Average Error: 1.0 → 0.0
Time: 51.5s
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{\frac{\frac{4}{3}}{\pi}}}{\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}} \cdot \frac{\frac{\sqrt{\frac{\frac{4}{3}}{\pi}}}{1 - v \cdot v}}{\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}\]

Error

Bits error versus v

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Initial simplification0.0

    \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.0

    \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}{\color{blue}{\left(\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\right) \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}}\]

  5. Applied *-un-lft-identity0.0

    \[\leadsto \frac{\frac{\frac{\frac{4}{3}}{\pi}}{\color{blue}{1 \cdot \left(1 - v \cdot v\right)}}}{\left(\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\right) \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}\]

  6. Applied add-sqr-sqrt0.0

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{\frac{\frac{4}{3}}{\pi}} \cdot \sqrt{\frac{\frac{4}{3}}{\pi}}}}{1 \cdot \left(1 - v \cdot v\right)}}{\left(\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\right) \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}\]

  7. Applied times-frac0.0

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{\frac{\frac{4}{3}}{\pi}}}{1} \cdot \frac{\sqrt{\frac{\frac{4}{3}}{\pi}}}{1 - v \cdot v}}}{\left(\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}\right) \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}\]

  8. Applied times-frac0.0

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{\frac{\frac{4}{3}}{\pi}}}{1}}{\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}} \cdot \frac{\frac{\sqrt{\frac{\frac{4}{3}}{\pi}}}{1 - v \cdot v}}{\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}}\]

  9. Simplified0.0

    \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{4}{3}}{\pi}}}{\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}} \cdot \frac{\frac{\sqrt{\frac{\frac{4}{3}}{\pi}}}{1 - v \cdot v}}{\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}\]

  10. Final simplification0.0

    \[\leadsto \frac{\sqrt{\frac{\frac{4}{3}}{\pi}}}{\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}} \cdot \sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}} \cdot \frac{\frac{\sqrt{\frac{\frac{4}{3}}{\pi}}}{1 - v \cdot v}}{\sqrt[3]{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}}\]

Runtime

Time bar (total: 51.5s)Debug logProfile

herbie shell --seed 2018348 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))