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Derivation

1. Split input into 2 regimes

2. if (/ 1 (* 2 a)) < 7.572626811676115e-270

   1. Initial program 35.3

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

   2. Simplified35.3

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

   4. Applied associate-*l/35.3

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

   5. Applied cbrt-div31.8

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

   if 7.572626811676115e-270 < (/ 1 (* 2 a))

   1. Initial program 33.7

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

   2. Simplified33.7

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

   4. Applied div-inv33.7

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

   5. Applied cbrt-prod30.5

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

   7. Applied add-sqr-sqrt30.5

      \[\leadsto \sqrt[3]{\color{blue}{\sqrt{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}}} - g} \cdot \sqrt[3]{\frac{1}{a \cdot 2}} + \sqrt[3]{\frac{\frac{-1}{2}}{a} \cdot \left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification31.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \le 7.572626811676115 \cdot 10^{-270}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{\left(g + h\right) \cdot \left(g - h\right)} - g}{a \cdot 2}} + \frac{\sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-1}{2}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2}} \cdot \sqrt[3]{\sqrt{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}} \cdot \sqrt{\sqrt{\left(g + h\right) \cdot \left(g - h\right)}} - g} + \sqrt[3]{\frac{\frac{-1}{2}}{a} \cdot \left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(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: 48.3s)Debug log