Average Error: 0.5 → 0.8
Time: 17.5s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left((4 \cdot \left((v \cdot v + \left({v}^{4}\right))_*\right) + -1)_*\right)\]

Error

Bits error versus v

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-cbrt-cube1.5

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

  6. Applied pow31.5

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

  7. Applied rem-cbrt-cube0.5

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

  8. Taylor expanded around 0 0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{4} + 4 \cdot {v}^{2}\right) - 1\right)}\]

  9. Simplified0.8

    \[\leadsto \cos^{-1} \color{blue}{\left((4 \cdot \left((v \cdot v + \left({v}^{4}\right))_*\right) + -1)_*\right)}\]

  10. Final simplification0.8

    \[\leadsto \cos^{-1} \left((4 \cdot \left((v \cdot v + \left({v}^{4}\right))_*\right) + -1)_*\right)\]

Reproduce

herbie shell --seed 2019030 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))