pocketfft_hdronly.h 109 KB
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/*
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This file is part of pocketfft.

Copyright (C) 2010-2019 Max-Planck-Society
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Copyright (C) 2019 Peter Bell

For the odd-sized DCT-IV transforms:
  Copyright (C) 2003, 2007-14 Matteo Frigo
  Copyright (C) 2003, 2007-14 Massachusetts Institute of Technology

Authors: Martin Reinecke, Peter Bell
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All rights reserved.

Redistribution and use in source and binary forms, with or without modification,
are permitted provided that the following conditions are met:

* Redistributions of source code must retain the above copyright notice, this
  list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright notice, this
  list of conditions and the following disclaimer in the documentation and/or
  other materials provided with the distribution.
* Neither the name of the copyright holder nor the names of its contributors may
  be used to endorse or promote products derived from this software without
  specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
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#ifndef POCKETFFT_HDRONLY_H
#define POCKETFFT_HDRONLY_H

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#ifndef __cplusplus
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#error This file is C++ and requires a C++ compiler.
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#endif

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#if !(__cplusplus >= 201103L || _MSVC_LANG+0L >= 201103L)
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#error This file requires at least C++11 support.
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#endif

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#ifndef POCKETFFT_CACHE_SIZE
#define POCKETFFT_CACHE_SIZE 16
#endif

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#include <cmath>
#include <cstring>
#include <cstdlib>
#include <stdexcept>
#include <memory>
#include <vector>
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#include <complex>
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#if POCKETFFT_CACHE_SIZE!=0
#include <array>
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#include <mutex>
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#endif
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#ifndef POCKETFFT_NO_MULTITHREADING
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#include <mutex>
#include <condition_variable>
#include <thread>
#include <queue>
#include <atomic>
#include <functional>
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#ifdef POCKETFFT_PTHREADS
#  include <pthread.h>
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#endif
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#endif
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#if defined(__GNUC__)
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#define POCKETFFT_NOINLINE __attribute__((noinline))
#define POCKETFFT_RESTRICT __restrict__
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#elif defined(_MSC_VER)
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#define POCKETFFT_NOINLINE __declspec(noinline)
#define POCKETFFT_RESTRICT __restrict
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#else
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#define POCKETFFT_NOINLINE
#define POCKETFFT_RESTRICT
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#endif

namespace pocketfft {

namespace detail {

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using namespace std;

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using shape_t = vector<size_t>;
using stride_t = vector<ptrdiff_t>;

constexpr bool FORWARD  = true,
               BACKWARD = false;

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// only enable vector support for gcc>=5.0 and clang>=5.0
#ifndef POCKETFFT_NO_VECTORS
#define POCKETFFT_NO_VECTORS
#if defined(__INTEL_COMPILER)
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// do nothing. This is necessary because this compiler also sets __GNUC__.
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#elif defined(__clang__)
#if __clang__>=5
#undef POCKETFFT_NO_VECTORS
#endif
#elif defined(__GNUC__)
#if __GNUC__>=5
#undef POCKETFFT_NO_VECTORS
#endif
#endif
#endif

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template<typename T> struct VLEN { static constexpr size_t val=1; };

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#ifndef POCKETFFT_NO_VECTORS
#if (defined(__AVX512F__))
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template<> struct VLEN<float> { static constexpr size_t val=16; };
template<> struct VLEN<double> { static constexpr size_t val=8; };
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#elif (defined(__AVX__))
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template<> struct VLEN<float> { static constexpr size_t val=8; };
template<> struct VLEN<double> { static constexpr size_t val=4; };
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#elif (defined(__SSE2__))
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template<> struct VLEN<float> { static constexpr size_t val=4; };
template<> struct VLEN<double> { static constexpr size_t val=2; };
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#elif (defined(__VSX__))
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template<> struct VLEN<float> { static constexpr size_t val=4; };
template<> struct VLEN<double> { static constexpr size_t val=2; };
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#else
#define POCKETFFT_NO_VECTORS
#endif
#endif

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template<typename T> class arr
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  {
  private:
    T *p;
    size_t sz;

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#if defined(POCKETFFT_NO_VECTORS)
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    static T *ralloc(size_t num)
      {
      if (num==0) return nullptr;
      void *res = malloc(num*sizeof(T));
      if (!res) throw bad_alloc();
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      return reinterpret_cast<T *>(res);
      }
    static void dealloc(T *ptr)
      { free(ptr); }
#elif __cplusplus >= 201703L
    static T *ralloc(size_t num)
      {
      if (num==0) return nullptr;
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      void *res = aligned_alloc(64,num*sizeof(T));
      if (!res) throw bad_alloc();
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      return reinterpret_cast<T *>(res);
      }
    static void dealloc(T *ptr)
      { free(ptr); }
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#else // portable emulation
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    static T *ralloc(size_t num)
      {
      if (num==0) return nullptr;
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      void *ptr = malloc(num*sizeof(T)+64);
      if (!ptr) throw bad_alloc();
      T *res = reinterpret_cast<T *>
        ((reinterpret_cast<size_t>(ptr) & ~(size_t(63))) + 64);
      (reinterpret_cast<void**>(res))[-1] = ptr;
      return res;
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      }
    static void dealloc(T *ptr)
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      { if (ptr) free((reinterpret_cast<void**>(ptr))[-1]); }
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#endif
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  public:
    arr() : p(0), sz(0) {}
    arr(size_t n) : p(ralloc(n)), sz(n) {}
    arr(arr &&other)
      : p(other.p), sz(other.sz)
      { other.p=nullptr; other.sz=0; }
    ~arr() { dealloc(p); }

    void resize(size_t n)
      {
      if (n==sz) return;
      dealloc(p);
      p = ralloc(n);
      sz = n;
      }

    T &operator[](size_t idx) { return p[idx]; }
    const T &operator[](size_t idx) const { return p[idx]; }

    T *data() { return p; }
    const T *data() const { return p; }

    size_t size() const { return sz; }
  };

template<typename T> struct cmplx {
  T r, i;
  cmplx() {}
  cmplx(T r_, T i_) : r(r_), i(i_) {}
  void Set(T r_, T i_) { r=r_; i=i_; }
  void Set(T r_) { r=r_; i=T(0); }
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  void Split(T &r_, T &i_) const { r_=r; i_=i; }
  void SplitConj(T &r_, T &i_) const { r_=r; i_=-i; }
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  cmplx &operator+= (const cmplx &other)
    { r+=other.r; i+=other.i; return *this; }
  template<typename T2>cmplx &operator*= (T2 other)
    { r*=other; i*=other; return *this; }
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  template<typename T2>cmplx &operator*= (const cmplx<T2> &other)
    {
    T tmp = r*other.r - i*other.i;
    i = r*other.i + i*other.r;
    r = tmp;
    return *this;
    }
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  template<typename T2>cmplx &operator+= (const cmplx<T2> &other)
    { r+=other.r; i+=other.i; return *this; }
  template<typename T2>cmplx &operator-= (const cmplx<T2> &other)
    { r-=other.r; i-=other.i; return *this; }
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  template<typename T2> auto operator* (const T2 &other) const
    -> cmplx<decltype(r*other)>
    { return {r*other, i*other}; }
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  template<typename T2> auto operator+ (const cmplx<T2> &other) const
    -> cmplx<decltype(r+other.r)>
    { return {r+other.r, i+other.i}; }
  template<typename T2> auto operator- (const cmplx<T2> &other) const
    -> cmplx<decltype(r+other.r)>
    { return {r-other.r, i-other.i}; }
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  template<typename T2> auto operator* (const cmplx<T2> &other) const
    -> cmplx<decltype(r+other.r)>
    { return {r*other.r-i*other.i, r*other.i + i*other.r}; }
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  template<bool fwd, typename T2> auto special_mul (const cmplx<T2> &other) const
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    -> cmplx<decltype(r+other.r)>
    {
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    using Tres = cmplx<decltype(r+other.r)>;
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    return fwd ? Tres(r*other.r+i*other.i, i*other.r-r*other.i)
               : Tres(r*other.r-i*other.i, r*other.i+i*other.r);
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    }
};
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template<typename T> inline void PM(T &a, T &b, T c, T d)
  { a=c+d; b=c-d; }
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template<typename T> inline void PMINPLACE(T &a, T &b)
  { T t = a; a+=b; b=t-b; }
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template<typename T> inline void MPINPLACE(T &a, T &b)
  { T t = a; a-=b; b=t+b; }
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template<typename T> cmplx<T> conj(const cmplx<T> &a)
  { return {a.r, -a.i}; }
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template<bool fwd, typename T, typename T2> void special_mul (const cmplx<T> &v1, const cmplx<T2> &v2, cmplx<T> &res)
  {
  res = fwd ? cmplx<T>(v1.r*v2.r+v1.i*v2.i, v1.i*v2.r-v1.r*v2.i)
            : cmplx<T>(v1.r*v2.r-v1.i*v2.i, v1.r*v2.i+v1.i*v2.r);
  }
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template<typename T> void ROT90(cmplx<T> &a)
  { auto tmp_=a.r; a.r=-a.i; a.i=tmp_; }
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template<bool fwd, typename T> void ROTX90(cmplx<T> &a)
  { auto tmp_= fwd ? -a.r : a.r; a.r = fwd ? a.i : -a.i; a.i=tmp_; }
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//
// twiddle factor section
//

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/** Approximate sin(pi*a) within the range [-0.25, 0.25] */
inline float sinpi0(float a)
  {
  // adapted from https://stackoverflow.com/questions/42792939/
  float s = a * a;
  float r =      -5.957031250000000000e-01f;
  r = fma (r, s,  2.550399541854858398e+00f);
  r = fma (r, s, -5.167724132537841797e+00f);
  r = (a * s) * r;
  return fma (a, 3.141592741012573242e+00f, r);
  }

/** Approximate sin(pi*a) within the range [-0.25, 0.25] */
inline double sinpi0(double a)
  {
  // adapted from https://stackoverflow.com/questions/42792939/
  double s = a * a;
  double r =      4.6151442520157035e-4;
  r = fma (r, s, -7.3700183130883555e-3);
  r = fma (r, s,  8.2145868949323936e-2);
  r = fma (r, s, -5.9926452893214921e-1);
  r = fma (r, s,  2.5501640398732688e+0);
  r = fma (r, s, -5.1677127800499516e+0);
  s = s * a;
  r = r * s;
  return fma (a, 3.1415926535897931e+0, r);
  }

/** Approximate cos(pi*x)-1 for x in [-0.25,0.25] */
inline float cosm1pi0(float a)
  {
  // adapted from https://stackoverflow.com/questions/42792939/
  float s = a * a;
  float r =        2.313842773437500000e-01f;
  r = fmaf (r, s, -1.335021972656250000e+00f);
  r = fmaf (r, s,  4.058703899383544922e+00f);
  r = fmaf (r, s, -4.934802055358886719e+00f);
  return r*s;
  }

/** Approximate cos(pi*x)-1 for x in [-0.25,0.25] */
inline double cosm1pi0(double a)
  {
  // adapted from https://stackoverflow.com/questions/42792939/
  double s = a * a;
  double r =     -1.0369917389758117e-4;
  r = fma (r, s,  1.9294935641298806e-3);
  r = fma (r, s, -2.5806887942825395e-2);
  r = fma (r, s,  2.3533063028328211e-1);
  r = fma (r, s, -1.3352627688538006e+0);
  r = fma (r, s,  4.0587121264167623e+0);
  r = fma (r, s, -4.9348022005446790e+0);
  return r*s;
  }

template <typename T> void sincosm1pi0(T a_, T *POCKETFFT_RESTRICT res)
  {
  if (sizeof(T)>sizeof(double)) // don't have the code for long double
    {
    constexpr T pi = T(3.141592653589793238462643383279502884197L);
    auto s = sin(pi*a_);
    res[1] = s;
    res[0] = (s*s)/(-sqrt((1-s)*(1+s))-1);
    return;
    }
  res[0] = T(cosm1pi0(double(a_)));
  res[1] = T(sinpi0(double(a_)));
  }

template <typename T> T sinpi(T a)
  {
  // reduce argument to primary approximation interval (-0.25, 0.25)
  auto r = nearbyint (a + a); // must use IEEE-754 "to nearest" rounding
  auto i = (int64_t)r;
  auto t = fma (T(-0.5), r, a);

  switch (i%4)
    {
    case 0:
      return sinpi0(t);
    case 1: case -3:
      return cosm1pi0(t) + T(1.);
    case 2: case -2:
      return T(0.) - sinpi0(t);
    case 3: case -1:
      return T(-1.) - cosm1pi0(t);
    }
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  throw runtime_error("cannot happen");
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  }

template <typename T> T cospi(T a)
  {
  // reduce argument to primary approximation interval (-0.25, 0.25)
  auto r = nearbyint (a + a); // must use IEEE-754 "to nearest" rounding
  auto i = (int64_t)r;
  auto t = fma (T(-0.5), r, a);

  switch (i%4)
    {
    case 0:
      return cosm1pi0(t) + T(1.);
    case 1: case -3:
      return T(0.) - sinpi0(t);
    case 2: case -2:
      return T(-1.) - cosm1pi0(t);
    case 3: case -1:
      return sinpi0(t);
    }
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  throw runtime_error("cannot happen");
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  }

inline long double cospi(long double a)
  {
  constexpr auto pi = 3.141592653589793238462643383279502884197L;
  return sizeof(long double) > sizeof(double) ? cos(a * pi) :
    static_cast<long double>(cospi(static_cast<double>(a)));
  }

inline long double sinpi(long double a)
  {
  constexpr auto pi = 3.141592653589793238462643383279502884197L;
  return sizeof(long double) > sizeof(double) ? sin(a * pi) :
    static_cast<long double>(sinpi(static_cast<double>(a)));
  }

template <typename T> void sincospi(T a, T *POCKETFFT_RESTRICT res)
  {
  // reduce argument to primary approximation interval (-0.25, 0.25)
  auto r = nearbyint (a + a); // must use IEEE-754 "to nearest" rounding
  auto i = (int64_t)r;
  auto t = fma (T(-0.5), r, a);

  auto c=cosm1pi0(t)+T(1.);
  auto s=sinpi0(t);

  // map results according to quadrant
  if (i & 2)
    {
    s = T(0.)-s;
    c = T(0.)-c;
    }
  if (i & 1)
    {
    swap(s, c);
    c = T(0.)-c;
    }
  res[0]=c;
  res[1]=s;
  }

inline void sincospi(long double a, long double *POCKETFFT_RESTRICT res)
  {
  if (sizeof(long double) > sizeof(double))
    {
    constexpr auto pi = 3.141592653589793238462643383279502884197L;
    res[0] = cos(pi * a);
    res[1] = sin(pi * a);
    }
  else
    {
    double sincos[2];
    sincospi(static_cast<double>(a), sincos);
    res[0] = static_cast<long double>(sincos[0]);
    res[1] = static_cast<long double>(sincos[1]);
    }
  }

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template<typename T> class sincos_2pibyn
  {
  private:
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    using Thigh = typename conditional<(sizeof(T)>sizeof(double)), T, double>::type;
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    arr<T> data;

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    POCKETFFT_NOINLINE void calc_first_octant(size_t den,
      T * POCKETFFT_RESTRICT res)
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      {
      size_t n = (den+4)>>3;
      if (n==0) return;
      res[0]=1.; res[1]=0.;
      if (n==1) return;
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      size_t l1 = size_t(sqrt(n));
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      arr<Thigh> tmp(2*l1);
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      for (size_t i=1; i<l1; ++i)
        {
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        sincosm1pi0(Thigh(2*i)/Thigh(den),&tmp[2*i]);
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        res[2*i  ] = T(tmp[2*i]+1);
        res[2*i+1] = T(tmp[2*i+1]);
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        }
      size_t start=l1;
      while(start<n)
        {
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        Thigh cs[2];
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        sincosm1pi0((Thigh(2*start))/Thigh(den),cs);
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        res[2*start] = T(cs[0]+1);
        res[2*start+1] = T(cs[1]);
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        size_t end = l1;
        if (start+end>n) end = n-start;
        for (size_t i=1; i<end; ++i)
          {
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          Thigh csx[2]={tmp[2*i], tmp[2*i+1]};
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          res[2*(start+i)] = T(((cs[0]*csx[0] - cs[1]*csx[1] + cs[0]) + csx[0]) + 1);
          res[2*(start+i)+1] = T((cs[0]*csx[1] + cs[1]*csx[0]) + cs[1] + csx[1]);
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          }
        start += l1;
        }
      }

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    void calc_first_quadrant(size_t n, T * POCKETFFT_RESTRICT res)
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      {
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      T * POCKETFFT_RESTRICT p = res+n;
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      calc_first_octant(n<<1, p);
      size_t ndone=(n+2)>>2;
      size_t i=0, idx1=0, idx2=2*ndone-2;
      for (; i+1<ndone; i+=2, idx1+=2, idx2-=2)
        {
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        res[idx1] = p[2*i  ]; res[idx1+1] = p[2*i+1];
        res[idx2] = p[2*i+3]; res[idx2+1] = p[2*i+2];
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        }
      if (i!=ndone)
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        { res[idx1] = p[2*i]; res[idx1+1] = p[2*i+1]; }
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      }

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    void calc_first_half(size_t n, T * POCKETFFT_RESTRICT res)
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      {
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      int ndone=int(n+1)>>1;
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      T * p = res+n-1;
      calc_first_octant(n<<2, p);
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      int i4=0, in=int(n), i=0;
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      for (; i4<=in-i4; ++i, i4+=4) // octant 0
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        { res[2*i] = p[2*i4]; res[2*i+1] = p[2*i4+1]; }
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      for (; i4-in <= 0; ++i, i4+=4) // octant 1
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        { auto xm = in-i4; res[2*i] = p[2*xm+1]; res[2*i+1] = p[2*xm]; }
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      for (; i4<=3*in-i4; ++i, i4+=4) // octant 2
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        { auto xm = i4-in; res[2*i] = -p[2*xm+1]; res[2*i+1] = p[2*xm]; }
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      for (; i<ndone; ++i, i4+=4) // octant 3
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        { auto xm = 2*in-i4; res[2*i] = -p[2*xm]; res[2*i+1] = p[2*xm+1]; }
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      }

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    void fill_first_quadrant(size_t n, T * POCKETFFT_RESTRICT res)
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      {
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      constexpr T hsqt2 = T(0.707106781186547524400844362104849L);
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      size_t quart = n>>2;
      if ((n&7)==0)
        res[quart] = res[quart+1] = hsqt2;
      for (size_t i=2, j=2*quart-2; i<quart; i+=2, j-=2)
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        { res[j] = res[i+1]; res[j+1] = res[i]; }
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      }

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    POCKETFFT_NOINLINE void fill_first_half(size_t n, T * POCKETFFT_RESTRICT res)
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      {
      size_t half = n>>1;
      if ((n&3)==0)
        for (size_t i=0; i<half; i+=2)
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          { res[i+half] = -res[i+1]; res[i+half+1] = res[i]; }
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      else
        for (size_t i=2, j=2*half-2; i<half; i+=2, j-=2)
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          { res[j] = -res[i]; res[j+1] = res[i+1]; }
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      }

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    void fill_second_half(size_t n, T * POCKETFFT_RESTRICT res)
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      {
      if ((n&1)==0)
        for (size_t i=0; i<n; ++i)
          res[i+n] = -res[i];
      else
        for (size_t i=2, j=2*n-2; i<n; i+=2, j-=2)
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          { res[j] = res[i]; res[j+1] = -res[i+1]; }
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      }

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    POCKETFFT_NOINLINE void sincos_2pibyn_half(size_t n, T * POCKETFFT_RESTRICT res)
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      {
      if ((n&3)==0)
        {
        calc_first_octant(n, res);
        fill_first_quadrant(n, res);
        fill_first_half(n, res);
        }
      else if ((n&1)==0)
        {
        calc_first_quadrant(n, res);
        fill_first_half(n, res);
        }
      else
        calc_first_half(n, res);
      }

  public:
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    POCKETFFT_NOINLINE sincos_2pibyn(size_t n, bool half)
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      : data(2*n)
      {
      sincos_2pibyn_half(n, data.data());
      if (!half) fill_second_half(n, data.data());
      }

    T operator[](size_t idx) const { return data[idx]; }
    const T *rdata() const { return data; }
    const cmplx<T> *cdata() const
      { return reinterpret_cast<const cmplx<T> *>(data.data()); }
  };

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struct util // hack to avoid duplicate symbols
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  {
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  static POCKETFFT_NOINLINE size_t largest_prime_factor (size_t n)
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    {
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    size_t res=1;
    while ((n&1)==0)
      { res=2; n>>=1; }
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    for (size_t x=3; x*x<=n; x+=2)
      while ((n%x)==0)
        { res=x; n/=x; }
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    if (n>1) res=n;
    return res;
    }

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  static POCKETFFT_NOINLINE double cost_guess (size_t n)
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    {
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    constexpr double lfp=1.1; // penalty for non-hardcoded larger factors
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    size_t ni=n;
    double result=0.;
    while ((n&1)==0)
      { result+=2; n>>=1; }
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    for (size_t x=3; x*x<=n; x+=2)
      while ((n%x)==0)
        {
        result+= (x<=5) ? double(x) : lfp*double(x); // penalize larger prime factors
        n/=x;
        }
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    if (n>1) result+=(n<=5) ? double(n) : lfp*double(n);
    return result*double(ni);
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    }

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  /* returns the smallest composite of 2, 3, 5, 7 and 11 which is >= n */
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  static POCKETFFT_NOINLINE size_t good_size_cmplx(size_t n)
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    {
    if (n<=12) return n;

    size_t bestfac=2*n;
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    for (size_t f11=1; f11<bestfac; f11*=11)
      for (size_t f117=f11; f117<bestfac; f117*=7)
        for (size_t f1175=f117; f1175<bestfac; f1175*=5)
          {
          size_t x=f1175;
          while (x<n) x*=2;
          for (;;)
            {
            if (x<n)
              x*=3;
            else if (x>n)
              {
              if (x<bestfac) bestfac=x;
              if (x&1) break;
              x>>=1;
              }
            else
              return n;
            }
          }
    return bestfac;
    }

  /* returns the smallest composite of 2, 3, 5 which is >= n */
  static POCKETFFT_NOINLINE size_t good_size_real(size_t n)
    {
    if (n<=6) return n;

    size_t bestfac=2*n;
    for (size_t f5=1; f5<bestfac; f5*=5)
      {
      size_t x = f5;
      while (x<n) x *= 2;
      for (;;)
        {
        if (x<n)
          x*=3;
        else if (x>n)
          {
          if (x<bestfac) bestfac=x;
          if (x&1) break;
          x>>=1;
          }
        else
          return n;
        }
      }
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    return bestfac;
    }
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  static size_t prod(const shape_t &shape)
    {
    size_t res=1;
    for (auto sz: shape)
      res*=sz;
    return res;
    }
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  static POCKETFFT_NOINLINE void sanity_check(const shape_t &shape,
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    const stride_t &stride_in, const stride_t &stride_out, bool inplace)
    {
    auto ndim = shape.size();
    if (ndim<1) throw runtime_error("ndim must be >= 1");
    if ((stride_in.size()!=ndim) || (stride_out.size()!=ndim))
      throw runtime_error("stride dimension mismatch");
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    if (inplace && (stride_in!=stride_out))
      throw runtime_error("stride mismatch");
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    }

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  static POCKETFFT_NOINLINE void sanity_check(const shape_t &shape,
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    const stride_t &stride_in, const stride_t &stride_out, bool inplace,
    const shape_t &axes)
    {
    sanity_check(shape, stride_in, stride_out, inplace);
    auto ndim = shape.size();
    shape_t tmp(ndim,0);
    for (auto ax : axes)
      {
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      if (ax>=ndim) throw invalid_argument("bad axis number");
      if (++tmp[ax]>1) throw invalid_argument("axis specified repeatedly");
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      }
    }

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  static POCKETFFT_NOINLINE void sanity_check(const shape_t &shape,
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    const stride_t &stride_in, const stride_t &stride_out, bool inplace,
    size_t axis)
    {
    sanity_check(shape, stride_in, stride_out, inplace);
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    if (axis>=shape.size()) throw invalid_argument("bad axis number");
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    }
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#ifdef POCKETFFT_NO_MULTITHREADING
  static size_t thread_count (size_t /*nthreads*/, const shape_t &/*shape*/,
    size_t /*axis*/, size_t /*vlen*/)
    { return 1; }
#else
  static size_t thread_count (size_t nthreads, const shape_t &shape,
    size_t axis, size_t vlen)
    {
    if (nthreads==1) return 1;
    size_t size = prod(shape);
    size_t parallel = size / (shape[axis] * vlen);
    if (shape[axis] < 1000)
      parallel /= 4;
    size_t max_threads = nthreads == 0 ?
      thread::hardware_concurrency() : nthreads;
    return max(size_t(1), min(parallel, max_threads));
    }
#endif
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  };
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namespace threading {
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#ifdef POCKETFFT_NO_MULTITHREADING

constexpr size_t thread_id = 0;
constexpr size_t num_threads = 1;

template <typename Func>
void thread_map(size_t /* nthreads */, Func f)
  { f(); }

#else

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thread_local size_t thread_id = 0;
thread_local size_t num_threads = 1;

class latch
  {
    atomic<size_t> num_left_;
    mutex mut_;
    condition_variable completed_;
    using lock_t = unique_lock<mutex>;

  public:
    latch(size_t n): num_left_(n) {}

    void count_down()
      {
      {
      lock_t lock(mut_);
      if (--num_left_)
        return;
      }
      completed_.notify_all();
      }

    void wait()
      {
      lock_t lock(mut_);
      completed_.wait(lock, [this]{ return is_ready(); });
      }
    bool is_ready() { return num_left_ == 0; }
  };

template <typename T> class concurrent_queue
  {
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    queue<T> q_;
    mutex mut_;
    condition_variable item_added_;
    bool shutdown_;
    using lock_t = unique_lock<mutex>;
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  public:
    concurrent_queue(): shutdown_(false) {}

    void push(T val)
      {
      {
      lock_t lock(mut_);
      if (shutdown_)
        throw runtime_error("Item added to queue after shutdown");
      q_.push(move(val));
      }
      item_added_.notify_one();
      }

    bool pop(T & val)
      {
      lock_t lock(mut_);
      item_added_.wait(lock, [this] { return (!q_.empty() || shutdown_); });
      if (q_.empty())
        return false;  // We are shutting down

      val = std::move(q_.front());
      q_.pop();
      return true;
      }

    void shutdown()
      {
      {
      lock_t lock(mut_);
      shutdown_ = true;
      }
      item_added_.notify_all();
      }
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    void restart() { shutdown_ = false; }
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  };

class thread_pool
  {
    concurrent_queue<function<void()>> work_queue_;
    vector<thread> threads_;

    void worker_main()
      {
      function<void()> work;
      while (work_queue_.pop(work))
        work();
      }

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    void create_threads()
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      {
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      size_t nthreads = threads_.size();
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      for (size_t i=0; i<nthreads; ++i)
        {
        try { threads_[i] = thread([this]{ worker_main(); }); }
        catch (...)
          {
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          shutdown();
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          throw;
          }
        }
      }

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  public:
    explicit thread_pool(size_t nthreads):
      threads_(nthreads)
      { create_threads(); }

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    thread_pool(): thread_pool(thread::hardware_concurrency()) {}

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    ~thread_pool() { shutdown(); }

    void submit(function<void()> work)
      {
      work_queue_.push(move(work));
      }

    void shutdown()
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      {
      work_queue_.shutdown();
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      for (auto &thread : threads_)
        if (thread.joinable())
          thread.join();
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      }

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    void restart()
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      {
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      work_queue_.restart();
      create_threads();
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      }
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  };
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thread_pool & get_pool()
  {
  static thread_pool pool;
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#ifdef POCKETFFT_PTHREADS
  static once_flag f;
  call_once(f,
    []{
    pthread_atfork(
      +[]{ get_pool().shutdown(); },  // prepare
      +[]{ get_pool().restart(); },   // parent
      +[]{ get_pool().restart(); }    // child
      );
    });
#endif

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  return pool;
  }

/** Map a function f over nthreads */
template <typename Func>
void thread_map(size_t nthreads, Func f)
  {
  if (nthreads == 0)
    nthreads = thread::hardware_concurrency();

  if (nthreads == 1)
    { f(); return; }

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  auto & pool = get_pool();
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  latch counter(nthreads);
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  exception_ptr ex;
  mutex ex_mut;
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  for (size_t i=0; i<nthreads; ++i)
    {
    pool.submit(
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      [&f, &counter, &ex, &ex_mut, i, nthreads] {
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      thread_id = i;
      num_threads = nthreads;
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      try { f(); }
      catch (...)
        {
        lock_guard<mutex> lock(ex_mut);
        ex = current_exception();
        }
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      counter.count_down();
      });
    }
  counter.wait();
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  if (ex)
    rethrow_exception(ex);
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  }
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#endif

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}

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//
// complex FFTPACK transforms
//

template<typename T0> class cfftp
  {
  private:
    struct fctdata
      {
      size_t fct;
      cmplx<T0> *tw, *tws;
      };

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    size_t length;
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    arr<cmplx<T0>> mem;
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    vector<fctdata> fact;
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    void add_factor(size_t factor)
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      { fact.push_back({factor, nullptr, nullptr}); }
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template<bool fwd, typename T> void pass2 (size_t ido, size_t l1,
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  const T * POCKETFFT_RESTRICT cc, T * POCKETFFT_RESTRICT ch,
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  const cmplx<T0> * POCKETFFT_RESTRICT wa) const
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  {
  constexpr size_t cdim=2;

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  auto CH = [ch,ido,l1](size_t a, size_t b, size_t c) -> T&
    { return ch[a+ido*(b+l1*c)]; };
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  auto CC = [cc,ido,cdim](size_t a, size_t b, size_t c) -> const T&
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    { return cc[a+ido*(b+cdim*c)]; };
  auto WA = [wa, ido](size_t x, size_t i)
    { return wa[i-1+x*(ido-1)]; };

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  if (ido==1)
    for (size_t k=0; k<l1; ++k)
      {
      CH(0,k,0) = CC(0,0,k)+CC(0,1,k);
      CH(0,k,1) = CC(0,0,k)-CC(0,1,k);
      }
  else
    for (size_t k=0; k<l1; ++k)
      {
      CH(0,k,0) = CC(0,0,k)+CC(0,1,k);
      CH(0,k,1) = CC(0,0,k)-CC(0,1,k);
      for (size_t i=1; i<ido; ++i)
        {
        CH(i,k,0) = CC(i,0,k)+CC(i,1,k);
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        special_mul<fwd>(CC(i,0,k)-CC(i,1,k),WA(0,i),CH(i,k,1));
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        }
      }
  }

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#define POCKETFFT_PREP3(idx) \
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        T t0 = CC(idx,0,k), t1, t2; \
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        PM (t1,t2,CC(idx,1,k),CC(idx,2,k)); \
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        CH(idx,k,0)=t0+t1;
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#define POCKETFFT_PARTSTEP3a(u1,u2,twr,twi) \
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        { \
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        T ca=t0+t1*twr; \
        T cb{-t2.i*twi, t2.r*twi}; \
        PM(CH(0,k,u1),CH(0,k,u2),ca,cb) ;\
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        }
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#define POCKETFFT_PARTSTEP3b(u1,u2,twr,twi) \
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        { \
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        T ca=t0+t1*twr; \
        T cb{-t2.i*twi, t2.r*twi}; \
        special_mul<fwd>(ca+cb,WA(u1-1,i),CH(i,k,u1)); \
        special_mul<fwd>(ca-cb,WA(u2-1,i),CH(i,k,u2)); \
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        }
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template<bool fwd, typename T> void pass3 (size_t ido, size_t l1,
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  const T * POCKETFFT_RESTRICT cc, T * POCKETFFT_RESTRICT ch,
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  const cmplx<T0> * POCKETFFT_RESTRICT wa) const
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  {
  constexpr size_t cdim=3;
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  constexpr T0 tw1r=-0.5,
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               tw1i= (fwd ? -1: 1) * T0(0.8660254037844386467637231707529362L);
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  auto CH = [ch,ido,l1](size_t a, size_t b, size_t c) -> T&
    { return ch[a+ido*(b+l1*c)]; };
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  auto CC = [cc,ido,cdim](size_t a, size_t b, size_t c) -> const T&
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    { return cc[a+ido*(b+cdim*c)]; };
  auto WA = [wa, ido](size_t x, size_t i)
    { return wa[i-1+x*(ido-1)]; };

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  if (ido==1)
    for (size_t k=0; k<l1; ++k)
      {
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      POCKETFFT_PREP3(0)
      POCKETFFT_PARTSTEP3a(1,2,tw1r,tw1i)
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      }
  else
    for (size_t k=0; k<l1; ++k)
      {
      {
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      POCKETFFT_PREP3(0)
      POCKETFFT_PARTSTEP3a(1,2,tw1r,tw1i)
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      }
      for (size_t i=1; i<ido; ++i)
        {
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        POCKETFFT_PREP3(i)
        POCKETFFT_PARTSTEP3b(1,2,tw1r,tw1i)
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        }
      }
  }

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#undef POCKETFFT_PARTSTEP3b
#undef POCKETFFT_PARTSTEP3a
#undef POCKETFFT_PREP3
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template<bool fwd, typename T> void pass4 (size_t ido, size_t l1,
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  const T * POCKETFFT_RESTRICT cc, T * POCKETFFT_RESTRICT ch,
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  const cmplx<T0> * POCKETFFT_RESTRICT wa) const
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  {
  constexpr size_t cdim=4;

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  auto CH = [ch,ido,l1](size_t a, size_t b, size_t c) -> T&
    { return ch[a+ido*(b+l1*c)]; };
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  auto CC = [cc,ido,cdim](size_t a, size_t b, size_t c) -> const T&
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    { return cc[a+ido*(b+cdim*c)]; };
  auto WA = [wa, ido](size_t x, size_t i)
    { return wa[i-1+x*(ido-1)]; };

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  if (ido==1)
    for (size_t k=0; k<l1; ++k)
      {
      T t1, t2, t3, t4;
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      PM(t2,t1,CC(0,0,k),CC(0,2,k));
      PM(t3,t4,CC(0,1,k),CC(0,3,k));
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      ROTX90<fwd>(t4);
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      PM(CH(0,k,0),CH(0,k,2),t2,t3);
      PM(CH(0,k,1),CH(0,k,3),t1,t4);
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      }
  else
    for (size_t k=0; k<l1; ++k)
      {
      {
      T t1, t2, t3, t4;
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      PM(t2,t1,CC(0,0,k),CC(0,2,k));
      PM(t3,t4,CC(0,1,k),CC(0,3,k));
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      ROTX90<fwd>(t4);
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      PM(CH(0,k,0),CH(0,k,2),t2,t3);
      PM(CH(0,k,1),CH(0,k,3),t1,t4);
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      }
      for (size_t i=1; i<ido; ++i)
        {
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        T t1, t2, t3, t4;
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        T cc0=CC(i,0,k), cc1=CC(i,1,k),cc2=CC(i,2,k),cc3=CC(i,3,k);
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        PM(t2,t1,cc0,cc2);
        PM(t3,t4,cc1,cc3);
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        ROTX90<fwd>(t4);
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        CH(i,k,0) = t2+t3;
        special_mul<fwd>(t1+t4,WA(0,i),CH(i,k,1));
        special_mul<fwd>(t2-t3,WA(1,i),CH(i,k,2));
        special_mul<fwd>(t1-t4,WA(2,i),CH(i,k,3));
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        }
      }
  }

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#define POCKETFFT_PREP5(idx) \
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        T t0 = CC(idx,0,k), t1, t2, t3, t4; \
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        PM (t1,t4,CC(idx,1,k),CC(idx,4,k)); \
        PM (t2,t3,CC(idx,2,k),CC(idx,3,k)); \
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        CH(idx,k,0).r=t0.r+t1.r+t2.r; \
        CH(idx,k,0).i=t0.i+t1.i+t2.i;

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#define POCKETFFT_PARTSTEP5a(u1,u2,twar,twbr,twai,twbi) \
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        { \
        T ca,cb; \
        ca.r=t0.r+twar*t1.r+twbr*t2.r; \
        ca.i=t0.i+twar*t1.i+twbr*t2.i; \
        cb.i=twai*t4.r twbi*t3.r; \
        cb.r=-(twai*t4.i twbi*t3.i); \
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        PM(CH(0,k,u1),CH(0,k,u2),ca,cb); \
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        }

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#define POCKETFFT_PARTSTEP5b(u1,u2,twar,twbr,twai,twbi) \
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        { \
        T ca,cb,da,db; \
        ca.r=t0.r+twar*t1.r+twbr*t2.r; \
        ca.i=t0.i+twar*t1.i+twbr*t2.i; \
        cb.i=twai*t4.r twbi*t3.r; \
        cb.r=-(twai*t4.i twbi*t3.i); \
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        special_mul<fwd>(ca+cb,WA(u1-1,i),CH(i,k,u1)); \
        special_mul<fwd>(ca-cb,WA(u2-1,i),CH(i,k,u2)); \
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        }
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template<bool fwd, typename T> void pass5 (size_t ido, size_t l1,
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  const T * POCKETFFT_RESTRICT cc, T * POCKETFFT_RESTRICT ch,
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  const cmplx<T0> * POCKETFFT_RESTRICT wa) const
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  {
  constexpr size_t cdim=5;
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sync  
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  constexpr T0 tw1r= T0(0.3090169943749474241022934171828191L),
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               tw1i= (fwd ? -1: 1) * T0(0.9510565162951535721164393333793821L),
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sync  
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               tw2r= T0(-0.8090169943749474241022934171828191L),
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               tw2i= (fwd ? -1: 1) * T0(0.5877852522924731291687059546390728L);
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  auto CH = [ch,ido,l1](size_t a, size_t b, size_t c) -> T&
    { return ch[a+ido*(b+l1*c)]; };
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  auto CC = [cc,ido,cdim](size_t a, size_t b, size_t c) -> const T&
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    { return cc[a+ido*(b+cdim*c)]; };
  auto WA = [wa, ido](size_t x, size_t i)
    { return wa[i-1+x*(ido-1)]; };

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  if (ido==1)
    for (size_t k=0; k<l1; ++k)
      {
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      POCKETFFT_PREP5(0)
      POCKETFFT_PARTSTEP5a(1,4,tw1r,tw2r,+tw1i,+tw2i)
      POCKETFFT_PARTSTEP5a(2,3,tw2r,tw1r,+tw2i,-tw1i)
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      }
  else
    for (size_t k=0; k<l1; ++k)
      {
      {
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      POCKETFFT_PREP5(0)
      POCKETFFT_PARTSTEP5a(1,4,tw1r,tw2r,+tw1i,+tw2i)
      POCKETFFT_PARTSTEP5a(2,3,tw2r,tw1r,+tw2i,-tw1i)
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      }
      for (size_t i=1; i<ido; ++i)
        {
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        POCKETFFT_PREP5(i)
        POCKETFFT_PARTSTEP5b(1,4,tw1r,tw2r,+tw1i,+tw2i)
        POCKETFFT_PARTSTEP5b(2,3,tw2r,tw1r,+tw2i,-tw1i)
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        }
      }
  }

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#undef POCKETFFT_PARTSTEP5b
#undef POCKETFFT_PARTSTEP5a
#undef POCKETFFT_PREP5
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#define POCKETFFT_PREP7(idx) \
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        T t1 = CC(idx,0,k), t2, t3, t4, t5, t6, t7; \
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        PM (t2,t7,CC(idx,1,k),CC(idx,6,k)); \
        PM (t3,t6,CC(idx,2,k),CC(idx,5,k)); \
        PM (t4,t5,CC(idx,3,k),CC(idx,4,k)); \