sse_mathfun.h

/* SIMD (SSE1+MMX or SSE2) implementation of sin, cos, exp and log

   Inspired by Intel Approximate Math library, and based on the
   corresponding algorithms of the cephes math library

   The default is to use the SSE1 version. If you define USE_SSE2 the
   the SSE2 intrinsics will be used in place of the MMX intrinsics. Do
   not expect any significant performance improvement with SSE2.
*/

/* Copyright (C) 2007  Julien Pommier

  This software is provided 'as-is', without any express or implied
  warranty.  In no event will the authors be held liable for any damages
  arising from the use of this software.

  Permission is granted to anyone to use this software for any purpose,
  including commercial applications, and to alter it and redistribute it
  freely, subject to the following restrictions:

  1. The origin of this software must not be misrepresented; you must not
     claim that you wrote the original software. If you use this software
     in a product, an acknowledgment in the product documentation would be
     appreciated but is not required.
  2. Altered source versions must be plainly marked as such, and must not be
     misrepresented as being the original software.
  3. This notice may not be removed or altered from any source distribution.

  (this is the zlib license)
*/

#include <xmmintrin.h>

/* yes I know, the top of this file is quite ugly */

#define USE_SSE2 // use SSE2 version

#ifdef _MSC_VER /* visual c++ */
# define ALIGN16_BEG __declspec(align(16))
# define ALIGN16_END 
#else /* gcc or icc */
# define ALIGN16_BEG
# define ALIGN16_END __attribute__((aligned(16)))
#endif

/* __m128 is ugly to write */
typedef __m128 v4sf;  // vector of 4 float (sse1)

#ifdef USE_SSE2
# include <emmintrin.h>
typedef __m128i v4si; // vector of 4 int (sse2)
#else
typedef __m64 v2si;   // vector of 2 int (mmx)
#endif

/* declare some SSE constants -- why can't I figure a better way to do that? */
#define _PS_CONST(Name, Val)                                            \
  static const ALIGN16_BEG float _ps_##Name[4] ALIGN16_END = { Val, Val, Val, Val }
#define _PI32_CONST(Name, Val)                                            \
  static const ALIGN16_BEG int _pi32_##Name[4] ALIGN16_END = { Val, Val, Val, Val }
#define _PS_CONST_TYPE(Name, Type, Val)                                 \
  static const ALIGN16_BEG Type _ps_##Name[4] ALIGN16_END = { Val, Val, Val, Val }

_PS_CONST(1  , 1.0f);
_PS_CONST(0p5, 0.5f);
/* the smallest non denormalized float number */
_PS_CONST_TYPE(min_norm_pos, int, 0x00800000);
_PS_CONST_TYPE(mant_mask, int, 0x7f800000);
_PS_CONST_TYPE(inv_mant_mask, int, ~0x7f800000);

_PS_CONST_TYPE(sign_mask, int, 0x80000000);
_PS_CONST_TYPE(inv_sign_mask, int, ~0x80000000);

_PI32_CONST(1, 1);
_PI32_CONST(inv1, ~1);
_PI32_CONST(2, 2);
_PI32_CONST(4, 4);
_PI32_CONST(0x7f, 0x7f);

_PS_CONST(cephes_SQRTHF, 0.707106781186547524);
_PS_CONST(cephes_log_p0, 7.0376836292E-2);
_PS_CONST(cephes_log_p1, - 1.1514610310E-1);
_PS_CONST(cephes_log_p2, 1.1676998740E-1);
_PS_CONST(cephes_log_p3, - 1.2420140846E-1);
_PS_CONST(cephes_log_p4, + 1.4249322787E-1);
_PS_CONST(cephes_log_p5, - 1.6668057665E-1);
_PS_CONST(cephes_log_p6, + 2.0000714765E-1);
_PS_CONST(cephes_log_p7, - 2.4999993993E-1);
_PS_CONST(cephes_log_p8, + 3.3333331174E-1);
_PS_CONST(cephes_log_q1, -2.12194440e-4);
_PS_CONST(cephes_log_q2, 0.693359375);

#if defined (__MINGW32__)

/* the ugly part below: many versions of gcc used to be completely buggy with respect to some intrinsics
   The movehl_ps is fixed in mingw 3.4.5, but I found out that all the _mm_cmp* intrinsics were completely
   broken on my mingw gcc 3.4.5 ...

   Note that the bug on _mm_cmp* does occur only at -O0 optimization level
*/

inline __m128 my_movehl_ps(__m128 a, const __m128 b) {
	asm (
			"movhlps %2,%0\n\t"
			: "=x" (a)
			: "0" (a), "x"(b)
	    );
	return a;                                 }
#warning "redefined _mm_movehl_ps (see gcc bug 21179)"
#define _mm_movehl_ps my_movehl_ps

inline __m128 my_cmplt_ps(__m128 a, const __m128 b) {
	asm (
			"cmpltps %2,%0\n\t"
			: "=x" (a)
			: "0" (a), "x"(b)
	    );
	return a;               
                  }
inline __m128 my_cmpgt_ps(__m128 a, const __m128 b) {
	asm (
			"cmpnleps %2,%0\n\t"
			: "=x" (a)
			: "0" (a), "x"(b)
	    );
	return a;               
}
inline __m128 my_cmpeq_ps(__m128 a, const __m128 b) {
	asm (
			"cmpeqps %2,%0\n\t"
			: "=x" (a)
			: "0" (a), "x"(b)
	    );
	return a;               
}
#warning "redefined _mm_cmpxx_ps functions..."
#define _mm_cmplt_ps my_cmplt_ps
#define _mm_cmpgt_ps my_cmpgt_ps
#define _mm_cmpeq_ps my_cmpeq_ps
#endif

#ifndef USE_SSE2
typedef union xmm_mm_union {
  __m128 xmm;
  __m64 mm[2];
} xmm_mm_union;

#define COPY_XMM_TO_MM(xmm_, mm0_, mm1_) {          \
    xmm_mm_union u; u.xmm = xmm_;                   \
    mm0_ = u.mm[0];                                 \
    mm1_ = u.mm[1];                                 \
}

#define COPY_MM_TO_XMM(mm0_, mm1_, xmm_) {                         \
    xmm_mm_union u; u.mm[0]=mm0_; u.mm[1]=mm1_; xmm_ = u.xmm;      \
  }

#endif // USE_SSE2

/* natural logarithm computed for 4 simultaneous float 
   return NaN for x <= 0
*/
v4sf log_ps(v4sf x) {
#ifdef USE_SSE2
  v4si emm0;
#else
  v2si mm0, mm1;
#endif
  v4sf one = *(v4sf*)_ps_1;

  v4sf invalid_mask = _mm_cmple_ps(x, _mm_setzero_ps());

  x = _mm_max_ps(x, *(v4sf*)_ps_min_norm_pos);  /* cut off denormalized stuff */

#ifndef USE_SSE2
  /* part 1: x = frexpf(x, &e); */
  COPY_XMM_TO_MM(x, mm0, mm1);
  mm0 = _mm_srli_pi32(mm0, 23);
  mm1 = _mm_srli_pi32(mm1, 23);
#else
  emm0 = _mm_srli_epi32(_mm_castps_si128(x), 23);
#endif
  /* keep only the fractional part */
  x = _mm_and_ps(x, *(v4sf*)_ps_inv_mant_mask);
  x = _mm_or_ps(x, *(v4sf*)_ps_0p5);

#ifndef USE_SSE2
  /* now e=mm0:mm1 contain the really base-2 exponent */
  mm0 = _mm_sub_pi32(mm0, *(v2si*)_pi32_0x7f);
  mm1 = _mm_sub_pi32(mm1, *(v2si*)_pi32_0x7f);
  v4sf e = _mm_cvtpi32x2_ps(mm0, mm1);
  _mm_empty(); /* bye bye mmx */
#else
  emm0 = _mm_sub_epi32(emm0, *(v4si*)_pi32_0x7f);
  v4sf e = _mm_cvtepi32_ps(emm0);
#endif

  e = _mm_add_ps(e, one);

  /* part2: 
     if( x < SQRTHF ) {
       e -= 1;
       x = x + x - 1.0;
     } else { x = x - 1.0; }
  */
  v4sf mask = _mm_cmplt_ps(x, *(v4sf*)_ps_cephes_SQRTHF);
  v4sf tmp = _mm_and_ps(x, mask);
  x = _mm_sub_ps(x, one);
  e = _mm_sub_ps(e, _mm_and_ps(one, mask));
  x = _mm_add_ps(x, tmp);


  v4sf z = _mm_mul_ps(x,x);

  v4sf y = *(v4sf*)_ps_cephes_log_p0;
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_log_p1);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_log_p2);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_log_p3);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_log_p4);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_log_p5);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_log_p6);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_log_p7);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_log_p8);
  y = _mm_mul_ps(y, x);

  y = _mm_mul_ps(y, z);
  

  tmp = _mm_mul_ps(e, *(v4sf*)_ps_cephes_log_q1);
  y = _mm_add_ps(y, tmp);


  tmp = _mm_mul_ps(z, *(v4sf*)_ps_0p5);
  y = _mm_sub_ps(y, tmp);

  tmp = _mm_mul_ps(e, *(v4sf*)_ps_cephes_log_q2);
  x = _mm_add_ps(x, y);
  x = _mm_add_ps(x, tmp);
  x = _mm_or_ps(x, invalid_mask); // negative arg will be NAN
  return x;
}

_PS_CONST(exp_hi,	88.3762626647949f);
_PS_CONST(exp_lo,	-88.3762626647949f);

_PS_CONST(cephes_LOG2EF, 1.44269504088896341);
_PS_CONST(cephes_exp_C1, 0.693359375);
_PS_CONST(cephes_exp_C2, -2.12194440e-4);

_PS_CONST(cephes_exp_p0, 1.9875691500E-4);
_PS_CONST(cephes_exp_p1, 1.3981999507E-3);
_PS_CONST(cephes_exp_p2, 8.3334519073E-3);
_PS_CONST(cephes_exp_p3, 4.1665795894E-2);
_PS_CONST(cephes_exp_p4, 1.6666665459E-1);
_PS_CONST(cephes_exp_p5, 5.0000001201E-1);

v4sf exp_ps(v4sf x) {
  v4sf tmp = _mm_setzero_ps(), fx;
#ifdef USE_SSE2
  v4si emm0;
#else
  v2si mm0, mm1;
#endif
  v4sf one = *(v4sf*)_ps_1;

  x = _mm_min_ps(x, *(v4sf*)_ps_exp_hi);
  x = _mm_max_ps(x, *(v4sf*)_ps_exp_lo);

  /* express exp(x) as exp(g + n*log(2)) */
  fx = _mm_mul_ps(x, *(v4sf*)_ps_cephes_LOG2EF);
  fx = _mm_add_ps(fx, *(v4sf*)_ps_0p5);

  /* how to perform a floorf with SSE: just below */
#ifndef USE_SSE2
  /* step 1 : cast to int */
  tmp = _mm_movehl_ps(tmp, fx);
  mm0 = _mm_cvttps_pi32(fx);
  mm1 = _mm_cvttps_pi32(tmp);
  /* step 2 : cast back to float */
  tmp = _mm_cvtpi32x2_ps(mm0, mm1);
#else
  emm0 = _mm_cvttps_epi32(fx);
  tmp  = _mm_cvtepi32_ps(emm0);
#endif
  /* if greater, substract 1 */
  v4sf mask = _mm_cmpgt_ps(tmp, fx);    
  mask = _mm_and_ps(mask, one);
  fx = _mm_sub_ps(tmp, mask);

  tmp = _mm_mul_ps(fx, *(v4sf*)_ps_cephes_exp_C1);
  v4sf z = _mm_mul_ps(fx, *(v4sf*)_ps_cephes_exp_C2);
  x = _mm_sub_ps(x, tmp);
  x = _mm_sub_ps(x, z);

  z = _mm_mul_ps(x,x);
  
  v4sf y = *(v4sf*)_ps_cephes_exp_p0;
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_exp_p1);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_exp_p2);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_exp_p3);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_exp_p4);
  y = _mm_mul_ps(y, x);
  y = _mm_add_ps(y, *(v4sf*)_ps_cephes_exp_p5);
  y = _mm_mul_ps(y, z);
  y = _mm_add_ps(y, x);
  y = _mm_add_ps(y, one);

  /* build 2^n */
#ifndef USE_SSE2
  z = _mm_movehl_ps(z, fx);
  mm0 = _mm_cvttps_pi32(fx);
  mm1 = _mm_cvttps_pi32(z);
  mm0 = _mm_add_pi32(mm0, *(v2si*)_pi32_0x7f);
  mm1 = _mm_add_pi32(mm1, *(v2si*)_pi32_0x7f);
  mm0 = _mm_slli_pi32(mm0, 23); 
  mm1 = _mm_slli_pi32(mm1, 23);
  
  v4sf pow2n; 
  COPY_MM_TO_XMM(mm0, mm1, pow2n);
  _mm_empty();
#else
  emm0 = _mm_cvttps_epi32(fx);
  emm0 = _mm_add_epi32(emm0, *(v4si*)_pi32_0x7f);
  emm0 = _mm_slli_epi32(emm0, 23);
  v4sf pow2n = _mm_castsi128_ps(emm0);
#endif
  y = _mm_mul_ps(y, pow2n);
  return y;
}

_PS_CONST(minus_cephes_DP1, -0.78515625);
_PS_CONST(minus_cephes_DP2, -2.4187564849853515625e-4);
_PS_CONST(minus_cephes_DP3, -3.77489497744594108e-8);
_PS_CONST(sincof_p0, -1.9515295891E-4);
_PS_CONST(sincof_p1,  8.3321608736E-3);
_PS_CONST(sincof_p2, -1.6666654611E-1);
_PS_CONST(coscof_p0,  2.443315711809948E-005);
_PS_CONST(coscof_p1, -1.388731625493765E-003);
_PS_CONST(coscof_p2,  4.166664568298827E-002);
_PS_CONST(cephes_FOPI, 1.27323954473516); // 4 / M_PI


/* evaluation of 4 sines at onces, using only SSE1+MMX intrinsics so
   it runs also on old athlons XPs and the pentium III of your grand
   mother.

   The code is the exact rewriting of the cephes sinf function.
   Precision is excellent as long as x < 8192 (I did not bother to
   take into account the special handling they have for greater values
   -- it does not return garbage for arguments over 8192, though, but
   the extra precision is missing).

   Note that it is such that sinf((float)M_PI) = 8.74e-8, which is the
   surprising but correct result.

   Performance is also surprisingly good, 1.33 times faster than the
   macos vsinf SSE2 function, and 1.5 times faster than the
   __vrs4_sinf of amd's ACML (which is only available in 64 bits). Not
   too bad for an SSE1 function (with no special tuning) !
   However the latter libraries probably have a much better handling of NaN,
   Inf, denormalized and other special arguments..

   On my core 1 duo, the execution of this function takes approximately 95 cycles.

   From what I have observed on the experiments with Intel AMath lib, switching to an
   SSE2 version would improve the perf by only 10%.

   Since it is based on SSE intrinsics, it has to be compiled at -O2 to
   deliver full speed.
*/
v4sf sin_ps(v4sf x) { // any x
  v4sf xmm1, xmm2 = _mm_setzero_ps(), xmm3, sign_bit, y;

#ifdef USE_SSE2
  v4si emm0, emm2;
#else
  v2si mm0, mm1, mm2, mm3;
#endif
  sign_bit = x;
  /* take the absolute value */
  x = _mm_and_ps(x, *(v4sf*)_ps_inv_sign_mask);
  /* extract the sign bit (upper one) */
  sign_bit = _mm_and_ps(sign_bit, *(v4sf*)_ps_sign_mask);
  
  /* scale by 4/Pi */
  y = _mm_mul_ps(x, *(v4sf*)_ps_cephes_FOPI);

  //printf("plop:"); print4(y); 
#ifdef USE_SSE2
  /* store the integer part of y in mm0 */
  emm2 = _mm_cvttps_epi32(y);
  /* j=(j+1) & (~1) (see the cephes sources) */
  emm2 = _mm_add_epi32(emm2, *(v4si*)_pi32_1);
  emm2 = _mm_and_si128(emm2, *(v4si*)_pi32_inv1);
  y = _mm_cvtepi32_ps(emm2);
  /* get the swap sign flag */
  emm0 = _mm_and_si128(emm2, *(v4si*)_pi32_4);
  emm0 = _mm_slli_epi32(emm0, 29);
  /* get the polynom selection mask 
     there is one polynom for 0 <= x <= Pi/4
     and another one for Pi/4<x<=Pi/2

     Both branches will be computed.
  */
  emm2 = _mm_and_si128(emm2, *(v4si*)_pi32_2);
  emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128());
  
  v4sf swap_sign_bit = _mm_castsi128_ps(emm0);
  v4sf poly_mask = _mm_castsi128_ps(emm2);
  sign_bit = _mm_xor_ps(sign_bit, swap_sign_bit);
#else
  /* store the integer part of y in mm0:mm1 */
  xmm2 = _mm_movehl_ps(xmm2, y);
  mm2 = _mm_cvttps_pi32(y);
  mm3 = _mm_cvttps_pi32(xmm2);
  /* j=(j+1) & (~1) (see the cephes sources) */
  mm2 = _mm_add_pi32(mm2, *(v2si*)_pi32_1);
  mm3 = _mm_add_pi32(mm3, *(v2si*)_pi32_1);
  mm2 = _mm_and_si64(mm2, *(v2si*)_pi32_inv1);
  mm3 = _mm_and_si64(mm3, *(v2si*)_pi32_inv1);
  y = _mm_cvtpi32x2_ps(mm2, mm3);
  /* get the swap sign flag */
  mm0 = _mm_and_si64(mm2, *(v2si*)_pi32_4);
  mm1 = _mm_and_si64(mm3, *(v2si*)_pi32_4);
  mm0 = _mm_slli_pi32(mm0, 29);
  mm1 = _mm_slli_pi32(mm1, 29);
  /* get the polynom selection mask */
  mm2 = _mm_and_si64(mm2, *(v2si*)_pi32_2);
  mm3 = _mm_and_si64(mm3, *(v2si*)_pi32_2);
  mm2 = _mm_cmpeq_pi32(mm2, _mm_setzero_si64());
  mm3 = _mm_cmpeq_pi32(mm3, _mm_setzero_si64());
  v4sf swap_sign_bit, poly_mask;
  COPY_MM_TO_XMM(mm0, mm1, swap_sign_bit);
  COPY_MM_TO_XMM(mm2, mm3, poly_mask);
  sign_bit = _mm_xor_ps(sign_bit, swap_sign_bit);
  _mm_empty(); /* good-bye mmx */
#endif
  
  /* The magic pass: "Extended precision modular arithmetic" 
     x = ((x - y * DP1) - y * DP2) - y * DP3; */
  xmm1 = *(v4sf*)_ps_minus_cephes_DP1;
  xmm2 = *(v4sf*)_ps_minus_cephes_DP2;
  xmm3 = *(v4sf*)_ps_minus_cephes_DP3;
  xmm1 = _mm_mul_ps(y, xmm1);
  xmm2 = _mm_mul_ps(y, xmm2);
  xmm3 = _mm_mul_ps(y, xmm3);
  x = _mm_add_ps(x, xmm1);
  x = _mm_add_ps(x, xmm2);
  x = _mm_add_ps(x, xmm3);

  /* Evaluate the first polynom  (0 <= x <= Pi/4) */
  y = *(v4sf*)_ps_coscof_p0;
  v4sf z = _mm_mul_ps(x,x);

  y = _mm_mul_ps(y, z);
  y = _mm_add_ps(y, *(v4sf*)_ps_coscof_p1);
  y = _mm_mul_ps(y, z);
  y = _mm_add_ps(y, *(v4sf*)_ps_coscof_p2);
  y = _mm_mul_ps(y, z);
  y = _mm_mul_ps(y, z);
  v4sf tmp = _mm_mul_ps(z, *(v4sf*)_ps_0p5);
  y = _mm_sub_ps(y, tmp);
  y = _mm_add_ps(y, *(v4sf*)_ps_1);
  
  /* Evaluate the second polynom  (Pi/4 <= x <= 0) */

  v4sf y2 = *(v4sf*)_ps_sincof_p0;
  y2 = _mm_mul_ps(y2, z);
  y2 = _mm_add_ps(y2, *(v4sf*)_ps_sincof_p1);
  y2 = _mm_mul_ps(y2, z);
  y2 = _mm_add_ps(y2, *(v4sf*)_ps_sincof_p2);
  y2 = _mm_mul_ps(y2, z);
  y2 = _mm_mul_ps(y2, x);
  y2 = _mm_add_ps(y2, x);

  /* select the correct result from the two polynoms */  
  xmm3 = poly_mask;
  y2 = _mm_and_ps(xmm3, y2); //, xmm3);
  y = _mm_andnot_ps(xmm3, y);
  y = _mm_add_ps(y,y2);
  /* update the sign */
  y = _mm_xor_ps(y, sign_bit);

  return y;
}

/* almost the same as sin_ps */
v4sf cos_ps(v4sf x) { // any x
  v4sf xmm1, xmm2 = _mm_setzero_ps(), xmm3, y;
#ifdef USE_SSE2
  v4si emm0, emm2;
#else
  v2si mm0, mm1, mm2, mm3;
#endif
  /* take the absolute value */
  x = _mm_and_ps(x, *(v4sf*)_ps_inv_sign_mask);
  
  /* scale by 4/Pi */
  y = _mm_mul_ps(x, *(v4sf*)_ps_cephes_FOPI);
  
#ifdef USE_SSE2
  /* store the integer part of y in mm0 */
  emm2 = _mm_cvttps_epi32(y);
  /* j=(j+1) & (~1) (see the cephes sources) */
  emm2 = _mm_add_epi32(emm2, *(v4si*)_pi32_1);
  emm2 = _mm_and_si128(emm2, *(v4si*)_pi32_inv1);
  y = _mm_cvtepi32_ps(emm2);

  emm2 = _mm_sub_epi32(emm2, *(v4si*)_pi32_2);
  
  /* get the swap sign flag */
  emm0 = _mm_andnot_si128(emm2, *(v4si*)_pi32_4);
  emm0 = _mm_slli_epi32(emm0, 29);
  /* get the polynom selection mask */
  emm2 = _mm_and_si128(emm2, *(v4si*)_pi32_2);
  emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128());
  
  v4sf sign_bit = _mm_castsi128_ps(emm0);
  v4sf poly_mask = _mm_castsi128_ps(emm2);
#else
  /* store the integer part of y in mm0:mm1 */
  xmm2 = _mm_movehl_ps(xmm2, y);
  mm2 = _mm_cvttps_pi32(y);
  mm3 = _mm_cvttps_pi32(xmm2);

  /* j=(j+1) & (~1) (see the cephes sources) */
  mm2 = _mm_add_pi32(mm2, *(v2si*)_pi32_1);
  mm3 = _mm_add_pi32(mm3, *(v2si*)_pi32_1);
  mm2 = _mm_and_si64(mm2, *(v2si*)_pi32_inv1);
  mm3 = _mm_and_si64(mm3, *(v2si*)_pi32_inv1);

  y = _mm_cvtpi32x2_ps(mm2, mm3);


  mm2 = _mm_sub_pi32(mm2, *(v2si*)_pi32_2);
  mm3 = _mm_sub_pi32(mm3, *(v2si*)_pi32_2);

  /* get the swap sign flag in mm0:mm1 and the 
     polynom selection mask in mm2:mm3 */

  mm0 = _mm_andnot_si64(mm2, *(v2si*)_pi32_4);
  mm1 = _mm_andnot_si64(mm3, *(v2si*)_pi32_4);
  mm0 = _mm_slli_pi32(mm0, 29);
  mm1 = _mm_slli_pi32(mm1, 29);

  mm2 = _mm_and_si64(mm2, *(v2si*)_pi32_2);
  mm3 = _mm_and_si64(mm3, *(v2si*)_pi32_2);

  mm2 = _mm_cmpeq_pi32(mm2, _mm_setzero_si64());
  mm3 = _mm_cmpeq_pi32(mm3, _mm_setzero_si64());

  v4sf sign_bit, poly_mask;
  COPY_MM_TO_XMM(mm0, mm1, sign_bit);
  COPY_MM_TO_XMM(mm2, mm3, poly_mask);
  _mm_empty(); /* good-bye mmx */
#endif
  /* The magic pass: "Extended precision modular arithmetic" 
     x = ((x - y * DP1) - y * DP2) - y * DP3; */
  xmm1 = *(v4sf*)_ps_minus_cephes_DP1;
  xmm2 = *(v4sf*)_ps_minus_cephes_DP2;
  xmm3 = *(v4sf*)_ps_minus_cephes_DP3;
  xmm1 = _mm_mul_ps(y, xmm1);
  xmm2 = _mm_mul_ps(y, xmm2);
  xmm3 = _mm_mul_ps(y, xmm3);
  x = _mm_add_ps(x, xmm1);
  x = _mm_add_ps(x, xmm2);
  x = _mm_add_ps(x, xmm3);
  
  /* Evaluate the first polynom  (0 <= x <= Pi/4) */
  y = *(v4sf*)_ps_coscof_p0;
  v4sf z = _mm_mul_ps(x,x);

  y = _mm_mul_ps(y, z);
  y = _mm_add_ps(y, *(v4sf*)_ps_coscof_p1);
  y = _mm_mul_ps(y, z);
  y = _mm_add_ps(y, *(v4sf*)_ps_coscof_p2);
  y = _mm_mul_ps(y, z);
  y = _mm_mul_ps(y, z);
  v4sf tmp = _mm_mul_ps(z, *(v4sf*)_ps_0p5);
  y = _mm_sub_ps(y, tmp);
  y = _mm_add_ps(y, *(v4sf*)_ps_1);
  
  /* Evaluate the second polynom  (Pi/4 <= x <= 0) */

  v4sf y2 = *(v4sf*)_ps_sincof_p0;
  y2 = _mm_mul_ps(y2, z);
  y2 = _mm_add_ps(y2, *(v4sf*)_ps_sincof_p1);
  y2 = _mm_mul_ps(y2, z);
  y2 = _mm_add_ps(y2, *(v4sf*)_ps_sincof_p2);
  y2 = _mm_mul_ps(y2, z);
  y2 = _mm_mul_ps(y2, x);
  y2 = _mm_add_ps(y2, x);

  /* select the correct result from the two polynoms */  
  xmm3 = poly_mask;
  y2 = _mm_and_ps(xmm3, y2); //, xmm3);
  y = _mm_andnot_ps(xmm3, y);
  y = _mm_add_ps(y,y2);
  /* update the sign */
  y = _mm_xor_ps(y, sign_bit);

  return y;
}

/* since sin_ps and cos_ps are almost identical, sincos_ps could replace both of them..
   it is almost as fast, and gives you a free cosine with your sine */
void sincos_ps(v4sf x, v4sf *s, v4sf *c) {
  v4sf xmm1, xmm2, xmm3 = _mm_setzero_ps(), sign_bit_sin, y;
#ifdef USE_SSE2
  v4si emm0, emm2, emm4;
#else
  v2si mm0, mm1, mm2, mm3, mm4, mm5;
#endif
  sign_bit_sin = x;
  /* take the absolute value */
  x = _mm_and_ps(x, *(v4sf*)_ps_inv_sign_mask);
  /* extract the sign bit (upper one) */
  sign_bit_sin = _mm_and_ps(sign_bit_sin, *(v4sf*)_ps_sign_mask);
  
  /* scale by 4/Pi */
  y = _mm_mul_ps(x, *(v4sf*)_ps_cephes_FOPI);
    
#ifdef USE_SSE2
  /* store the integer part of y in emm2 */
  emm2 = _mm_cvttps_epi32(y);

  /* j=(j+1) & (~1) (see the cephes sources) */
  emm2 = _mm_add_epi32(emm2, *(v4si*)_pi32_1);
  emm2 = _mm_and_si128(emm2, *(v4si*)_pi32_inv1);
  y = _mm_cvtepi32_ps(emm2);

  emm4 = emm2;

  /* get the swap sign flag for the sine */
  emm0 = _mm_and_si128(emm2, *(v4si*)_pi32_4);
  emm0 = _mm_slli_epi32(emm0, 29);
  v4sf swap_sign_bit_sin = _mm_castsi128_ps(emm0);

  /* get the polynom selection mask for the sine*/
  emm2 = _mm_and_si128(emm2, *(v4si*)_pi32_2);
  emm2 = _mm_cmpeq_epi32(emm2, _mm_setzero_si128());
  v4sf poly_mask = _mm_castsi128_ps(emm2);
#else
  /* store the integer part of y in mm2:mm3 */
  xmm3 = _mm_movehl_ps(xmm3, y);
  mm2 = _mm_cvttps_pi32(y);
  mm3 = _mm_cvttps_pi32(xmm3);

  /* j=(j+1) & (~1) (see the cephes sources) */
  mm2 = _mm_add_pi32(mm2, *(v2si*)_pi32_1);
  mm3 = _mm_add_pi32(mm3, *(v2si*)_pi32_1);
  mm2 = _mm_and_si64(mm2, *(v2si*)_pi32_inv1);
  mm3 = _mm_and_si64(mm3, *(v2si*)_pi32_inv1);

  y = _mm_cvtpi32x2_ps(mm2, mm3);

  mm4 = mm2;
  mm5 = mm3;

  /* get the swap sign flag for the sine */
  mm0 = _mm_and_si64(mm2, *(v2si*)_pi32_4);
  mm1 = _mm_and_si64(mm3, *(v2si*)_pi32_4);
  mm0 = _mm_slli_pi32(mm0, 29);
  mm1 = _mm_slli_pi32(mm1, 29);
  v4sf swap_sign_bit_sin;
  COPY_MM_TO_XMM(mm0, mm1, swap_sign_bit_sin);

  /* get the polynom selection mask for the sine */

  mm2 = _mm_and_si64(mm2, *(v2si*)_pi32_2);
  mm3 = _mm_and_si64(mm3, *(v2si*)_pi32_2);
  mm2 = _mm_cmpeq_pi32(mm2, _mm_setzero_si64());
  mm3 = _mm_cmpeq_pi32(mm3, _mm_setzero_si64());
  v4sf poly_mask;
  COPY_MM_TO_XMM(mm2, mm3, poly_mask);
#endif

  /* The magic pass: "Extended precision modular arithmetic" 
     x = ((x - y * DP1) - y * DP2) - y * DP3; */
  xmm1 = *(v4sf*)_ps_minus_cephes_DP1;
  xmm2 = *(v4sf*)_ps_minus_cephes_DP2;
  xmm3 = *(v4sf*)_ps_minus_cephes_DP3;
  xmm1 = _mm_mul_ps(y, xmm1);
  xmm2 = _mm_mul_ps(y, xmm2);
  xmm3 = _mm_mul_ps(y, xmm3);
  x = _mm_add_ps(x, xmm1);
  x = _mm_add_ps(x, xmm2);
  x = _mm_add_ps(x, xmm3);

#ifdef USE_SSE2
  emm4 = _mm_sub_epi32(emm4, *(v4si*)_pi32_2);
  emm4 = _mm_andnot_si128(emm4, *(v4si*)_pi32_4);
  emm4 = _mm_slli_epi32(emm4, 29);
  v4sf sign_bit_cos = _mm_castsi128_ps(emm4);
#else
  /* get the sign flag for the cosine */
  mm4 = _mm_sub_pi32(mm4, *(v2si*)_pi32_2);
  mm5 = _mm_sub_pi32(mm5, *(v2si*)_pi32_2);
  mm4 = _mm_andnot_si64(mm4, *(v2si*)_pi32_4);
  mm5 = _mm_andnot_si64(mm5, *(v2si*)_pi32_4);
  mm4 = _mm_slli_pi32(mm4, 29);
  mm5 = _mm_slli_pi32(mm5, 29);
  v4sf sign_bit_cos;
  COPY_MM_TO_XMM(mm4, mm5, sign_bit_cos);
  _mm_empty(); /* good-bye mmx */
#endif

  sign_bit_sin = _mm_xor_ps(sign_bit_sin, swap_sign_bit_sin);

  
  /* Evaluate the first polynom  (0 <= x <= Pi/4) */
  v4sf z = _mm_mul_ps(x,x);
  y = *(v4sf*)_ps_coscof_p0;

  y = _mm_mul_ps(y, z);
  y = _mm_add_ps(y, *(v4sf*)_ps_coscof_p1);
  y = _mm_mul_ps(y, z);
  y = _mm_add_ps(y, *(v4sf*)_ps_coscof_p2);
  y = _mm_mul_ps(y, z);
  y = _mm_mul_ps(y, z);
  v4sf tmp = _mm_mul_ps(z, *(v4sf*)_ps_0p5);
  y = _mm_sub_ps(y, tmp);
  y = _mm_add_ps(y, *(v4sf*)_ps_1);
  
  /* Evaluate the second polynom  (Pi/4 <= x <= 0) */

  v4sf y2 = *(v4sf*)_ps_sincof_p0;
  y2 = _mm_mul_ps(y2, z);
  y2 = _mm_add_ps(y2, *(v4sf*)_ps_sincof_p1);
  y2 = _mm_mul_ps(y2, z);
  y2 = _mm_add_ps(y2, *(v4sf*)_ps_sincof_p2);
  y2 = _mm_mul_ps(y2, z);
  y2 = _mm_mul_ps(y2, x);
  y2 = _mm_add_ps(y2, x);

  /* select the correct result from the two polynoms */  
  xmm3 = poly_mask;
  v4sf ysin2 = _mm_and_ps(xmm3, y2);
  v4sf ysin1 = _mm_andnot_ps(xmm3, y);
  y2 = _mm_sub_ps(y2,ysin2);
  y = _mm_sub_ps(y, ysin1);

  xmm1 = _mm_add_ps(ysin1,ysin2);
  xmm2 = _mm_add_ps(y,y2);
 
  /* update the sign */
  *s = _mm_xor_ps(xmm1, sign_bit_sin);
  *c = _mm_xor_ps(xmm2, sign_bit_cos);
}