Average Error: 0.0 → 0.0
Time: 2.4m
Precision: 64
Internal Precision: 320
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\left(\frac{\sqrt{2} \cdot \frac{1}{2}}{16} \cdot \left(\left(1 - v \cdot v\right) \cdot (\left(v \cdot -3\right) \cdot v + 1)_*\right)\right) \cdot \left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{(\left(v \cdot -3\right) \cdot v + 1)_*}\right)}\]

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. 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-cbrt-cube0.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Applied cbrt-unprod0.0

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

  8. Applied cbrt-unprod0.0

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

  9. Simplified0.0

    \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{(\left(v \cdot -3\right) \cdot v + 1)_*} \cdot \left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right)\right) \cdot \left(\frac{\frac{1}{2} \cdot \sqrt{2}}{16} \cdot \left(\left(1 - v \cdot v\right) \cdot (\left(v \cdot -3\right) \cdot v + 1)_*\right)\right)}}\]

  10. Final simplification0.0

    \[\leadsto \sqrt[3]{\left(\frac{\sqrt{2} \cdot \frac{1}{2}}{16} \cdot \left(\left(1 - v \cdot v\right) \cdot (\left(v \cdot -3\right) \cdot v + 1)_*\right)\right) \cdot \left(\left(\left(1 - v \cdot v\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{(\left(v \cdot -3\right) \cdot v + 1)_*}\right)}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018249 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))