Average Error: 0.4 → 0.6
Time: 24.1s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\sqrt[3]{\left((\left(\cos \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right) \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5)_* \cdot (\left(\cos \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right) \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5)_*\right) \cdot (\left(\cos \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right) \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5)_*}\]

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

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019004 +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: 22.9s)Debug log

sample208.0ms

Algorithm
intervals

simplify118.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

118.0ms
(+ (* (* (/ 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

localize36.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.3b
(* 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)

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

6.0ms
(* (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:

974.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
410.0ms
(fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)
346.0ms
(pow (* -2 (log u1)) 0.5)
85.0ms
(* PI (* 2 u2))

simplify662.0ms

Counts
18 → 60
Calls

18 calls. Slowest were:

211.0ms
(- (+ 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))))
135.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
83.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)

prune818.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

1.0b
(exp (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.3b
(* PI (* 2 u2))
0.1b
(pow (* -2 (log u1)) 0.5)

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
4 → 52
Calls

4 calls. Slowest were:

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

series2.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

906.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
570.0ms
(exp (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))
416.0ms
(pow (* -2 (log u1)) 0.5)
77.0ms
(* PI (* 2 u2))

simplify1.1s

Counts
23 → 64
Calls

23 calls. Slowest were:

608.0ms
(- (exp (+ 0.5 (* 1/6 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5)))) (* 1/3 (* (* (pow u2 2) (* (exp (+ 0.5 (* 1/6 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5)))) (pow PI 2))) (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5))))
111.0ms
(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))

prune974.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 0.0b

localize23.0ms

Local error

Found 4 expressions with local error:

0.6b
(cbrt (* (* (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)))
0.5b
(* (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)

rewrite36.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

13.0ms
(cbrt (* (* (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)))
8.0ms
(* (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))
7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series2.9s

Counts
4 → 12
Calls

4 calls. Slowest were:

940.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
928.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
562.0ms
(* (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))
436.0ms
(cbrt (* (* (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)))

simplify2.4s

Counts
33 → 76
Calls

33 calls. Slowest were:

515.0ms
(* (* (* (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)) (* (* (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)) (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)))
271.0ms
(pow (+ (* 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) 2)
257.0ms
(pow (+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5) 2)

prune1.2s

Pruning

7 alts after pruning (6 fresh and 1 done)

Merged error: 0b

localize14.0ms

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)
0.3b
(* PI (* 2 u2))
0.1b
(log1p (* (pow (* -2 (log u1)) 0.5) 1/6))

rewrite5.0ms

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:

1.2s
(expm1 (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))
1.2s
(log1p (* (pow (* -2 (log u1)) 0.5) 1/6))
946.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
72.0ms
(* PI (* 2 u2))

simplify419.0ms

Counts
17 → 59
Calls

17 calls. Slowest were:

75.0ms
(log (+ (* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5)) 1))
67.0ms
(* 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))

prune1.1s

Pruning

7 alts after pruning (5 fresh and 2 done)

Merged error: 0b

regimes498.0ms

Accuracy

0% (0.5b remaining)

Error of 0.6b against oracle of 0.1b and baseline of 0.6b

bsearch5.0ms

end0.0ms

sample3.2s

Algorithm
intervals