Average Error: 0.4 → 0.5
Time: 33.4s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1}{\frac{0.5 - \cos \left(\left(\pi \cdot u2\right) \cdot 2\right) \cdot \left(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right)}{0.5 \cdot 0.5 - \left(\cos \left(\left(\pi \cdot u2\right) \cdot 2\right) \cdot \left(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right)\right) \cdot \left(\cos \left(\left(\pi \cdot u2\right) \cdot 2\right) \cdot \left(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right)\right)}}\]

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 flip-+0.4

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

  6. Applied clear-num0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019022 
(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: 32.3s)Debug log

sample141.0ms

Algorithm
intervals
Results
80.0ms256×body80valid
8.0ms256×pre80true

simplify94.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
94.0ms
(+ (* (* (/ 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

localize38.0ms

Local error

Found 4 expressions with local error:

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

rewrite11.0ms

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
5.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
3.0ms
(+ 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)

series1.8s

Counts
4 → 12
Calls
4 calls:
Slowest
940.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
475.0ms
(+ 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
314.0ms
(pow (* -2 (log u1)) 0.5)
51.0ms
(* PI u2)

simplify436.0ms

Counts
18 → 53
Calls
18 calls:
Slowest
134.0ms
(- (+ 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))))
77.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
64.0ms
(+ (* 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) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
41.0ms
(+ (* 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)

prune696.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 0.0b

localize35.0ms

Local error

Found 4 expressions with local error:

39.3b
(cbrt (* (* (* PI u2) (* PI u2)) (* PI u2)))
0.4b
(* (pow (* -2 (log u1)) 0.5) 1/6)
0.3b
(* (* PI u2) (* PI u2))
0.2b
(* PI u2)

rewrite34.0ms

Algorithm
rewrite-expression-head
Rules
9×pow1
6×add-exp-log
6×add-cbrt-cube
5×add-cube-cbrt
5×associate-*l*
5×*-un-lft-identity
5×add-sqr-sqrt
4×add-log-exp
3×*-commutative
1×cbrt-unprod
1×rem-cbrt-cube
1×pow-prod-up
1×prod-exp
1×pow-prod-down
1×pow3
1×associate-*r*
1×pow1/3
1×pow-plus
1×unpow-prod-down
1×pow2
1×cbrt-prod
Counts
4 → 46
Calls
4 calls:
Slowest
20.0ms
(cbrt (* (* (* PI u2) (* PI u2)) (* PI u2)))
7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
6.0ms
(* (* PI u2) (* PI u2))
0.0ms
(* PI u2)

series1.0s

Counts
4 → 12
Calls
4 calls:
Slowest
855.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
82.0ms
(* (* PI u2) (* PI u2))
44.0ms
(cbrt (* (* (* PI u2) (* PI u2)) (* PI u2)))
37.0ms
(* PI u2)

simplify873.0ms

Counts
22 → 58
Calls
22 calls:
Slowest
614.0ms
(* (* (* (* PI u2) (* PI u2)) (* PI u2)) (* (* (* PI u2) (* PI u2)) (* PI u2)))
64.0ms
(* -1 (* u2 (* (cbrt -1) PI)))
45.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
27.0ms
(* 1/6 (pow (* (pow -1 1.0) (* (pow -2 1.0) (pow (log (/ 1 u1)) 1.0))) 0.5))
21.0ms
(cbrt (* (* PI u2) (* PI u2)))

prune834.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 0.0b

localize17.0ms

Local error

Found 4 expressions with local error:

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

rewrite95.0ms

Algorithm
rewrite-expression-head
Rules
31×pow1
18×add-exp-log
18×add-cbrt-cube
15×add-cube-cbrt
15×*-un-lft-identity
15×add-sqr-sqrt
13×pow-prod-down
12×times-frac
9×associate-*l*
8×cbrt-unprod
8×prod-exp
4×difference-of-squares
4×add-log-exp
4×associate-/l*
4×pow-prod-up
3×*-commutative
3×associate-/r*
2×associate-/l/
2×flip--
2×associate-/r/
2×flip3--
2×pow-plus
2×unpow-prod-down
1×div-inv
1×associate-*r*
1×div-exp
1×div-sub
1×frac-2neg
1×clear-num
1×cbrt-undiv
1×pow2
Counts
4 → 89
Calls
4 calls:
Slowest
68.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
11.0ms
(/ (- (* 0.5 0.5) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))) (- 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series3.0s

Counts
4 → 12
Calls
4 calls:
Slowest
905.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
884.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
774.0ms
(/ (- (* 0.5 0.5) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))) (- 0.5 (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
444.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))

simplify10.4s

Counts
79 → 101
Calls
79 calls:
Slowest
717.0ms
(* (* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
642.0ms
(- (* (* 0.5 0.5) (* 0.5 0.5)) (* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))))
373.0ms
(+ (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))) (log (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
340.0ms
(+ (log (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))))
328.0ms
(+ (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))) (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))))

prune1.8s

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize21.0ms

Local error

Found 4 expressions with local error:

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

rewrite87.0ms

Algorithm
rewrite-expression-head
Rules
31×pow1
16×add-exp-log
16×add-cbrt-cube
13×pow-prod-down
13×associate-*l*
8×cbrt-unprod
8×prod-exp
7×add-cube-cbrt
7×*-un-lft-identity
7×add-sqr-sqrt
4×add-log-exp
4×pow-prod-up
4×*-commutative
3×unpow-prod-down
2×pow-plus
1×associate-*r*
1×pow2
Counts
4 → 65
Calls
4 calls:
Slowest
75.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
3.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
3.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)

series3.0s

Counts
4 → 12
Calls
4 calls:
Slowest
832.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
817.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
814.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
522.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))

simplify3.6s

Counts
45 → 77
Calls
45 calls:
Slowest
710.0ms
(* (* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
320.0ms
(+ (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))) (log (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))))
294.0ms
(+ (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))) (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))))
294.0ms
(+ (log (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2)))) (+ (log (* (pow (* -2 (log u1)) 0.5) 1/6)) (log (cos (* (* PI u2) 2)))))
222.0ms
(* (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))) (* (* (pow (* -2 (log u1)) 0.5) 1/6) (cos (* (* PI u2) 2))))

prune1.5s

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 0.0b

regimes444.0ms

Accuracy

0% (0.5b remaining)

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

bsearch6.0ms

end0.0ms

sample2.4s

Algorithm
intervals
Results
2.0s8000×body80valid
297.0ms8000×pre80true