Automatic Generation of Efficient Codes from Mathematical Descriptions of Stencil Computation Takayuki Muranushi 1 Seiya Nishizawa 1 Hirofumi Tomita 1
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1 Automatic Generation of Efficient Codes from Mathematical Descriptions of Stencil Computation Takayuki Muranushi 1 Seiya Nishizawa 1 Hirofumi Tomita 1 Keigo Nitadori 1 Masaki Iwasawa 1 Yutaka Maruyama 1 Hisashi Yashiro 1 Yoshifumi Nakamura 1 Hideyuki Hotta 2 Junichiro Makino 3 Natsuki Hosono 4 Hikaru Inoue 5 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 Sep 22, / 37
2 Programming Language Formura T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
3 Programming language Formura Domain specific language for stencil computaion T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
4 T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
5 Good news of Formura 1/2 1:184 Petaflops (11.62% of the peak) on 663,552 cores T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
6 Good news of Formura 1/2 ACM Gordon Bell Prize Finalist T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
7 Good news of Formura X 3 i=1 i (v i ) ddt_ = - fun i ( * v i) T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
8 Formura is a functional programming language is implemented in a functional programming language (Haskell) T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
9 Backend: How we generate efficient codes Backend: How we generate efficient codes T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
10 Backend: How we generate efficient codes Stencil Computation T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
11 Backend: How we generate efficient codes Byte / Flops of hardwares are decreasing T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
12 Backend: How we generate efficient codes Naive implementation of stencil computation The optimal B F = 2H e C e T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
13 Backend: How we generate efficient codes Temporal Blocking The optimal B F = 2H e C e 1 N F 1 + 2dN s A N T T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
14 Backend: How we generate efficient codes Decompose & fuse array computations in space-time manifest :: a[i] b[i] = a[i -1] + a[i] + a[i +1] manifest :: c[i] = b[i -1] * b[i] * b[i +1] d[i] = c[i -1] + c[i] + c[i +1] manifest :: e[i] = d[i -1] * d[i] * d[i +1] T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
15 Backend: How we generate efficient codes T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
16 Backend: How we generate efficient codes In which language shall we code? T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
17 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 Sep 22, / 37
18 Backend: How we generate efficient codes T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
19 Backend: How we generate efficient codes T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
20 Backend: How we generate efficient codes Paraiso: a bad sell T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
21 Backend: How we generate efficient codes Our team T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
22 Formura : a standalone DSL Formura : a standalone DSL T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
23 Formura : a standalone DSL Design principle of Formura Simple enough Rich enough T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
24 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 Sep 22, / 37
25 Formura : a standalone DSL Tuples are functions (a,b) 1 = b (f,(h,p,c)) 1 2 = c T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
26 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 + dt /4 * ddt ( sys_0 ) sys_q3 = sys_0 + dt /3 * ddt ( sys_q4 ) sys_q2 = sys_0 + dt /2 * ddt ( sys_q3 ) sys_next = sys_0 + dt * ddt ( sys_q2 ) in sys_next T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
27 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 Sep 22, / 37
28 Formura : a standalone DSL Nabla and = (ddx,ddy, ddz ) = fun ( e) e 0 + e 1 + e 2 T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
29 Formura : a standalone DSL Evaluation of formura expression i ( * v i) T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
30 Formura : a standalone DSL Evaluation of formura expression i ( * v i) = fun ( e) e 0 + e 1 + e 2 T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
31 Formura : a standalone DSL Evaluation of formura expression i ( * v i) = fun ( e) e 0 + e 1 + e 2! i ( * v i)) 0 + i ( * v i)) 1 + i ( * v i)) 2 T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
32 Formura : a standalone DSL Evaluation of formura expression i ( * v i)) 0 T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
33 Formura : a standalone DSL Evaluation of formura expression i ( * v i)) 0 ( * v 0)) T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
34 Formura : a standalone DSL Evaluation of formura expression i ( * v i)) 0 ( * v = (ddx,ddy, ddz ) v = (vx,vy,vz) (a,b,c) 0 = a T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
35 Formura : a standalone DSL Evaluation of formura expression i ( * v i)) 0 ( * v = (ddx,ddy, ddz ) v = (vx,vy,vz) (a,b,c) 0 = a! ddx ( * vx) T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
36 Formura : a standalone DSL Evaluation of formura expression ddx ( * vx) T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
37 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 T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
38 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 Sep 22, / 37
39 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 T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
40 Formura : a standalone DSL Evaluation of formura expression i ( * v i) T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
41 Formura : a standalone DSL Evaluation of formura expression 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 Sep 22, / 37
42 Formura : a standalone DSL Evaluation of formura expression 3X i=1 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 Sep 22, / 37
43 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 Sep 22, / 37
44 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 Sep 22, / 37
45 Conclusion Conclusion Pflops Formura T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
46 Bibliography Bibliography I Bourgoin, M., Chailloux, E., & Lamotte, J.-L. 2012, Parallel Processing Letters, 22, 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, Oliveira, B. C. d. S., Mu, S.-C., & You, S.-H. 2015, in Proceedings of the 8th ACM SIGPLAN Symposium on Haskell, ACM, Rompf, T. 2012, PhD thesis, ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Svensson, J. 2011, PhD thesis, Chalmers University of Technology T. Muranushi et al. (RIKEN AICS) Formura Sep 22, / 37
参考資料配布 PRESS RELEASE 2016 年 12 月 2 日理化学研究所千葉大学神戸大学京都大学富士通株式会社 式が書ければ 京 が使える - 高度なプログラムを自動生成できる新言語 Formura を開発 - 要旨理化学研究所 ( 理研 ) 計算科学研究機構コデザイン推進チームの村主崇
PRESS RELEASE 2016 年 12 月 2 日理化学研究所千葉大学神戸大学京都大学富士通株式会社 式が書ければ 京 が使える - 高度なプログラムを自動生成できる新言語 Formura を開発 - 要旨理化学研究所 ( 理研 ) 計算科学研究機構コデザイン推進チームの村主崇行特別研究員らと 千葉大学の堀田英之特任助教 神戸大学の牧野淳一郎教授 京都大学の細野七月特任助教 富士通株式会社の井上晃マネージャーらの共同研究グループ
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