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Typing the Numeric Tower Vincent St-Amour, Sam Tobin-Hochstadt, - PowerPoint PPT Presentation

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Typing the Numeric Tower Vincent St-Amour, Sam Tobin-Hochstadt, Matthew Flatt, Matthias Felleisen PLT PADL 2012 - January 24th, 2012 Approaches to numerics Traditional Java, C(++), Fortran, ... Type Classes Haskell, Clean, ...

  1. Typing the Numeric Tower Vincent St-Amour, Sam Tobin-Hochstadt, Matthew Flatt, Matthias Felleisen PLT PADL 2012 - January 24th, 2012

  3. Approaches to numerics • Traditional Java, C(++), Fortran, ... • Type Classes Haskell, Clean, ... • Numeric Tower Racket, Scheme, Smalltalk, ...

  4. Approaches to numerics • Traditional Java, C(++), Fortran, ... • Type Classes Haskell, Clean, ... • Numeric Tower Racket, Scheme, Smalltalk, ...

  5. Approaches to numerics • Traditional Java, C(++), Fortran, ... • Type Classes Racket, Scheme, Smalltalk, ... } Haskell, Clean, ... Typed Numeric Tower Typed Racket • Numeric Tower

  6. Our criteria • Ease of expression • Domain fidelity • Static checking • Performance

  7. Our benchmarks

  8. Type Classes p = (2^31) - 1 a = 7^5 x = unsafePerformIO $ newIORef 42 genRandom min max = do old <- readIORef x writeIORef x (mod (a * old) p) new <- readIORef x return $ min + (((max-min) * (fromInteger new)) / (fromInteger p))

  9. Type Classes p = (2^31) - 1 a = 7^5 x = unsafePerformIO $ newIORef 42 genRandom min max = do old <- readIORef x writeIORef x (mod (a * old) p) new <- readIORef x return $ min + (((max-min) * (fromInteger new)) / (fromInteger p)) genRandom 2.3 7.4 -- => 2.3016764082953687

  10. Type Classes p = (2^31) - 1 a = 7^5 x = unsafePerformIO $ newIORef 42 genRandom min max = do old <- readIORef x writeIORef x (mod (a * old) p) new <- readIORef x return $ min + (((max-min) * (fromInteger new)) / (fromInteger p)) genRandom 2.3 7.4 -- => 2.3016764082953687 genRandom (toRational 2) (toRational 7) -- => 9927678409 % 2147483647

  11. Type Classes Ease of expression p = (2^31) - 1 a = 7^5 x = unsafePerformIO $ newIORef 42 genRandom min max = do old <- readIORef x writeIORef x (mod (a * old) p) new <- readIORef x return $ min + (((max-min) * (fromInteger new)) / (fromInteger p)) genRandom 2.3 7.4 -- => 2.3016764082953687 genRandom (toRational 2) (toRational 7) -- => 9927678409 % 2147483647

  12. Type Classes Ease of expression Domain fidelity p = (2^31) - 1 a = 7^5 x = unsafePerformIO $ newIORef 42 genRandom min max = do old <- readIORef x writeIORef x (mod (a * old) p) new <- readIORef x return $ min + (((max-min) * (fromInteger new)) / (fromInteger p)) genRandom 2.3 7.4 -- => 2.3016764082953687 genRandom (toRational 2) (toRational 7) -- => 9927678409 % 2147483647

  13. Type Classes Ease of expression Domain fidelity Static checking p = (2^31) - 1 a = 7^5 x = unsafePerformIO $ newIORef 42 genRandom min max = do old <- readIORef x writeIORef x (mod (a * old) p) new <- readIORef x return $ min + (((max-min) * (fromInteger new)) / (fromInteger p)) genRandom 2.3 7.4 -- => 2.3016764082953687 genRandom (toRational 2) (toRational 7) -- => 9927678409 % 2147483647

  14. Type Classes Ease of expression Domain fidelity Static checking Performance p = (2^31) - 1 a = 7^5 x = unsafePerformIO $ newIORef 42 genRandom min max = do old <- readIORef x writeIORef x (mod (a * old) p) new <- readIORef x return $ min + (((max-min) * (fromInteger new)) / (fromInteger p)) genRandom 2.3 7.4 -- => 2.3016764082953687 genRandom (toRational 2) (toRational 7) -- => 9927678409 % 2147483647

  15. Numeric Tower (define p (- (expt 2 31) 1)) (define A (expt 7 5)) (define x 42) ; state of the PRNG (define (gen-random min max) (set! x (modulo (* A x) p)) (+ min (/ (* (- max min) x) p)))

  16. Numeric Tower (define p (- (expt 2 31) 1)) (define A (expt 7 5)) (define x 42) ; state of the PRNG (define (gen-random min max) (set! x (modulo (* A x) p)) (+ min (/ (* (- max min) x) p))) (gen-random 2.3 7.4) ; => 2.3016764082953687

  17. Numeric Tower (define p (- (expt 2 31) 1)) (define A (expt 7 5)) (define x 42) ; state of the PRNG (define (gen-random min max) (set! x (modulo (* A x) p)) (+ min (/ (* (- max min) x) p))) (gen-random 2.3 7.4) ; => 2.3016764082953687 (gen-random 2 7) ; => 9927678409/2147483647

  18. Numeric Tower Ease of expression (define p (- (expt 2 31) 1)) (define A (expt 7 5)) (define x 42) ; state of the PRNG (define (gen-random min max) (set! x (modulo (* A x) p)) (+ min (/ (* (- max min) x) p))) (gen-random 2.3 7.4) ; => 2.3016764082953687 (gen-random 2 7) ; => 9927678409/2147483647

  19. Numeric Tower Ease of expression Domain fidelity (define p (- (expt 2 31) 1)) (define A (expt 7 5)) (define x 42) ; state of the PRNG (define (gen-random min max) (set! x (modulo (* A x) p)) (+ min (/ (* (- max min) x) p))) (gen-random 2.3 7.4) ; => 2.3016764082953687 (gen-random 2 7) ; => 9927678409/2147483647

  20. Numeric Tower Ease of expression Domain fidelity Static checking (define p (- (expt 2 31) 1)) (define A (expt 7 5)) (define x 42) ; state of the PRNG (define (gen-random min max) (set! x (modulo (* A x) p)) (+ min (/ (* (- max min) x) p))) (gen-random 2.3 7.4) ; => 2.3016764082953687 (gen-random 2 7) ; => 9927678409/2147483647

  21. Numeric Tower Ease of expression Domain fidelity Static checking Performance (define p (- (expt 2 31) 1)) (define A (expt 7 5)) (define x 42) ; state of the PRNG (define (gen-random min max) (set! x (modulo (* A x) p)) (+ min (/ (* (- max min) x) p))) (gen-random 2.3 7.4) ; => 2.3016764082953687 (gen-random 2 7) ; => 9927678409/2147483647

  22. Type Classes q :: (Floating a) => a -> a -> a -> a q a b c = (-b + sqrt (b**2 - 4*a*c)) / (2*a)

  23. Type Classes q :: (Floating a) => a -> a -> a -> a q a b c = (-b + sqrt (b**2 - 4*a*c)) / (2*a) q 2.4 36.2 (-7.5) -- => 0.20441209042786124

  24. Type Classes q :: (Floating a) => a -> a -> a -> a q a b c = (-b + sqrt (b**2 - 4*a*c)) / (2*a) q 2.4 36.2 (-7.5) -- => 0.20441209042786124 q (fromInteger 1) (fromInteger 3) (fromInteger (-4)) -- => 1.0

  25. Type Classes q :: (Floating a) => a -> a -> a -> a q a b c = (-b + sqrt (b**2 - 4*a*c)) / (2*a) q 2.4 36.2 (-7.5) -- => 0.20441209042786124 q (fromInteger 1) (fromInteger 3) (fromInteger (-4)) -- => 1.0 q (fromInteger 1) (fromInteger 3) (fromInteger 3) -- => NaN

  26. Type Classes Ease of expression q :: (Floating a) => a -> a -> a -> a q a b c = (-b + sqrt (b**2 - 4*a*c)) / (2*a) q 2.4 36.2 (-7.5) -- => 0.20441209042786124 q (fromInteger 1) (fromInteger 3) (fromInteger (-4)) -- => 1.0 q (fromInteger 1) (fromInteger 3) (fromInteger 3) -- => NaN

  27. Type Classes Ease of expression Domain fidelity q :: (Floating a) => a -> a -> a -> a q a b c = (-b + sqrt (b**2 - 4*a*c)) / (2*a) q 2.4 36.2 (-7.5) -- => 0.20441209042786124 q (fromInteger 1) (fromInteger 3) (fromInteger (-4)) -- => 1.0 q (fromInteger 1) (fromInteger 3) (fromInteger 3) -- => NaN

  28. Type Classes Ease of expression Domain fidelity Static checking q :: (Floating a) => a -> a -> a -> a q a b c = (-b + sqrt (b**2 - 4*a*c)) / (2*a) q 2.4 36.2 (-7.5) -- => 0.20441209042786124 q (fromInteger 1) (fromInteger 3) (fromInteger (-4)) -- => 1.0 q (fromInteger 1) (fromInteger 3) (fromInteger 3) -- => NaN

  29. Type Classes Ease of expression Domain fidelity Static checking Performance q :: (Floating a) => a -> a -> a -> a q a b c = (-b + sqrt (b**2 - 4*a*c)) / (2*a) q 2.4 36.2 (-7.5) -- => 0.20441209042786124 q (fromInteger 1) (fromInteger 3) (fromInteger (-4)) -- => 1.0 q (fromInteger 1) (fromInteger 3) (fromInteger 3) -- => NaN

  30. Numeric Tower (define (q a b c) (/ (+ (- b) (sqrt (- (sqr b) (* 4 a c)))) (* 2 a)))

  31. Numeric Tower (define (q a b c) (/ (+ (- b) (sqrt (- (sqr b) (* 4 a c)))) (* 2 a))) (q 2.4 36.2 -7.5) ; => 0.20441209042786124

  32. Numeric Tower (define (q a b c) (/ (+ (- b) (sqrt (- (sqr b) (* 4 a c)))) (* 2 a))) (q 2.4 36.2 -7.5) ; => 0.20441209042786124 (q 1 3 -4) ; => 1 ; =>

  33. Numeric Tower (define (q a b c) (/ (+ (- b) (sqrt (- (sqr b) (* 4 a c)))) (* 2 a))) (q 2.4 36.2 -7.5) ; => 0.20441209042786124 (q 1 3 -4) ; => 1 ; => (q 1 3 3) ; => -1.5+0.8660254037844386i ; =>

  34. Numeric Tower Ease of expression (define (q a b c) (/ (+ (- b) (sqrt (- (sqr b) (* 4 a c)))) (* 2 a))) (q 2.4 36.2 -7.5) ; => 0.20441209042786124 (q 1 3 -4) ; => 1 ; => (q 1 3 3) ; => -1.5+0.8660254037844386i ; =>

  35. Numeric Tower Ease of expression Domain fidelity (define (q a b c) (/ (+ (- b) (sqrt (- (sqr b) (* 4 a c)))) (* 2 a))) (q 2.4 36.2 -7.5) ; => 0.20441209042786124 (q 1 3 -4) ; => 1 ; => (q 1 3 3) ; => -1.5+0.8660254037844386i ; =>

  36. Numeric Tower Ease of expression Domain fidelity Static checking (define (q a b c) (/ (+ (- b) (sqrt (- (sqr b) (* 4 a c)))) (* 2 a))) (q 2.4 36.2 -7.5) ; => 0.20441209042786124 (q 1 3 -4) ; => 1 ; => (q 1 3 3) ; => -1.5+0.8660254037844386i ; =>

  37. Numeric Tower Ease of expression Domain fidelity Static checking Performance (define (q a b c) (/ (+ (- b) (sqrt (- (sqr b) (* 4 a c)))) (* 2 a))) (q 2.4 36.2 -7.5) ; => 0.20441209042786124 (q 1 3 -4) ; => 1 ; => (q 1 3 3) ; => -1.5+0.8660254037844386i ; =>

  38. Type Classes Numeric Tower PRNG Quad PRNG Quad Ease of expression Domain fidelity Static checking Performance

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