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Derivation

1. Initial program 0.6

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

3. Applied acos-asin0.6

   \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
4. Using strategy rm

5. Applied add-sqr-sqrt1.5

   \[\leadsto \color{blue}{\sqrt{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\]
6. Using strategy rm

7. Applied flip3--1.5

   \[\leadsto \sqrt{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt{\color{blue}{\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}}\]

8. Applied sqrt-div0.6

   \[\leadsto \sqrt{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \color{blue}{\frac{\sqrt{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}^{3}}}{\sqrt{\frac{\pi}{2} \cdot \frac{\pi}{2} + \left(\sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) + \frac{\pi}{2} \cdot \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}}\]

9. Final simplification0.6

   \[\leadsto \sqrt{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)} \cdot \frac{\sqrt{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)}^{3}}}{\sqrt{\left(\sin^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right) \cdot \sin^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right) + \sin^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right) \cdot \frac{\pi}{2}\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}}\]

Runtime

Time bar (total: 48.3s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))