How to multiply two quaternions with minimal instructions?

Question

After some thought, I came up with the following code for multiplying two quaternions using SSE:

#include <pmmintrin.h> /* SSE3 intrinsics */

/* multiplication of two quaternions (x, y, z, w) x (a, b, c, d) */

__m128 _mm_cross4_ps(__m128 xyzw, __m128 abcd)
{
    /* The product of two quaternions is:                                 */
    /* (X,Y,Z,W) = (xd+yc-zb+wa, -xc+yd+za+wb, xb-ya+zd+wc, -xa-yb-zc+wd) */

    __m128 wzyx = _mm_shuffle_ps(xyzw, xyzw, _MM_SHUFFLE(0,1,2,3));
    __m128 baba = _mm_shuffle_ps(abcd, abcd, _MM_SHUFFLE(0,1,0,1));
    __m128 dcdc = _mm_shuffle_ps(abcd, abcd, _MM_SHUFFLE(2,3,2,3));

    /* variable names below are for parts of componens of result (X,Y,Z,W) */
    /* nX stands for -X and similarly for the other components             */

    /* znxwy  = (xb - ya, zb - wa, wd - zc, yd - xc) */
    __m128 ZnXWY = _mm_hsub_ps(_mm_mul_ps(xyzw, baba), _mm_mul_ps(wzyx, dcdc));

    /* xzynw  = (xd + yc, zd + wc, wb + za, yb + xa) */
    __m128 XZYnW = _mm_hadd_ps(_mm_mul_ps(xyzw, dcdc), _mm_mul_ps(wzyx, baba));

    /* _mm_shuffle_ps(XZYnW, ZnXWY, _MM_SHUFFLE(3,2,1,0)) */
    /*      = (xd + yc, zd + wc, wd - zc, yd - xc)        */
    /* _mm_shuffle_ps(ZnXWY, XZYnW, _MM_SHUFFLE(2,3,0,1)) */
    /*      = (zb - wa, xb - ya, yb + xa, wb + za)        */

    /* _mm_addsub_ps adds elements 1 and 3 and subtracts elements 0 and 2, so we get: */
    /* _mm_addsub_ps(*, *) = (xd+yc-zb+wa, xb-ya+zd+wc, wd-zc+yb+xa, yd-xc+wb+za)     */

    __m128 XZWY = _mm_addsub_ps(_mm_shuffle_ps(XZYnW, ZnXWY, _MM_SHUFFLE(3,2,1,0)),
                                _mm_shuffle_ps(ZnXWY, XZYnW, _MM_SHUFFLE(2,3,0,1)));

    /* now we only need to shuffle the components in place and return the result      */
    return _mm_shuffle_ps(XZWY, XZWY, _MM_SHUFFLE(2,1,3,0));

    /* operations: 6 shuffles, 4 multiplications, 3 compound additions/subtractions   */
}

I expected the assembly to have a minimal amount of instructions. However, when I compile it to assembly with gcc -msse3 -S, the resulting function has 67 instructions:

    .text   
    .globl __Z13_mm_cross4_psU8__vectorfS_
__Z13_mm_cross4_psU8__vectorfS_:
LFB594:
    pushq   %rbp
LCFI0:
    movq    %rsp, %rbp
LCFI1:
    subq    $232, %rsp
    movaps  %xmm0, -336(%rbp)
    movaps  %xmm1, -352(%rbp)
    movaps  -336(%rbp), %xmm0
    movaps  -336(%rbp), %xmm1
    shufps  $27, %xmm1, %xmm0
    movaps  %xmm0, -16(%rbp)
    movaps  -352(%rbp), %xmm0
    movaps  -352(%rbp), %xmm1
    shufps  $17, %xmm1, %xmm0
    movaps  %xmm0, -32(%rbp)
    movaps  -352(%rbp), %xmm0
    movaps  -352(%rbp), %xmm1
    shufps  $187, %xmm1, %xmm0
    movaps  %xmm0, -48(%rbp)
    movaps  -16(%rbp), %xmm0
    movaps  %xmm0, -112(%rbp)
    movaps  -48(%rbp), %xmm0
    movaps  %xmm0, -128(%rbp)
    movaps  -128(%rbp), %xmm0
    movaps  -112(%rbp), %xmm1
    mulps   %xmm1, %xmm0
    movaps  -336(%rbp), %xmm1
    movaps  %xmm1, -144(%rbp)
    movaps  -32(%rbp), %xmm1
    movaps  %xmm1, -160(%rbp)
    movaps  -160(%rbp), %xmm1
    movaps  -144(%rbp), %xmm2
    mulps   %xmm2, %xmm1
    movaps  %xmm1, -176(%rbp)
    movaps  %xmm0, -192(%rbp)
    movaps  -176(%rbp), %xmm0
    hsubps  -192(%rbp), %xmm0
    movaps  %xmm0, -64(%rbp)
    movaps  -16(%rbp), %xmm0
    movaps  %xmm0, -208(%rbp)
    movaps  -32(%rbp), %xmm0
    movaps  %xmm0, -224(%rbp)
    movaps  -224(%rbp), %xmm0
    movaps  -208(%rbp), %xmm1
    mulps   %xmm1, %xmm0
    movaps  -336(%rbp), %xmm1
    movaps  %xmm1, -240(%rbp)
    movaps  -48(%rbp), %xmm1
    movaps  %xmm1, -256(%rbp)
    movaps  -256(%rbp), %xmm1
    movaps  -240(%rbp), %xmm2
    mulps   %xmm2, %xmm1
    movaps  %xmm1, -272(%rbp)
    movaps  %xmm0, -288(%rbp)
    movaps  -272(%rbp), %xmm0
    haddps  -288(%rbp), %xmm0
    movaps  %xmm0, -80(%rbp)
    movaps  -64(%rbp), %xmm0
    movaps  -80(%rbp), %xmm1
    shufps  $177, %xmm1, %xmm0
    movaps  -80(%rbp), %xmm1
    movaps  -64(%rbp), %xmm2
    shufps  $228, %xmm2, %xmm1
    movaps  %xmm1, -304(%rbp)
    movaps  %xmm0, -320(%rbp)
    movaps  -304(%rbp), %xmm0
    addsubps        -320(%rbp), %xmm0
    movaps  %xmm0, -96(%rbp)
    movaps  -96(%rbp), %xmm0
    movaps  -96(%rbp), %xmm1
    shufps  $156, %xmm1, %xmm0
    leave
LCFI2:
    ret

What am I doing wrong? There must a better way to shuffle the elements around without using so many movaps instructions.

Answer 1

Never mind. If I compile the code with gcc -msse3 -O1 -S instead, I get the following:

    .text
    .align 4,0x90
    .globl __Z13_mm_cross4_psU8__vectorfS_
__Z13_mm_cross4_psU8__vectorfS_:
LFB644:
    movaps  %xmm0, %xmm5
    movaps  %xmm1, %xmm3
    movaps  %xmm0, %xmm2
    shufps  $27, %xmm0, %xmm5
    movaps  %xmm5, %xmm4
    shufps  $17, %xmm1, %xmm3
    shufps  $187, %xmm1, %xmm1
    mulps   %xmm3, %xmm2
    mulps   %xmm1, %xmm4
    mulps   %xmm5, %xmm3
    mulps   %xmm1, %xmm0
    hsubps  %xmm4, %xmm2
    haddps  %xmm3, %xmm0
    movaps  %xmm2, %xmm1
    shufps  $177, %xmm0, %xmm1
    shufps  $228, %xmm2, %xmm0
    addsubps        %xmm1, %xmm0
    shufps  $156, %xmm0, %xmm0
    ret

That's only 18 instructions now. That's what I expected in the beginning. Oops.