Average Error: 0.4 → 0.6
Time: 26.6s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[(\left(\cos \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right) \cdot \left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \left(\frac{1}{6} \cdot \sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}}\right)\right) + 0.5)_*\]

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-sqr-sqrt0.6

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

  5. Applied associate-*l*0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019005 +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: 24.7s)Debug log

sample231.0ms

Algorithm
intervals

simplify114.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

114.0ms
(+ (* (* (/ 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: 0.5b

localize40.0ms

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

rewrite30.0ms

Algorithm
rewrite-expression-head
Counts
4 → 48
Calls

4 calls. Slowest were:

20.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
3.0ms
(pow (* -2 (log u1)) 0.5)
2.0ms
(* PI (* 2 u2))

series2.0s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.1s
(* (pow (* -2 (log u1)) 0.5) 1/6)
458.0ms
(fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)
373.0ms
(pow (* -2 (log u1)) 0.5)
79.0ms
(* PI (* 2 u2))

simplify458.0ms

Counts
18 → 60
Calls

18 calls. Slowest were:

152.0ms
(- (+ 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))))
70.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
62.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)

prune942.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0.0b

localize13.0ms

Local error

Found 4 expressions with local error:

0.3b
(* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)
0.3b
(* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))
0.2b
(* PI (* 2 u2))
0.1b
(pow (* -2 (log u1)) 0.5)

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 62
Calls

4 calls. Slowest were:

18.0ms
(* (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)
2.0ms
(pow (* -2 (log u1)) 0.5)

series2.2s

Counts
4 → 12
Calls

4 calls. Slowest were:

903.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)
888.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))
362.0ms
(pow (* -2 (log u1)) 0.5)
79.0ms
(* PI (* 2 u2))

simplify710.0ms

Counts
30 → 74
Calls

30 calls. Slowest were:

307.0ms
(* (* (* (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)))
119.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
53.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune1.2s

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 0b

localize29.0ms

Local error

Found 4 expressions with local error:

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

rewrite93.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

43.0ms
(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)))
29.0ms
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))
12.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series3.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

1.0s
(* (* (pow (* -2 (log u1)) 0.5) 1/6) (* (pow (* -2 (log u1)) 0.5) 1/6))
951.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
941.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
908.0ms
(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)))

simplify1.1s

Counts
27 → 70
Calls

27 calls. Slowest were:

658.0ms
(* (* (* (* (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)))
79.0ms
(+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (* (pow (* -2 (log u1)) 0.5) 1/6)))
68.0ms
(* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cbrt 1/216))

prune1.4s

Pruning

11 alts after pruning (9 fresh and 2 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

0.5b
(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)

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
4 → 64
Calls

4 calls. Slowest were:

10.0ms
(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)))
4.0ms
(* (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))
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series2.8s

Counts
4 → 12
Calls

4 calls. Slowest were:

986.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
931.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
478.0ms
(* (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))
401.0ms
(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)))

simplify2.6s

Counts
33 → 76
Calls

33 calls. Slowest were:

560.0ms
(* (* (* (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)))
353.0ms
(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)
223.0ms
(- (+ 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))))

prune1.3s

Pruning

11 alts after pruning (8 fresh and 3 done)

Merged error: 0b

regimes527.0ms

Accuracy

0% (0.6b remaining)

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

bsearch4.0ms

end0.0ms

sample3.0s

Algorithm
intervals