Average Error: 0.5 → 0.5
Time: 32.5s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{(\left(v \cdot v\right) \cdot -5 + 1)_*}{(v \cdot v + -1)_*}\right)\]

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

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

  4. Applied clear-num0.5

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

  5. Taylor expanded around -inf 0.5

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

  6. Simplified0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))