Average Error: 60.5 → 59.3
Time: 55.7s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]
↓
\[\cos^{-1} \left({\left(\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)} \cdot \left(\log \left(\sqrt[3]{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}} \cdot \sqrt[3]{e^{\sqrt{\log \left(e^{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right)\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]

Error

Bits error versus a

Derivation

  1. Initial program 60.5

    \[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]
  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-sqr-sqrt59.6

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

  6. Applied exp-prod59.6

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

  7. Applied log-pow59.3

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

  9. Applied add-cube-cbrt59.3

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

  10. Applied log-prod59.3

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

  12. Applied add-log-exp59.3

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

  13. Final simplification59.3

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

Reproduce

herbie shell --seed 2019010 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))

Details

Time bar (total: 53.9s)Debug log

sample1.9s

Algorithm
intervals

simplify26.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
25.0ms
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))

prune15.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 60.7b

localize29.0ms

Local error

Found 2 expressions with local error:

59.8b
(fmod (cosh a) (* a a))
1.2b
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))

rewrite19.0ms

Algorithm
rewrite-expression-head
Rules
2×add-log-exp
2×log1p-expm1-u
2×add-cube-cbrt
2×add-exp-log
2×add-cbrt-cube
2×*-un-lft-identity
2×pow1
2×expm1-log1p-u
2×add-sqr-sqrt
1×acos-asin
Counts
2 → 19
Calls
2 calls:
Slowest
1.0ms
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))
1.0ms
(fmod (cosh a) (* a a))

series12.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
9.0ms
(fmod (cosh a) (* a a))
2.0ms
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))

simplify37.0ms

Counts
6 → 25
Calls
6 calls:
Slowest
13.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))
8.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))
8.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))
3.0ms
(fmod (cosh a) (pow a 2))
2.0ms
(fmod (cosh a) (pow a 2))

prune281.0ms

Pruning

6 alts after pruning (5 fresh and 1 done)

Merged error: 59.5b

localize22.0ms

Local error

Found 3 expressions with local error:

59.8b
(fmod (cosh a) (* a a))
10.1b
(log (exp (fmod (cosh a) (* a a))))
1.2b
(acos (pow (log (exp (fmod (cosh a) (* a a)))) (log1p a)))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
5×add-cube-cbrt
5×*-un-lft-identity
5×add-sqr-sqrt
4×log-pow
4×pow1
3×add-log-exp
3×log1p-expm1-u
3×add-exp-log
3×add-cbrt-cube
3×exp-prod
3×log-prod
3×expm1-log1p-u
1×rem-log-exp
1×acos-asin
Counts
3 → 36
Calls
3 calls:
Slowest
4.0ms
(log (exp (fmod (cosh a) (* a a))))
2.0ms
(acos (pow (log (exp (fmod (cosh a) (* a a)))) (log1p a)))
0.0ms
(fmod (cosh a) (* a a))

series15.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
7.0ms
(log (exp (fmod (cosh a) (* a a))))
5.0ms
(acos (pow (log (exp (fmod (cosh a) (* a a)))) (log1p a)))
4.0ms
(fmod (cosh a) (* a a))

simplify107.0ms

Counts
16 → 45
Calls
16 calls:
Slowest
18.0ms
(log (* (cbrt (exp (fmod (cosh a) (* a a)))) (cbrt (exp (fmod (cosh a) (* a a))))))
15.0ms
(log (sqrt (exp (fmod (cosh a) (* a a)))))
12.0ms
(sqrt (fmod (cosh a) (* a a)))
12.0ms
(cbrt (fmod (cosh a) (* a a)))
9.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))

prune719.0ms

Pruning

8 alts after pruning (8 fresh and 0 done)

Merged error: 59.5b

localize33.0ms

Local error

Found 4 expressions with local error:

59.8b
(fmod (cosh a) (* a a))
59.8b
(fmod (cosh a) (* a a))
9.3b
(log (exp (sqrt (fmod (cosh a) (* a a)))))
1.2b
(acos (pow (* (sqrt (fmod (cosh a) (* a a))) (log (exp (sqrt (fmod (cosh a) (* a a)))))) (log1p a)))

rewrite13.0ms

Algorithm
rewrite-expression-head
Rules
7×add-cube-cbrt
7×log-pow
7×*-un-lft-identity
7×add-sqr-sqrt
6×exp-prod
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×add-exp-log
4×add-cbrt-cube
4×expm1-log1p-u
3×sqrt-prod
3×log-prod
1×rem-log-exp
1×acos-asin
Counts
4 → 48
Calls
4 calls:
Slowest
7.0ms
(acos (pow (* (sqrt (fmod (cosh a) (* a a))) (log (exp (sqrt (fmod (cosh a) (* a a)))))) (log1p a)))
5.0ms
(log (exp (sqrt (fmod (cosh a) (* a a)))))
0.0ms
(fmod (cosh a) (* a a))
0.0ms
(fmod (cosh a) (* a a))

series34.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
21.0ms
(log (exp (sqrt (fmod (cosh a) (* a a)))))
5.0ms
(acos (pow (* (sqrt (fmod (cosh a) (* a a))) (log (exp (sqrt (fmod (cosh a) (* a a)))))) (log1p a)))
4.0ms
(fmod (cosh a) (* a a))
3.0ms
(fmod (cosh a) (* a a))

simplify181.0ms

Counts
22 → 60
Calls
22 calls:
Slowest
20.0ms
(log (* (cbrt (exp (sqrt (fmod (cosh a) (* a a))))) (cbrt (exp (sqrt (fmod (cosh a) (* a a)))))))
15.0ms
(log (sqrt (exp (sqrt (fmod (cosh a) (* a a))))))
14.0ms
(sqrt (sqrt (fmod (cosh a) (* a a))))
14.0ms
(sqrt (cbrt (fmod (cosh a) (* a a))))
14.0ms
(sqrt (sqrt (fmod (cosh a) (* a a))))

prune831.0ms

Pruning

7 alts after pruning (7 fresh and 0 done)

Merged error: 59.5b

localize23.0ms

Local error

Found 4 expressions with local error:

59.8b
(fmod (cosh a) (* a a))
59.8b
(fmod (cosh a) (* a a))
59.8b
(fmod (cosh a) (* a a))
59.8b
(fmod (cosh a) (* a a))

rewrite1.0ms

Algorithm
rewrite-expression-head
Rules
4×add-log-exp
4×log1p-expm1-u
4×add-cube-cbrt
4×add-exp-log
4×add-cbrt-cube
4×*-un-lft-identity
4×pow1
4×expm1-log1p-u
4×add-sqr-sqrt
Counts
4 → 36
Calls
4 calls:
Slowest
0.0ms
(fmod (cosh a) (* a a))
0.0ms
(fmod (cosh a) (* a a))
0.0ms
(fmod (cosh a) (* a a))
0.0ms
(fmod (cosh a) (* a a))

series8.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
2.0ms
(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))

simplify37.0ms

Counts
12 → 48
Calls
12 calls:
Slowest
6.0ms
(fmod (cosh a) (pow a 2))
3.0ms
(fmod (cosh a) (pow a 2))
3.0ms
(fmod (cosh a) (pow a 2))
3.0ms
(fmod (cosh a) (pow a 2))
3.0ms
(fmod (cosh a) (pow a 2))

prune940.0ms

Pruning

8 alts after pruning (7 fresh and 1 done)

Merged error: 59.4b

regimes161.0ms

Accuracy

0% (0.1b remaining)

Error of 59.3b against oracle of 59.1b and baseline of 59.3b

bsearch3.0ms

end0.0ms

sample48.4s

Algorithm
intervals