Average Error: 0.0 → 0.0
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{(\left(v \cdot -3\right) \cdot v + 1)_*} \cdot \left(\sqrt{2} - \sqrt{2} \cdot \left(v \cdot v\right)\right)}{4}\]

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. Initial simplification0.0

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

  4. Applied add-log-exp0.0

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

  6. Applied associate-*r/0.0

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

  7. Applied associate-*l/0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018257 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))