Average Error: 35.3 → 35.9
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\frac{\frac{-1}{2}}{a} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g\right)} + \sqrt[3]{\frac{\sqrt{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g} \cdot \sqrt{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{a \cdot 2}}\]

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 35.3

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

  2. Initial simplification35.3

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

  4. Applied add-sqr-sqrt35.9

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

  5. Final simplification35.9

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(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))))))))