Average Error: 1.0 → 0.0
Time: 2.0m
Precision: 64
Internal Precision: 320
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\left|\sqrt[3]{(\left(v \cdot v\right) \cdot \left(-6\right) + 2)_*}\right|} \cdot \frac{\frac{\frac{4}{\pi \cdot 3}}{1 - v \cdot v}}{\sqrt{\sqrt[3]{(\left(v \cdot v\right) \cdot \left(-6\right) + 2)_*}}}\]

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

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied sqrt-prod0.0

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

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

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

  7. Applied times-frac0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))