Error

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. Simplified0.0

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

4. Applied add-cube-cbrt1.0

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

5. Applied sqrt-prod1.0

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

6. Applied add-sqr-sqrt0.0

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

7. Applied add-cube-cbrt0.0

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

8. Applied times-frac0.0

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

9. Applied times-frac0.0

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

10. Simplified0.0

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

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

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

13. Applied div-inv1.0

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

14. Applied times-frac0.0

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

15. Simplified0.0

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

16. Final simplification0.0

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

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))