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Derivation

1. Initial program 0.5

   \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
2. Using strategy rm

3. Applied add-log-exp0.6

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

5. Applied add-sqr-sqrt1.5

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

6. Taylor expanded around 0 0.5

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}\]

7. Final simplification0.5

   \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\]

Runtime

Time bar (total: 26.2s)Debug logProfile

herbie shell --seed 2018290 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))