Average Error: 0.0 → 0.0
Time: 1.4m
Precision: 64
Internal Precision: 320
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt{2}}{4} \cdot \left(\left|\sqrt[3]{(\left(v \cdot 3\right) \cdot \left(-v\right) + 1)_*}\right| \cdot \sqrt{\sqrt[3]{1 - \left(v \cdot v\right) \cdot 3}}\right)\right) \cdot \left(1 - v \cdot v\right)\]

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-cube-cbrt0.0

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

  4. Applied sqrt-prod0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))