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Derivation

1. Initial program 35.6

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

2. Initial simplification35.6

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

4. Applied add-cube-cbrt35.6

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

6. Applied add-cube-cbrt35.6

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

8. Applied add-cube-cbrt35.6

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

9. Final simplification35.6

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018250 
(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))))))))