Average Error: 0.0 → 0.0
Time: 34.5s
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{(\left(-9 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right) + 1)_*} \cdot \frac{1 - {v}^{4}}{\frac{4}{\sqrt{2}}}}{\sqrt{\left(v \cdot v\right) \cdot 3 + 1} \cdot \left(1 + v \cdot v\right)}\]

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. Using strategy rm

  3. Applied flip--0.0

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

  4. Applied flip--0.0

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

  5. Applied sqrt-div0.0

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

  6. Applied associate-*r/0.0

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

  7. Applied frac-times0.0

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

  8. Simplified0.0

    \[\leadsto \frac{\color{blue}{\frac{1 - {v}^{4}}{\frac{4}{\sqrt{2}}} \cdot \sqrt{(\left(-9 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right) + 1)_*}}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)}\]

  9. Final simplification0.0

    \[\leadsto \frac{\sqrt{(\left(-9 \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot v\right) + 1)_*} \cdot \frac{1 - {v}^{4}}{\frac{4}{\sqrt{2}}}}{\sqrt{\left(v \cdot v\right) \cdot 3 + 1} \cdot \left(1 + v \cdot v\right)}\]

Reproduce

herbie shell --seed 2019030 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))