Average Error: 0.0 → 0.0
Time: 2.2m
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2}}{4} \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}\right)\]

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 associate-/r/0.0

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

  5. Applied associate-*l*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))