Average Error: 0.4 → 0.5
Time: 19.9s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[(\left(\cos \left(\sqrt{\left(u2 \cdot 2\right) \cdot \pi} \cdot \sqrt{\left(u2 \cdot 2\right) \cdot \pi}\right)\right) \cdot \left((e^{\log_* (1 + \frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5})} - 1)^*\right) + 0.5)_*\]

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 add-sqr-sqrt0.4

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

  6. Applied expm1-log1p-u0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019022 +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: 18.6s)Debug log

sample218.0ms

Algorithm
intervals
Results
46.0ms256×body80valid
9.0ms256×pre80true

simplify99.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
99.0ms
(+ (* (* (/ 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

localize76.0ms

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)

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×log1p-expm1-u
4×associate-*l*
4×add-cbrt-cube
4×expm1-log1p-u
2×*-commutative
2×unpow-prod-down
1×fma-udef
1×pow-exp
1×associate-*r*
1×pow-pow
1×pow-to-exp
Counts
4 → 48
Calls
4 calls:
Slowest
7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
1.0ms
(pow (* -2 (log u1)) 0.5)
1.0ms
(* PI (* 2 u2))
0.0ms
(fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)

series1.8s

Counts
4 → 12
Calls
4 calls:
Slowest
981.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
406.0ms
(fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)
378.0ms
(pow (* -2 (log u1)) 0.5)
77.0ms
(* PI (* 2 u2))

simplify630.0ms

Counts
18 → 60
Calls
18 calls:
Slowest
183.0ms
(- (+ 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))))
125.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
99.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)
50.0ms
(+ (* 1/6 (* (cos (* 2 (* u2 PI))) (pow (* (pow -1 1.0) (* (pow -2 1.0) (pow (log (/ 1 u1)) 1.0))) 0.5))) 0.5)
38.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune895.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize22.0ms

Local error

Found 4 expressions with local error:

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

rewrite25.0ms

Algorithm
rewrite-expression-head
Rules
9×pow1
8×associate-*l*
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-exp-log
6×associate-*r*
6×add-cbrt-cube
4×add-log-exp
4×log1p-expm1-u
4×*-commutative
4×pow1/2
4×expm1-log1p-u
2×sqrt-prod
2×pow-prod-up
2×pow-prod-down
1×cbrt-unprod
1×prod-exp
1×sqrt-unprod
1×pow-plus
1×unpow-prod-down
1×rem-square-sqrt
1×pow2
Counts
4 → 64
Calls
4 calls:
Slowest
13.0ms
(* (sqrt (* PI (* 2 u2))) (sqrt (* PI (* 2 u2))))
7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
2.0ms
(* PI (* 2 u2))
2.0ms
(* PI (* 2 u2))

series1.1s

Counts
4 → 12
Calls
4 calls:
Slowest
862.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
91.0ms
(* (sqrt (* PI (* 2 u2))) (sqrt (* PI (* 2 u2))))
66.0ms
(* PI (* 2 u2))
54.0ms
(* PI (* 2 u2))

simplify960.0ms

Counts
33 → 76
Calls
33 calls:
Slowest
272.0ms
(* (* PI (* 2 u2)) (* PI (* 2 u2)))
251.0ms
(* (* PI (* 2 u2)) (* PI (* 2 u2)))
128.0ms
(* (* (* (sqrt (* PI (* 2 u2))) (sqrt (* PI (* 2 u2)))) (sqrt (* PI (* 2 u2)))) (* (* (sqrt (* PI (* 2 u2))) (sqrt (* PI (* 2 u2)))) (sqrt (* PI (* 2 u2)))))
35.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
21.0ms
(* (pow (sqrt 2) 2) (* u2 PI))

prune1.2s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize12.0ms

Local error

Found 4 expressions with local error:

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

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
9×pow1
8×associate-*l*
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
6×add-exp-log
6×add-cbrt-cube
5×associate-*r*
4×add-log-exp
4×log1p-expm1-u
4×pow1/2
4×expm1-log1p-u
3×*-commutative
2×sqrt-prod
2×pow-prod-up
2×pow-prod-down
1×expm1-udef
1×cbrt-unprod
1×prod-exp
1×sqrt-unprod
1×pow-plus
1×unpow-prod-down
1×rem-square-sqrt
1×expm1-log1p
1×pow2
Counts
4 → 64
Calls
4 calls:
Slowest
7.0ms
(* (sqrt (* PI (* 2 u2))) (sqrt (* PI (* 2 u2))))
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
1.0ms
(* PI (* 2 u2))
0.0ms
(expm1 (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))

series2.1s

Counts
4 → 12
Calls
4 calls:
Slowest
1.1s
(expm1 (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))
856.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
76.0ms
(* (sqrt (* PI (* 2 u2))) (sqrt (* PI (* 2 u2))))
60.0ms
(* PI (* 2 u2))

simplify1.1s

Counts
34 → 76
Calls
34 calls:
Slowest
309.0ms
(* (* PI (* 2 u2)) (* PI (* 2 u2)))
293.0ms
(* (* PI (* 2 u2)) (* PI (* 2 u2)))
112.0ms
(* (* (* (sqrt (* PI (* 2 u2))) (sqrt (* PI (* 2 u2)))) (sqrt (* PI (* 2 u2)))) (* (* (sqrt (* PI (* 2 u2))) (sqrt (* PI (* 2 u2)))) (sqrt (* PI (* 2 u2)))))
64.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
45.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune1.2s

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize7.0ms

Local error

Found 4 expressions with local error:

0.5b
(expm1 (log1p (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)
0.2b
(* PI (* 2 u2))
0.1b
(log1p (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×expm1-log1p-u
5×add-sqr-sqrt
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×associate-*l*
4×add-cbrt-cube
4×pow1
2×*-commutative
1×expm1-udef
1×log1p-expm1
1×associate-*r*
1×unpow-prod-down
1×expm1-log1p
1×log1p-udef
Counts
4 → 47
Calls
4 calls:
Slowest
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
1.0ms
(* PI (* 2 u2))
0.0ms
(log1p (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))
0.0ms
(expm1 (log1p (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)))

series2.0s

Counts
4 → 12
Calls
4 calls:
Slowest
944.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
474.0ms
(log1p (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))
471.0ms
(expm1 (log1p (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)))
67.0ms
(* PI (* 2 u2))

simplify798.0ms

Counts
17 → 59
Calls
17 calls:
Slowest
203.0ms
(- (log (+ 1.5 (* 1/6 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5)))) (* 1/3 (* (/ (* (pow u2 2) (pow PI 2)) (+ 1.5 (* 1/6 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5)))) (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5))))
195.0ms
(- (+ 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))))
79.0ms
(log (+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 1.5))
66.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)
54.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune908.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

regimes550.0ms

Accuracy

0% (0.5b remaining)

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

bsearch4.0ms

end0.0ms

sample2.8s

Algorithm
intervals
Results
2.0s8000×body80valid
678.0ms8000×pre80true