   { return __builtin_cprojf(__z); }
1897
1898   inline __complex__ double
1899   __complex_proj(__complex__ double __z)
1900   { return __builtin_cproj(__z); }
1901
1902   inline __complex__ long double
1903   __complex_proj(const __complex__ long double& __z)
1904   { return __builtin_cprojl(__z); }
1905
1906   template<typename _Tp>
1907     inline std::complex<_Tp>
1908     proj(const std::complex<_Tp>& __z)
1909     { return __complex_proj(__z.__rep()); }
1910 #else
1911   template<typename _Tp>
1912     inline std::complex<_Tp>
1913     proj(const std::complex<_Tp>& __z)
1914     { return __complex_proj(__z); }
1915 #endif
1916
1917   // DR 1137.
1918   template<typename _Tp>
1919     inline typename __gnu_cxx::__promote<_Tp>::__type
1920     proj(_Tp __x)
1921     { return __x; }
1922
1923   template<typename _Tp>
1924     inline typename __gnu_cxx::__promote<_Tp>::__type
1925     conj(_Tp __x)
1926     { return __x; }
1927
1928 _GLIBCXX_END_NAMESPACE_VERSION
1929 } // namespace
1930
1931 #endif  // C++11
1932
1933 #endif  /* _GLIBCXX_COMPLEX */