Average Error: 0.4 → 0.6
Time: 26.5s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{{\left(\log u1 \cdot -2\right)}^{0.5}} \cdot \left(\frac{1}{6} \cdot \sqrt{{\left(\log u1 \cdot -2\right)}^{0.5}}\right)\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5\]

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-sqr-sqrt0.6

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

  5. Applied associate-*l*0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019005 
(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: 25.0s)Debug log

sample386.0ms

Algorithm
intervals

simplify320.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

319.0ms
(+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5)

prune18.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.5b

localize45.0ms

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))))

rewrite19.0ms

Algorithm
rewrite-expression-head
Counts
4 → 41
Calls

4 calls. Slowest were:

5.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
3.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
1.0ms
(pow (* -2 (log u1)) 0.5)

series1.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

826.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
460.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
397.0ms
(pow (* -2 (log u1)) 0.5)
53.0ms
(* PI u2)

simplify533.0ms

Counts
18 → 53
Calls

18 calls. Slowest were:

216.0ms
(- (+ 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))))
95.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
64.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)

prune775.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)
0.3b
(* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))
0.2b
(* PI u2)
0.1b
(pow (* -2 (log u1)) 0.5)

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 53
Calls

4 calls. Slowest were:

22.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))
4.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)
2.0ms
(pow (* -2 (log u1)) 0.5)

series2.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

885.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))
824.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)
347.0ms
(pow (* -2 (log u1)) 0.5)
41.0ms
(* PI u2)

simplify763.0ms

Counts
30 → 65
Calls

30 calls. Slowest were:

397.0ms
(* (* (* (sqrt (pow (* -2 (log u1)) 0.5)) (sqrt (pow (* -2 (log u1)) 0.5))) (sqrt (pow (* -2 (log u1)) 0.5))) (* (* (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)))
74.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
66.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune871.0ms

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0b

localize16.0ms

Local error

Found 4 expressions with local error:

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

rewrite26.0ms

Algorithm
rewrite-expression-head
Counts
4 → 60
Calls

4 calls. Slowest were:

17.0ms
(* (* (sqrt (pow (* -2 (log u1)) 0.5)) (sqrt (pow (* -2 (log u1)) 0.5))) 1/6)
7.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) (sqrt (pow (* -2 (log u1)) 0.5)))
1.0ms
(pow (* -2 (log u1)) 0.5)

series2.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.0s
(* (* (sqrt (pow (* -2 (log u1)) 0.5)) (sqrt (pow (* -2 (log u1)) 0.5))) 1/6)
915.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) (sqrt (pow (* -2 (log u1)) 0.5)))
366.0ms
(pow (* -2 (log u1)) 0.5)
41.0ms
(* PI u2)

simplify420.0ms

Counts
40 → 72
Calls

40 calls. Slowest were:

72.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
68.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
45.0ms
(pow (pow (pow (pow (pow (pow (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 1.0) 1.0) 1.0) 1.0) 1.0) 1.0) 0.5)

prune1.0s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 0b

localize33.0ms

Local error

Found 4 expressions with local error:

0.8b
(pow (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) 3)
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
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)

rewrite89.0ms

Algorithm
rewrite-expression-head
Counts
4 → 71
Calls

4 calls. Slowest were:

69.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
10.0ms
(pow (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) 3)
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series3.1s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.0s
(* (pow (* -2 (log u1)) 0.5) 1/6)
957.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
559.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
557.0ms
(pow (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) 3)

simplify5.3s

Counts
50 → 83
Calls

50 calls. Slowest were:

683.0ms
(* (* (* (* (* (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)))))
402.0ms
(+ (+ (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)))))
364.0ms
(+ (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)))))

prune1.5s

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 0b

regimes522.0ms

Accuracy

0% (0.6b remaining)

Error of 0.6b against oracle of 0.0b and baseline of 0.6b

bsearch2.0ms

end0.0ms

sample3.1s

Algorithm
intervals