8 Bit LERP with SSE

Question

I have been trying to figure out the best way to use AMD64 SIMD instructions to implement a lerp to be used with large sets of u8 values but I can't seem to figure out the correct instructions without requiring all the SIMD extensions.

The formula I am working with right now is

u8* a;
u8* b;
u8* result;
size_t count;
u16 total;
u16 progress;

u32 invertedProgress = total - progress;
for(size_t i = 0; i < count; i++){
    result[i] = (u8)((b[i] * progress + a[i] * invertedProgress) / total);
}

I am thinking it would look something like:

u8* a;
u8* b;
u8* result;
size_t count;
u16 total;
u16 progress;

__m128i mmxZero;
__m128i mmxProgress;
__m128i mmxInvertedProgress;
__m128i mmxProductA;
__m128i mmxProductB;

mmxZero = _mm_xor_ps(zero, zero); // Is there a clear?

mmxProgress = Fill with progress;

mmxTotal = Fill with total;

mmxInvertedProgress = mmxTotal;
mmxInvertedProgress = _mm_unpacklo_epi8(mmxInvertedProgres, mmxZero);
mmxInvertedProgress = _mm_sub_epi8(mmxTotal, progress);

for(size_t i = 0; i < count; i += 8){
    mmxProductA = load A;
    // u8 -> u16
    mmxProductA = _mm_unpacklo_epi8(mmxProductA, mmxZero);
    
    mmxProductB = load B;
    // u8 -> u16
    mmxProductB = _mm_unpacklo_epi8(mmxProductB, mmxZero);

    // a * (total - progress)
    mmxProductA = _mm_mullo_epi16(mmxProductA, mmxInvertedProgress);
    // b * progress
    mmxProductB = _mm_mullo_epi16(mmxProductB, mmxProgress);

    // a * (total - progress) + b * progress
    mmxProductA = _mm_add_epi16(mmxProductA, mmxProductB);
    // (a * (total - progress) + b * progress) / total
    mmxProductA = _mm_div_epi16(mmxProductA, mmxTotal);

    mmxProductA = saturated u16 -> u8; 
    store result = maxProductA;
}

There are a couple of things here that I just could not seem to find digging around in the guide, mostly related to loading and storing values.

I know there are some newer instructions that can do larger amounts at the same time, this initial implementation is supposed to work on older chips.

For this example I am also ignoring alignment and the potential for a buffer overrun, I figured that was a little out of scope for the question.

Answer 1

Good question. As you found out, SSE has no integer divide instruction, and (unlike ARM NEON) it doesn’t have multiplication or FMA for bytes.

Here’s what I usually do instead. The code below splits vectors into even/odd bytes, uses 16-bit multiplication instructions to scale separately, then merges them back into bytes.

// Linear interpolation is based on the following formula: x*(1-s) + y*s which can equivalently be written as x + s(y-x).
class LerpBytes
{
    // Multipliers are fixed point numbers in 16-bit lanes of these vectors, in 1.8 format
    __m128i mulX, mulY;

public:

    LerpBytes( uint16_t progress, uint16_t total )
    {
        // The source and result are bytes.
        // Multipliers only need 1.8 fixed point format, anything above that is wasteful.
        assert( total > 0 );
        assert( progress >= 0 );
        assert( progress <= total );

        const uint32_t fp = (uint32_t)progress * 0x100 / total;
        mulY = _mm_set1_epi16( (short)fp );
        mulX = _mm_set1_epi16( (short)( 0x100 - fp ) );
    }

    __m128i lerp( __m128i x, __m128i y ) const
    {
        const __m128i lowMask = _mm_set1_epi16( 0xFF );

        // Split both vectors into even/odd bytes in 16-bit lanes
        __m128i lowX = _mm_and_si128( x, lowMask );
        __m128i highX = _mm_srli_epi16( x, 8 );
        __m128i lowY = _mm_and_si128( y, lowMask );
        __m128i highY = _mm_srli_epi16( y, 8 );

        // That multiply instruction has relatively high latency, 3-5 cycles.
        // We're lucky to have 4 vectors to handle.
        lowX = _mm_mullo_epi16( lowX, mulX );
        lowY = _mm_mullo_epi16( lowY, mulY );
        highX = _mm_mullo_epi16( highX, mulX );
        highY = _mm_mullo_epi16( highY, mulY );

        // Add the products
        __m128i low = _mm_adds_epu16( lowX, lowY );
        __m128i high = _mm_adds_epu16( highX, highY );

        // Pack them back into bytes.
        // The multiplier was 1.8 fixed point, trimming the lowest byte off both vectors.
        low = _mm_srli_epi16( low, 8 );
        high = _mm_andnot_si128( lowMask, high );
        return _mm_or_si128( low, high );
    }
};

static void print( const char* what, __m128i v )
{
    printf( "%s:\t", what );
    alignas( 16 ) std::array<uint8_t, 16> arr;
    _mm_store_si128( ( __m128i * )arr.data(), v );
    for( uint8_t b : arr )
        printf( " %02X", (int)b );
    printf( "\n" );
}

int main()
{
    const __m128i x = _mm_setr_epi32( 0x33221100, 0x77665544, 0xBBAA9988, 0xFFEEDDCC );
    const __m128i y = _mm_setr_epi32( 0xCCDDEEFF, 0x8899AABB, 0x44556677, 0x00112233 );
    LerpBytes test( 0, 1 );
    print( "zero", test.lerp( x, y ) );
    test = LerpBytes( 1, 1 );
    print( "one", test.lerp( x, y ) );
    test = LerpBytes( 1, 2 );
    print( "half", test.lerp( x, y ) );
    test = LerpBytes( 1, 3 );
    print( "1/3", test.lerp( x, y ) );
    test = LerpBytes( 1, 4 );
    print( "1/4", test.lerp( x, y ) );
    return 0;
}