Error

Derivation

1. Initial program 34.9

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

2. Simplified34.9

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

4. Applied div-inv34.9

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

5. Applied *-un-lft-identity34.9

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

6. Applied *-un-lft-identity34.9

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

7. Applied distribute-lft-out--34.9

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

8. Applied times-frac34.9

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

9. Applied cbrt-prod33.0

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

11. Applied add-cube-cbrt33.1

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

12. Applied add-cube-cbrt33.1

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

13. Applied times-frac33.1

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

14. Applied cbrt-prod31.5

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

15. Simplified31.5

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

17. Applied add-sqr-sqrt31.5

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

18. Applied sqrt-prod31.5

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

19. Final simplification31.5

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

Reproduce

herbie shell --seed 2019094 
(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))))))))