How to employ "hidden" parallelism in CPU:s of today to improve the performance of CPU intensive algorithms

1. Better performance
through Superscalarity
Mårten Rånge

2. How many GigaFlops?
i5 6600K 3.5 GHz
(4x cores)

3. ~224 GigaFlops

4. 64 Flops/cycle

6. Zn+1 = Zn
2 + C (1)
Z0 = C (2)

7. (x,y)

8. (x,y) + (c,d)

9. (x+c,y+d)

10. (x,y)2

11. (x2 - y2,2xy)

12. r
aZk
Z0
2
2a
r2
Z1 = Z0
2 + C
C
|R| = 2
Zl
Zm
Z0
Zn+1 = Zn
2 + C

14. constexpr auto max_iter = 50U;
auto mandelbrot (double cx, double cy) {
auto x = cx ;
auto y = cy ;
auto iter = max_iter;
for (; iter > 0; --iter) {
auto x2 = x*x;
auto y2 = y*y;
if (x2 + y2 > 4) return iter;
y = 2*x*y + cy ;
x = x2 - y2 + cx ;
}
return iter;
}
r2 = x2 + y2
y
x
r
(x,y)2 = (x2 - y2,2xy)
Zn+1 = Zn
2 + C

15. SIMD

16. a = b+c

17. (a0,a1)=(b0,b1)+(c0,c1)

18. 0 1 2 3
4 5 6 7
4 6 8 10
+

19. AVX
8 flops/instruction

20. constexpr auto max_iter = 50U;
auto mandelbrot (double cx, double cy) {
auto x = cx ;
auto y = cy ;
auto iter = max_iter;
for (; iter > 0; --iter) {
auto x2 = x*x;
auto y2 = y*y;
if (x2 + y2 > 4) return iter;
y = 2*x*y + cy ;
x = x2 - y2 + cx ;
}
return iter;
}

21. auto mandelbrot (__m256 cx, __m256 cy) {
auto x = cx;
auto y = cy;
int cmp_mask = 0 ;
for (auto iter = max_iter; iter > 0; --iter) {
auto x2 = x*x;
auto y2 = y*y;
auto r2 = x2 + y2;
auto _4 = float8 (4.0F);
cmp_mask = r2 <= _4;
if (!cmp_mask) return 0;
auto xy = x*y;
y = xy + xy + cy;
x = x2 - y2 + cx;
}
return cmp_mask;
}

22. Minimize CPU stalls

23. opcode Latency Throughput
vmulps 5 1
vaddps 3 1
vsubps 3 1
vcmpps 3 1
vmovmskps 1 1

24. Task<float>

25. auto mandelbrot (__m256 cx, __m256 cy) {
auto x = cx;
auto y = cy;
int cmp_mask = 0 ;
for (auto iter = max_iter; iter > 0; --iter) {
auto x2 = x*x;
auto y2 = y*y;
auto r2 = x2 + y2;
auto _4 = float8 (4.0F);
cmp_mask = r2 <= _4;
if (!cmp_mask) return 0;
auto xy = x*y;
y = xy + xy + cy;
x = x2 - y2 + cx;
}
return cmp_mask;
}

26. x2[0] = x[0]*x[0];
y2[0] = y[0]*y[0];
r2[0] = x2[0] + y2[0];
x2[1] = x[1]*x[1];
y2[1] = y[1]*y[1];
r2[1] = x2[1] + y2[1];
auto _4 = float8 (4.0);
cmp_mask = r2[0] <= _4 | ((r2[1] <= _4) << 8);

27. x2[0] = x[0]*x[0];
y2[0] = y[0]*y[0];
r2[0] = x2[0] + y2[0];
x2[1] = x[1]*x[1];
y2[1] = y[1]*y[1];
r2[1] = x2[1] + y2[1];
r2[0] = x2[0] + y2[0];
auto _4 = float8 (4.0);
cmp_mask = r2[0] <= _4 | ((r2[1] <= _4) << 8);

28. x2[0] = x[0]*x[0]
y2[0] = y[0]*y[0]
r2[0] = x2[0]+y2[0]
x2[1] = x[1]*x[1]
y2[1] = y[1]*y[1]
r2[1] = x2[1]+y2[1]
Instructionqueue
FU
x2[0]
y2[0]
r2[0]
x2[1]
y2[1]
r2[1]
Resultqueue

29. Shouldn’t compilers
do this for us?

30. constexpr auto max_iter = 50U;
auto mandelbrot (double cx, double cy) {
auto x = cx ;
auto y = cy ;
auto iter = max_iter;
for (; iter > 0; --iter) {
auto x2 = x*x;
auto y2 = y*y;
if (x2 + y2 > 4) return iter;
y = 2*x*y + cy ;
x = x2 - y2 + cx ;
}
return iter;
}

31. auto mandelbrot (__m256 cx, __m256 cy) {
auto x = cx;
auto y = cy;
int cmp_mask = 0 ;
for (auto iter = max_iter; iter > 0; --iter) {
auto x2 = x*x;
auto y2 = y*y;
auto r2 = x2 + y2;
auto _4 = float8 (4.0F);
cmp_mask = r2 <= _4;
if (!cmp_mask) return 0;
auto xy = x*y;
y = xy + xy + cy;
x = x2 - y2 + cx;
}
return cmp_mask;
}
Uses the mathematical properties of mandelbrot
Uses knowledge that inf and NaN <= 4 is false

32. AVX512
&
Hyper-threading

33. constexpr auto max_iter = 50U;
auto mandelbrot (double cx, double cy) {
auto x = cx ;
auto y = cy ;
auto iter = max_iter;
for (; iter > 0; --iter) {
auto x2 = x*x;
auto y2 = y*y;
if (x2 + y2 > 4) return iter;
y = 2*x*y + cy ;
x = x2 - y2 + cx ;
}
return iter;
}

34. Questions?