Average Error: 0.0 → 0.0
Time: 4.5m
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*} \cdot \left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{(\left(\frac{4}{\sqrt{2}}\right) \cdot \left(v \cdot v\right) + \left(\frac{4}{\sqrt{2}}\right))_*}\]

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 *-un-lft-identity0.0

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

  5. Applied sqrt-prod0.0

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

  6. Applied associate-/l*0.0

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

  7. Applied associate-*l/0.0

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

  8. Applied frac-times0.0

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

  9. Simplified0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))