Average Error: 0.5 → 0.7
Time: 26.2s
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left((4 \cdot \left((v \cdot v + \left({v}^{4}\right))_*\right) + -1)_*\right)\]

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. Initial simplification0.5

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Runtime

Time bar (total: 26.2s)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))