Average Error: 60.5 → 59.3
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left({\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}^{\left(\log_* (1 + a)\right)}\right)\]
↓
\[\log \left(e^{\cos^{-1} \left({\left(\log \left(e^{\sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}}\right) \cdot \sqrt{\left(\left(\cosh a\right) \bmod \left(a \cdot a\right)\right)}\right)}^{\left(\log_* (1 + a)\right)}\right)}\right)\]

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. Initial simplification60.5

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

  4. Applied add-log-exp59.6

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

  6. Applied add-sqr-sqrt59.6

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

  7. Applied exp-prod59.6

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

  8. Applied log-pow59.3

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

  10. Applied add-log-exp59.3

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

  11. Final simplification59.3

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

Runtime

Time bar (total: 1.2m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes59.359.359.20.10%
herbie shell --seed 2018353 +o rules:numerics
(FPCore (a)
  :name "Random Jason Timeout Test 012"
  (acos (pow (fmod (cosh a) (* a a)) (log1p a))))