Average Error: 0.0 → 0.0
Time: 1.9m
Precision: 64
Internal Precision: 384
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\frac{\frac{2}{4}}{4 \cdot 4} \cdot \left(e^{\log \left(\sqrt{2} - 3 \cdot \left(\sqrt{2} \cdot {v}^{2}\right)\right)} \cdot \sqrt{(\left(v \cdot v\right) \cdot \left(-3\right) + 1)_*}\right)} \cdot \left(1 - v \cdot v\right)\]

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 add-cbrt-cube0.0

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

  4. Applied add-cbrt-cube0.0

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

  5. Applied cbrt-unprod0.0

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

  6. Applied simplify0.0

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

  7. Taylor expanded around 0 0.0

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

  9. Applied add-exp-log0.0

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))