Average Error: 0.0 → 0.0
Time: 1.9m
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\left(\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{4} - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{4}\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{1 + v \cdot v}\right) \cdot \sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}\]

Error

Bits error versus v

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{4}{1 - v \cdot v}} \cdot \sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}}\]
  3. Using strategy rm

  4. Applied flip--0.0

    \[\leadsto \frac{\sqrt{2}}{\frac{4}{\color{blue}{\frac{1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)}{1 + v \cdot v}}}} \cdot \sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}\]

  5. Applied associate-/r/0.0

    \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{4}{1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)}} \cdot \sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}\]

  6. Applied add-cube-cbrt0.0

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}{\frac{4}{1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)} \cdot \sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}\]

  7. Applied times-frac0.0

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{4}{1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)}} \cdot \frac{\sqrt[3]{\sqrt{2}}}{1 + v \cdot v}\right)} \cdot \sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}\]

  8. Simplified0.0

    \[\leadsto \left(\color{blue}{\left(\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{4} - \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{4} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right)} \cdot \frac{\sqrt[3]{\sqrt{2}}}{1 + v \cdot v}\right) \cdot \sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}\]

  9. Final simplification0.0

    \[\leadsto \left(\left(\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{4} - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{4}\right) \cdot \frac{\sqrt[3]{\sqrt{2}}}{1 + v \cdot v}\right) \cdot \sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))