Average Error: 60.3 → 59.1
Time: 55.5s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]
↓
\[\cos^{-1} \left({\left(\left(\log \left(\sqrt{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}\right) + \left(\log \left(\sqrt{\sqrt{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}}\right) + \log \left(\sqrt{\sqrt{e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}}}\right)\right)\right) \cdot \sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)\]

Error

Bits error versus a

Derivation

  1. Initial program 60.3

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

  3. Applied add-log-exp59.4

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

  5. Applied add-sqr-sqrt59.4

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

  6. Applied exp-prod59.4

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

  7. Applied log-pow59.1

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

  9. Applied add-sqr-sqrt59.1

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

  10. Applied log-prod59.1

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

  12. Applied add-sqr-sqrt59.1

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

  13. Applied sqrt-prod59.1

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

  14. Applied log-prod59.1

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

  15. Final simplification59.1

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

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))

Details

Time bar (total: 53.6s)Debug log

sample1.9s

Algorithm
intervals
Results
1053×(pre true 80)
540×(body nan 80)
252×(body exit 10240)
108×(body real 2560)
77×(body real 1280)
38×(body real 640)
18×(body real 320)
12×(body real 160)
4×(body real 80)
2×(body nan 5120)
1×(body nan 1280)
1×(body nan 320)

simplify27.0ms

Counts
1 → 1
Calls
1 calls:
Slowest
26.0ms
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 60.1b

localize39.0ms

Local error

Found 2 expressions with local error:

59.4b
(fmod (cosh a) (* a a))
2.1b
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))

rewrite6.0ms

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

series18.0ms

Counts
2 → 6
Calls
2 calls:
Slowest
14.0ms
(fmod (cosh a) (* a a))
4.0ms
(acos (pow (fmod (cosh a) (* a a)) (log1p a)))

simplify52.0ms

Counts
6 → 25
Calls
6 calls:
Slowest
25.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))
12.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))
7.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))
2.0ms
(fmod (cosh a) (pow a 2))
2.0ms
(fmod (cosh a) (pow a 2))

prune295.0ms

Pruning

5 alts after pruning (5 fresh and 0 done)

Merged error: 58.8b

localize17.0ms

Local error

Found 3 expressions with local error:

59.4b
(fmod (cosh a) (* a a))
10.8b
(log (exp (fmod (cosh a) (* a a))))
2.1b
(acos (pow (log (exp (fmod (cosh a) (* a a)))) (log1p a)))

rewrite3.0ms

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
2.0ms
(log (exp (fmod (cosh a) (* a a))))
1.0ms
(acos (pow (log (exp (fmod (cosh a) (* a a)))) (log1p a)))
0.0ms
(fmod (cosh a) (* a a))

series7.0ms

Counts
3 → 9
Calls
3 calls:
Slowest
2.0ms
(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))))

simplify99.0ms

Counts
16 → 45
Calls
16 calls:
Slowest
14.0ms
(cbrt (fmod (cosh a) (* a a)))
12.0ms
(sqrt (fmod (cosh a) (* a a)))
12.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))
12.0ms
(log (sqrt (exp (fmod (cosh a) (* a a)))))
9.0ms
(log (* (cbrt (exp (fmod (cosh a) (* a a)))) (cbrt (exp (fmod (cosh a) (* a a))))))

prune639.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 58.8b

localize16.0ms

Local error

Found 4 expressions with local error:

59.4b
(fmod (cosh a) (* a a))
59.4b
(fmod (cosh a) (* a a))
10.0b
(log (exp (sqrt (fmod (cosh a) (* a a)))))
2.1b
(acos (pow (* (sqrt (fmod (cosh a) (* a a))) (log (exp (sqrt (fmod (cosh a) (* a a)))))) (log1p a)))

rewrite9.0ms

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
5.0ms
(log (exp (sqrt (fmod (cosh a) (* a a)))))
2.0ms
(acos (pow (* (sqrt (fmod (cosh a) (* a a))) (log (exp (sqrt (fmod (cosh a) (* a a)))))) (log1p a)))
0.0ms
(fmod (cosh a) (* a a))
0.0ms
(fmod (cosh a) (* a a))

series26.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
16.0ms
(log (exp (sqrt (fmod (cosh a) (* a a)))))
4.0ms
(fmod (cosh a) (* a a))
3.0ms
(fmod (cosh a) (* a a))
3.0ms
(acos (pow (* (sqrt (fmod (cosh a) (* a a))) (log (exp (sqrt (fmod (cosh a) (* a a)))))) (log1p a)))

simplify181.0ms

Counts
22 → 60
Calls
22 calls:
Slowest
17.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))
17.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))
17.0ms
(acos (pow (fmod (cosh a) (pow a 2)) (log1p a)))
12.0ms
(sqrt (cbrt (fmod (cosh a) (* a a))))
11.0ms
(sqrt (fmod (cosh a) (pow a 2)))

prune962.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 58.8b

localize40.0ms

Local error

Found 4 expressions with local error:

59.4b
(fmod (cosh a) (* a a))
59.4b
(fmod (cosh a) (* a a))
59.4b
(fmod (cosh a) (* a a))
10.1b
(log (sqrt (exp (sqrt (fmod (cosh a) (* a a))))))

rewrite6.0ms

Algorithm
rewrite-expression-head
Rules
6×add-cube-cbrt
6×*-un-lft-identity
6×log-prod
6×add-sqr-sqrt
5×add-exp-log
5×pow1
4×add-log-exp
4×log1p-expm1-u
4×add-cbrt-cube
4×expm1-log1p-u
3×sqrt-prod
2×log-pow
1×rem-log-exp
1×pow1/2
Counts
4 → 45
Calls
4 calls:
Slowest
4.0ms
(log (sqrt (exp (sqrt (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))

series216.0ms

Counts
4 → 12
Calls
4 calls:
Slowest
207.0ms
(log (sqrt (exp (sqrt (fmod (cosh a) (* a a))))))
3.0ms
(fmod (cosh a) (* a a))
3.0ms
(fmod (cosh a) (* a a))
3.0ms
(fmod (cosh a) (* a a))

simplify133.0ms

Counts
19 → 57
Calls
19 calls:
Slowest
15.0ms
(sqrt (exp (sqrt (fmod (cosh a) (* a a)))))
14.0ms
(log (sqrt (sqrt (exp (sqrt (fmod (cosh a) (* a a)))))))
10.0ms
(log (sqrt (sqrt (exp (sqrt (fmod (cosh a) (* a a)))))))
10.0ms
(log (* (cbrt (sqrt (exp (sqrt (fmod (cosh a) (* a a)))))) (cbrt (sqrt (exp (sqrt (fmod (cosh a) (* a a))))))))
9.0ms
(log (sqrt (* (cbrt (exp (sqrt (fmod (cosh a) (* a a))))) (cbrt (exp (sqrt (fmod (cosh a) (* a a))))))))

prune926.0ms

Pruning

9 alts after pruning (9 fresh and 0 done)

Merged error: 58.8b

regimes331.0ms

Accuracy

0% (0.1b remaining)

Error of 59.1b against oracle of 59.0b and baseline of 59.1b

bsearch4.0ms

end0.0ms

sample47.6s

Algorithm
intervals
Results
31956×(pre true 80)
16058×(body nan 80)
7852×(body exit 10240)
3194×(body real 2560)
2504×(body real 1280)
1226×(body real 640)
593×(body real 320)
356×(body real 160)
127×(body real 80)
10×(body nan 160)
9×(body nan 2560)
8×(body nan 1280)
8×(body nan 320)
7×(body nan 5120)
4×(body nan 640)