Average Error: 0.5 → 0.5
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v + -1}\right)\]

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. Using strategy rm

  3. Applied add-sqr-sqrt0.5

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

  4. Applied associate-/l*0.6

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

  5. Taylor expanded around 0 0.5

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

  6. Simplified0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019050 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))