Average Error: 0.4 → 0.6
Time: 24.5s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[e^{\log \left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right) + 0.5\right)}\]

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-log0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019008 
(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: 23.2s)Debug log

sample228.0ms

Algorithm
intervals

simplify92.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

92.0ms
(+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5)

prune14.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize42.0ms

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 41
Calls

4 calls. Slowest were:

5.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
4.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
1.0ms
(pow (* -2 (log u1)) 0.5)

series1.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

926.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
522.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
334.0ms
(pow (* -2 (log u1)) 0.5)
31.0ms
(* PI u2)

simplify528.0ms

Counts
18 → 53
Calls

18 calls. Slowest were:

142.0ms
(- (+ 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))))
116.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
89.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)

prune719.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Counts
4 → 51
Calls

4 calls. Slowest were:

7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
4.0ms
(exp (log (+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))))
1.0ms
(log (+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))

series1.9s

Counts
4 → 12
Calls

4 calls. Slowest were:

860.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
500.0ms
(log (+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
467.0ms
(exp (log (+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))))
50.0ms
(* PI u2)

simplify1.7s

Counts
30 → 63
Calls

30 calls. Slowest were:

238.0ms
(log (+ (pow 0.5 3) (pow (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) 3)))
190.0ms
(log (- (* 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))))))
183.0ms
(exp (log (+ (pow 0.5 3) (pow (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) 3))))

prune1.0s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0b

localize14.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (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
(sqrt (pow (* -2 (log u1)) 0.5))

rewrite31.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

22.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))
6.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)
2.0ms
(sqrt (pow (* -2 (log u1)) 0.5))

series2.9s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.0s
(* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)
939.0ms
(sqrt (pow (* -2 (log u1)) 0.5))
891.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))
39.0ms
(* PI u2)

simplify709.0ms

Counts
32 → 67
Calls

32 calls. Slowest were:

309.0ms
(* (* (* (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)))
69.0ms
(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) 0.25)
66.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune1.0s

Pruning

9 alts after pruning (7 fresh and 2 done)

Merged error: 0b

localize20.0ms

Local error

Found 4 expressions with local error:

0.5b
(cbrt (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6)))
0.5b
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))
0.4b
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6))
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)

rewrite131.0ms

Algorithm
rewrite-expression-head
Counts
4 → 61
Calls

4 calls. Slowest were:

59.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6))
41.0ms
(cbrt (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6)))
26.0ms
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))

series3.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

985.0ms
(cbrt (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6)))
917.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
916.0ms
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))
876.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6))

simplify2.4s

Counts
41 → 73
Calls

41 calls. Slowest were:

546.0ms
(* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6)))
526.0ms
(* (* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6)))
414.0ms
(* (* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6))) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6)))

prune978.0ms

Pruning

9 alts after pruning (6 fresh and 3 done)

Merged error: 0b

regimes390.0ms

Accuracy

0% (0.5b remaining)

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

bsearch4.0ms

end0.0ms

sample2.9s

Algorithm
intervals