Automatic Generation of Efficient Codes from Mathematical Descriptions of Stencil Computation Takayuki Muranushi1 Seiya Nishizawa1 Hirofumi Tomita1 Keigo Nitadori1 Masaki Iwasawa1 Yutaka Maruyama1 Hisashi Yashiro1 Yoshifumi Nakamura1 Hideyuki Hotta2 Junichiro Makino3 Natsuki Hosono4 Hikaru Inoue5 1 RIKEN Advanced Institute for Computational Science 2 Chiba University 3 Kobe University 4 Kyoto University 5 Fujitsu Ltd.

Sep 22, 2016 for FHPC 2016 workshop / ICFP’16 Nara, Japan T. Muranushi et al. (RIKEN AICS)

Formura

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Programming Language

Formura T. Muranushi et al. (RIKEN AICS)

Formura

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Programming language Formura

Domain specific language for stencil computaion

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Formura

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T. Muranushi et al. (RIKEN AICS)

Formura

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Good news of Formura 1/2

1:184 Petaflops (11.62% of the peak) on 663,552 cores

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Formura

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Good news of Formura 1/2

ACM Gordon Bell Prize Finalist

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Formura

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Good news of Formura 2/2

@ @t ddt_  = -

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=

3 @ X (vi) @x i=1 i

fun ( i ) @ i (  * v i )

Formura

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Formura

is a functional programming language is implemented in a functional programming language (Haskell)

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Formura

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Backend: How we generate efficient codes

Backend: How we generate efficient codes

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Formura

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Backend: How we generate efficient codes

Stencil Computation

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Backend: How we generate efficient codes

Byte / Flops of hardwares are decreasing

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Formura

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Backend: How we generate efficient codes

Naive implementation of stencil computation

2He The optimal B = F C e

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Formura

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Backend: How we generate efficient codes

Temporal Blocking

The optimal

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B F

0 1 2 He @ 1 2 dNs A = C N + N e

Formura

F

T

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Backend: How we generate efficient codes

Decompose & fuse array computations in space-time

manifest :: a [ i ] b[i] manifest :: c [ i ] d[i] manifest :: e [ i ]

T. Muranushi et al. (RIKEN AICS)

= = = =

a [i -1] b [i -1] c [i -1] d [i -1]

Formura

+ * + *

a[i] b[i] c[i] d[i]

+ * + *

a [ i +1] b [ i +1] c [ i +1] d [ i +1]

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Backend: How we generate efficient codes

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Formura

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Backend: How we generate efficient codes

In which language shall we code?

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Formura

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Backend: How we generate efficient codes

Paraiso : a DSL embedded in Haskell (Muranushi, 2012) among Nikola (Mainland & Morrisett, 2010), Obsidian (Svensson, 2011), Accelerate (Chakravarty et al., 2011), SPOC (Bourgoin et al., 2012), NOVA (Collins et al., 2014), and LMS series (Rompf, 2012). T. Muranushi et al. (RIKEN AICS)

Formura

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Backend: How we generate efficient codes

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Formura

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Backend: How we generate efficient codes

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Formura

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Backend: How we generate efficient codes

Paraiso: a bad sell

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Formura

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Backend: How we generate efficient codes

Our team

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Formura

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Formura : a standalone DSL

Formura : a standalone DSL

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Formura

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Formura : a standalone DSL

Design principle of Formura

 Simple

enough  Rich enough

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Formura

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Formura : a standalone DSL

Syntax of Formura # dimension declaration dimension :: 3 # array declaration double [] :: vx , vy , vz # array computation A2 [i ,j , k ] = A [i -1] + A [ i +1] # Tuple v = ( vx , vy , vz ) # Lambda expression tripe = fun ( x ) 3 * x

T. Muranushi et al. (RIKEN AICS)

Formura

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Formura : a standalone DSL

Tuples are functions (a , b ) 1 = b (f ,( h ,p , c )) 1 2 = c

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Formura

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Formura : a standalone DSL

Inferred promotion to tuples and functions x + (a , b ) = ( x +a , x + b ) (x , y ) + (a , b ) = ( x +a , y + b ) (x , y ) + (a ,b , c ) = ? (f + g) x = f x + g x ( f + g + 1) x = f x + g x + 1 rk4 = fun ( ddt ) \ fun ( sys_0 ) let \ sys_q4 = sys_0 + sys_q3 = sys_0 + sys_q2 = sys_0 + sys_next = sys_0 in sys_next T. Muranushi et al. (RIKEN AICS)

dt /4 dt /3 dt /2 + dt Formura

* * * *

ddt ( sys_0 ) ddt ( sys_q4 ) ddt ( sys_q3 ) ddt ( sys_q2 ) Sep 22, 2016

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Formura : a standalone DSL

Differentiation Operators ddx = fun ( a ) ( a [ i +1/2 , j , k ] - a [i -1/2 , j , k ])/ dx ddy = fun ( a ) ( a [i , j +1/2 , k ] - a [i ,j -1/2 , k ])/ dy ddz = fun ( a ) ( a [i ,j , k +1/2] - a [i ,j ,k -1/2])/ dz

T. Muranushi et al. (RIKEN AICS)

Formura

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Formura : a standalone DSL

Nabla and Summation @ = ( ddx , ddy , ddz )  = fun ( e ) e 0 + e 1 + e 2

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Formura : a standalone DSL

Evaluation of formura expression



fun(i) @ i ( * v i)

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Formura : a standalone DSL

Evaluation of formura expression





fun(i) @ i ( * v i)

= fun ( e ) e 0 + e 1 + e 2

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Formura

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Formura : a standalone DSL

Evaluation of formura expression





fun(i) @ i ( * v i)

= fun ( e ) e 0 + e 1 + e 2

!

(fun(i) @ i ( * v i)) 0 + (fun(i) @ i ( * v i)) 1 + (fun(i) @ i ( * v i)) 2

T. Muranushi et al. (RIKEN AICS)

Formura

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Formura : a standalone DSL

Evaluation of formura expression

(fun(i) @ i ( * v i)) 0

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Formura

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Formura : a standalone DSL

Evaluation of formura expression

!

(fun(i) @ i ( * v i)) 0 @ 0 ( * v 0))

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Formura

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Formura : a standalone DSL

Evaluation of formura expression

!

(fun(i) @ i ( * v i)) 0 @ 0 ( * v 0))

@ = ( ddx , ddy , ddz ) v = ( vx , vy , vz ) (a ,b , c ) 0 = a

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Formura

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Formura : a standalone DSL

Evaluation of formura expression

!

(fun(i) @ i ( * v i)) 0 @ 0 ( * v 0))

!

ddx ( * vx)

@ = ( ddx , ddy , ddz ) v = ( vx , vy , vz ) (a ,b , c ) 0 = a

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Formura : a standalone DSL

Evaluation of formura expression

ddx ( * vx)

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Formura

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Formura : a standalone DSL

Evaluation of formura expression

ddx ( * vx) ddx = fun ( a ) ( a [ i +1/2 , j , k ] - a [i -1/2 , j , k ])/ dx

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Formura

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Formura : a standalone DSL

Evaluation of formura expression

ddx ( * vx) ddx = fun ( a ) ( a [ i +1/2 , j , k ] - a [i -1/2 , j , k ])/ dx

!

(( * vx)[i+1/2,j,k] ( * vx)[i-1/2,j,k])/dx

T. Muranushi et al. (RIKEN AICS)

Formura

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Formura : a standalone DSL

Evaluation of formura expression

ddx ( * vx) ddx = fun ( a ) ( a [ i +1/2 , j , k ] - a [i -1/2 , j , k ])/ dx

!

(( * vx)[i+1/2,j,k] ( * vx)[i-1/2,j,k])/dx ! ([i+1/2,j,k] * vx[i+1/2,j,k] [i-1/2,j,k] * vx[i-1/2,j,k])/dx

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Formura

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Formura : a standalone DSL

Evaluation of formura expression



fun(i) @ i ( * v i)

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Formura

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Formura : a standalone DSL

Evaluation of formura expression



fun(i) @ i ( * v i) ! ([i+1/2,j,k] * vx[i+1/2,j,k] [i-1/2,j,k] * vx[i-1/2,j,k])/dx + ([i,j+1/2,k] * vy[i,j+1/2,k] [i,j-1/2,k] * vy[i,j-1/2,k])/dy + ([i,j,k+1/2] * vz[i,j,k+1/2] [i,j,k-1/2] * vz[i,j,k-1/2])/dz

T. Muranushi et al. (RIKEN AICS)

Formura

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Formura : a standalone DSL

Evaluation of formura expression



!

3 @ X (vi) i=1 @xi

fun ( i ) @ i (  * v i )

([i+1/2,j,k] * vx[i+1/2,j,k] [i-1/2,j,k] * vx[i-1/2,j,k])/dx + ([i,j+1/2,k] * vy[i,j+1/2,k] [i,j-1/2,k] * vy[i,j-1/2,k])/dy + ([i,j,k+1/2] * vz[i,j,k+1/2] [i,j,k-1/2] * vz[i,j,k-1/2])/dz

T. Muranushi et al. (RIKEN AICS)

Formura

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Formura : a standalone DSL

More to talk about Modular Reifiable Matching (MRM)(Oliveira et al., 2015) + Pattern synoynm solves “expression problem” Details of code transformation paths Varieties of temporal blocking methods How we have gave proof to certain types of temporal blocking methods

T. Muranushi et al. (RIKEN AICS)

Formura

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Conclusion

Conclusion

Functional programming is a good choice for user interface ! weather scientists and astronomers can use it is crucial in implementing all the program transformations ! achieves high performance T. Muranushi et al. (RIKEN AICS)

Formura

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Conclusion

Conclusion

1.184 Pflops Formura T. Muranushi et al. (RIKEN AICS)

Formura

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Bibliography

Bibliography I Bourgoin, M., Chailloux, E., & Lamotte, J.-L. 2012, Parallel Processing Letters, 22, 1240007 Chakravarty, M. M., Keller, G., Lee, S., McDonell, T. L., & Grover, V. 2011, in Proceedings of the sixth workshop on Declarative aspects of multicore programming, ACM, 3–14 Collins, A., Grewe, D., Grover, V., Lee, S., & Susnea, A. 2014, in Proceedings of ACM SIGPLAN International Workshop on Libraries, Languages, and Compilers for Array Programming, ACM, 8 Mainland, G., & Morrisett, G. 2010in , ACM, 67–78 Oliveira, B. C. d. S., Mu, S.-C., & You, S.-H. 2015, in Proceedings of the 8th ACM SIGPLAN Symposium on Haskell, ACM, 82–93 ´ ´ ERALE ´ Rompf, T. 2012, PhD thesis, ECOLE POLYTECHNIQUE FED DE LAUSANNE Svensson, J. 2011, PhD thesis, Chalmers University of Technology T. Muranushi et al. (RIKEN AICS)

Formura

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