pffft.c

/* Copyright (c) 2011  Julien Pommier ( pommier@modartt.com )

   Based on original fortran 77 code from FFTPACKv4 from NETLIB
   (http://www.netlib.org/fftpack), authored by Dr Paul Swarztrauber
   of NCAR, in 1985.

   As confirmed by the NCAR fftpack software curators, the following
   FFTPACKv5 license applies to FFTPACKv4 sources. My changes are
   released under the same terms.

   FFTPACK license:

   http://www.cisl.ucar.edu/css/software/fftpack5/ftpk.html

   Copyright (c) 2004 the University Corporation for Atmospheric
   Research ("UCAR"). All rights reserved. Developed by NCAR's
   Computational and Information Systems Laboratory, UCAR,
   www.cisl.ucar.edu.

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

   - Neither the names of NCAR's Computational and Information Systems
   Laboratory, the University Corporation for Atmospheric Research,
   nor the names of its sponsors or contributors may be used to
   endorse or promote products derived from this Software without
   specific prior written permission.  

   - Redistributions of source code must retain the above copyright
   notices, this list of conditions, and the disclaimer below.

   - Redistributions in binary form must reproduce the above copyright
   notice, this list of conditions, and the disclaimer below in the
   documentation and/or other materials provided with the
   distribution.

   THIS SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
   EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO THE WARRANTIES OF
   MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
   NONINFRINGEMENT. IN NO EVENT SHALL THE CONTRIBUTORS OR COPYRIGHT
   HOLDERS BE LIABLE FOR ANY CLAIM, INDIRECT, INCIDENTAL, SPECIAL,
   EXEMPLARY, OR CONSEQUENTIAL DAMAGES OR OTHER LIABILITY, WHETHER IN AN
   ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
   CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS WITH THE
   SOFTWARE.


   PFFFT : a Pretty Fast FFT.

   This file is largerly based on the original FFTPACK implementation, modified in
   order to take advantage of SIMD instructions of modern CPUs.
*/

/*
  ChangeLog: 
  - 2011/10/02, version 1: This is the very first release of this file.
*/

#include "pffft.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <assert.h>

/* detect compiler flavour */
#if defined(_MSC_VER)
#  define COMPILER_MSVC
#elif defined(__GNUC__)
#  define COMPILER_GCC
#endif

#if defined(COMPILER_GCC)
#  define ALWAYS_INLINE(return_type) return_type __attribute__ ((always_inline))
#  define NEVER_INLINE(return_type) return_type __attribute__ ((noinline))
#  define RESTRICT __restrict
#  define VLA_ARRAY_ON_STACK(type__, varname__, size__) type__ varname__[size__];
#elif defined(COMPILER_MSVC)
#  define ALWAYS_INLINE(return_type) __forceinline return_type
#  define NEVER_INLINE(return_type) __declspec(noinline) return_type
#  define RESTRICT __restrict
#  define VLA_ARRAY_ON_STACK(type__, varname__, size__) type__ *varname__ = (v4sf*)_alloca(size__ * sizeof(type__))
#endif


/* 
   vector support macros: the rest of the code is independant of
   SSE/Altivec/NEON -- adding support for other platforms with 4-element
   vectors should be limited to these macros 
*/


// define PFFFT_SIMD_DISABLE if you want to use scalar code instead of simd code
//#define PFFFT_SIMD_DISABLE

/*
   Altivec support macros 
*/
#if !defined(PFFFT_SIMD_DISABLE) && (defined(__ppc__) || defined(__ppc64__))
typedef vector float v4sf;
#  define SIMD_SZ 4
#  define VZERO() ((vector float) vec_splat_u8(0))
#  define VMUL(a,b) vec_madd(a,b, VZERO())
#  define VADD(a,b) vec_add(a,b)
#  define VMADD(a,b,c) vec_madd(a,b,c)
#  define VSUB(a,b) vec_sub(a,b)
inline v4sf ld_ps1(const float *p) { v4sf v=vec_lde(0,p); return vec_splat(vec_perm(v, v, vec_lvsl(0, p)), 0); }
#  define LD_PS1(p) ld_ps1(&p)
#  define INTERLEAVE2(in1, in2, out1, out2) { v4sf tmp__ = vec_mergeh(in1, in2); out2 = vec_mergel(in1, in2); out1 = tmp__; }
#  define UNINTERLEAVE2(in1, in2, out1, out2) {                           \
    vector unsigned char vperm1 =  (vector unsigned char)(0,1,2,3,8,9,10,11,16,17,18,19,24,25,26,27); \
    vector unsigned char vperm2 =  (vector unsigned char)(4,5,6,7,12,13,14,15,20,21,22,23,28,29,30,31); \
    v4sf tmp__ = vec_perm(in1, in2, vperm1); out2 = vec_perm(in1, in2, vperm2); out1 = tmp__; \
  }
#  define VTRANSPOSE4(x0,x1,x2,x3) {              \
    v4sf y0 = vec_mergeh(x0, x2);               \
    v4sf y1 = vec_mergel(x0, x2);               \
    v4sf y2 = vec_mergeh(x1, x3);               \
    v4sf y3 = vec_mergel(x1, x3);               \
    x0 = vec_mergeh(y0, y2);                    \
    x1 = vec_mergel(y0, y2);                    \
    x2 = vec_mergeh(y1, y3);                    \
    x3 = vec_mergel(y1, y3);                    \
  }
#  define VSWAPHL(a,b) vec_perm(a,b, (vector unsigned char)(16,17,18,19,20,21,22,23,8,9,10,11,12,13,14,15))
#  define VALIGNED(ptr) ((((long)(ptr)) & 0xF) == 0)

/*
  SSE1 support macros
*/
#elif !defined(PFFFT_SIMD_DISABLE) && (defined(__x86_64__) || defined(_M_X64) || defined(i386) || defined(_M_IX86))

#include <xmmintrin.h>
typedef __m128 v4sf;
#  define SIMD_SZ 4 // 4 floats by simd vector -- this is pretty much hardcoded in the preprocess/finalize functions anyway so you will have to work if you want to enable AVX with its 256-bit vectors.
#  define VZERO() _mm_setzero_ps()
#  define VMUL(a,b) _mm_mul_ps(a,b)
#  define VADD(a,b) _mm_add_ps(a,b)
#  define VMADD(a,b,c) _mm_add_ps(_mm_mul_ps(a,b), c)
#  define VSUB(a,b) _mm_sub_ps(a,b)
#  define LD_PS1(p) _mm_set1_ps(p)
#  define INTERLEAVE2(in1, in2, out1, out2) { v4sf tmp__ = _mm_unpacklo_ps(in1, in2); out2 = _mm_unpackhi_ps(in1, in2); out1 = tmp__; }
#  define UNINTERLEAVE2(in1, in2, out1, out2) { v4sf tmp__ = _mm_shuffle_ps(in1, in2, _MM_SHUFFLE(2,0,2,0)); out2 = _mm_shuffle_ps(in1, in2, _MM_SHUFFLE(3,1,3,1)); out1 = tmp__; }
#  define VTRANSPOSE4(x0,x1,x2,x3) _MM_TRANSPOSE4_PS(x0,x1,x2,x3)
#  define VSWAPHL(a,b) _mm_shuffle_ps(b, a, _MM_SHUFFLE(3,2,1,0))
#  define VALIGNED(ptr) ((((long)(ptr)) & 0xF) == 0)

/*
  ARM NEON support macros
*/
#elif !defined(PFFFT_SIMD_DISABLE) && defined(__arm__) 
#  include <arm_neon.h>
typedef float32x4_t v4sf;
#  define SIMD_SZ 4
#  define VZERO() vdupq_n_f32(0)
#  define VMUL(a,b) vmulq_f32(a,b)
#  define VADD(a,b) vaddq_f32(a,b)
#  define VMADD(a,b,c) vmlaq_f32(c,a,b)
#  define VSUB(a,b) vsubq_f32(a,b)
#  define LD_PS1(p) vld1q_dup_f32(&(p))
#  define INTERLEAVE2(in1, in2, out1, out2) { float32x4x2_t tmp__ = vzipq_f32(in1,in2); out1=tmp__.val[0]; out2=tmp__.val[1]; }
#  define UNINTERLEAVE2(in1, in2, out1, out2) { float32x4x2_t tmp__ = vuzpq_f32(in1,in2); out1=tmp__.val[0]; out2=tmp__.val[1]; }
#  define VTRANSPOSE4_(x0,x1,x2,x3) {                                    \
    float32x4x2_t t0_ = vzipq_f32(x0, x2);                              \
    float32x4x2_t t1_ = vzipq_f32(x1, x3);                              \
    float32x4x2_t u0_ = vzipq_f32(t0_.val[0], t1_.val[0]);              \
    float32x4x2_t u1_ = vzipq_f32(t0_.val[1], t1_.val[1]);              \
    x0 = u0_.val[0]; x1 = u0_.val[1]; x2 = u1_.val[0]; x3 = u1_.val[1]; \
  }
// marginally faster version
#  define VTRANSPOSE4(x0,x1,x2,x3) { asm("vtrn.32 %q0, %q1;\n vtrn.32 %q2,%q3\n vswp %f0,%e2\n vswp %f1,%e3" : "+w"(x0), "+w"(x1), "+w"(x2), "+w"(x3)::); }
#  define VSWAPHL(a,b) vcombine_f32(vget_low_f32(b), vget_high_f32(a))
#  define VALIGNED(ptr) ((((long)(ptr)) & 0x3) == 0)
#else
#  if !defined(PFFFT_SIMD_DISABLE)
#    warning "building with simd disabled !\n";
#    define PFFFT_SIMD_DISABLE // fallback to scalar code
#  endif
#endif

// fallback mode for situations where SSE/Altivec are not available, use scalar mode instead
#ifdef PFFFT_SIMD_DISABLE
typedef float v4sf;
#  define SIMD_SZ 1
#  define VZERO() 0.f
#  define VMUL(a,b) ((a)*(b))
#  define VADD(a,b) ((a)+(b))
#  define VMADD(a,b,c) ((a)*(b)+(c))
#  define VSUB(a,b) ((a)-(b))
#  define LD_PS1(p) (p)
#  define VALIGNED(ptr) ((((long)(ptr)) & 0x3) == 0)
#endif

// shortcuts for complex multiplcations
#define VCPLXMUL(ar,ai,br,bi) { v4sf tmp; tmp=VMUL(ar,bi); ar=VMUL(ar,br); ar=VSUB(ar,VMUL(ai,bi)); ai=VMUL(ai,br); ai=VADD(ai,tmp); }
#define VCPLXMULCONJ(ar,ai,br,bi) { v4sf tmp; tmp=VMUL(ar,bi); ar=VMUL(ar,br); ar=VADD(ar,VMUL(ai,bi)); ai=VMUL(ai,br); ai=VSUB(ai,tmp); }

#if !defined(PFFFT_SIMD_DISABLE)
typedef union v4sf_union {
  v4sf  v;
  float f[4];
} v4sf_union;

#include <string.h>

#define assertv4(v,f0,f1,f2,f3) assert(v.f[0] == (f0) && v.f[1] == (f1) && v.f[2] == (f2) && v.f[3] == (f3))

/* detect bugs with the vector support macros */
void validate_pffft_simd() {
  float f[16] = { 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 };
  v4sf_union a0, a1, a2, a3, t, u; 
  memcpy(a0.f, f, 4*sizeof(float));
  memcpy(a1.f, f+4, 4*sizeof(float));
  memcpy(a2.f, f+8, 4*sizeof(float));
  memcpy(a3.f, f+12, 4*sizeof(float));

  t = a0; u = a1; t.v = VZERO();
  printf("VZERO=[%2g %2g %2g %2g]\n", t.f[0], t.f[1], t.f[2], t.f[3]); assertv4(t, 0, 0, 0, 0);
  t.v = VADD(a1.v, a2.v);
  printf("VADD(4:7,8:11)=[%2g %2g %2g %2g]\n", t.f[0], t.f[1], t.f[2], t.f[3]); assertv4(t, 12, 14, 16, 18);
  t.v = VMUL(a1.v, a2.v);
  printf("VMUL(4:7,8:11)=[%2g %2g %2g %2g]\n", t.f[0], t.f[1], t.f[2], t.f[3]); assertv4(t, 32, 45, 60, 77);
  t.v = VMADD(a1.v, a2.v,a0.v);
  printf("VMADD(4:7,8:11,0:3)=[%2g %2g %2g %2g]\n", t.f[0], t.f[1], t.f[2], t.f[3]); assertv4(t, 32, 46, 62, 80);

  INTERLEAVE2(a1.v,a2.v,t.v,u.v);
  printf("INTERLEAVE2(4:7,8:11)=[%2g %2g %2g %2g] [%2g %2g %2g %2g]\n", t.f[0], t.f[1], t.f[2], t.f[3], u.f[0], u.f[1], u.f[2], u.f[3]);
  assertv4(t, 4, 8, 5, 9); assertv4(u, 6, 10, 7, 11);
  UNINTERLEAVE2(a1.v,a2.v,t.v,u.v);
  printf("UNINTERLEAVE2(4:7,8:11)=[%2g %2g %2g %2g] [%2g %2g %2g %2g]\n", t.f[0], t.f[1], t.f[2], t.f[3], u.f[0], u.f[1], u.f[2], u.f[3]);
  assertv4(t, 4, 6, 8, 10); assertv4(u, 5, 7, 9, 11);

  t.v=LD_PS1(f[15]);
  printf("LD_PS1(15)=[%2g %2g %2g %2g]\n", t.f[0], t.f[1], t.f[2], t.f[3]);
  assertv4(t, 15, 15, 15, 15);
  t.v = VSWAPHL(a1.v, a2.v);
  printf("VSWAPHL(4:7,8:11)=[%2g %2g %2g %2g]\n", t.f[0], t.f[1], t.f[2], t.f[3]);
  assertv4(t, 8, 9, 6, 7);
  VTRANSPOSE4(a0.v, a1.v, a2.v, a3.v);
  printf("VTRANSPOSE4(0:3,4:7,8:11,12:15)=[%2g %2g %2g %2g] [%2g %2g %2g %2g] [%2g %2g %2g %2g] [%2g %2g %2g %2g]\n", 
         a0.f[0], a0.f[1], a0.f[2], a0.f[3], a1.f[0], a1.f[1], a1.f[2], a1.f[3], 
         a2.f[0], a2.f[1], a2.f[2], a2.f[3], a3.f[0], a3.f[1], a3.f[2], a3.f[3]); 
  assertv4(a0, 0, 4, 8, 12); assertv4(a1, 1, 5, 9, 13); assertv4(a2, 2, 6, 10, 14); assertv4(a3, 3, 7, 11, 15);
}
#endif //!PFFFT_SIMD_DISABLE

/* SSE and co like 16-bytes aligned pointers */
#define MALLOC_V4SF_ALIGNMENT 64 // with a 64-byte alignment, we are even aligned on L2 cache lines...
void *pffft_aligned_malloc(size_t nb_bytes) {
  void *p0, *p;
  if (!(p0 = malloc(nb_bytes + MALLOC_V4SF_ALIGNMENT))) return (void *) 0;
  p = (void *) (((size_t) p0 + MALLOC_V4SF_ALIGNMENT) & (~((size_t) (MALLOC_V4SF_ALIGNMENT-1))));
  *((void **) p - 1) = p0;
  return p;
}

void pffft_aligned_free(void *p) {
  if (p) free(*((void **) p - 1));
}

int pffft_simd_size() { return SIMD_SZ; }

/*
  passf2 and passb2 has been merged here, fsign = -1 for passf2, +1 for passb2
*/
static NEVER_INLINE(void) passf2_ps(int ido, int l1, const v4sf *cc, v4sf *ch, const float *wa1, float fsign) {
  int k, i;
  int l1ido = l1*ido;
  if (ido <= 2) {
    for (k=0; k < l1ido; k += ido, ch += ido, cc+= 2*ido) {
      ch[0]         = VADD(cc[0], cc[ido+0]);
      ch[l1ido]     = VSUB(cc[0], cc[ido+0]);
      ch[1]         = VADD(cc[1], cc[ido+1]);
      ch[l1ido + 1] = VSUB(cc[1], cc[ido+1]);
    }
  } else {
    for (k=0; k < l1ido; k += ido, ch += ido, cc += 2*ido) {
      for (i=0; i<ido-1; i+=2) {
        v4sf tr2 = VSUB(cc[i+0], cc[i+ido+0]);
        v4sf ti2 = VSUB(cc[i+1], cc[i+ido+1]);
        v4sf wr = LD_PS1(wa1[i]), wi = VMUL(LD_PS1(fsign), LD_PS1(wa1[i+1]));
        ch[i]   = VADD(cc[i+0], cc[i+ido+0]);
        ch[i+1] = VADD(cc[i+1], cc[i+ido+1]);
        VCPLXMUL(tr2, ti2, wr, wi);
        ch[i+l1ido]   = tr2;
        ch[i+l1ido+1] = ti2;
      }
    }
  }
}

/*
  passf3 and passb3 has been merged here, fsign = -1 for passf3, +1 for passb3
*/
static NEVER_INLINE(void) passf3_ps(int ido, int l1, const v4sf *cc, v4sf *ch,
                                    const float *wa1, const float *wa2, float fsign) {
  static const float taur = -0.5f;
  float taui = 0.866025403784439f*fsign;
  int i, k;
  v4sf tr2, ti2, cr2, ci2, cr3, ci3, dr2, di2, dr3, di3;
  int l1ido = l1*ido;
  float wr1, wi1, wr2, wi2;
  assert(ido > 2);
  for (k=0; k< l1ido; k += ido, cc+= 3*ido, ch +=ido) {
    for (i=0; i<ido-1; i+=2) {
      tr2 = VADD(cc[i+ido], cc[i+2*ido]);
      cr2 = VADD(cc[i], VMUL(LD_PS1(taur),tr2));
      ch[i]    = VADD(cc[i], tr2);
      ti2 = VADD(cc[i+ido+1], cc[i+2*ido+1]);
      ci2 = VADD(cc[i    +1], VMUL(LD_PS1(taur),ti2));
      ch[i+1]  = VADD(cc[i+1], ti2);
      cr3 = VMUL(LD_PS1(taui), VSUB(cc[i+ido], cc[i+2*ido]));
      ci3 = VMUL(LD_PS1(taui), VSUB(cc[i+ido+1], cc[i+2*ido+1]));
      dr2 = VSUB(cr2, ci3);
      dr3 = VADD(cr2, ci3);
      di2 = VADD(ci2, cr3);
      di3 = VSUB(ci2, cr3);
      wr1=wa1[i], wi1=fsign*wa1[i+1], wr2=wa2[i], wi2=fsign*wa2[i+1]; 
      VCPLXMUL(dr2, di2, LD_PS1(wr1), LD_PS1(wi1));
      ch[i+l1ido] = dr2; 
      ch[i+l1ido + 1] = di2;
      VCPLXMUL(dr3, di3, LD_PS1(wr2), LD_PS1(wi2));
      ch[i+2*l1ido] = dr3;
      ch[i+2*l1ido+1] = di3;
    }
  }
} /* passf3 */

static NEVER_INLINE(void) passf4_ps(int ido, int l1, const v4sf *cc, v4sf *ch,
                                    const float *wa1, const float *wa2, const float *wa3, float fsign) {
  /* isign == -1 for forward transform and +1 for backward transform */

  int i, k;
  v4sf ci2, ci3, ci4, cr2, cr3, cr4, ti1, ti2, ti3, ti4, tr1, tr2, tr3, tr4;
  int l1ido = l1*ido;
  if (ido == 2) {
    for (k=0; k < l1ido; k += ido, ch += ido, cc += 4*ido) {
      tr1 = VSUB(cc[0], cc[2*ido + 0]);
      tr2 = VADD(cc[0], cc[2*ido + 0]);
      ti1 = VSUB(cc[1], cc[2*ido + 1]);
      ti2 = VADD(cc[1], cc[2*ido + 1]);
      ti4 = VMUL(VSUB(cc[1*ido + 0], cc[3*ido + 0]), LD_PS1(fsign));
      tr4 = VMUL(VSUB(cc[3*ido + 1], cc[1*ido + 1]), LD_PS1(fsign));
      tr3 = VADD(cc[ido + 0], cc[3*ido + 0]);
      ti3 = VADD(cc[ido + 1], cc[3*ido + 1]);

      ch[0*l1ido + 0] = VADD(tr2, tr3);
      ch[0*l1ido + 1] = VADD(ti2, ti3);
      ch[1*l1ido + 0] = VADD(tr1, tr4);
      ch[1*l1ido + 1] = VADD(ti1, ti4);
      ch[2*l1ido + 0] = VSUB(tr2, tr3);
      ch[2*l1ido + 1] = VSUB(ti2, ti3);        
      ch[3*l1ido + 0] = VSUB(tr1, tr4);
      ch[3*l1ido + 1] = VSUB(ti1, ti4);
    }
  } else {
    for (k=0; k < l1ido; k += ido, ch+=ido, cc += 4*ido) {
      for (i=0; i<ido-1; i+=2) {
        float wr1, wi1, wr2, wi2, wr3, wi3;
        tr1 = VSUB(cc[i + 0], cc[i + 2*ido + 0]);
        tr2 = VADD(cc[i + 0], cc[i + 2*ido + 0]);
        ti1 = VSUB(cc[i + 1], cc[i + 2*ido + 1]);
        ti2 = VADD(cc[i + 1], cc[i + 2*ido + 1]);
        tr4 = VMUL(VSUB(cc[i + 3*ido + 1], cc[i + 1*ido + 1]), LD_PS1(fsign));
        ti4 = VMUL(VSUB(cc[i + 1*ido + 0], cc[i + 3*ido + 0]), LD_PS1(fsign));
        tr3 = VADD(cc[i + ido + 0], cc[i + 3*ido + 0]);
        ti3 = VADD(cc[i + ido + 1], cc[i + 3*ido + 1]);

        ch[i] = VADD(tr2, tr3);
        cr3    = VSUB(tr2, tr3);
        ch[i + 1] = VADD(ti2, ti3);
        ci3 = VSUB(ti2, ti3);

        cr2 = VADD(tr1, tr4);
        cr4 = VSUB(tr1, tr4);
        ci2 = VADD(ti1, ti4);
        ci4 = VSUB(ti1, ti4);
        wr1=wa1[i], wi1=fsign*wa1[i+1];
        VCPLXMUL(cr2, ci2, LD_PS1(wr1), LD_PS1(wi1));
        wr2=wa2[i], wi2=fsign*wa2[i+1]; 
        ch[i + l1ido] = cr2;
        ch[i + l1ido + 1] = ci2;

        VCPLXMUL(cr3, ci3, LD_PS1(wr2), LD_PS1(wi2));
        wr3=wa3[i], wi3=fsign*wa3[i+1]; 
        ch[i + 2*l1ido] = cr3;
        ch[i + 2*l1ido + 1] = ci3;

        VCPLXMUL(cr4, ci4, LD_PS1(wr3), LD_PS1(wi3));
        ch[i + 3*l1ido] = cr4;
        ch[i + 3*l1ido + 1] = ci4;
      }
    }
  }
} /* passf4 */

static NEVER_INLINE(void) radf2_ps(int ido, int l1, const v4sf * RESTRICT cc, v4sf * RESTRICT ch, const float *wa1) {
  static const float minus_one = -1.f;
  int i, k, l1ido = l1*ido;
  for (k=0; k < l1ido; k += ido) {
    v4sf a = cc[k], b = cc[k + l1ido];
    ch[2*k] = VADD(a, b);
    ch[2*(k+ido)-1] = VSUB(a, b);
  }
  if (ido < 2) return;
  if (ido != 2) {
    for (k=0; k < l1ido; k += ido) {
      for (i=2; i<ido; i+=2) {
        v4sf tr2 = cc[i - 1 + k + l1ido], ti2 = cc[i + k + l1ido];
        v4sf br = cc[i - 1 + k], bi = cc[i + k];
        VCPLXMULCONJ(tr2, ti2, LD_PS1(wa1[i - 2]), LD_PS1(wa1[i - 1])); 
        ch[i + 2*k] = VADD(bi, ti2);
        ch[2*(k+ido) - i] = VSUB(ti2, bi);
        ch[i - 1 + 2*k] = VADD(br, tr2);
        ch[2*(k+ido) - i -1] = VSUB(br, tr2);
      }
    }
    if (ido % 2 == 1) return;
  }
  for (k=0; k < l1ido; k += ido) {
    ch[2*k + ido] = VMUL(LD_PS1(minus_one), cc[ido-1 + k + l1ido]);
    ch[2*k + ido-1] = cc[k + ido-1];
  }
} /* radf2 */


static NEVER_INLINE(void) radb2_ps(int ido, int l1, const v4sf *cc, v4sf *ch, const float *wa1) {
  static const float minus_two=-2;
  int i, k, l1ido = l1*ido;
  v4sf a,b,c,d, tr2, ti2;
  for (k=0; k < l1ido; k += ido) {
    a = cc[2*k]; b = cc[2*(k+ido) - 1];
    ch[k] = VADD(a, b);
    ch[k + l1ido] =VSUB(a, b);
  }
  if (ido < 2) return;
  if (ido != 2) {
    for (k = 0; k < l1ido; k += ido) {
      for (i = 2; i < ido; i += 2) {
        a = cc[i-1 + 2*k]; b = cc[2*(k + ido) - i - 1];
        c = cc[i+0 + 2*k]; d = cc[2*(k + ido) - i + 0];
        ch[i-1 + k] = VADD(a, b);
        tr2 = VSUB(a, b);
        ch[i+0 + k] = VSUB(c, d);
        ti2 = VADD(c, d);
        VCPLXMUL(tr2, ti2, LD_PS1(wa1[i - 2]), LD_PS1(wa1[i - 1]));
        ch[i-1 + k + l1ido] = tr2;
        ch[i+0 + k + l1ido] = ti2;
      }
    }
    if (ido % 2 == 1) return;
  }
  for (k = 0; k < l1ido; k += ido) {
    a = cc[2*k + ido-1]; b = cc[2*k + ido];
    ch[k + ido-1] = VADD(a,a);
    ch[k + ido-1 + l1ido] = VMUL(LD_PS1(minus_two), b);
  }
} /* radb2 */

static void radf3_ps(int ido, int l1, const v4sf * RESTRICT cc, v4sf * RESTRICT ch,
                     const float *wa1, const float *wa2) {
  static const float taur = -0.5f;
  static const float taui = 0.866025403784439f;
  int i, k, ic;
  v4sf ci2, di2, di3, cr2, dr2, dr3, ti2, ti3, tr2, tr3, wr1, wi1, wr2, wi2;
  for (k=0; k<l1; k++) {
    cr2 = VADD(cc[(k + l1)*ido], cc[(k + 2*l1)*ido]);
    ch[3*k*ido] = VADD(cc[k*ido], cr2);
    ch[(3*k+2)*ido] = VMUL(LD_PS1(taui), VSUB(cc[(k + l1*2)*ido], cc[(k + l1)*ido]));
    ch[ido-1 + (3*k + 1)*ido] = VADD(cc[k*ido], VMUL(LD_PS1(taur), cr2));
  }
  if (ido == 1) return;
  for (k=0; k<l1; k++) {
    for (i=2; i<ido; i+=2) {
      ic = ido - i;
      wr1 = LD_PS1(wa1[i - 2]); wi1 = LD_PS1(wa1[i - 1]);
      dr2 = cc[i - 1 + (k + l1)*ido]; di2 = cc[i + (k + l1)*ido];
      VCPLXMULCONJ(dr2, di2, wr1, wi1);

      wr2 = LD_PS1(wa2[i - 2]); wi2 = LD_PS1(wa2[i - 1]);
      dr3 = cc[i - 1 + (k + l1*2)*ido]; di3 = cc[i + (k + l1*2)*ido];
      VCPLXMULCONJ(dr3, di3, wr2, wi2);
        
      cr2 = VADD(dr2, dr3);
      ci2 = VADD(di2, di3);
      ch[i - 1 + 3*k*ido] = VADD(cc[i - 1 + k*ido], cr2);
      ch[i + 3*k*ido] = VADD(cc[i + k*ido], ci2);
      tr2 = VADD(cc[i - 1 + k*ido], VMUL(LD_PS1(taur), cr2));
      ti2 = VADD(cc[i + k*ido], VMUL(LD_PS1(taur), ci2));
      tr3 = VMUL(LD_PS1(taui), VSUB(di2, di3));
      ti3 = VMUL(LD_PS1(taui), VSUB(dr3, dr2));
      ch[i - 1 + (3*k + 2)*ido] = VADD(tr2, tr3);
      ch[ic - 1 + (3*k + 1)*ido] = VSUB(tr2, tr3);
      ch[i + (3*k + 2)*ido] = VADD(ti2, ti3);
      ch[ic + (3*k + 1)*ido] = VSUB(ti3, ti2);
    }
  }
} /* radf3 */


static void radb3_ps(int ido, int l1, const v4sf *RESTRICT cc, v4sf *RESTRICT ch,
                     const float *wa1, const float *wa2)
{
  static const float taur = -0.5f;
  static const float taui = 0.866025403784439f;
  static const float taui_2 = 0.866025403784439f*2;
  int i, k, ic;
  v4sf ci2, ci3, di2, di3, cr2, cr3, dr2, dr3, ti2, tr2;
  for (k=0; k<l1; k++) {
    tr2 = cc[ido-1 + (3*k + 1)*ido]; tr2 = VADD(tr2,tr2);
    cr2 = VMADD(LD_PS1(taur), tr2, cc[3*k*ido]);
    ch[k*ido] = VADD(cc[3*k*ido], tr2);
    ci3 = VMUL(LD_PS1(taui_2), cc[(3*k + 2)*ido]);
    ch[(k + l1)*ido] = VSUB(cr2, ci3);
    ch[(k + 2*l1)*ido] = VADD(cr2, ci3);
  }
  if (ido == 1) return;
  for (k=0; k<l1; k++) {
    for (i=2; i<ido; i+=2) {
      ic = ido - i;
      tr2 = VADD(cc[i - 1 + (3*k + 2)*ido], cc[ic - 1 + (3*k + 1)*ido]);
      cr2 = VMADD(LD_PS1(taur), tr2, cc[i - 1 + 3*k*ido]);
      ch[i - 1 + k*ido] = VADD(cc[i - 1 + 3*k*ido], tr2);
      ti2 = VSUB(cc[i + (3*k + 2)*ido], cc[ic + (3*k + 1)*ido]);
      ci2 = VMADD(LD_PS1(taur), ti2, cc[i + 3*k*ido]);
      ch[i + k*ido] = VADD(cc[i + 3*k*ido], ti2);
      cr3 = VMUL(LD_PS1(taui), VSUB(cc[i - 1 + (3*k + 2)*ido], cc[ic - 1 + (3*k + 1)*ido]));
      ci3 = VMUL(LD_PS1(taui), VADD(cc[i + (3*k + 2)*ido], cc[ic + (3*k + 1)*ido]));
      dr2 = VSUB(cr2, ci3);
      dr3 = VADD(cr2, ci3);
      di2 = VADD(ci2, cr3);
      di3 = VSUB(ci2, cr3);
      VCPLXMUL(dr2, di2, LD_PS1(wa1[i-2]), LD_PS1(wa1[i-1]));
      ch[i - 1 + (k + l1)*ido] = dr2;
      ch[i + (k + l1)*ido] = di2;
      VCPLXMUL(dr3, di3, LD_PS1(wa2[i-2]), LD_PS1(wa2[i-1]));
      ch[i - 1 + (k + 2*l1)*ido] = dr3;
      ch[i + (k + 2*l1)*ido] = di3;
    }
  }
} /* radb3 */


static NEVER_INLINE(void) radf4_ps(int ido, int l1, const v4sf *RESTRICT cc, v4sf * RESTRICT ch,
                                   const float * RESTRICT wa1, const float * RESTRICT wa2, const float * RESTRICT wa3)
{
  static const float minus_hsqt2 = (float)-0.7071067811865475;
  int i, k, l1ido = l1*ido;
  {
    const v4sf *RESTRICT cc_ = cc, * RESTRICT cc_end = cc + l1ido; 
    v4sf * RESTRICT ch_ = ch;
    while (cc < cc_end) {
      // this loop represents between 25% and 40% of total radf4_ps cost !
      v4sf a0 = cc[0], a1 = cc[l1ido];
      v4sf a2 = cc[2*l1ido], a3 = cc[3*l1ido];
      v4sf tr1 = VADD(a1, a3);
      v4sf tr2 = VADD(a0, a2);
      ch[2*ido-1] = VSUB(a0, a2);
      ch[2*ido  ] = VSUB(a3, a1);
      ch[0      ] = VADD(tr1, tr2);
      ch[4*ido-1] = VSUB(tr2, tr1);
      cc += ido; ch += 4*ido;
    }
    cc = cc_; ch = ch_;
  }
  if (ido < 2) return;
  if (ido != 2) {
    for (k = 0; k < l1ido; k += ido) {
      const v4sf * RESTRICT pc = (v4sf*)(cc + 1 + k);
      for (i=2; i<ido; i += 2, pc += 2) {
        int ic = ido - i;
        v4sf wr, wi, cr2, ci2, cr3, ci3, cr4, ci4;
        v4sf tr1, ti1, tr2, ti2, tr3, ti3, tr4, ti4;

        cr2 = pc[1*l1ido+0];
        ci2 = pc[1*l1ido+1];
        wr=LD_PS1(wa1[i - 2]);
        wi=LD_PS1(wa1[i - 1]);
        VCPLXMULCONJ(cr2,ci2,wr,wi);

        cr3 = pc[2*l1ido+0];
        ci3 = pc[2*l1ido+1];
        wr = LD_PS1(wa2[i-2]); 
        wi = LD_PS1(wa2[i-1]);
        VCPLXMULCONJ(cr3, ci3, wr, wi);

        cr4 = pc[3*l1ido];
        ci4 = pc[3*l1ido+1];
        wr = LD_PS1(wa3[i-2]); 
        wi = LD_PS1(wa3[i-1]);
        VCPLXMULCONJ(cr4, ci4, wr, wi);

        /* at this point, on SSE, five of "cr2 cr3 cr4 ci2 ci3 ci4" should be loaded in registers */

        tr1 = VADD(cr2,cr4);
        tr4 = VSUB(cr4,cr2); 
        tr2 = VADD(pc[0],cr3);
        tr3 = VSUB(pc[0],cr3);
        ch[i - 1 + 4*k] = VADD(tr1,tr2);
        ch[ic - 1 + 4*k + 3*ido] = VSUB(tr2,tr1); // at this point tr1 and tr2 can be disposed
        ti1 = VADD(ci2,ci4);
        ti4 = VSUB(ci2,ci4);
        ch[i - 1 + 4*k + 2*ido] = VADD(ti4,tr3);
        ch[ic - 1 + 4*k + 1*ido] = VSUB(tr3,ti4); // dispose tr3, ti4
        ti2 = VADD(pc[1],ci3);
        ti3 = VSUB(pc[1],ci3);
        ch[i + 4*k] = VADD(ti1, ti2);
        ch[ic + 4*k + 3*ido] = VSUB(ti1, ti2);
        ch[i + 4*k + 2*ido] = VADD(tr4, ti3);
        ch[ic + 4*k + 1*ido] = VSUB(tr4, ti3);
      }
    }
    if (ido % 2 == 1) return;
  }
  for (k=0; k<l1ido; k += ido) {
    v4sf a = cc[ido-1 + k + l1ido], b = cc[ido-1 + k + 3*l1ido];
    v4sf c = cc[ido-1 + k], d = cc[ido-1 + k + 2*l1ido];
    v4sf ti1 = VMUL(LD_PS1(minus_hsqt2), VADD(a, b));
    v4sf tr1 = VMUL(LD_PS1(minus_hsqt2), VSUB(b, a));
    ch[ido-1 + 4*k] = VADD(tr1, c);
    ch[ido-1 + 4*k + 2*ido] = VSUB(c, tr1);
    ch[4*k + 1*ido] = VSUB(ti1, d); 
    ch[4*k + 3*ido] = VADD(ti1, d); 
  }
} /* radf4 */


static NEVER_INLINE(void) radb4_ps(int ido, int l1, const v4sf * RESTRICT cc, v4sf * RESTRICT ch,
                                   const float * RESTRICT wa1, const float * RESTRICT wa2, const float *RESTRICT wa3)
{
  static const float minus_sqrt2 = (float)-1.414213562373095;
  static const float two = 2.f;
  int i, k, l1ido = l1*ido;
  v4sf ci2, ci3, ci4, cr2, cr3, cr4, ti1, ti2, ti3, ti4, tr1, tr2, tr3, tr4;
  {
    const v4sf *RESTRICT cc_ = cc, * RESTRICT ch_end = ch + l1ido; 
    v4sf *ch_ = ch;
    while (ch < ch_end) {
      v4sf a = cc[0], b = cc[4*ido-1];
      v4sf c = cc[2*ido], d = cc[2*ido-1];
      tr3 = VMUL(LD_PS1(two),d);
      tr2 = VADD(a,b);
      tr1 = VSUB(a,b);
      tr4 = VMUL(LD_PS1(two),c);
      ch[0*l1ido] = VADD(tr2, tr3);
      ch[2*l1ido] = VSUB(tr2, tr3);
      ch[1*l1ido] = VSUB(tr1, tr4);
      ch[3*l1ido] = VADD(tr1, tr4);
      
      cc += 4*ido; ch += ido;
    }
    cc = cc_; ch = ch_;
  }
  if (ido < 2) return;
  if (ido != 2) {
    for (k = 0; k < l1ido; k += ido) {
      const v4sf * RESTRICT pc = (v4sf*)(cc - 1 + 4*k);
      v4sf * RESTRICT ph = (v4sf*)(ch + k + 1);
      for (i = 2; i < ido; i += 2) {

        tr1 = VSUB(pc[i], pc[4*ido - i]);
        tr2 = VADD(pc[i], pc[4*ido - i]);
        ti4 = VSUB(pc[2*ido + i], pc[2*ido - i]);
        tr3 = VADD(pc[2*ido + i], pc[2*ido - i]);
        ph[0] = VADD(tr2, tr3);
        cr3 = VSUB(tr2, tr3);

        ti3 = VSUB(pc[2*ido + i + 1], pc[2*ido - i + 1]);
        tr4 = VADD(pc[2*ido + i + 1], pc[2*ido - i + 1]);
        cr2 = VSUB(tr1, tr4);
        cr4 = VADD(tr1, tr4);

        ti1 = VADD(pc[i + 1], pc[4*ido - i + 1]);
        ti2 = VSUB(pc[i + 1], pc[4*ido - i + 1]);

        ph[1] = VADD(ti2, ti3); ph += l1ido;
        ci3 = VSUB(ti2, ti3);
        ci2 = VADD(ti1, ti4);
        ci4 = VSUB(ti1, ti4);
        VCPLXMUL(cr2, ci2, LD_PS1(wa1[i-2]), LD_PS1(wa1[i-1]));
        ph[0] = cr2;
        ph[1] = ci2; ph += l1ido;
        VCPLXMUL(cr3, ci3, LD_PS1(wa2[i-2]), LD_PS1(wa2[i-1]));
        ph[0] = cr3;
        ph[1] = ci3; ph += l1ido;
        VCPLXMUL(cr4, ci4, LD_PS1(wa3[i-2]), LD_PS1(wa3[i-1]));
        ph[0] = cr4;
        ph[1] = ci4; ph = ph - 3*l1ido + 2;
      }
    }
    if (ido % 2 == 1) return;
  }
  for (k=0; k < l1ido; k+=ido) {
    int i0 = 4*k + ido;
    v4sf c = cc[i0-1], d = cc[i0 + 2*ido-1];
    v4sf a = cc[i0+0], b = cc[i0 + 2*ido+0];
    tr1 = VSUB(c,d);
    tr2 = VADD(c,d);
    ti1 = VADD(b,a);
    ti2 = VSUB(b,a);
    ch[ido-1 + k + 0*l1ido] = VADD(tr2,tr2);
    ch[ido-1 + k + 1*l1ido] = VMUL(LD_PS1(minus_sqrt2), VSUB(ti1, tr1));
    ch[ido-1 + k + 2*l1ido] = VADD(ti2, ti2);
    ch[ido-1 + k + 3*l1ido] = VMUL(LD_PS1(minus_sqrt2), VADD(ti1, tr1));
  }
} /* radb4 */

static NEVER_INLINE(v4sf *) rfftf1_ps(int n, const v4sf *input_readonly, v4sf *work1, v4sf *work2, 
                                      const float *wa, const int *ifac) {  
  v4sf *in  = (v4sf*)input_readonly;
  v4sf *out = (in == work2 ? work1 : work2);
  int nf = ifac[1], k1;
  int l2 = n;
  int iw = n-1;
  assert(in != out && work1 != work2);
  for (k1 = 1; k1 <= nf; ++k1) {
    int kh = nf - k1;
    int ip = ifac[kh + 2];
    int l1 = l2 / ip;
    int ido = n / l2;
    iw -= (ip - 1)*ido;
    switch (ip) {
      case 4: {
        int ix2 = iw + ido;
        int ix3 = ix2 + ido;
        radf4_ps(ido, l1, in, out, &wa[iw], &wa[ix2], &wa[ix3]);
      } break;
      case 3: {
        int ix2 = iw + ido;
        radf3_ps(ido, l1, in, out, &wa[iw], &wa[ix2]);
      } break;
      case 2:
        radf2_ps(ido, l1, in, out, &wa[iw]);
        break;
      default:
        assert(0);
        break;
    }
    l2 = l1;
    if (out == work2) {
      out = work1; in = work2;
    } else {
      out = work2; in = work1;
    }
  }
  return in; /* this is in fact the output .. */
} /* rfftf1 */

static NEVER_INLINE(v4sf *) rfftb1_ps(int n, const v4sf *input_readonly, v4sf *work1, v4sf *work2, 
                                      const float *wa, const int *ifac) {  
  v4sf *in  = (v4sf*)input_readonly;
  v4sf *out = (in == work2 ? work1 : work2);
  int nf = ifac[1], k1;
  int l1 = 1;
  int iw = 0;
  assert(in != out);
  for (k1=1; k1<=nf; k1++) {
    int ip = ifac[k1 + 1];
    int l2 = ip*l1;
    int ido = n / l2;
    switch (ip) {
      case 4: {
        int ix2 = iw + ido;
        int ix3 = ix2 + ido;
        radb4_ps(ido, l1, in, out, &wa[iw], &wa[ix2], &wa[ix3]);
      } break;
      case 3: {
        int ix2 = iw + ido;
        radb3_ps(ido, l1, in, out, &wa[iw], &wa[ix2]);
      } break;
      case 2:
        radb2_ps(ido, l1, in, out, &wa[iw]);
        break;
      default:
        assert(0);
        break;
    }
    l1 = l2;
    iw += (ip - 1)*ido;

    if (out == work2) {
      out = work1; in = work2;
    } else {
      out = work2; in = work1;
    }
  }
  return in; /* this is in fact the output .. */
}

static int decompose(int n, int *ifac, const int ntryh[3]) {
  int nl = n, nf = 0, i, j = 0;
  for (j=0; j < 3; ++j) {
    int ntry = ntryh[j];
    while (nl != 1) {
      int nq = nl / ntry;
      int nr = nl - ntry * nq;
      if (nr == 0) {
        ifac[2+nf++] = ntry;
        nl = nq;
        if (ntry == 2 && nf != 1) {
          for (i = 2; i <= nf; ++i) {
            int ib = nf - i + 2;
            ifac[ib + 1] = ifac[ib];
          }
          ifac[2] = 2;
        }
      } else break;
    }
  }
  ifac[0] = n;
  ifac[1] = nf;  
  return nf;
}



static void rffti1_ps(int n, float *wa, int *ifac)
{
  static const int ntryh[3] = { 4,2,3 };
  int k1, j, ii;

  int nf = decompose(n,ifac,ntryh);
  float argh = (2*M_PI) / n;
  int is = 0;
  int nfm1 = nf - 1;
  int l1 = 1;
  if (nfm1 == 0) return;
  for (k1 = 1; k1 <= nfm1; k1++) {
    int ip = ifac[k1 + 1];
    int ld = 0;
    int l2 = l1*ip;
    int ido = n / l2;
    int ipm = ip - 1;
    for (j = 1; j <= ipm; ++j) {
      float argld;
      int i = is, fi=0;
      ld += l1;
      argld = ld*argh;
      for (ii = 3; ii <= ido; ii += 2) {
        i += 2;
        fi += 1;
        wa[i - 2] = cos(fi*argld);
        wa[i - 1] = sin(fi*argld);
      }
      is += ido;
    }
    l1 = l2;
  }
} /* rffti1 */

void cffti1_ps(int n, float *wa, int *ifac)
{
  static const int ntryh[3] = { 3,4,2 };
  int k1, j, ii;

  int nf = decompose(n,ifac,ntryh);
  float argh = (2*M_PI)/(float)n;
  int i = 1;
  int l1 = 1;
  for (k1=1; k1<=nf; k1++) {
    int ip = ifac[k1+1];
    int ld = 0;
    int l2 = l1*ip;
    int ido = n / l2;
    int idot = ido + ido + 2;
    int ipm = ip - 1;
    for (j=1; j<=ipm; j++) {
      float argld;
      int i1 = i, fi = 0;
      wa[i-1] = 1;
      wa[i] = 0;
      ld += l1;
      argld = ld*argh;
      for (ii = 4; ii <= idot; ii += 2) {
        i += 2;
        fi += 1;
        wa[i-1] = cos(fi*argld);
        wa[i] = sin(fi*argld);
      }
      if (ip > 5) {
        wa[i1-1] = wa[i-1];
        wa[i1] = wa[i];
      }
    }
    l1 = l2;
  }
} /* cffti1 */


v4sf *cfftf1_ps(int n, const v4sf *input_readonly, v4sf *work1, v4sf *work2, const float *wa, const int *ifac, int isign) {
  v4sf *in  = (v4sf*)input_readonly;
  v4sf *out = (in == work2 ? work1 : work2); 
  int nf = ifac[1], k1;
  int l1 = 1;
  int iw = 0;
  assert(in != out && work1 != work2);
  for (k1=2; k1<=nf+1; k1++) {
    int ip = ifac[k1];
    int l2 = ip*l1;
    int ido = n / l2;
    int idot = ido + ido;
    switch (ip) {
      case 4: {
        int ix2 = iw + idot;
        int ix3 = ix2 + idot;
        passf4_ps(idot, l1, in, out, &wa[iw], &wa[ix2], &wa[ix3], isign);
      } break;
      case 2: {
        passf2_ps(idot, l1, in, out, &wa[iw], isign);
      } break;
      case 3: {
        int ix2 = iw + idot;
        passf3_ps(idot, l1, in, out, &wa[iw], &wa[ix2], isign);
      } break;
      default:
        assert(0);
    }
    l1 = l2;
    iw += (ip - 1)*idot;
    if (out == work2) {
      out = work1; in = work2;
    } else {
      out = work2; in = work1;
    }
  }

  return in; /* this is in fact the output .. */
}


struct PFFFT_Setup {
  int     N;
  int     Ncvec; // nb of complex simd vectors (N/4 if PFFFT_COMPLEX, N/8 if PFFFT_REAL)
  int ifac[15];
  pffft_transform_t transform;
  v4sf *data; // allocated room for twiddle coefs
  float *e;    // points into 'data' , N/4*3 elements
  float *twiddle; // points into 'data', N/4 elements
};

PFFFT_Setup *pffft_new_setup(int N, pffft_transform_t transform) {
  PFFFT_Setup *s = (PFFFT_Setup*)malloc(sizeof(PFFFT_Setup));
  int k, m;
  if (transform == PFFFT_REAL) { assert(N >= 32); }
  if (transform == PFFFT_COMPLEX) { assert(N >= 16); }
  //assert((N % 32) == 0);
  s->N = N;
  s->transform = transform;  
  /* nb of complex simd vectors */
  s->Ncvec = (transform == PFFFT_REAL ? N/2 : N)/SIMD_SZ;
  s->data = (v4sf*)pffft_aligned_malloc(2*s->Ncvec * sizeof(v4sf));
  s->e = (float*)s->data;
  s->twiddle = (float*)(s->data + (2*s->Ncvec*(SIMD_SZ-1))/SIMD_SZ);  

  if (transform == PFFFT_REAL) {
    for (k=0; k < s->Ncvec; ++k) {
      int i = k/SIMD_SZ;
      int j = k%SIMD_SZ;
      for (m=0; m < SIMD_SZ-1; ++m) {
        float A = -2*M_PI*(m+1)*k / N;
        s->e[(2*(i*3 + m) + 0) * SIMD_SZ + j] = cos(A);
        s->e[(2*(i*3 + m) + 1) * SIMD_SZ + j] = sin(A);
      }
    }
    rffti1_ps(N/SIMD_SZ, s->twiddle, s->ifac);
  } else {
    for (k=0; k < s->Ncvec; ++k) {
      int i = k/SIMD_SZ;
      int j = k%SIMD_SZ;
      for (m=0; m < SIMD_SZ-1; ++m) {
        float A = -2*M_PI*(m+1)*k / N;
        s->e[(2*(i*3 + m) + 0)*SIMD_SZ + j] = cos(A);
        s->e[(2*(i*3 + m) + 1)*SIMD_SZ + j] = sin(A);
      }
    }
    cffti1_ps(N/SIMD_SZ, s->twiddle, s->ifac);
  }
  return s;
}


void pffft_destroy_setup(PFFFT_Setup *s) {
  pffft_aligned_free(s->data);
  free(s);
}

#if !defined(PFFFT_SIMD_DISABLE)

/* [0 0 1 2 3 4 5 6 7 8] -> [0 8 7 6 5 4 3 2 1] */
static void reversed_copy(int N, const v4sf *in, int in_stride, v4sf *out) {
  v4sf g0, g1;
  int k;
  INTERLEAVE2(in[0], in[1], g0, g1); in += in_stride;
  
  *--out = VSWAPHL(g0, g1); // [g0l, g0h], [g1l g1h] -> [g1l, g0h]
  for (k=1; k < N; ++k) {
    v4sf h0, h1;
    INTERLEAVE2(in[0], in[1], h0, h1); in += in_stride;
    *--out = VSWAPHL(g1, h0);
    *--out = VSWAPHL(h0, h1);
    g1 = h1;
  }
  *--out = VSWAPHL(g1, g0);
}

static void unreversed_copy(int N, const v4sf *in, v4sf *out, int out_stride) {
  v4sf g0, g1, h0, h1;
  int k;
  g0 = g1 = in[0]; ++in;
  for (k=1; k < N; ++k) {
    h0 = *in++; h1 = *in++;
    g1 = VSWAPHL(g1, h0);
    h0 = VSWAPHL(h0, h1);
    UNINTERLEAVE2(h0, g1, out[0], out[1]); out += out_stride;
    g1 = h1;
  }
  h0 = *in++; h1 = g0;
  g1 = VSWAPHL(g1, h0);
  h0 = VSWAPHL(h0, h1);
  UNINTERLEAVE2(h0, g1, out[0], out[1]);
}

void pffft_zreorder(PFFFT_Setup *setup, const float *in, float *out, pffft_direction_t direction) {
  int k, N = setup->N, Ncvec = setup->Ncvec;
  const v4sf *vin = (const v4sf*)in;
  v4sf *vout = (v4sf*)out;
  assert(in != out);
  if (setup->transform == PFFFT_REAL) {
    int k, dk = N/32;
    if (direction == PFFFT_FORWARD) {
      for (k=0; k < dk; ++k) {
        INTERLEAVE2(vin[k*8 + 0], vin[k*8 + 1], vout[2*(0*dk + k) + 0], vout[2*(0*dk + k) + 1]);
        INTERLEAVE2(vin[k*8 + 4], vin[k*8 + 5], vout[2*(2*dk + k) + 0], vout[2*(2*dk + k) + 1]);
      }
      reversed_copy(dk, vin+2, 8, (v4sf*)(out + N/2));
      reversed_copy(dk, vin+6, 8, (v4sf*)(out + N));
    } else {
      for (k=0; k < dk; ++k) {
        UNINTERLEAVE2(vin[2*(0*dk + k) + 0], vin[2*(0*dk + k) + 1], vout[k*8 + 0], vout[k*8 + 1]);
        UNINTERLEAVE2(vin[2*(2*dk + k) + 0], vin[2*(2*dk + k) + 1], vout[k*8 + 4], vout[k*8 + 5]);
      }
      unreversed_copy(dk, (v4sf*)(in + N/4), (v4sf*)(out + N - 6*SIMD_SZ), -8);
      unreversed_copy(dk, (v4sf*)(in + 3*N/4), (v4sf*)(out + N - 2*SIMD_SZ), -8);
    }
  } else {
    if (direction == PFFFT_FORWARD) {
      for (k=0; k < Ncvec; ++k) { 
        int kk = (k/4) + (k%4)*(Ncvec/4);
        INTERLEAVE2(vin[k*2], vin[k*2+1], vout[kk*2], vout[kk*2+1]);
      }
    } else {
      for (k=0; k < Ncvec; ++k) { 
        int kk = (k/4) + (k%4)*(Ncvec/4);
        UNINTERLEAVE2(vin[kk*2], vin[kk*2+1], vout[k*2], vout[k*2+1]);
      }
    }
  }
}

void pffft_cplx_finalize(int Ncvec, const v4sf *in, v4sf *out, const v4sf *e) {
  int k, dk = Ncvec/SIMD_SZ; // number of 4x4 matrix blocks
  v4sf r0, i0, r1, i1, r2, i2, r3, i3;
  v4sf sr0, dr0, sr1, dr1, si0, di0, si1, di1;
  assert(in != out);
  for (k=0; k < dk; ++k) {    
    r0 = in[8*k+0]; i0 = in[8*k+1];
    r1 = in[8*k+2]; i1 = in[8*k+3];
    r2 = in[8*k+4]; i2 = in[8*k+5];
    r3 = in[8*k+6]; i3 = in[8*k+7];
    VTRANSPOSE4(r0,r1,r2,r3);
    VTRANSPOSE4(i0,i1,i2,i3);
    VCPLXMUL(r1,i1,e[k*6+0],e[k*6+1]);
    VCPLXMUL(r2,i2,e[k*6+2],e[k*6+3]);
    VCPLXMUL(r3,i3,e[k*6+4],e[k*6+5]);

    sr0 = VADD(r0,r2); dr0 = VSUB(r0, r2);
    sr1 = VADD(r1,r3); dr1 = VSUB(r1, r3);
    si0 = VADD(i0,i2); di0 = VSUB(i0, i2);
    si1 = VADD(i1,i3); di1 = VSUB(i1, i3);

    /*
      transformation for each column is:
      
      [1   1   1   1   0   0   0   0]   [r0]
      [1   0  -1   0   0  -1   0   1]   [r1]
      [1  -1   1  -1   0   0   0   0]   [r2]
      [1   0  -1   0   0   1   0  -1]   [r3]
      [0   0   0   0   1   1   1   1] * [i0]
      [0   1   0  -1   1   0  -1   0]   [i1]
      [0   0   0   0   1  -1   1  -1]   [i2]
      [0  -1   0   1   1   0  -1   0]   [i3]    
    */
    
    r0 = VADD(sr0, sr1); i0 = VADD(si0, si1);
    r1 = VADD(dr0, di1); i1 = VSUB(di0, dr1);
    r2 = VSUB(sr0, sr1); i2 = VSUB(si0, si1);
    r3 = VSUB(dr0, di1); i3 = VADD(di0, dr1);
  
    *out++ = r0; *out++ = i0; *out++ = r1; *out++ = i1;
    *out++ = r2; *out++ = i2; *out++ = r3; *out++ = i3;
  }
}

void pffft_cplx_preprocess(int Ncvec, const v4sf *in, v4sf *out, const v4sf *e) {
  int k, dk = Ncvec/SIMD_SZ; // number of 4x4 matrix blocks
  v4sf r0, i0, r1, i1, r2, i2, r3, i3;
  v4sf sr0, dr0, sr1, dr1, si0, di0, si1, di1;
  assert(in != out);
  for (k=0; k < dk; ++k) {    
    r0 = in[8*k+0]; i0 = in[8*k+1];
    r1 = in[8*k+2]; i1 = in[8*k+3];
    r2 = in[8*k+4]; i2 = in[8*k+5];
    r3 = in[8*k+6]; i3 = in[8*k+7];

    sr0 = VADD(r0,r2); dr0 = VSUB(r0, r2);
    sr1 = VADD(r1,r3); dr1 = VSUB(r1, r3);
    si0 = VADD(i0,i2); di0 = VSUB(i0, i2);
    si1 = VADD(i1,i3); di1 = VSUB(i1, i3);

    r0 = VADD(sr0, sr1); i0 = VADD(si0, si1);
    r1 = VSUB(dr0, di1); i1 = VADD(di0, dr1);
    r2 = VSUB(sr0, sr1); i2 = VSUB(si0, si1);
    r3 = VADD(dr0, di1); i3 = VSUB(di0, dr1);

    VCPLXMULCONJ(r1,i1,e[k*6+0],e[k*6+1]);
    VCPLXMULCONJ(r2,i2,e[k*6+2],e[k*6+3]);
    VCPLXMULCONJ(r3,i3,e[k*6+4],e[k*6+5]);

    VTRANSPOSE4(r0,r1,r2,r3);
    VTRANSPOSE4(i0,i1,i2,i3);

    *out++ = r0; *out++ = i0; *out++ = r1; *out++ = i1;
    *out++ = r2; *out++ = i2; *out++ = r3; *out++ = i3;
  }
}


ALWAYS_INLINE(void) pffft_real_finalize_4x4(const v4sf *in0, const v4sf *in1, const v4sf *in,
                            const v4sf *e, v4sf *out) {
  v4sf r0, i0, r1, i1, r2, i2, r3, i3;
  v4sf sr0, dr0, sr1, dr1, si0, di0, si1, di1;
  r0 = *in0; i0 = *in1;
  r1 = *in++; i1 = *in++; r2 = *in++; i2 = *in++; r3 = *in++; i3 = *in++;
  VTRANSPOSE4(r0,r1,r2,r3);
  VTRANSPOSE4(i0,i1,i2,i3);
 
  /*
    transformation for each column is:

    [1   1   1   1   0   0   0   0]   [r0]
    [1   0  -1   0   0  -1   0   1]   [r1]
    [1   0  -1   0   0   1   0  -1]   [r2]
    [1  -1   1  -1   0   0   0   0]   [r3]
    [0   0   0   0   1   1   1   1] * [i0]
    [0  -1   0   1  -1   0   1   0]   [i1]
    [0  -1   0   1   1   0  -1   0]   [i2]
    [0   0   0   0  -1   1  -1   1]   [i3]    
  */
  
  //cerr << "matrix initial, before e , REAL:\n 1: " << r0 << "\n 1: " << r1 << "\n 1: " << r2 << "\n 1: " << r3 << "\n";
  //cerr << "matrix initial, before e, IMAG :\n 1: " << i0 << "\n 1: " << i1 << "\n 1: " << i2 << "\n 1: " << i3 << "\n";

  VCPLXMUL(r1,i1,e[0],e[1]);
  VCPLXMUL(r2,i2,e[2],e[3]);
  VCPLXMUL(r3,i3,e[4],e[5]);

  //cerr << "matrix initial, real part:\n 1: " << r0 << "\n 1: " << r1 << "\n 1: " << r2 << "\n 1: " << r3 << "\n";
  //cerr << "matrix initial, imag part:\n 1: " << i0 << "\n 1: " << i1 << "\n 1: " << i2 << "\n 1: " << i3 << "\n";

  sr0 = VADD(r0,r2); dr0 = VSUB(r0,r2); 
  sr1 = VADD(r1,r3); dr1 = VSUB(r3,r1);
  si0 = VADD(i0,i2); di0 = VSUB(i0,i2); 
  si1 = VADD(i1,i3); di1 = VSUB(i3,i1);

  r0 = VADD(sr0, sr1);
  r3 = VSUB(sr0, sr1);
  i0 = VADD(si0, si1);
  i3 = VSUB(si1, si0);
  r1 = VADD(dr0, di1);
  r2 = VSUB(dr0, di1);
  i1 = VSUB(dr1, di0);
  i2 = VADD(dr1, di0);

  *out++ = r0;
  *out++ = i0;
  *out++ = r1;
  *out++ = i1;
  *out++ = r2;
  *out++ = i2;
  *out++ = r3;
  *out++ = i3;

}

static NEVER_INLINE(void) pffft_real_finalize(int Ncvec, const v4sf *in, v4sf *out, const v4sf *e) {
  int k, dk = Ncvec/SIMD_SZ; // number of 4x4 matrix blocks
  /* fftpack order is f0r f1r f1i f2r f2i ... f(n-1)r f(n-1)i f(n)r */

  v4sf_union cr, ci, *uout = (v4sf_union*)out;
  v4sf save = in[7], zero=VZERO();
  float xr0, xi0, xr1, xi1, xr2, xi2, xr3, xi3;
  static const float s = M_SQRT2/2;

  cr.v = in[0]; ci.v = in[Ncvec*2-1];
  assert(in != out);
  pffft_real_finalize_4x4(&zero, &zero, in+1, e, out);

  /*
    [cr0 cr1 cr2 cr3 ci0 ci1 ci2 ci3]

    [Xr(1)]  ] [1   1   1   1   0   0   0   0]
    [Xr(N/4) ] [0   0   0   0   1   s   0  -s]
    [Xr(N/2) ] [1   0  -1   0   0   0   0   0]
    [Xr(3N/4)] [0   0   0   0   1  -s   0   s]
    [Xi(1)   ] [1  -1   1  -1   0   0   0   0]
    [Xi(N/4) ] [0   0   0   0   0  -s  -1  -s]
    [Xi(N/2) ] [0  -1   0   1   0   0   0   0]
    [Xi(3N/4)] [0   0   0   0   0  -s   1  -s]
  */

  xr0=(cr.f[0]+cr.f[2]) + (cr.f[1]+cr.f[3]); uout[0].f[0] = xr0;
  xi0=(cr.f[0]+cr.f[2]) - (cr.f[1]+cr.f[3]); uout[1].f[0] = xi0;
  xr2=(cr.f[0]-cr.f[2]);                     uout[4].f[0] = xr2;
  xi2=(cr.f[3]-cr.f[1]);                     uout[5].f[0] = xi2;
  xr1= ci.f[0] + s*(ci.f[1]-ci.f[3]);        uout[2].f[0] = xr1;
  xi1=-ci.f[2] - s*(ci.f[1]+ci.f[3]);        uout[3].f[0] = xi1;
  xr3= ci.f[0] - s*(ci.f[1]-ci.f[3]);        uout[6].f[0] = xr3;
  xi3= ci.f[2] - s*(ci.f[1]+ci.f[3]);        uout[7].f[0] = xi3; 

  for (k=1; k < dk; ++k) {
    v4sf save_next = in[8*k+7];
    pffft_real_finalize_4x4(&save, &in[8*k+0], in + 8*k+1,
                           e + k*6, out + k*8);
    save = save_next;
  }

}

ALWAYS_INLINE(void) pffft_real_preprocess_4x4(const v4sf *in, 
                                             const v4sf *e, v4sf *out, int first) {
  v4sf r0=in[0], i0=in[1], r1=in[2], i1=in[3], r2=in[4], i2=in[5], r3=in[6], i3=in[7];
  /*
    transformation for each column is:

    [1   1   1   1   0   0   0   0]   [r0]
    [1   0   0  -1   0  -1  -1   0]   [r1]
    [1  -1  -1   1   0   0   0   0]   [r2]
    [1   0   0  -1   0   1   1   0]   [r3]
    [0   0   0   0   1  -1   1  -1] * [i0]
    [0  -1   1   0   1   0   0   1]   [i1]
    [0   0   0   0   1   1  -1  -1]   [i2]
    [0   1  -1   0   1   0   0   1]   [i3]    
  */

  v4sf sr0 = VADD(r0,r3), dr0 = VSUB(r0,r3); 
  v4sf sr1 = VADD(r1,r2), dr1 = VSUB(r1,r2);
  v4sf si0 = VADD(i0,i3), di0 = VSUB(i0,i3); 
  v4sf si1 = VADD(i1,i2), di1 = VSUB(i1,i2);

  r0 = VADD(sr0, sr1);
  r2 = VSUB(sr0, sr1);
  r1 = VSUB(dr0, si1);
  r3 = VADD(dr0, si1);
  i0 = VSUB(di0, di1);
  i2 = VADD(di0, di1);
  i1 = VSUB(si0, dr1);
  i3 = VADD(si0, dr1);

  VCPLXMULCONJ(r1,i1,e[0],e[1]);
  VCPLXMULCONJ(r2,i2,e[2],e[3]);
  VCPLXMULCONJ(r3,i3,e[4],e[5]);

  VTRANSPOSE4(r0,r1,r2,r3);
  VTRANSPOSE4(i0,i1,i2,i3);

  if (!first) {
    *out++ = r0;
    *out++ = i0;
  }
  *out++ = r1;
  *out++ = i1;
  *out++ = r2;
  *out++ = i2;
  *out++ = r3;
  *out++ = i3;
}

static NEVER_INLINE(void) pffft_real_preprocess(int Ncvec, const v4sf *in, v4sf *out, const v4sf *e) {
  int k, dk = Ncvec/SIMD_SZ; // number of 4x4 matrix blocks
  /* fftpack order is f0r f1r f1i f2r f2i ... f(n-1)r f(n-1)i f(n)r */

  v4sf_union Xr, Xi, *uout = (v4sf_union*)out;
  float cr0, ci0, cr1, ci1, cr2, ci2, cr3, ci3;
  static const float s = M_SQRT2;
  assert(in != out);
  for (k=0; k < 4; ++k) {
    Xr.f[k] = ((float*)in)[8*k];
    Xi.f[k] = ((float*)in)[8*k+4];
  }

  pffft_real_preprocess_4x4(in, e, out+1, 1); // will write only 6 values

  /*
    [Xr0 Xr1 Xr2 Xr3 Xi0 Xi1 Xi2 Xi3]

    [cr0] [1   0   2   0   1   0   0   0]
    [cr1] [1   0   0   0  -1   0  -2   0]
    [cr2] [1   0  -2   0   1   0   0   0]
    [cr3] [1   0   0   0  -1   0   2   0]
    [ci0] [0   2   0   2   0   0   0   0]
    [ci1] [0   s   0  -s   0  -s   0  -s]
    [ci2] [0   0   0   0   0  -2   0   2]
    [ci3] [0  -s   0   s   0  -s   0  -s]
  */
  for (k=1; k < dk; ++k) {    
    pffft_real_preprocess_4x4(in+8*k, e + k*6, out-1+k*8, 0);
  }

  cr0=(Xr.f[0]+Xi.f[0]) + 2*Xr.f[2]; uout[0].f[0] = cr0;
  cr1=(Xr.f[0]-Xi.f[0]) - 2*Xi.f[2]; uout[0].f[1] = cr1;
  cr2=(Xr.f[0]+Xi.f[0]) - 2*Xr.f[2]; uout[0].f[2] = cr2;
  cr3=(Xr.f[0]-Xi.f[0]) + 2*Xi.f[2]; uout[0].f[3] = cr3;
  ci0= 2*(Xr.f[1]+Xr.f[3]);                       uout[2*Ncvec-1].f[0] = ci0;
  ci1= s*(Xr.f[1]-Xr.f[3]) - s*(Xi.f[1]+Xi.f[3]); uout[2*Ncvec-1].f[1] = ci1;
  ci2= 2*(Xi.f[3]-Xi.f[1]);                       uout[2*Ncvec-1].f[2] = ci2;
  ci3=-s*(Xr.f[1]-Xr.f[3]) - s*(Xi.f[1]+Xi.f[3]); uout[2*Ncvec-1].f[3] = ci3;
}


void pffft_transform_internal(PFFFT_Setup *setup, const float *finput, float *foutput, v4sf *scratch,
                             pffft_direction_t direction, int ordered) {
  int k, Ncvec   = setup->Ncvec;
  int nf_odd = (setup->ifac[1] & 1);

  // temporary buffer is allocated on the stack if the scratch pointer is NULL
  int stack_allocate = (scratch == 0 ? Ncvec*2 : 1);
  VLA_ARRAY_ON_STACK(v4sf, scratch_on_stack, stack_allocate);
  if (scratch == 0) scratch = scratch_on_stack;

  const v4sf *vinput = (const v4sf*)finput;
  v4sf *voutput      = (v4sf*)foutput;
  v4sf *buff[2]      = { voutput, scratch };
  int ib = (nf_odd ^ ordered ? 1 : 0);

  assert(VALIGNED(finput) && VALIGNED(foutput));

  //assert(finput != foutput);
  if (direction == PFFFT_FORWARD) {
    ib = !ib;
    if (setup->transform == PFFFT_REAL) { 
      ib = (rfftf1_ps(Ncvec*2, vinput, buff[ib], buff[!ib],
                      setup->twiddle, &setup->ifac[0]) == buff[0] ? 0 : 1);      
      pffft_real_finalize(Ncvec, buff[ib], buff[!ib], (v4sf*)setup->e);
    } else {
      v4sf *tmp = buff[ib];
      for (k=0; k < Ncvec; ++k) {
        UNINTERLEAVE2(vinput[k*2], vinput[k*2+1], tmp[k*2], tmp[k*2+1]);
      }
      ib = (cfftf1_ps(Ncvec, buff[ib], buff[!ib], buff[ib], 
                      setup->twiddle, &setup->ifac[0], -1) == buff[0] ? 0 : 1);
      pffft_cplx_finalize(Ncvec, buff[ib], buff[!ib], (v4sf*)setup->e);
    }
    if (ordered) {
      pffft_zreorder(setup, (float*)buff[!ib], (float*)buff[ib], PFFFT_FORWARD);       
    } else ib = !ib;
  } else {
    if (vinput == buff[ib]) { 
      ib = !ib; // may happen when finput == foutput
    }
    if (ordered) {
      pffft_zreorder(setup, (float*)vinput, (float*)buff[ib], PFFFT_BACKWARD); 
      vinput = buff[ib]; ib = !ib;
    }
    if (setup->transform == PFFFT_REAL) {
      pffft_real_preprocess(Ncvec, vinput, buff[ib], (v4sf*)setup->e);
      ib = (rfftb1_ps(Ncvec*2, buff[ib], buff[0], buff[1], 
                      setup->twiddle, &setup->ifac[0]) == buff[0] ? 0 : 1);
    } else {
      pffft_cplx_preprocess(Ncvec, vinput, buff[ib], (v4sf*)setup->e);
      ib = (cfftf1_ps(Ncvec, buff[ib], buff[0], buff[1], 
                      setup->twiddle, &setup->ifac[0], +1) == buff[0] ? 0 : 1);
      for (k=0; k < Ncvec; ++k) {
        INTERLEAVE2(buff[ib][k*2], buff[ib][k*2+1], buff[ib][k*2], buff[ib][k*2+1]);
      }
    }
  }
  
  if (buff[ib] != voutput) {
    /* extra copy required -- this situation should only happen when finput == foutput */
    assert(finput==foutput);
    for (k=0; k < Ncvec; ++k) {
      v4sf a = buff[ib][2*k], b = buff[ib][2*k+1];
      voutput[2*k] = a; voutput[2*k+1] = b;
    }
    ib = !ib;
  }
  assert(buff[ib] == voutput);
}

void pffft_zconvolve_accumulate(PFFFT_Setup *s, const float *a, const float *b, float *ab, float scaling) {
  int i, Ncvec = s->Ncvec;
  const v4sf * RESTRICT va = (const v4sf*)a;
  const v4sf * RESTRICT vb = (const v4sf*)b;
  v4sf * RESTRICT vab = (v4sf*)ab;

#ifdef __arm__
  __builtin_prefetch(va);
  __builtin_prefetch(vb);
  __builtin_prefetch(vab);
  __builtin_prefetch(va+2);
  __builtin_prefetch(vb+2);
  __builtin_prefetch(vab+2);
  __builtin_prefetch(va+4);
  __builtin_prefetch(vb+4);
  __builtin_prefetch(vab+4);
  __builtin_prefetch(va+6);
  __builtin_prefetch(vb+6);
  __builtin_prefetch(vab+6);
#endif

  float ar, ai, br, bi, abr, abi;
  v4sf vscal = LD_PS1(scaling);

  assert(VALIGNED(a) && VALIGNED(b) && VALIGNED(ab));
  ar = ((v4sf_union*)va)[0].f[0];
  ai = ((v4sf_union*)va)[1].f[0];
  br = ((v4sf_union*)vb)[0].f[0];
  bi = ((v4sf_union*)vb)[1].f[0];
  abr = ((v4sf_union*)vab)[0].f[0];
  abi = ((v4sf_union*)vab)[1].f[0];
 
#ifdef __arm__
#  if 1 // inline asm version
  const float *a_ = a, *b_ = b; float *ab_ = ab;
  int N = Ncvec;
  asm volatile("mov         r8, %2                  \n"
               "vdup.f32    q15, %4                 \n"
               "1:                                  \n"
               "pld         [%0,#64]                \n"
               "pld         [%1,#64]                \n"
               "pld         [%2,#64]                \n"
               "pld         [%0,#96]                \n"
               "pld         [%1,#96]                \n"
               "pld         [%2,#96]                \n"
               "vld1.f32    {q0,q1},   [%0,:128]!         \n"
               "vld1.f32    {q4,q5},   [%1,:128]!         \n"
               "vld1.f32    {q2,q3},   [%0,:128]!         \n"
               "vld1.f32    {q6,q7},   [%1,:128]!         \n"
               "vld1.f32    {q8,q9},   [r8,:128]!          \n"
               
               "vmul.f32    q10, q0, q4             \n"
               "vmul.f32    q11, q0, q5             \n"
               "vmul.f32    q12, q2, q6             \n" 
               "vmul.f32    q13, q2, q7             \n"                 
               "vmls.f32    q10, q1, q5             \n"
               "vmla.f32    q11, q1, q4             \n"
               "vld1.f32    {q0,q1}, [r8,:128]!     \n"
               "vmls.f32    q12, q3, q7             \n"
               "vmla.f32    q13, q3, q6             \n"
               "vmla.f32    q8, q10, q15            \n"
               "vmla.f32    q9, q11, q15            \n"
               "vmla.f32    q0, q12, q15            \n"
               "vmla.f32    q1, q13, q15            \n"
               "vst1.f32    {q8,q9},[%2,:128]!    \n"
               "vst1.f32    {q0,q1},[%2,:128]!    \n"
               "subs        %3, #2                  \n"
               "bne         1b                      \n"
               : "+r"(a_), "+r"(b_), "+r"(ab_), "+r"(N) : "r"(scaling) : "r8", "q0","q1","q2","q3","q4","q5","q6","q7","q8","q9", "q10","q11","q12","q13","q15","memory");

#  else // neon instrinsics version, 30% slower that the asm one with gcc 4.6
  v4sf a1r, a1i, b1r, b1i;
  v4sf a2r, a2i, b2r, b2i;
  v4sf ab1r, ab1i, ab2r, ab2i;
  for (i=0; i < Ncvec; i += 2) {
    __builtin_prefetch(va+8);
    __builtin_prefetch(va+10);

    a1r = *va++; a1i = *va++;
    a2r = *va++; a2i = *va++;
    b1r = *vb++; b1i = *vb++;
    b2r = *vb++; b2i = *vb++;
    ab1r = vab[0]; ab1i = vab[1];
    ab2r = vab[2]; ab2i = vab[3];
    
    v4sf z1r = VMUL(a1r, b1r);
    v4sf z2r = VMUL(a2r, b2r);
    v4sf z1i = VMUL(a1r, b1i);
    v4sf z2i = VMUL(a2r, b2i);

    __builtin_prefetch(vb+4);
    __builtin_prefetch(vb+6);

    z1r = vmlsq_f32(z1r, a1i, b1i);
    z2r = vmlsq_f32(z2r, a2i, b2i);
    z1i = vmlaq_f32(z1i, a1i, b1r);
    z2i = vmlaq_f32(z2i, a2i, b2r);

    __builtin_prefetch(vab+4);
    __builtin_prefetch(vab+6);

    ab1r = vmlaq_f32(ab1r, z1r, vscal);
    ab2r = vmlaq_f32(ab2r, z2r, vscal);
    ab1i = vmlaq_f32(ab1i, z1i, vscal);
    ab2i = vmlaq_f32(ab2i, z2i, vscal);

    *vab++ = ab1r; *vab++ = ab1i;
    *vab++ = ab2r; *vab++ = ab2i;
  }
#  endif 
    
#else // not ARM, no need to use a special routine
  for (i=0; i < Ncvec; i += 2) {
    v4sf ar, ai, br, bi;
    ar = va[2*i+0]; ai = va[2*i+1];
    br = vb[2*i+0]; bi = vb[2*i+1];
    VCPLXMUL(ar, ai, br, bi);
    vab[2*i+0] = VMADD(ar, vscal, vab[2*i+0]);
    vab[2*i+1] = VMADD(ai, vscal, vab[2*i+1]);
    ar = va[2*i+2]; ai = va[2*i+3];
    br = vb[2*i+2]; bi = vb[2*i+3];
    VCPLXMUL(ar, ai, br, bi);
    vab[2*i+2] = VMADD(ar, vscal, vab[2*i+2]);
    vab[2*i+3] = VMADD(ai, vscal, vab[2*i+3]);
  }
#endif
  if (s->transform == PFFFT_REAL) {
    ((v4sf_union*)vab)[0].f[0] = abr + ar*br*scaling;
    ((v4sf_union*)vab)[1].f[0] = abi + ai*bi*scaling;
  }
}


#else // defined(PFFFT_SIMD_DISABLE)

// standard routine using scalar floats, without SIMD stuff.

#define pffft_zreorder_nosimd pffft_zreorder
void pffft_zreorder_nosimd(PFFFT_Setup *setup, const float *in, float *out, pffft_direction_t direction) {
  int k, N = setup->N;
  if (setup->transform == PFFFT_COMPLEX) {
    for (k=0; k < 2*N; ++k) out[k] = in[k];
    return;
  }
  else if (direction == PFFFT_FORWARD) {
    float x_N = in[N-1];
    for (k=N-1; k > 1; --k) out[k] = in[k-1]; 
    out[0] = in[0];
    out[1] = x_N;
  } else {
    float x_N = in[1];
    for (k=1; k < N-1; ++k) out[k] = in[k+1]; 
    out[0] = in[0];
    out[N-1] = x_N;
  }
}

#define pffft_transform_internal_nosimd pffft_transform_internal
void pffft_transform_internal_nosimd(PFFFT_Setup *setup, const float *input, float *output, float *scratch,
                                    pffft_direction_t direction, int ordered) {
  int Ncvec   = setup->Ncvec;
  int nf_odd = (setup->ifac[1] & 1);

  // temporary buffer is allocated on the stack if the scratch pointer is NULL
  int stack_allocate = (scratch == 0 ? Ncvec*2 : 1);
  VLA_ARRAY_ON_STACK(v4sf, scratch_on_stack, stack_allocate);
  if (scratch == 0) scratch = scratch_on_stack;

  float *buff[2] = { output, scratch };
  int ib;
  if (setup->transform == PFFFT_COMPLEX) ordered = 0; // it is always ordered.
  ib = (nf_odd ^ ordered ? 1 : 0);

  if (direction == PFFFT_FORWARD) {
    if (setup->transform == PFFFT_REAL) { 
      ib = (rfftf1_ps(Ncvec*2, input, buff[ib], buff[!ib],
                      setup->twiddle, &setup->ifac[0]) == buff[0] ? 0 : 1);      
    } else {
      ib = (cfftf1_ps(Ncvec, input, buff[ib], buff[!ib], 
                      setup->twiddle, &setup->ifac[0], -1) == buff[0] ? 0 : 1);
    }
    if (ordered) {
      pffft_zreorder(setup, buff[ib], buff[!ib], PFFFT_FORWARD); ib = !ib;
    }
  } else {    
    if (input == buff[ib]) { 
      ib = !ib; // may happen when finput == foutput
    }
    if (ordered) {
      pffft_zreorder(setup, input, buff[!ib], PFFFT_BACKWARD); 
      input = buff[!ib];
    }
    if (setup->transform == PFFFT_REAL) {
      ib = (rfftb1_ps(Ncvec*2, input, buff[ib], buff[!ib], 
                      setup->twiddle, &setup->ifac[0]) == buff[0] ? 0 : 1);
    } else {
      ib = (cfftf1_ps(Ncvec, input, buff[ib], buff[!ib], 
                      setup->twiddle, &setup->ifac[0], +1) == buff[0] ? 0 : 1);
    }
  }
  if (buff[ib] != output) {
    int k;
    // extra copy required -- this situation should happens only when finput == foutput
    assert(input==output);
    for (k=0; k < Ncvec; ++k) {
      float a = buff[ib][2*k], b = buff[ib][2*k+1];
      output[2*k] = a; output[2*k+1] = b;
    }
    ib = !ib;
  }
  assert(buff[ib] == output);
}

#define pffft_zconvolve_accumulate_nosimd pffft_zconvolve_accumulate
void pffft_zconvolve_accumulate_nosimd(PFFFT_Setup *s, const float *a, const float *b,
                                       float *ab, float scaling) {
  int i, Ncvec = s->Ncvec;

  if (s->transform == PFFFT_REAL) {
    // take care of the fftpack ordering
    ab[0] += a[0]*b[0]*scaling;
    ab[2*Ncvec-1] += a[2*Ncvec-1]*b[2*Ncvec-1]*scaling;
    ++ab; ++a; ++b; --Ncvec;
  }
  for (i=0; i < Ncvec; ++i) {
    float ar, ai, br, bi;
    ar = a[2*i+0]; ai = a[2*i+1];
    br = b[2*i+0]; bi = b[2*i+1];
    VCPLXMUL(ar, ai, br, bi);
    ab[2*i+0] += ar*scaling;
    ab[2*i+1] += ai*scaling;
  }
}

#endif // defined(PFFFT_SIMD_DISABLE)

void pffft_transform(PFFFT_Setup *setup, const float *input, float *output, float *work, pffft_direction_t direction) {
  pffft_transform_internal(setup, input, output, (v4sf*)work, direction, 0);
}

void pffft_transform_ordered(PFFFT_Setup *setup, const float *input, float *output, float *work, pffft_direction_t direction) {
  pffft_transform_internal(setup, input, output, (v4sf*)work, direction, 1);
}