Average Error: 0.4 → 0.4
Time: 19.6s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right) + 0.5\]

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 2019018 
(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: 19.0s)Debug log

sample181.0ms

Algorithm
intervals
Results
95.0ms256×body80valid
11.0ms256×pre80true

simplify97.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
97.0ms
(+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5)

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize32.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.3b
(* 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))))

rewrite12.0ms

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
(* (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)
0.0ms
(* PI u2)

series2.0s

Counts
4 → 12
Calls
4 calls:
Slowest
994.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
536.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
385.0ms
(pow (* -2 (log u1)) 0.5)
51.0ms
(* PI u2)

simplify813.0ms

Counts
18 → 53
Calls
18 calls:
Slowest
492.0ms
(- (+ 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))))
89.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
60.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)
53.0ms
(+ (* 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)
51.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune665.0ms

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

rewrite19.0ms

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
8.0ms
(exp (log (* (pow (* -2 (log u1)) 0.5) 1/6)))
6.0ms
(log (* (pow (* -2 (log u1)) 0.5) 1/6))
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.0ms
(* PI u2)

series2.8s

Counts
4 → 12
Calls
4 calls:
Slowest
986.0ms
(exp (log (* (pow (* -2 (log u1)) 0.5) 1/6)))
958.0ms
(log (* (pow (* -2 (log u1)) 0.5) 1/6))
810.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
46.0ms
(* PI u2)

simplify500.0ms

Counts
23 → 55
Calls
23 calls:
Slowest
85.0ms
(log (* 1/6 (pow (* (pow -1 1.0) (* (pow -2 1.0) (pow (log (/ 1 u1)) 1.0))) 0.5)))
67.0ms
(log (* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5)))
64.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
50.0ms
(exp (log (pow (* -2 (log u1)) 0.5)))
45.0ms
(exp (sqrt (log (* (pow (* -2 (log u1)) 0.5) 1/6))))

prune908.0ms

Pruning

4 alts after pruning (4 fresh and 0 done)

Merged error: 0b

localize2.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.3b
(* 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))))

rewrite12.0ms

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
(* (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)
0.0ms
(* PI u2)

series1.9s

Counts
4 → 12
Calls
4 calls:
Slowest
995.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
528.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
354.0ms
(pow (* -2 (log u1)) 0.5)
50.0ms
(* PI u2)

simplify564.0ms

Counts
18 → 53
Calls
18 calls:
Slowest
196.0ms
(- (+ 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))))
117.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
67.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)
42.0ms
(+ (* 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)
38.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune718.0ms

Pruning

4 alts after pruning (3 fresh and 1 done)

Merged error: 0b

localize26.0ms

Local error

Found 4 expressions with local error:

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

rewrite62.0ms

Algorithm
rewrite-expression-head
Rules
14×associate-*l*
9×add-cube-cbrt
9×*-un-lft-identity
9×pow1
9×add-sqr-sqrt
8×sqrt-prod
6×add-exp-log
6×add-cbrt-cube
5×exp-prod
4×add-log-exp
3×*-commutative
2×log-pow
2×pow-prod-down
2×unpow-prod-down
1×rem-exp-log
1×cbrt-unprod
1×prod-exp
1×exp-sum
1×associate-*r*
1×log-prod
Counts
4 → 56
Calls
4 calls:
Slowest
42.0ms
(exp (log (* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))))
15.0ms
(* (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)
0.0ms
(* PI u2)

series2.9s

Counts
4 → 12
Calls
4 calls:
Slowest
992.0ms
(exp (log (* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))))
936.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)
881.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))
56.0ms
(* PI u2)

simplify823.0ms

Counts
34 → 68
Calls
34 calls:
Slowest
413.0ms
(* (* (* (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)))
65.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
55.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
38.0ms
(exp (sqrt (log (* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)))))
33.0ms
(* 1/6 (pow (* (pow -1 1.0) (* (pow -2 1.0) (pow (log (/ 1 u1)) 1.0))) 0.5))

prune1.1s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

regimes237.0ms

Accuracy

0% (0.3b remaining)

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

bsearch2.0ms

end0.0ms

sample2.6s

Algorithm
intervals
Results
2.2s8000×body80valid
314.0ms8000×pre80true