Average Error: 0.4 → 0.5
Time: 20.4s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[(\left(\cos \left(\left(u2 \cdot 2\right) \cdot \pi\right)\right) \cdot \left((e^{\log_* (1 + {\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6})} - 1)^*\right) + 0.5)_*\]

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 expm1-log1p-u0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019010 +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: 18.9s)Debug log

sample202.0ms

Algorithm
intervals

simplify123.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
122.0ms
(+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5)

prune20.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.3b

localize35.0ms

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)

rewrite17.0ms

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
9.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
2.0ms
(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)

series1.9s

Counts
4 → 12
Calls
4 calls:
Slowest
986.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
433.0ms
(fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)
365.0ms
(pow (* -2 (log u1)) 0.5)
86.0ms
(* PI (* 2 u2))

simplify524.0ms

Counts
18 → 60
Calls
18 calls:
Slowest
179.0ms
(- (+ 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))))
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)
79.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
38.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
34.0ms
(+ (* 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)

prune1.4s

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 0.0b

localize28.0ms

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)
0.2b
(* PI (* 2 u2))
0.1b
(pow (* -2 (log u1)) 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×expm1-udef
1×pow-exp
1×associate-*r*
1×pow-pow
1×expm1-log1p
1×pow-to-exp
Counts
4 → 49
Calls
4 calls:
Slowest
8.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
2.0ms
(pow (* -2 (log u1)) 0.5)
2.0ms
(* PI (* 2 u2))
0.0ms
(expm1 (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))

series2.3s

Counts
4 → 12
Calls
4 calls:
Slowest
1.1s
(expm1 (log1p (* (pow (* -2 (log u1)) 0.5) 1/6)))
823.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
333.0ms
(pow (* -2 (log u1)) 0.5)
63.0ms
(* PI (* 2 u2))

simplify289.0ms

Counts
19 → 61
Calls
19 calls:
Slowest
87.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
46.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
35.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))
20.0ms
(* (log (* -2 (log u1))) 0.5)
18.0ms
(* 1/6 (pow (* (pow -1 1.0) (* (pow -2 1.0) (pow (log (/ 1 u1)) 1.0))) 0.5))

prune932.0ms

Pruning

9 alts after pruning (8 fresh and 1 done)

Merged error: 0.0b

localize8.0ms

Local error

Found 4 expressions with local error:

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

rewrite9.0ms

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
4.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
2.0ms
(exp (log (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)))
1.0ms
(* PI (* 2 u2))
1.0ms
(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
941.0ms
(* (pow (* -2 (log u1)) 0.5) 1/6)
443.0ms
(log (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5))
403.0ms
(exp (log (fma (cos (* PI (* 2 u2))) (* (pow (* -2 (log u1)) 0.5) 1/6) 0.5)))
63.0ms
(* PI (* 2 u2))

simplify1.0s

Counts
27 → 68
Calls
27 calls:
Slowest
193.0ms
(- (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))))
180.0ms
(- (+ 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))))
103.0ms
(log (+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5))
72.0ms
(+ (* 1/6 (* (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5) (cos (* 2 (* u2 PI))))) 0.5)
64.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune1.1s

Pruning

10 alts after pruning (8 fresh and 2 done)

Merged error: 0b

localize26.0ms

Local error

Found 4 expressions with local error:

0.4b
(* (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)

rewrite33.0ms

Algorithm
rewrite-expression-head
Rules
14×associate-*l*
8×sqrt-prod
8×add-cube-cbrt
8×*-un-lft-identity
8×add-sqr-sqrt
7×add-exp-log
7×pow1
6×add-cbrt-cube
4×add-log-exp
4×log1p-expm1-u
4×expm1-log1p-u
3×*-commutative
3×unpow-prod-down
2×associate-*r*
1×cbrt-unprod
1×prod-exp
1×pow-exp
1×pow-prod-down
1×pow-pow
1×pow-to-exp
Counts
4 → 62
Calls
4 calls:
Slowest
22.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))
8.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)
1.0ms
(pow (* -2 (log u1)) 0.5)
1.0ms
(* PI (* 2 u2))

series2.2s

Counts
4 → 12
Calls
4 calls:
Slowest
942.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6)
859.0ms
(* (sqrt (pow (* -2 (log u1)) 0.5)) (* (sqrt (pow (* -2 (log u1)) 0.5)) 1/6))
342.0ms
(pow (* -2 (log u1)) 0.5)
56.0ms
(* PI (* 2 u2))

simplify776.0ms

Counts
30 → 74
Calls
30 calls:
Slowest
347.0ms
(* (* (* (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)))
101.0ms
(pow (* -2 (- (log -1) (log (/ -1 u1)))) 0.5)
56.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.25))
35.0ms
(* 1/6 (pow (* (pow -1 1.0) (* (pow -2 1.0) (pow (log (/ 1 u1)) 1.0))) 0.25))
33.0ms
(* 1/6 (pow (* (pow (- (log -1) (log (/ -1 u1))) 1.0) (pow -2 1.0)) 0.5))

prune1.1s

Pruning

10 alts after pruning (7 fresh and 3 done)

Merged error: 0b

regimes462.0ms

Accuracy

0% (0.5b remaining)

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

bsearch6.0ms

end0.0ms

sample2.6s

Algorithm
intervals