Automatically improving floating point code

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1 Automatically improving floating point code

2 Scientists Write Code Every scientist needs to write code Analyze data Simulate models Control experiments

3 Scientists Write Code Every scientist needs to write code Analyze data Simulate models Control experiments They have little computer science training

4 How scientific code is written 1. Come up with mathematical formula

5 How scientific code is written 1. Come up with mathematical formula 2. Write as floating-point code f(x) sqrt(x + 1) - sqrt(x)

6 How scientific code is written f(x) sqrt(x + 1) - sqrt(x) 3. Test code f(1) =

7 How scientific code is written f(x) sqrt(x + 1) - sqrt(x) 3. Test code f(1) = Publish

8 But try a few other values... f( ) = 7.451e-9

9 But try a few other values... f( ) = 7.451e-9 That s a 16% error!

10 Outline Why did this happen? How can we fix it? How does Casio help?

11 Outline Why did this happen? How can we fix it? How does Casio help?

12 Why did this happen? Because floating-point is imprecise!

13 Why did this happen? Because floating-point is imprecise! We want something more constructive.

14 Why did this happen? Because floating-point is imprecise! We want something more constructive. Rounding

15 Why did this happen? Because floating-point is imprecise! We want something more constructive. Rounding Let s try to be a bit more precise

16 Let s try to be a bit more precise

17 Let s try to be a bit more precise Rounding error

18 Let s try to be a bit more precise % error % error.03% error Rounding error

19 Let s try to be a bit more precise % error % error.03% error Rounding error Error in output proportional to size of input.

20 For our example

21 For our example

22 For our example

23 For our example

24 So, in summary Code is numerically imprecise The subtraction is the culprit Figuring out why was hard

25 Outline Why did this happen? How can we fix it? How does Casio help?

26 How can we fix it? Run with higher precision? Software floating point is slow. Add correction terms? Very hard to do; very error-prone

27 How can we fix it? Better idea: rephrase the program Compute the same thing in a different way. Somehow get rid of the subtraction

28 One way of rephrasing it

29 One way of rephrasing it

30 One way of rephrasing it

31 One way of rephrasing it

32 One way of rephrasing it

33 Implementing this rephrasing f(x) = 1/(sqrt(x + 1) + sqrt(x)) This version has effectively no error.

34 Implementing this rephrasing f(x) = 1/(sqrt(x + 1) + sqrt(x)) This version has effectively no error. But it s a harder to understand: Is this computing the right function?

35 So, in summary Problem solved Algebraic ingenuity required Fixing it was hard

36 Outline Why did this happen? How can we fix it? How does Casio help?

37 How does Casio help? Diagnosing and fixing precision problems is hard

38 How does Casio help? Diagnosing and fixing precision problems is hard Casio automatically Computes error Finds better code

39 How does Casio help? Diagnosing and fixing precision problems is hard Casio automatically Computes error Finds better code Resulting expression is the same over the reals but better over the floats

40 Automatically computing error Compute exact answers with arbitrary precision Use program analysis tools to find bad subexpressions

41 Automatically improving code Small database of mathematical identities e.g. a - a = 0 e.g. a 2 - b 2 = (a + b) (a - b) Apply identities to the problem subexpression recursive goal-directed rewrite Evaluate resulting code versus exact answer

42 Inner Loop Equation Input Distribution

43 Inner Loop Equation Computed Output Input Distribution Exact Output Arbitrary Precision

44 Inner Loop Equation Computed Output Bad Subexpressions Input Distribution Exact Output Arbitrary Precision Program Analysis

45 Inner Loop Rewrite Database Equation Computed Output Bad Subexpressions Candidate Programs Input Distribution Exact Output Arbitrary Precision Program Analysis Rewrite Generation

46 Inner Loop Rewrite Database Equation Computed Output Bad Subexpressions Candidate Programs Best Candidates Input Distribution Exact Output Arbitrary Precision Program Analysis Rewrite Generation Ranking

47 Domain Knowledge Simplification Inferring branch conditions Periodicity analysis

48 > (improve '(λ (x) (- (sqrt (+ x 1)) (sqrt x))) 3)

49 > (improve '(λ (x) (- (sqrt (+ x 1)) (sqrt x))) 3) (λ (x) (if (< x ) (- (sqrt (+ x 1)) (sqrt x)) (/ 1 (+ (sqrt x) (sqrt (+ x 1))))))

50 > (improve '(λ (x) (- (sqrt (+ x 1)) (sqrt x))) 3) (λ (x) (if (< x ) (- (sqrt (+ x 1)) (sqrt x)) (/ 1 (+ (sqrt x) (sqrt (+ x 1)))))) Improvement by an average of 23.2 bits

51

52

53 Future work Extracting floating point computation from code More domain knowledge Provide explanation of what Casio did Unsound rewrites

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