我尝试将以下代码翻译成AVX内在函数以提高性能：
for (int alpha = 0; alpha < 4; alpha++) {
for (int k = 0; k < 3; k++) {
for (int beta = 0; beta < 4; beta++) {
for (int l = 0; l < 4 ; l++) {
d2_phi[(alpha*3+k)*16 + beta*4+l] =
- (d2_phi[(alpha*3+k)*16 + beta*dim+l]
+ b[k] * ( lam_12[ beta][alpha] * a[l]
+ lam_22[alpha][ beta] * b[l]
+ lam_23[alpha][ beta] * rjk[l] )
+ rjk[k] * ( lam_13[ beta][alpha] * a[l]
+ lam_23[ beta][alpha] * b[l]
+ lam_33[alpha][ beta] * rjk[l] )
) / sqrt_gamma;
}
}
}
}
并尝试以下方式：
// load sqrt_gamma, because it is constant
__m256d ymm7 = _mm256_broadcast_sd(&sqrt_gamma);
for (int alpha=0; alpha < 4; alpha++) {
for (int k=0; k < 3; k++) {
// Load values that are only dependent on k
__m256d ymm9 = _mm256_broadcast_sd(b+k); // all b[k]
__m256d ymm8 = _mm256_broadcast_sd(rjk+k); // all rjk[k]
for (int beta=0; beta < 4; beta++) {
// Load the lambdas, because they will stay the same for nine iterations
__m256d ymm15 = _mm256_broadcast_sd(lam_12_p + 4*beta + alpha); // all lam_12[ beta][alpha]
__m256d ymm14 = _mm256_broadcast_sd(lam_22_p + 4*alpha + beta); // all lam_22[alpha][ beta]
__m256d ymm13 = _mm256_broadcast_sd(lam_23_p + 4*alpha + beta); // all lam_23[alpha][ beta]
__m256d ymm12 = _mm256_broadcast_sd(lam_13_p + 4*beta + alpha); // all lam_13[ beta][alpha]
__m256d ymm11 = _mm256_broadcast_sd(lam_23_p + 4*beta + alpha); // all lam_23[ beta][alpha]
__m256d ymm10 = _mm256_broadcast_sd(lam_33_p + 4*alpha + beta); // lam_33[alpha][ beta]
// Load the values that depend on the innermost loop, which is removed do to AVX
__m256d ymm6 =_mm256_load_pd(a); // a[i] until a[l+3]
__m256d ymm5 =_mm256_load_pd(b); // b[i] until b[l+3]
__m256d ymm4 =_mm256_load_pd(rjk); // rjk[i] until rjk[l+3]
//__m256d ymm3 =_mm256_load_pd(d2_phi_p + (alpha*3+k)*16 + beta*dim); // d2_phi[(alpha*3+k)*12 + beta*dim] until d2_phi[(alpha*3+k)*12 + beta*dim +3]
__m256d ymm3 =_mm256_load_pd(d2_phi_p + 4*s);
// Block that is later on multiplied with b[k]
__m256d ymm2 = _mm256_mul_pd(ymm15, ymm6); // lam_12[ beta][alpha] * a[l]
__m256d ymm1 = _mm256_mul_pd(ymm14, ymm5); // lam_22[alpha][ beta] * b[l];
__m256d ymm0 = _mm256_add_pd(ymm2, ymm1); // lam_12[ beta][alpha] * a[l] + lam_22[alpha][ beta]*b[l];
ymm2 = _mm256_mul_pd(ymm13, ymm4); // lam_23[alpha][ beta] * rjk[l]
ymm0 = _mm256_add_pd(ymm2, ymm0); // lam_12[ beta][alpha] * a[l] + lam_22[alpha][ beta]*b[l] + lam_23[alpha][ beta] * b[i];
ymm0 = _mm256_mul_pd(ymm9, ymm0); // b[k] * (first sum of three)
// Block that is later on multiplied with rjk[k]
ymm2 = _mm256_mul_pd(ymm12, ymm6); // lam_13[ beta][alpha] * a[l]
ymm1 = _mm256_mul_pd(ymm11, ymm5); // lam_23[ beta][alpha] * b[l]
ymm2 = _mm256_add_pd(ymm2, ymm1); // lam_13[ beta][alpha] * a[l] + lam_22[alpha][ beta]*b[l];
ymm1 = _mm256_mul_pd(ymm10, ymm4); // lam_33[alpha][ beta] * rjk[l]
ymm2 = _mm256_add_pd(ymm2, ymm1); // lam_13[ beta][alpha] * a[l] + lam_22[alpha][ beta]*b[l] + lam_33[alpha][ beta] *rjk[l]
ymm2 = _mm256_mul_pd(ymm2, ymm8); // rjk[k] * (second sum of three)
ymm0 = _mm256_add_pd(ymm0, ymm2); // add to temporal result in ymm0
ymm0 = _mm256_add_pd(ymm3, ymm0); // Old value of d2 Phi;
ymm0 = _mm256_div_pd(ymm0, ymm7); // all divided by sqrt_gamma
_mm256_store_pd(d2_phi_p + (alpha*3+k)*16 + beta*dim, ymm0);
}
}
}
但表现不好。它甚至比英特尔编译器生成的自动矢量化代码慢。我尝试了以下事项：
__declspec(align(64)) _mm256_stream_pd 当我查看创建的汇编代码时，我看到，自动代码每次迭代都会获取所有参数（而不是像我一样，仅在它们所属的循环中）。它还包含更多算术运算。作为最后一点，最后的商店只需要我的一半时间（我重复代码片段1000000次），我不明白这个原因。 （我使用英特尔VTune放大器来查看装配和花费的时间。）
提前感谢您的帮助！
答案 0 :(得分：3)
请注意，VDIVPD很昂贵 - 它的典型延迟/吞吐量大约为20-40个周期（具体数字取决于CPU）。因此，我建议的一个立即改变是将除以常数转换为乘法，因为VMULPD只有几个周期的延迟和单周期吞吐量：
// load 1 / sqrt_gamma, because it is constant
const double re_sqrt_gamma = 1.0 / sqrt_gamma;
__m256d ymm7 = _mm256_broadcast_sd(&re_sqrt_gamma);
...
ymm0 = _mm256_mul_pd(ymm0, ymm7); // all divided by sqrt_gamma
答案 1 :(得分：3)
我将此作为第二个答案，因为它是一个不同的，更广泛的优化。关键是change the order of the loops通过将许多负载和算术运算从最里面的循环中提升出来来减少冗余操作的数量。
原始循环结构：
for (int alpha=0; alpha < 4; alpha++) {
for (int k=0; k < 3; k++) {
for (int beta=0; beta < 4; beta++) {
for (int l=0; l < 4 ; l++) {
新循环结构：
for (int alpha=0; alpha < 4; alpha++) {
for (int beta=0; beta < 4; beta++) {
for (int k=0; k < 3; k++) {
for (int l=0; l < 4 ; l++) {
完整测试并优化了您的功能实现：
static void foo(
double lam_11[4][4],
double lam_12[4][4],
double lam_13[4][4],
double lam_22[4][4],
double lam_23[4][4],
double lam_33[4][4],
const double rjk[4],
const double a[4],
const double b[4],
const double sqrt_gamma,
const double SPab,
const double d1_phi[16],
double d2_phi[192])
{
const double re_sqrt_gamma = 1.0 / sqrt_gamma;
memset(d2_phi, 0.0, 192*sizeof(double));
const __m256d ymm6 = _mm256_load_pd(a); // load the whole 4-vector 'a' into register
{
// load SPab, because it is constant
const __m256d ymm0 = _mm256_broadcast_sd(&SPab);
const __m256d ymm7 = _mm256_load_pd(b); // load the whole 4-vector 'b' into register
const __m256d ymm8 = _mm256_load_pd(rjk); // load the whole 4-vector 'rjk' into register
for (int alpha=0; alpha < 4; alpha++)
{
for (int beta=0; beta < 4; beta++)
{
// Load the three lambdas to all
const __m256d ymm3 = _mm256_broadcast_sd(&lam_11[alpha][beta]);
const __m256d ymm4 = _mm256_broadcast_sd(&lam_12[alpha][beta]);
const __m256d ymm5 = _mm256_broadcast_sd(&lam_13[alpha][beta]);
const __m256d ymm9 = _mm256_load_pd(d1_phi + beta*4);
// Do the three Multiplications
const __m256d ymm13 = _mm256_mul_pd(ymm4,ymm7); // lam_12[alpha][ beta] * b[l] = PROD2
const __m256d ymm14 = _mm256_mul_pd(ymm5,ymm8); // lam_13[alpha][ beta] * rjk[l] = PROD3
const __m256d ymm15 = _mm256_mul_pd(ymm3,ymm6); // lam_11[alpha][ beta] * a[l] = PROD1
__m256d ymm12 = _mm256_add_pd(ymm15, ymm13); // PROD1 + PROD2 = PROD12
ymm12 = _mm256_add_pd(ymm12, ymm14); // PROD12 + PROD3 = PROD123
double* addr = d2_phi + alpha*3*16 + beta*dim;
for (int k=0; k < 3; k++)
{
const __m256d ymm1 = _mm256_broadcast_sd(&d1_phi[alpha*dim + k]); // load d1_phi[alpha*dim+k] to all
const __m256d ymm2 = _mm256_broadcast_sd(&a[k]); // load a[k] to all
const __m256d ymm10 = _mm256_mul_pd(ymm0, ymm1); // SPab * d1_phi[alpha*dim+k] = PRE
const __m256d ymm11 = _mm256_mul_pd(ymm10, ymm9); // PRE * d1_phi[beta*dim+l] = SUM1
__m256d ymm12t = _mm256_mul_pd(ymm12, ymm2); // a[k] * PROD123 = SUM2
ymm12t = _mm256_add_pd(ymm11, ymm12t); // SUM1 + SUM2
_mm256_store_pd(addr, ymm12t);
addr+=16;
}
}
}
}
{
const __m256d ymm4 =_mm256_load_pd(rjk); // rjk[i] until rjk[l+3]
const __m256d ymm5 =_mm256_load_pd(b); // b[l] until b[l+3]
// load sqrt_gamma, because it is constant
const __m256d ymm7 = _mm256_broadcast_sd(&re_sqrt_gamma);
for (int alpha=0; alpha < 4; alpha++)
{
for (int beta=0; beta < 4; beta++)
{
// Load the lambdas, because they will stay the same for nine iterations
const __m256d ymm15 = _mm256_broadcast_sd(&lam_12[beta][alpha]); // all lam_12[ beta][alpha]
const __m256d ymm14 = _mm256_broadcast_sd(&lam_22[alpha][beta]); // all lam_22[alpha][ beta]
const __m256d ymm13 = _mm256_broadcast_sd(&lam_23[alpha][beta]); // all lam_23[alpha][ beta]
const __m256d ymm12 = _mm256_broadcast_sd(&lam_13[beta][alpha]); // all lam_13[ beta][alpha]
const __m256d ymm11 = _mm256_broadcast_sd(&lam_23[beta][alpha]); // all lam_23[ beta][alpha]
const __m256d ymm10 = _mm256_broadcast_sd(&lam_33[alpha][beta]); // lam_33[alpha][ beta]
__m256d ymm0, ymm1, ymm2;
// Block that is later on multiplied with b[k]
ymm2 = _mm256_mul_pd(ymm15 , ymm6); // lam_12[ beta][alpha] * a[l]
ymm1 = _mm256_mul_pd(ymm14 , ymm5); // lam_22[alpha][ beta] * b[l];
ymm0 = _mm256_add_pd(ymm2, ymm1); // lam_12[ beta][alpha]* a[l] + lam_22[alpha][ beta]*b[l];
ymm2 = _mm256_mul_pd(ymm13 , ymm4); // lam_23[alpha][ beta] * rjk[l]
ymm0 = _mm256_add_pd(ymm2, ymm0); // lam_12[ beta][alpha]* a[l] + lam_22[alpha][ beta]*b[l] + lam_23[alpha][ beta] * b[i];
// Block that is later on multiplied with rjk[k]
ymm2 = _mm256_mul_pd(ymm12 , ymm6); // lam_13[ beta][alpha] * a[l]
ymm1 = _mm256_mul_pd(ymm11 , ymm5); // lam_23[ beta][alpha] * b[l]
ymm2 = _mm256_add_pd(ymm2, ymm1); // lam_13[ beta][alpha] * a[l] + lam_22[alpha][ beta]*b[l];
ymm1 = _mm256_mul_pd(ymm10 , ymm4); // lam_33[alpha][ beta] * rjk[l]
ymm2 = _mm256_add_pd(ymm2 , ymm1); // lam_13[ beta][alpha] * a[l] + lam_22[alpha][ beta]*b[l] + lam_33[alpha][ beta] *rjk[l]
double* addr = d2_phi + alpha*3*16 + beta*dim;
for (int k=0; k < 3; k++)
{
// Load values that are only dependent on k
const __m256d ymm9 = _mm256_broadcast_sd(b+k); // all b[k]
const __m256d ymm8 = _mm256_broadcast_sd(rjk+k); // all rjk[k]
// Load the values that depend on the innermost loop, which is removed do to AVX
const __m256d ymm3 =_mm256_load_pd(addr);
__m256d ymm0t, ymm1t, ymm2t;
// Block that is later on multiplied with b[k]
ymm0t = _mm256_mul_pd(ymm9 , ymm0); // b[k] * (first sum of three)
// Block that is later on multiplied with rjk[k]
ymm1t = _mm256_mul_pd(ymm2 , ymm8); // rjk[k] * (second sum of three)
ymm2t = _mm256_add_pd(ymm0t, ymm1t); // add to temporal result in ymm0
ymm1t = _mm256_add_pd(ymm3, ymm2t); // Old value of d2 Phi;
ymm2t = _mm256_mul_pd(ymm1t, ymm7); // all divided by sqrt_gamma
ymm1t = _mm256_xor_pd(ymm2t, SIGNMASK);
_mm256_store_pd(addr, ymm1t);
addr += 16;
}
}
}
}
}
原始AVX代码使用您的测试工具运行大约500毫秒，新版本运行大约200毫秒，因此吞吐量提高了2.5倍。
在此处使用原始代码和优化代码更新了测试工具的版本：http://pastebin.com/yMPbYPjb