Average Error: 0.4 → 0.5
Time: 23.7s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[(\left(\cos \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right) \cdot \left(e^{\log_* (1 + {\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6})} - 1\right) + 0.5)_*\]

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 expm1-log1p-u0.5

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

  6. Applied expm1-udef0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019008 +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: 22.1s)Debug log

sample192.0ms

Algorithm
intervals

simplify99.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

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

prune25.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize29.0ms

Local error

Found 4 expressions with local error:

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

rewrite21.0ms

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))

series1.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

901.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
403.0ms
(pow (* -2 (log u1)) 0.5)
398.0ms
(fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)
60.0ms
(* PI (* 2 u2))

simplify608.0ms

Counts
18 → 60
Calls

18 calls. Slowest were:

218.0ms
(- (+ 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))))
112.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
65.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)

prune813.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize29.0ms

Local error

Found 4 expressions with local error:

0.3b
(fma (* (pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5) (cos (* (* u2 2) PI))) 1/6 0.5)
0.2b
(* (* u2 2) PI)
0.1b
(pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5)
0.0b
(* (pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5) (cos (* (* u2 2) PI)))

rewrite33.0ms

Algorithm
rewrite-expression-head
Counts
4 → 68
Calls

4 calls. Slowest were:

27.0ms
(* (pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5) (cos (* (* u2 2) PI)))
3.0ms
(pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5)
2.0ms
(* (* u2 2) PI)

series2.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.0s
(pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5)
486.0ms
(* (pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5) (cos (* (* u2 2) PI)))
462.0ms
(fma (* (pow (* (pow (log u1) 1.0) (pow -2 1.0)) 0.5) (cos (* (* u2 2) PI))) 1/6 0.5)
50.0ms
(* (* u2 2) PI)

simplify2.5s

Counts
38 → 80
Calls

38 calls. Slowest were:

190.0ms
(- (+ 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))))
171.0ms
(- (pow (pow (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 1.0) 1.0) 0.5) (* 2 (* (* (pow u2 2) (pow PI 2)) (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5))))
150.0ms
(+ (* (+ (log (pow (log u1) 1.0)) (log (pow -2 1.0))) 0.5) (log (cos (* (* u2 2) PI))))

prune1.2s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize11.0ms

Local error

Found 4 expressions with local error:

0.4b
(expm1 (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.2b
(* PI (* 2 u2))
0.2b
(log1p (* (pow (* -2 (log u1)) 0.5) 1/6))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
1.0ms
(* PI (* 2 u2))
0.0ms
(log1p (* (pow (* -2 (log u1)) 0.5) 1/6))

series3.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.3s
(expm1 (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))
1.1s
(log1p (* (pow (* -2 (log u1)) 0.5) 1/6))
906.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
56.0ms
(* PI (* 2 u2))

simplify350.0ms

Counts
17 → 59
Calls

17 calls. Slowest were:

66.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
60.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
50.0ms
(log (+ (* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5)) 1))

prune1.0s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.3b
(exp (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))
0.2b
(* PI (* 2 u2))
0.2b
(log1p (* (pow (* -2 (log u1)) 0.5) 1/6))

rewrite7.0ms

Algorithm
rewrite-expression-head
Counts
4 → 50
Calls

4 calls. Slowest were:

4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
1.0ms
(exp (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))
1.0ms
(* PI (* 2 u2))

series3.3s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.2s
(exp (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))
1.1s
(log1p (* (pow (* -2 (log u1)) 0.5) 1/6))
929.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
63.0ms
(* PI (* 2 u2))

simplify342.0ms

Counts
20 → 62
Calls

20 calls. Slowest were:

74.0ms
(+ (* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5)) 1)
64.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
38.0ms
(log (+ (* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5)) 1))

prune1.0s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0b

regimes484.0ms

Accuracy

0% (0.5b remaining)

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

bsearch2.0ms

end0.0ms

sample3.0s

Algorithm
intervals