Average Error: 0.4 → 0.4
Time: 31.2s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right) + 0.5\]

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  2. Simplified0.4

    \[\leadsto \color{blue}{0.5 + \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)}\]
  3. Using strategy rm

  4. Applied add-exp-log0.5

    \[\leadsto 0.5 + \color{blue}{e^{\log \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)}} \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]
  5. Using strategy rm

  6. Applied rem-exp-log0.4

    \[\leadsto 0.5 + \color{blue}{\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]

  7. Final simplification0.4

    \[\leadsto \left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right) + 0.5\]

Reproduce

herbie shell --seed 2019016 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))

Details

Time bar (total: 30.3s)Debug log

sample363.0ms

Algorithm
intervals
Results
256×(pre true 80)
256×(body real 80)

simplify273.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
273.0ms
(+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5)

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize46.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.2b
(* PI u2)
0.1b
(pow (* -2 (log u1)) 0.5)
0.0b
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×add-exp-log
5×*-un-lft-identity
5×pow1
5×add-sqr-sqrt
4×add-log-exp
4×associate-*l*
4×add-cbrt-cube
2×*-commutative
2×unpow-prod-down
1×flip-+
1×pow-exp
1×flip3-+
1×pow-pow
1×+-commutative
1×pow-to-exp
Counts
4 → 41
Calls
4 calls:
Slowest
6.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
5.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
1.0ms
(pow (* -2 (log u1)) 0.5)
0.0ms
(* PI u2)

series1.9s

Counts
4 → 12
Calls
4 calls:
Slowest
1.0s
(* (pow (* -2 (log u1)) 0.5) 1/6)
510.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
346.0ms
(pow (* -2 (log u1)) 0.5)
34.0ms
(* PI u2)

simplify560.0ms

Counts
18 → 53
Calls
18 calls:
Slowest
156.0ms
(- (+ 0.5 (* 1/6 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5))) (* 1/3 (* (* (pow u2 2) (pow PI 2)) (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5))))
115.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
63.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
57.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)
57.0ms
(+ (* 1/6 (* (pow (* (pow (log (/ 1 u1)) 1.0) (* (pow -2 1.0) (pow -1 1.0))) 0.5) (cos (* 2 (* u2 PI))))) 0.5)

prune898.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

0.5b
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
0.4b
(/ (- (* 0.5 0.5) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))) (- 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)

rewrite115.0ms

Algorithm
rewrite-expression-head
Rules
31×pow1
18×add-exp-log
18×add-cbrt-cube
15×add-cube-cbrt
15×*-un-lft-identity
15×add-sqr-sqrt
13×pow-prod-down
12×times-frac
9×associate-*l*
8×cbrt-unprod
8×prod-exp
4×difference-of-squares
4×add-log-exp
4×associate-/l*
4×pow-prod-up
3×*-commutative
3×associate-/r*
2×associate-/l/
2×flip--
2×associate-/r/
2×flip3--
2×pow-plus
2×unpow-prod-down
1×div-inv
1×associate-*r*
1×div-exp
1×div-sub
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow2
Counts
4 → 89
Calls
4 calls:
Slowest
93.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
14.0ms
(/ (- (* 0.5 0.5) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))) (- 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series3.0s

Counts
4 → 12
Calls
4 calls:
Slowest
947.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
854.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
753.0ms
(/ (- (* 0.5 0.5) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))) (- 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
423.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))

simplify10.6s

Counts
79 → 101
Calls
79 calls:
Slowest
743.0ms
(* (* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
666.0ms
(- (* (* 0.5 0.5) (* 0.5 0.5)) (* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))))
392.0ms
(+ (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))) (log (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
330.0ms
(+ (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))) (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))))
328.0ms
(+ (log (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))))

prune1.8s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0b

localize12.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.3b
(exp (log (* (pow (* -2 (log u1)) 0.5) 1/6)))
0.3b
(log (* (pow (* -2 (log u1)) 0.5) 1/6))
0.2b
(* PI u2)

rewrite15.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
5×add-exp-log
4×add-log-exp
4×associate-*l*
4×add-cbrt-cube
4×exp-prod
2×*-commutative
2×log-pow
2×log-prod
1×rem-exp-log
1×exp-sum
1×rem-log-exp
1×unpow-prod-down
Counts
4 → 43
Calls
4 calls:
Slowest
6.0ms
(log (* (pow (* -2 (log u1)) 0.5) 1/6))
5.0ms
(exp (log (* (pow (* -2 (log u1)) 0.5) 1/6)))
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.0ms
(* PI u2)

series2.9s

Counts
4 → 12
Calls
4 calls:
Slowest
1.1s
(log (* (pow (* -2 (log u1)) 0.5) 1/6))
926.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
898.0ms
(exp (log (* (pow (* -2 (log u1)) 0.5) 1/6)))
37.0ms
(* PI u2)

simplify559.0ms

Counts
23 → 55
Calls
23 calls:
Slowest
127.0ms
(log (* 1/6 (pow (* (pow -1 1.0) (* (pow -2 1.0) (pow (log (/ 1 u1)) 1.0))) 0.5)))
58.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
51.0ms
(exp (* (cbrt (log (* (pow (* -2 (log u1)) 0.5) 1/6))) (cbrt (log (* (pow (* -2 (log u1)) 0.5) 1/6)))))
50.0ms
(exp (log (pow (* -2 (log u1)) 0.5)))
47.0ms
(log (* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5)))

prune735.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize2.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.2b
(* PI u2)
0.1b
(pow (* -2 (log u1)) 0.5)
0.0b
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))

rewrite16.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×add-exp-log
5×*-un-lft-identity
5×pow1
5×add-sqr-sqrt
4×add-log-exp
4×associate-*l*
4×add-cbrt-cube
2×*-commutative
2×unpow-prod-down
1×flip-+
1×pow-exp
1×flip3-+
1×pow-pow
1×+-commutative
1×pow-to-exp
Counts
4 → 41
Calls
4 calls:
Slowest
7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
6.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
2.0ms
(pow (* -2 (log u1)) 0.5)
0.0ms
(* PI u2)

series1.9s

Counts
4 → 12
Calls
4 calls:
Slowest
945.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
509.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
373.0ms
(pow (* -2 (log u1)) 0.5)
55.0ms
(* PI u2)

simplify477.0ms

Counts
18 → 53
Calls
18 calls:
Slowest
169.0ms
(- (+ 0.5 (* 1/6 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5))) (* 1/3 (* (* (pow u2 2) (pow PI 2)) (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5))))
88.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
59.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)
48.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
37.0ms
(+ (* 1/6 (* (pow (* (pow (log (/ 1 u1)) 1.0) (* (pow -2 1.0) (pow -1 1.0))) 0.5) (cos (* 2 (* u2 PI))))) 0.5)

prune867.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

regimes321.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch4.0ms

end0.0ms

sample2.9s

Algorithm
intervals
Results
8000×(pre true 80)
8000×(body real 80)