Average Error: 60.6 → 59.5
Time: 1.0m
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]
↓
\[\cos^{-1} \left({\left(\left(\log \left(\sqrt{e^{\sqrt[3]{\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right) \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)\]

Error

Bits error versus a

Derivation

  1. Initial program 60.6

    \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]
  2. Using strategy rm

  3. Applied add-log-exp59.6

    \[\leadsto \cos^{-1} \left({\color{blue}{\left(\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)\right)}}^{\left(\log_* (1 + a)\right)}\right)\]
  4. Using strategy rm

  5. Applied add-cube-cbrt59.6

    \[\leadsto \cos^{-1} \left({\left(\log \left(e^{\color{blue}{\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right) \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]

  6. Applied exp-prod59.6

    \[\leadsto \cos^{-1} \left({\left(\log \color{blue}{\left({\left(e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)}^{\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)\]

  7. Applied log-pow59.5

    \[\leadsto \cos^{-1} \left({\color{blue}{\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \log \left(e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right)\right)}}^{\left(\log_* (1 + a)\right)}\right)\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt59.5

    \[\leadsto \cos^{-1} \left({\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \log \color{blue}{\left(\sqrt{e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}} \cdot \sqrt{e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)\]

  10. Applied log-prod59.5

    \[\leadsto \cos^{-1} \left({\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \color{blue}{\left(\log \left(\sqrt{e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)\]
  11. Using strategy rm

  12. Applied add-log-exp59.5

    \[\leadsto \cos^{-1} \left({\left(\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{\color{blue}{\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]

  13. Final simplification59.5

    \[\leadsto \cos^{-1} \left({\left(\left(\log \left(\sqrt{e^{\sqrt[3]{\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right) \cdot \sqrt[3]{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))

Details

Time bar (total: 58.3s)Debug log

sample1.8s

Algorithm
intervals

simplify38.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

37.0ms
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 59.6b

localize38.0ms

Local error

Found 4 expressions with local error:

59.1b
(fmod (cosh a) (* a a))
2.7b
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))
0.0b
(pow (fmod (cosh a) (* a a)) (log1p a))
0.0b
(log1p a)

rewrite10.0ms

Algorithm
rewrite-expression-head
Counts
4 → 47
Calls

4 calls. Slowest were:

4.0ms
(pow (fmod (cosh a) (* a a)) (log1p a))
2.0ms
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))
2.0ms
(fmod (cosh a) (* a a))

series105.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

53.0ms
(pow (fmod (cosh a) (* a a)) (log1p a))
39.0ms
(log1p a)
9.0ms
(fmod (cosh a) (* a a))
3.0ms
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))

simplify2.1s

Counts
20 → 59
Calls

20 calls. Slowest were:

570.0ms
(- (+ (exp (* -1 (* (log (fmod (cosh a) (pow a 2))) (log (/ 1 a))))) (+ (/ (* (log (fmod (cosh a) (pow a 2))) (exp (* -1 (* (log (fmod (cosh a) (pow a 2))) (log (/ 1 a)))))) a) (* 1/2 (/ (* (pow (log (fmod (cosh a) (pow a 2))) 2) (exp (* -1 (* (log (fmod (cosh a) (pow a 2))) (log (/ 1 a)))))) (pow a 2))))) (* 1/2 (/ (* (log (fmod (cosh a) (pow a 2))) (exp (* -1 (* (log (fmod (cosh a) (pow a 2))) (log (/ 1 a)))))) (pow a 2))))
422.0ms
(- (+ (/ 1 a) (log -1)) (+ (* 1/2 (/ 1 (pow a 2))) (log (/ -1 a))))
323.0ms
(- (+ a (* 1/3 (pow a 3))) (* 1/2 (pow a 2)))

prune621.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 57.7b

localize11.0ms

Local error

Found 4 expressions with local error:

59.1b
(fmod (cosh a) (* a a))
12.7b
(log (exp (fmod (cosh a) (* a a))))
2.7b
(acos (pow (log (exp (fmod (cosh a) (* a a)))) (log1p a)))
0.0b
(pow (log (exp (fmod (cosh a) (* a a)))) (log1p a))

rewrite6.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

3.0ms
(pow (log (exp (fmod (cosh a) (* a a)))) (log1p a))
2.0ms
(log (exp (fmod (cosh a) (* a a))))
1.0ms
(acos (pow (log (exp (fmod (cosh a) (* a a)))) (log1p a)))

series56.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

48.0ms
(pow (log (exp (fmod (cosh a) (* a a)))) (log1p a))
4.0ms
(acos (pow (log (exp (fmod (cosh a) (* a a)))) (log1p a)))
2.0ms
(fmod (cosh a) (* a a))
2.0ms
(log (exp (fmod (cosh a) (* a a))))

simplify1.1s

Counts
31 → 70
Calls

31 calls. Slowest were:

445.0ms
(- (+ (exp (* -1 (* (log (fmod (cosh a) (pow a 2))) (log (/ 1 a))))) (+ (/ (* (log (fmod (cosh a) (pow a 2))) (exp (* -1 (* (log (fmod (cosh a) (pow a 2))) (log (/ 1 a)))))) a) (* 1/2 (/ (* (pow (log (fmod (cosh a) (pow a 2))) 2) (exp (* -1 (* (log (fmod (cosh a) (pow a 2))) (log (/ 1 a)))))) (pow a 2))))) (* 1/2 (/ (* (log (fmod (cosh a) (pow a 2))) (exp (* -1 (* (log (fmod (cosh a) (pow a 2))) (log (/ 1 a)))))) (pow a 2))))
252.0ms
(- (+ (/ (* (log (fmod (cosh a) (pow a 2))) (pow (fmod (cosh a) (pow a 2)) (- (log -1) (log (/ -1 a))))) a) (+ (pow (fmod (cosh a) (pow a 2)) (- (log -1) (log (/ -1 a)))) (* 1/2 (/ (* (pow (log (fmod (cosh a) (pow a 2))) 2) (pow (fmod (cosh a) (pow a 2)) (- (log -1) (log (/ -1 a))))) (pow a 2))))) (* 1/2 (/ (* (log (fmod (cosh a) (pow a 2))) (pow (fmod (cosh a) (pow a 2)) (- (log -1) (log (/ -1 a))))) (pow a 2))))
189.0ms
(- (+ (* 1/2 (* (pow a 2) (pow (log (fmod (cosh a) (pow a 2))) 2))) (+ (* a (log (fmod (cosh a) (pow a 2)))) 1)) (* 1/2 (* (pow a 2) (log (fmod (cosh a) (pow a 2))))))

prune889.0ms

Pruning

11 alts after pruning (11 fresh and 0 done)

Merged error: 57.7b

localize39.0ms

Local error

Found 4 expressions with local error:

59.1b
(fmod (cosh a) (* a a))
59.1b
(fmod (cosh a) (* a a))
59.1b
(fmod (cosh a) (* a a))
12.3b
(log (exp (* (cbrt (fmod (cosh a) (* a a))) (cbrt (fmod (cosh a) (* a a))))))

rewrite3.0ms

Algorithm
rewrite-expression-head
Counts
4 → 43
Calls

4 calls. Slowest were:

2.0ms
(log (exp (* (cbrt (fmod (cosh a) (* a a))) (cbrt (fmod (cosh a) (* a a))))))
0.0ms
(fmod (cosh a) (* a a))
0.0ms
(fmod (cosh a) (* a a))

series179.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

173.0ms
(log (exp (* (cbrt (fmod (cosh a) (* a a))) (cbrt (fmod (cosh a) (* a a))))))
2.0ms
(fmod (cosh a) (* a a))
2.0ms
(fmod (cosh a) (* a a))
2.0ms
(fmod (cosh a) (* a a))

simplify151.0ms

Counts
18 → 55
Calls

18 calls. Slowest were:

18.0ms
(pow (pow (fmod (cosh a) (pow a 2)) 2) 1/3)
17.0ms
(pow (pow (fmod (cosh a) (pow a 2)) 2) 1/3)
17.0ms
(pow (pow (fmod (cosh a) (pow a 2)) 2) 1/3)

prune921.0ms

Pruning

10 alts after pruning (10 fresh and 0 done)

Merged error: 57.7b

localize23.0ms

Local error

Found 4 expressions with local error:

59.1b
(fmod (cosh a) (* a a))
59.1b
(fmod (cosh a) (* a a))
59.1b
(fmod (cosh a) (* a a))
59.1b
(fmod (cosh a) (* a a))

rewrite1.0ms

Algorithm
rewrite-expression-head
Counts
4 → 36
Calls

4 calls. Slowest were:

0.0ms
(fmod (cosh a) (* a a))
0.0ms
(fmod (cosh a) (* a a))
0.0ms
(fmod (cosh a) (* a a))

series10.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

5.0ms
(fmod (cosh a) (* a a))
2.0ms
(fmod (cosh a) (* a a))
2.0ms
(fmod (cosh a) (* a a))
2.0ms
(fmod (cosh a) (* a a))

simplify55.0ms

Counts
12 → 48
Calls

12 calls. Slowest were:

5.0ms
(fmod (cosh a) (pow a 2))
5.0ms
(fmod (cosh a) (pow a 2))
5.0ms
(fmod (cosh a) (pow a 2))

prune930.0ms

Pruning

11 alts after pruning (10 fresh and 1 done)

Merged error: 57.7b

regimes343.0ms

Accuracy

0% (0.8b remaining)

Error of 59.5b against oracle of 58.7b and baseline of 59.5b

bsearch6.0ms

end0.0ms

sample48.8s

Algorithm
intervals