Average Error: 35.4 → 35.6
Time: 2.2m
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{a \cdot 2}} + \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}}{\frac{a \cdot 2}{\sqrt[3]{-\left(g + g\right)}}}}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 35.4

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

  2. Applied simplify35.4

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

  4. Applied add-cube-cbrt35.5

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

  6. Applied add-cube-cbrt35.5

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

  7. Taylor expanded around inf 35.6

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

  8. Applied simplify35.6

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018296 
(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))))))))