Average Error: 0.6 → 1.0
Time: 56.8s
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{{\left(\frac{\pi}{2}\right)}^{3} - {\left(\sin^{-1} \left(\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 + v}}{v - 1}\right)\right)}^{3}}{\left(\sin^{-1} \left(\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 + v}}{v - 1}\right) \cdot \frac{\pi}{2} + \sin^{-1} \left(\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 + v}}{v - 1}\right) \cdot \sin^{-1} \left(\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 + v}}{v - 1}\right)\right) + \frac{\pi}{2} \cdot \frac{\pi}{2}}\]

Error

Bits error versus v

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Results

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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. Initial simplification0.6

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

  4. Applied acos-asin0.6

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

  6. Applied difference-of-sqr-11.0

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

  7. Applied associate-/r*1.0

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

  9. Applied flip3--1.0

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

  10. Final simplification1.0

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

Runtime

Time bar (total: 56.8s)Debug logProfile

herbie shell --seed 2018242 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))