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Derivation

1. Initial program 34.6

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

2. Initial simplification34.6

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

4. Applied div-inv34.6

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

5. Applied cbrt-prod32.7

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

7. Applied associate-*l/32.7

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

8. Applied cbrt-div31.0

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

10. Applied add-cube-cbrt31.0

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

11. Final simplification31.0

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

Reproduce

herbie shell --seed 2018362 
(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))))))))

Details

Time bar (total: 27.6s)Debug log