Average Error: 0.0 → 0.0
Time: 1.0m
Precision: 64
Internal Precision: 384
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{\left(\left(1 - \left(v \cdot v\right) \cdot 3\right) \cdot \left(\frac{\frac{2}{4}}{4} \cdot \frac{\sqrt{2}}{4}\right)\right) \cdot \sqrt{1 - \left(\sqrt[3]{\left(v \cdot v\right) \cdot 3} \cdot \sqrt[3]{\left(v \cdot v\right) \cdot 3}\right) \cdot \sqrt[3]{\left(v \cdot v\right) \cdot 3}}} \cdot \left(1 - v \cdot v\right)\]

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

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

  8. Applied add-cube-cbrt0.0

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

Runtime

Time bar (total: 1.0m)Debug logProfile

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