Average Error: 0.4 → 0.5
Time: 28.4s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[(\left(\cos \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right) \cdot \left(\sqrt[3]{\left(\sqrt{{\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{1.0} \cdot \frac{1}{36}} \cdot \sqrt{{\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{1.0} \cdot \frac{1}{36}}\right) \cdot \left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right)}\right) + 0.5)_*\]

Error

Bits error versus u1

Bits error versus u2

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

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

  4. Applied add-cbrt-cube0.6

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

  5. Taylor expanded around 0 0.5

    \[\leadsto (\left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right)\right) \cdot \left(\sqrt[3]{\color{blue}{\left(\frac{1}{36} \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{1.0}\right)} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)}\right) + 0.5)_*\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.5

    \[\leadsto (\left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right)\right) \cdot \left(\sqrt[3]{\color{blue}{\left(\sqrt{\frac{1}{36} \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{1.0}} \cdot \sqrt{\frac{1}{36} \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{1.0}}\right)} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)}\right) + 0.5)_*\]

  8. Final simplification0.5

    \[\leadsto (\left(\cos \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right) \cdot \left(\sqrt[3]{\left(\sqrt{{\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{1.0} \cdot \frac{1}{36}} \cdot \sqrt{{\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{1.0} \cdot \frac{1}{36}}\right) \cdot \left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right)}\right) + 0.5)_*\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(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: 26.5s)Debug log

sample225.0ms

Algorithm
intervals

simplify190.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune28.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize65.0ms

Local error

Found 4 expressions with local error:

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

rewrite12.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

8.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
1.0ms
(pow (* -2 (log u1)) 0.5)
1.0ms
(* PI (* 2 u2))

series1.9s

Counts
4 → 12
Calls

4 calls. Slowest were:

973.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
429.0ms
(fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)
409.0ms
(pow (* -2 (log u1)) 0.5)
71.0ms
(* PI (* 2 u2))

simplify540.0ms

Counts
18 → 60
Calls

18 calls. Slowest were:

163.0ms
(- (+ 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))))
99.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
95.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)

prune866.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize18.0ms

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)

rewrite55.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

31.0ms
(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)))
14.0ms
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))
5.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series3.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.1s
(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)))
948.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
900.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
871.0ms
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))

simplify1.1s

Counts
27 → 70
Calls

27 calls. Slowest were:

566.0ms
(* (* (* (* (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)))
72.0ms
(* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cbrt 1/216))
66.0ms
(+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (* (pow (* -2 (log u1)) 0.5) 1/6)))

prune1.3s

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize54.0ms

Local error

Found 4 expressions with local error:

0.5b
(cbrt (* (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)) (* (pow (* -2 (log u1)) 0.5) 1/6)))
0.4b
(* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.4b
(* (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)) (* (pow (* -2 (log u1)) 0.5) 1/6))

rewrite55.0ms

Algorithm
rewrite-expression-head
Counts
4 → 54
Calls

4 calls. Slowest were:

26.0ms
(* (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)) (* (pow (* -2 (log u1)) 0.5) 1/6))
19.0ms
(cbrt (* (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)) (* (pow (* -2 (log u1)) 0.5) 1/6)))
5.0ms
(* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))

series3.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

998.0ms
(cbrt (* (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)) (* (pow (* -2 (log u1)) 0.5) 1/6)))
909.0ms
(* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))
902.0ms
(* (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)) (* (pow (* -2 (log u1)) 0.5) 1/6))
859.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

simplify1.3s

Counts
23 → 66
Calls

23 calls. Slowest were:

671.0ms
(* (* (* (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)) (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6)) (* (pow (* -2 (log u1)) 0.5) 1/6)))
237.0ms
(+ (log (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (log (* (pow (* -2 (log u1)) 0.5) 1/6)))
71.0ms
(* (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)) (* (pow (* -2 (log u1)) 0.5) 1/6))

prune1.4s

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize33.0ms

Local error

Found 4 expressions with local error:

0.5b
(cbrt (* (* (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)))) (* (pow (* -2 (log u1)) 0.5) 1/6)))
0.5b
(* (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))))
0.4b
(* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))
0.4b
(* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))

rewrite116.0ms

Algorithm
rewrite-expression-head
Counts
4 → 67
Calls

4 calls. Slowest were:

57.0ms
(cbrt (* (* (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)))) (* (pow (* -2 (log u1)) 0.5) 1/6)))
40.0ms
(* (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))))
9.0ms
(* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))

series3.7s

Counts
4 → 12
Calls

4 calls. Slowest were:

984.0ms
(* (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))))
980.0ms
(cbrt (* (* (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)))) (* (pow (* -2 (log u1)) 0.5) 1/6)))
858.0ms
(* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))
855.0ms
(* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))

simplify726.0ms

Counts
39 → 79
Calls

39 calls. Slowest were:

192.0ms
(* (* (* (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)))) (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)))) (* (* (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0))) (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)))) (sqrt (* 1/36 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0)))))
67.0ms
(* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cbrt 1/216))
65.0ms
(* 1/36 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 1.0))

prune1.7s

Pruning

10 alts after pruning (9 fresh and 1 done)

Merged error: 0.0b

regimes699.0ms

Accuracy

0% (0.5b remaining)

Error of 0.5b against oracle of 0.0b and baseline of 0.5b

bsearch4.0ms

end0.0ms

sample3.0s

Algorithm
intervals