1. APL Optimization Techniques Eugene Ying Senior Software Developer Fiserv, Inc. September 14, 2012 1

2. Topics Component File Fragmentation The Match Function The Inner Product Storing Numbers in a Native File The Outer Product File I/O Optimization CPU Optimization 2

3. A Component File where each Component Contains 100 Rows of Data Updating component 2 with 150 rows of data comp 2 file is fragmented Updating component 2 with 50 rows of data 3

4. Suppose your data will not have more than 500 rows of data. To minimize the chance of fragmentation, you allocate 500 rows of data for each component. Initializing a Component File (500 10⍴' ')⎕FAPPEND TIE ⍝ Component 1 (500 4⍴0)⎕FAPPEND TIE ⍝ Component 2 (500 20⍴' ')⎕FAPPEND TIE ⍝ Component 3 (500 5⍴0)⎕FAPPEND TIE ⍝ Component 4 (500 15⍴' ')⎕FAPPEND TIE ⍝ Component 5 4

5. Initializing a Component File Intended Initialization Actual Initialization comp 1comp 2 comp 1 comp 3 comp 4 comp 5 comp 3comp 2 comp 4 characters numbers Numeric Components are greatly under-allocated in size 5

6. Storage Sizes of APL Numbers BOOLEAN←1000⍴1 ⎕SIZE 'BOOLEAN' 144 INTEGER1←1000⍴2 ⎕SIZE 'INTEGER1' 1016 INTEGER2←1000⍴128 ⎕SIZE 'INTEGER2 ' 2016 INTEGER4←1000⍴32768 ⎕SIZE 'INTEGER4' 4016 FLOAT8←1000⍴0.1 ⎕SIZE 'FLOAT8' 8016 6

7. The Default APL Number 0 X←1000⍴0 ⎕SIZE 'X' 144 X←1000⍴0.1-0.1 ⎕SIZE 'X' 144 X←1000⍴0×0.1 ⎕SIZE 'X' 144 X←1000↑0⍴0.1 ⎕SIZE 'X' 144 X←0×1000⍴0.1 ⎕SIZE 'X' 144 7

8. F64_0←1⊃11 645 ⎕DR 1000⍴0 ⍝ Floating pt # 0 ⎕SIZE 'F64_0' 8016 B32_999←1⊃163 323 ⎕DR 1000⍴999 ⍝ Binary-32 # 999 ⎕SIZE 'B32_999' 4032 B16_2←1⊃83 163 ⎕DR 1000⍴2 ⍝ Binary-16 # 2 ⎕SIZE 'B16_2' 2032 B8_0←1⊃11 83 ⎕DR 1000⍴0 ⍝ Binary-8 # 0 ⎕SIZE 'B8_0' 1016 How Do You Create A Vector of Integer Zeros or A Vector of Floating Point Zeros? 8

9. Declaring Numbers Using a Defined Function to Preserve Numeric Type F64←64 DCL 1000⍴0 ⍝ Floating pt # 0 ⎕SIZE 'F64' 8016 I32←32 DCL 1000⍴999 ⍝ Binary-32 # 999 ⎕SIZE 'I32' 4032 I16←16 DCL 1000⍴2 ⍝ Binary-16 # 2 ⎕SIZE 'I16' 2032 I8←8 DCL 1000⍴0 ⍝ Binary-8 # 0 ⎕SIZE 'I8' 1016 9

10. The DCL (Declare) Function [0] Z←X DCL Y;D;R [1] ⍝ Declare a floating point or integer array so that each [2] ⍝ item occupies the number of bits requested by the X argument [3] ⍝ X: # of bits that each number in the array will occupy [4] ⍝ 8 for 8-bit (1-byte) integer (¯128 to 127) [5] ⍝ 16 for 16-bit (2-byte) integer (¯32768 to 32767) [6] ⍝ 32 for 32-bit (4-byte) integer (¯2147483648 to 2147483647) [7] ⍝ 64 for 64-bit (8-byte) floating point # [8] ⍝ Y: Numeric array declared [9] ⍝ Z: Numeric array that occupies the space you requested [10] [11] D←⎕DR Y ⍝ Current data type of Y [12] :Select ⍬⍴X [13] :Case 8 ⋄ R←83 [14] :Case 16 ⋄ R←163 [15] :Case 32 ⋄ R←323 [16] :Case 64 ⋄ R←645 [17] :Else ⋄ ∘ ⍝ Stop if requested data type not supported [18] :EndSelect [19] →(D>R)↑'∘' ⍝ Stop if numeric overflow [20] Z←1⊃(D,R)⎕DR Y ⍝ Convert to requested data type 10

11. For more accurate initialization: Initialization as Intended (500 10⍴' ')⎕FAPPEND TIE ⍝ Component 1 (64 DCL 500 4⍴0)⎕FAPPEND TIE⍝ Component 2 (500 20⍴' ')⎕FAPPEND TIE ⍝ Component 3 (32 DCL 500 5⍴0)⎕FAPPEND TIE ⍝ Component 4 (500 15⍴' ')⎕FAPPEND TIE ⍝ Component 5 11

12. Changing the Floating Point 0 Z1000←64 DCL 1000⍴0⍝ 1,000 Floating pt 0 ⎕SIZE 'Z1000' 8016 Z2000←2000↑Z1000⍝ 2,000 Floating pt 0 ⎕SIZE 'Z2000' 268 Z2000←64 DCL 2000⍴0⍝ 2,000 Floating pt 0 ⎕SIZE 'Z2000' 16016 12

13. The internal representation of the result R←X ⎕ DR Y is guaranteed to remain unmodified until it is re-assigned (or partially re-assigned) with the result of any function (ref: Dyalog Apl Reference Manual Chapter 6) Precaution Do not change a Declared array and then re-use it. If you need another similar array but of different dimensions, you should declare the new one from scratch. Reason: 13

14. Storing Numbers in a Native File 14

15. Blanks and commas are the most frequently used separators for numbers stored in a text file. Index Generator is also frequently used. N1←'40001 40002' Storing Numbers as Characters N3←'40000+⍳2' N2←'40001,40002' :For I :In ⍳10000 X←⍎N1 Y←⍎N2 Z←⍎N3 :EndFor ⍝ Elapsed time = 72 ms ⍝ Elapsed time = 89 ms ⍝ Elapsed time = 94 ms The character strings are executed to retrieve the numbers 15

16. :For I :In ⍳100 X←⍎N1 Y←⍎N2 Z←⍎N3 :EndFor ⍝ Run Time 96 ms ⍝ Run Time 661 ms Storing 1,000 Numbers as Characters ⍝ Run Time 504 ms N1←⍕N N2←N1 ((N2=' ')/N2)←',' N3←¯1↓,'(',(⍕⍪¯1+(1000⍴1 0)/N),500 5⍴'+⍳2),' N←4000+(1500⍴1 1 0)/⍳1500 ⍝ (4000+⍳2),(4003+⍳2),... Comma separated Index generated ⍝ 4001,4002,4004,4005,... comma separated ⍝ 4001 4002 4004 4005... space separated 16

17. Space Wasted by Trailing Blanks Character Matrix with 2 records Record 1 can be compressed a little bit by the Index Generator so that record 2 has less trailing blanks But in a nested vector, record 2 naturally has no trailing blanks 2911029106299111291132911529114 29246 (29109+ ⍳ 2),29106,(29112+ ⍳ 2),29115 29246 2911029106299111291132911529114 29246 17

18. File I/O Optimization Suggestions Use the DCL function to Declare arrays to initialize the numeric components of a component file, otherwise the numeric components are under- allocated in size and the component file becomes fragmented too quickly. To store purely numeric data in a native file, do not use commas to separate the numbers, even though CSV format is very popular, because APL commas are being executed as primitive functions. 18

19. Outer Product 19

20. Replacing Outer Product by Indexing Y←⍳32000 :For I :In ⍳5 L←1≠+/Y∘.=Y M←Y∊((⍳⍴Y)≠Y⍳Y)/Y :EndFor ⎕WA 2656824552 X←1≠+/D∘.=D←⍳33000 LIMIT ERROR ⎕WA 270924 ⍝ 10,000 times smaller WS X←D∊((⍳⍴D)≠D⍳D)/D←⍳33000 ⍝ No LIMIT ERROR ⍝ 1,000 times faster ⍝ 21724 ms ⍝ 20 ms 20

21. Replacing Outer Product by Simple Logic M←100000↑50000⍴⍳13 :For I :In ⍳1000 L←1≠×/×M∘.-1 12 N←(M≥1)^M≤12 :EndFor M←100000↑50000⍴⍳13 ⎕WA 1397828 L←1≠×/×M∘.-1 12 WS FULL ⎕WA 37832 L←(M≥1)^M≤12 ⍝ 40 times smaller WS ⍝ No WS FULL ⍝ 9210 ms ⍝ 813 ms ⍝ 10 times faster 21

22. Replacing Outer Product by a Loop :For J :In ⍳10 X←+/((⍳⍴A)∘.≥⍳⍴A)^A∘.<B Y←⍬ :For I :In ⍳⍴B Y,←+/A[I]<I↑B :EndFor ⎕WA 2047735492 X←+/((⍳⍴A)∘.≥⍳⍴A)^A∘.<B LIMIT ERROR ⎕WA 405316 X←⍬ :For I :In ⍳⍴B X,←+/A[I]<I↑B :EndFor ⍝ 3 times faster ⍝ 5,000 times smaller workspace A←32800?32800 B←20000+32800?32800 ⍝ No LIMIT ERROR ⍝ 75810 ms ⍝ 26422 ms 22

23. Inner Product 23

24. Matrix on the (wrong) Side of the Expression Requiring a Matrix Transpose 'ABC'^.=⍉((1↑⍴D),3)↑D (((1↑⍴D),3)↑D)^.='ABC' ⍝ Transpose needed ⍝ Transpose not needed 24 “one less pair of parentheses”

25. Transposed Inner Product VECTOR^.=⍉MATRIX Y←10000 6⍴⎕A :For I :In ⍳10000 L←'EFGHIJ'^.=⍉Y M←Y^.='EFGHIJ' :EndFor MATRIX^.=VECTOR ⍝ 14561 ms ⍝ 2302 ms 25 vs

26. Array Comparisons 26

27. Comparing Array Contents with a scalar ^/M^.=' ' or ^/^/M=' ' or M≡(⍴M)⍴' ' M←1000 1000⍴⎕AV 27

28. Character Comparison Efficiency M←1000 1000⍴⎕AV :For I :In ⍳10000 {}^/M^.=' ' {}^/^/M=' ' {}M≡(⍴M)⍴' ' :EndFor ⍝ 9108 ms ⍝ 9060 ms ⍝ 587 ms 28

29. Numeric Comparison Efficiency M←1000 1000⍴ ⍳10000 :For I :In ⍳10000 {}^/M^.=0 {}^/^/M=0 {}M≡(⍴M)⍴0 :EndFor ⍝ 12254 ms ⍝ 12201 ms ⍝ 52 ms 29

30. Comparing Vectors A←10000?10000 B←10000?10000 C←A^.=B :For I :In ⍳10000 {}A^.=B {}A≡B :EndFor C←A≡B ⍝ 1244 ms ⍝ 135 ms 30

31. Comparing Vectors of Unequal Lengths A←10000?10000 B←9999?9999 C←A^.=B LENGTH ERROR C←A^.=B ^ 31

32. Comparing Vectors of Unequal Lengths L←(⍴A)⌈⍴B C←(L↑A)^.=L↑B or :If C←(⍴A)=⍴B :AndIf C←A^.=B :EndIf or C←A≡B To avoid LENGTH ERROR 32

33. Checking the Return Code of a Function →(¯1∊DATA←FUNCTION_1)/ERR But there are still many functions written such that the result returned can be either the data or the return code. Nowadays, many functions are written such that a 2-item nested vector is returned where one item contains the result and another item contains the return code. E.g. if ¯1 returned by a function means an error has occurred; then we need to be very careful with the use of the ∊ membership function. 33

34. Example of Function Return Code A popular IBM APL utility function to read text file is called ∆FM (File Matrix I/O). When ∆FM reads a text file and encounters an error, instead of returning the data, it returns an error code of 28. Thus many programmers would write the text file I/O coding in the following way. →(28∊DATA←∆FM 'file.csv')/ERR 34

35. Example of Return Code Inefficiency Y←∆FM 'file.csv' ⎕SIZE'Y' 9979076 ⍴Y 72312 138 :For I :In ⍳1000 {}28∊Y {}28≡Y :EndFor ⍝ 3208521 ms ⍝ 4 ms 35

36. CPU Optimization Suggestions When an elegant outer product generates a sparse matrix that causes LIMIT ERROR, WS FULL, or computational slow down, replace the outer product by a simpler but not so elegant expression. Example of code elegance: 1≠×/×M∘.-1 12 vs (M≥1)^M≤12 Try to avoid unnecessary transpose of a matrix when you perform an inner product of a matrix with a vector. Remember that in some cases, the match function can run much faster than the inner product or the membership function. 36

37. The End 37 Eugene Ying Fiserv, Inc.