Error

Derivation

1. Initial program 35.1

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

2. Simplified35.1

   \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\frac{a}{\frac{1}{2}}}}}\]
3. Using strategy rm

4. Applied div-inv35.1

   \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{\frac{a}{\frac{1}{2}}}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\frac{a}{\frac{1}{2}}}}\]

5. Applied cbrt-prod33.1

   \[\leadsto \color{blue}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\frac{a}{\frac{1}{2}}}}\]
6. Using strategy rm

7. Applied add-cube-cbrt33.2

   \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\color{blue}{\left(\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}\right) \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}}}\]

8. Applied *-un-lft-identity33.2

   \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{\left(-g\right) - \color{blue}{1 \cdot \sqrt{g \cdot g - h \cdot h}}}{\left(\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}\right) \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}}\]

9. Applied *-un-lft-identity33.2

   \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{\color{blue}{1 \cdot \left(-g\right)} - 1 \cdot \sqrt{g \cdot g - h \cdot h}}{\left(\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}\right) \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}}\]

10. Applied distribute-lft-out--33.2

    \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{\color{blue}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\left(\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}\right) \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}}\]

11. Applied times-frac33.2

    \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\color{blue}{\frac{1}{\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}} \cdot \frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\sqrt[3]{\frac{a}{\frac{1}{2}}}}}}\]

12. Applied cbrt-prod31.3

    \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \color{blue}{\sqrt[3]{\frac{1}{\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}} \cdot \sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\sqrt[3]{\frac{a}{\frac{1}{2}}}}}}\]
13. Using strategy rm

14. Applied div-inv31.3

    \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{1}{\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}} \cdot \sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\sqrt[3]{\color{blue}{a \cdot \frac{1}{\frac{1}{2}}}}}}\]

15. Applied cbrt-prod31.3

    \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{1}{\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}} \cdot \sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\color{blue}{\sqrt[3]{a} \cdot \sqrt[3]{\frac{1}{\frac{1}{2}}}}}}\]

16. Applied *-un-lft-identity31.3

    \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{1}{\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}} \cdot \sqrt[3]{\frac{\color{blue}{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{a} \cdot \sqrt[3]{\frac{1}{\frac{1}{2}}}}}\]

17. Applied times-frac31.3

    \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{1}{\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}} \cdot \sqrt[3]{\color{blue}{\frac{1}{\sqrt[3]{a}} \cdot \frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\sqrt[3]{\frac{1}{\frac{1}{2}}}}}}\]

18. Applied cbrt-prod31.4

    \[\leadsto \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{1}{\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}} \cdot \color{blue}{\left(\sqrt[3]{\frac{1}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\sqrt[3]{\frac{1}{\frac{1}{2}}}}}\right)}\]

19. Final simplification31.4

    \[\leadsto \sqrt[3]{\frac{1}{\sqrt[3]{\frac{a}{\frac{1}{2}}} \cdot \sqrt[3]{\frac{a}{\frac{1}{2}}}}} \cdot \left(\sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\sqrt[3]{2}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a}}}\right) + \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{\frac{a}{\frac{1}{2}}}}\]

Reproduce

herbie shell --seed 2019051 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  (+ (cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))