Error

Derivation

1. Initial program 0.5

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]

2. Initial simplification0.5

   \[\leadsto \cos^{-1} \left(\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right)\]
3. Using strategy rm

4. Applied add-exp-log0.5

   \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right)\right)}}\]
5. Using strategy rm

6. Applied expm1-log1p-u0.5

   \[\leadsto e^{\log \left(\cos^{-1} \left(\frac{\color{blue}{(e^{\log_* (1 + (\left(-5 \cdot v\right) \cdot v + 1)_*)} - 1)^*}}{(v \cdot v + -1)_*}\right)\right)}\]
7. Using strategy rm

8. Applied add-cube-cbrt1.5

   \[\leadsto e^{\log \color{blue}{\left(\left(\sqrt[3]{\cos^{-1} \left(\frac{(e^{\log_* (1 + (\left(-5 \cdot v\right) \cdot v + 1)_*)} - 1)^*}{(v \cdot v + -1)_*}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{(e^{\log_* (1 + (\left(-5 \cdot v\right) \cdot v + 1)_*)} - 1)^*}{(v \cdot v + -1)_*}\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(\frac{(e^{\log_* (1 + (\left(-5 \cdot v\right) \cdot v + 1)_*)} - 1)^*}{(v \cdot v + -1)_*}\right)}\right)}}\]

9. Applied log-prod2.4

   \[\leadsto e^{\color{blue}{\log \left(\sqrt[3]{\cos^{-1} \left(\frac{(e^{\log_* (1 + (\left(-5 \cdot v\right) \cdot v + 1)_*)} - 1)^*}{(v \cdot v + -1)_*}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{(e^{\log_* (1 + (\left(-5 \cdot v\right) \cdot v + 1)_*)} - 1)^*}{(v \cdot v + -1)_*}\right)}\right) + \log \left(\sqrt[3]{\cos^{-1} \left(\frac{(e^{\log_* (1 + (\left(-5 \cdot v\right) \cdot v + 1)_*)} - 1)^*}{(v \cdot v + -1)_*}\right)}\right)}}\]

10. Simplified1.5

    \[\leadsto e^{\color{blue}{\left(\log \left(\sqrt[3]{\cos^{-1} \left(\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}\right)}\right) + \log \left(\sqrt[3]{\cos^{-1} \left(\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}\right)}\right)\right)} + \log \left(\sqrt[3]{\cos^{-1} \left(\frac{(e^{\log_* (1 + (\left(-5 \cdot v\right) \cdot v + 1)_*)} - 1)^*}{(v \cdot v + -1)_*}\right)}\right)}\]
11. Using strategy rm

12. Applied add-cube-cbrt0.5

    \[\leadsto e^{\left(\log \left(\sqrt[3]{\cos^{-1} \left(\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}\right)}\right) + \color{blue}{\left(\sqrt[3]{\log \left(\sqrt[3]{\cos^{-1} \left(\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}\right)}\right)} \cdot \sqrt[3]{\log \left(\sqrt[3]{\cos^{-1} \left(\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}\right)}\right)}\right) \cdot \sqrt[3]{\log \left(\sqrt[3]{\cos^{-1} \left(\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}\right)}\right)}}\right) + \log \left(\sqrt[3]{\cos^{-1} \left(\frac{(e^{\log_* (1 + (\left(-5 \cdot v\right) \cdot v + 1)_*)} - 1)^*}{(v \cdot v + -1)_*}\right)}\right)}\]

13. Final simplification0.5

    \[\leadsto e^{\left(\sqrt[3]{\log \left(\sqrt[3]{\cos^{-1} \left(\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}\right)}\right)} \cdot \left(\sqrt[3]{\log \left(\sqrt[3]{\cos^{-1} \left(\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}\right)}\right)} \cdot \sqrt[3]{\log \left(\sqrt[3]{\cos^{-1} \left(\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}\right)}\right)}\right) + \log \left(\sqrt[3]{\cos^{-1} \left(\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}\right)}\right)\right) + \log \left(\sqrt[3]{\cos^{-1} \left(\frac{(e^{\log_* (1 + (\left(-5 \cdot v\right) \cdot v + 1)_*)} - 1)^*}{(v \cdot v + -1)_*}\right)}\right)}\]

Runtime

Time bar (total: 32.0s)Debug logProfile

herbie shell --seed 2018254 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))