Average Error: 35.6 → 32.5
Time: 54.1s
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g} \cdot \sqrt[3]{\frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\left(\sqrt{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} + g} \cdot \sqrt{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} + g}\right) \cdot \frac{-1}{2}}}{\sqrt[3]{a}}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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 div-inv35.6

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

  5. Applied cbrt-prod33.6

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

  7. Applied associate-*l/33.6

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

  8. Applied cbrt-div31.9

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

  10. Applied add-sqr-sqrt32.5

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

  11. Final simplification32.5

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

Runtime

Time bar (total: 54.1s)Debug logProfile

herbie shell --seed 2018277 
(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))))))))