Average Error: 0.4 → 0.4
Time: 19.6s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[(\left(\cos \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right) \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5)_*\]

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-exp-log0.5

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

  6. Applied rem-exp-log0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019016 +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: 18.8s)Debug log

sample121.0ms

Algorithm
intervals
Results
256×(pre true 80)
256×(body real 80)

simplify165.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
164.0ms
(+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5)

prune31.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.4b

localize53.0ms

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)

rewrite21.0ms

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
12.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
3.0ms
(pow (* -2 (log u1)) 0.5)
2.0ms
(* PI (* 2 u2))
0.0ms
(fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)

series2.2s

Counts
4 → 12
Calls
4 calls:
Slowest
1.3s
(* (pow (* -2 (log u1)) 0.5) 1/6)
438.0ms
(fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)
398.0ms
(pow (* -2 (log u1)) 0.5)
66.0ms
(* PI (* 2 u2))

simplify592.0ms

Counts
18 → 60
Calls
18 calls:
Slowest
197.0ms
(- (+ 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))))
122.0ms
(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)
53.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 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)

prune763.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 0.0b

localize14.0ms

Local error

Found 4 expressions with local error:

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

rewrite27.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×*-un-lft-identity
6×pow1
6×add-sqr-sqrt
5×add-exp-log
4×add-log-exp
4×log1p-expm1-u
4×associate-*l*
4×add-cbrt-cube
4×exp-prod
4×expm1-log1p-u
2×*-commutative
2×log-pow
2×log-prod
1×rem-exp-log
1×exp-sum
1×associate-*r*
1×rem-log-exp
1×unpow-prod-down
Counts
4 → 52
Calls
4 calls:
Slowest
10.0ms
(exp (log (* (pow (* -2 (log u1)) 0.5) 1/6)))
7.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
7.0ms
(log (* (pow (* -2 (log u1)) 0.5) 1/6))
2.0ms
(* PI (* 2 u2))

series3.0s

Counts
4 → 12
Calls
4 calls:
Slowest
997.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
952.0ms
(log (* (pow (* -2 (log u1)) 0.5) 1/6))
930.0ms
(exp (log (* (pow (* -2 (log u1)) 0.5) 1/6)))
75.0ms
(* PI (* 2 u2))

simplify717.0ms

Counts
23 → 64
Calls
23 calls:
Slowest
140.0ms
(log (* 1/6 (pow (* (pow -1 1.0) (* (pow -2 1.0) (pow (log (/ 1 u1)) 1.0))) 0.5)))
94.0ms
(log (* 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))
56.0ms
(exp (* (cbrt (log (* (pow (* -2 (log u1)) 0.5) 1/6))) (cbrt (log (* (pow (* -2 (log u1)) 0.5) 1/6)))))
56.0ms
(log (* 1/6 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5)))

prune951.0ms

Pruning

5 alts after pruning (4 fresh and 1 done)

Merged error: 0b

localize2.0ms

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)

rewrite14.0ms

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
8.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
3.0ms
(pow (* -2 (log u1)) 0.5)
2.0ms
(* 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
933.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
447.0ms
(fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)
375.0ms
(pow (* -2 (log u1)) 0.5)
53.0ms
(* PI (* 2 u2))

simplify632.0ms

Counts
18 → 60
Calls
18 calls:
Slowest
251.0ms
(- (+ 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))))
118.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
74.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)
62.0ms
(+ (* 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)
35.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune824.0ms

Pruning

5 alts after pruning (3 fresh and 2 done)

Merged error: 0b

localize15.0ms

Local error

Found 4 expressions with local error:

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

rewrite17.0ms

Algorithm
rewrite-expression-head
Rules
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
6×log-prod
6×pow1
5×add-exp-log
4×add-log-exp
4×log1p-expm1-u
4×associate-*l*
4×add-cbrt-cube
4×exp-prod
4×expm1-log1p-u
3×exp-sum
2×*-commutative
2×log-pow
1×rem-exp-log
1×associate-*r*
1×rem-log-exp
1×unpow-prod-down
Counts
4 → 56
Calls
4 calls:
Slowest
8.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
4.0ms
(exp (log (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)))
2.0ms
(* PI (* 2 u2))
2.0ms
(log (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))

series1.9s

Counts
4 → 12
Calls
4 calls:
Slowest
922.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
548.0ms
(log (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))
371.0ms
(exp (log (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)))
57.0ms
(* PI (* 2 u2))

simplify896.0ms

Counts
27 → 68
Calls
27 calls:
Slowest
235.0ms
(- (log (+ 0.5 (* 1/6 (pow (* (pow -2 1.0) (pow (log u1) 1.0)) 0.5)))) (* 1/3 (* (/ (* (pow u2 2) (pow PI 2)) (+ 0.5 (* 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))))
151.0ms
(- (+ 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))))
84.0ms
(log (+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5))
55.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)
49.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune1.1s

Pruning

5 alts after pruning (2 fresh and 3 done)

Merged error: 0b

regimes314.0ms

Accuracy

0% (0.4b remaining)

Error of 0.4b against oracle of 0.0b and baseline of 0.4b

bsearch4.0ms

end0.0ms

sample2.7s

Algorithm
intervals
Results
8000×(pre true 80)
8000×(body real 80)