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. Simplified0.5

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

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

8. Applied expm1-log1p-u0.5

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

9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))

Details

Time bar (total: 17.9s)Debug log