Average Error: 0.4 → 0.6
Time: 30.6s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt[3]{\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right)\right)} + 0.5\]

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-cbrt-cube0.6

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

  5. Final simplification0.6

    \[\leadsto \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt[3]{\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right)\right)} + 0.5\]

Reproduce

herbie shell --seed 2019007 
(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: 29.4s)Debug log

sample344.0ms

Algorithm
intervals

simplify132.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

131.0ms
(+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5)

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize39.0ms

Local error

Found 4 expressions with local error:

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

series1.9s

Counts
4 → 12
Calls

4 calls. Slowest were:

992.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
536.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
342.0ms
(pow (* -2 (log u1)) 0.5)
35.0ms
(* PI u2)

simplify493.0ms

Counts
18 → 53
Calls

18 calls. Slowest were:

151.0ms
(- (+ 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))))
103.0ms
(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) (cos (* 2 (* u2 PI))))) 0.5)

prune708.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

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

rewrite75.0ms

Algorithm
rewrite-expression-head
Counts
4 → 65
Calls

4 calls. Slowest were:

61.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
5.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series3.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

945.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
942.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
872.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
517.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))

simplify4.0s

Counts
45 → 77
Calls

45 calls. Slowest were:

627.0ms
(* (* (* (* (* (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)))))
368.0ms
(+ (+ (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)))))
350.0ms
(+ (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)))))

prune1.4s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize16.0ms

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

rewrite102.0ms

Algorithm
rewrite-expression-head
Counts
4 → 55
Calls

4 calls. Slowest were:

90.0ms
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (cos (* (* PI u2) 2)) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series3.4s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.0s
(* (pow (* -2 (log u1)) 0.5) 1/6)
945.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
918.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
557.0ms
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (cos (* (* PI u2) 2)) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))

simplify2.9s

Counts
32 → 67
Calls

32 calls. Slowest were:

430.0ms
(+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (* (cos (* (* PI u2) 2)) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))))
386.0ms
(+ (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))))))
365.0ms
(+ (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))))))

prune1.1s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0b

localize23.0ms

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

rewrite51.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

21.0ms
(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)))
15.0ms
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))
7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series3.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

973.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
972.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
912.0ms
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))
847.0ms
(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)))

simplify1.2s

Counts
27 → 62
Calls

27 calls. Slowest were:

691.0ms
(* (* (* (* (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)))
86.0ms
(+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (* (pow (* -2 (log u1)) 0.5) 1/6)))
59.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune973.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0b

regimes538.0ms

Accuracy

0% (0.5b remaining)

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

bsearch5.0ms

end0.0ms

sample3.0s

Algorithm
intervals