Creating double-precision integer multiplication with a quad-precision result from single-precision multiplication with a double-precision result using intrinsics (part 1)
Some time ago, I derived how to create double-precision integer multiplication with a quad-precision result from single-precision multiplication with a double-precision result, but I expressed it in assembly language. This time, I’ll express it in C using intrinsics.
Using intrinsics rather than inline assembly language is slightly more portable, since all the compiler toolchains that implement the intrinsics agree on what the intrinsics mean. They disagree on the members of a __m128i, but at least they agree on the intrinsics.
First, a straightforward translation of the assembly language code to intrinsics:
__m128i Multiply64x64To128(uint64_t x, uint64_t y)
{
// movq xmm0, x // xmm0 = { 0, 0, A, B } = { *, *, A, B }
auto x00AB = _mm_loadl_epi64((__m128i*) &x);
// movq xmm1, y // xmm1 = { 0, 0, C, D } = { *, *, C, D }
auto x00CD = _mm_loadl_epi64((__m128i*) &y);
// punpckldq xmm0, xmm0 // xmm0 = { A, A, B, B } = { *, A, *, B }
auto xAABB = _mm_unpacklo_epi32(x00AB, x00AB);
// punpckldq xmm1, xmm1 // xmm1 = { C, C, D, D } = { *, C, *, D }
auto xCCDD = _mm_unpacklo_epi32(x00CD, x00CD);
// pshufd xmm2, xmm1, _MM_SHUFFLE(1, 0, 3, 2)
// // xmm2 = { D, D, C, C } = { *, D, *, C }
auto xDDCC = _mm_shuffle_epi32(xCCDD, _MM_SHUFFLE(1, 0, 3, 2));
// pmuludq xmm1, xmm0 // xmm1 = { AC, BD } // provisional result
auto result = _mm_mul_epu32(xAABB, xCCDD);
// pmuludq xmm2, xmm0 // xmm2 = { AD, BC } // cross-terms
auto crossterms = _mm_mul_epu32(xAABB, xDDCC);
// mov eax, crossterms[0]
// add result[4], eax
auto carry = _addcarry_u32(0,
result.m128i_u32[1],
crossterms.m128i_u32[0],
&result.m128i_u32[1]);
// mov edx, crossterms[4] // edx:eax = BC
// adc result[8], edx
carry = _addcarry_u32(carry,
result.m128i_u32[2],
crossterms.m128i_u32[1],
&result.m128i_u32[2]);
// adc result[12], 0 // add the first cross-term
_addcarry_u32(carry,
result.m128i_u32[3],
0,
&result.m128i_u32[3]);
// mov eax, crossterms[8]
// add result[4], eax
carry = _addcarry_u32(0,
result.m128i_u32[1],
crossterms.m128i_u32[2],
&result.m128i_u32[1]);
// mov edx, crossterms[12] // edx:eax = AD
// adc result[8], edx
carry = _addcarry_u32(carry,
result.m128i_u32[2],
crossterms.m128i_u32[3],
&result.m128i_u32[2]);
// adc result[12], 0 // add the second cross-term
_addcarry_u32(carry,
result.m128i_u32[3],
0,
&result.m128i_u32[3]);
return result;
}
The Microsoft Visual Studio compiler produces the following:
; standard function prologue
push ebp
mov ebp, esp
and esp, -16 ; room for _result on the stack
sub esp, 24
; load up x and y
movq xmm1, QWORD PTR _y$[ebp]
movq xmm2, QWORD PTR _x$[ebp]
; duplicate x
punpckldq xmm1, xmm1
; make another copy of the duplicated x
movaps xmm0, xmm1
; duplicate y
punpckldq xmm2, xmm2
; multiply main terms, result in xmm0
pmuludq xmm0, xmm2
; shuffle and multiply cross terms, cross-terms in xmm1
pshufd xmm1, xmm1, 141
pmuludq xmm1, xmm2
; Now the adjustments for cross-terms
; save a register
push esi
; save result to memory
movaps XMMWORD PTR _result$[esp+32], xmm0
; load up result[2] into esi and result[3] into ecx
mov esi, DWORD PTR _result$[esp+40]
mov ecx, DWORD PTR _result$[esp+44]
; load result[1] into edx by shifting
psrldq xmm0, 4
movd edx, xmm0
; xmm0 holds cross-terms
movaps xmm0, xmm1
; load crossterms[0] into eax
movd eax, xmm1
; prepare to load crossterms[1] into eax by shifting
psrldq xmm0, 4
; add crossterms[0] to result[1]
add edx, eax
; load crossterms[1] into eax
movd eax, xmm0
; xmm0 holds cross-terms (again)
movaps xmm0, xmm1
; prepare to load crossterms[3] into eax by shifting
psrldq xmm1, 12
; prepare to load crossterms[2] into eax by shifting
psrldq xmm0, 8
; add crossterms[1] to result[2], with carry
adc esi, eax
; load crossterms[2] into eax
movd eax, xmm0
; propagate carry into result[3]
adc ecx, 0
; add crossterms[2] to result[1]
add edx, eax
; load crossterms[3] into eax
movd eax, xmm1
; save final result[1]
mov DWORD PTR _result$[esp+36], edx
; add crossterms[3] to result[2]
adc esi, eax
; save final result[2]
mov DWORD PTR _result$[esp+40], esi
; propagate carry into result[3]
adc ecx, 0
; save final result[3]
mov DWORD PTR _result$[esp+44], ecx
; load final result
movaps xmm0, XMMWORD PTR _result$[esp+32]
; clean up stack and return
pop esi
mov esp, ebp
pop ebp
ret
I was impressed that the compiler was able to convert our direct accesses to the internal members of the __m128i into corresponding shifts and extractions. (Since this code was compiled with only SSE2 support, the compiler could not use the pextrd instruction to do the extraction.)
Even with this very lame conversion, the compiler does quite a good job of optimiznig the code. The opening instructions match our handwritten assembly almost exactly; the second half doesn’t match as well, but that’s because the compiler was able to replace many of our memory accesses with register accesses.
The compiler was able to optimize our inline assembly!
We’ll take this as inspiration for trying to get rid of all the memory accesses. The story continues next time.
Bonus chatter: In the three years since I wrote the original article, nobody picked up on the fact that I got the parameters to _MM_SHUFFLE wrong.
Bonus bonus chatter: I could have reduced the dependency chain a bit by tweaking the calculation of xDDCC:
// pshufd xmm2, xmm1, _MM_SHUFFLE(0, 0, 1, 1);
// // xmm2 = { D, D, C, C } = { *, D, *, C }
auto xDDCC = _mm_shuffle_epi32(x00CD, _MM_SHUFFLE(0, 0, 1, 1));
// punpckldq xmm1, xmm1 // xmm1 = { C, C, D, D } = { *, C, *, D }
auto xCCDD = _mm_unpacklo_epi32(x00CD, x00CD);
Basing the calculation of xDDCC on x00CD rather than 0xCCDD removes one instruction from the dependency chain.
Bonus bonus bonus chatter: I chose to use punpckldq instead of pshufd to calculate xCCDD because punpckldq encodes one byte shorter.
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