Average Error: 35.2 → 35.6
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{a \cdot 2}} + \sqrt[3]{\frac{\frac{-1}{2}}{a} \cdot \left(\sqrt{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} + g} \cdot \sqrt{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} + g}\right)}\]

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 35.2

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

  2. Initial simplification35.1

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

  4. Applied add-sqr-sqrt35.6

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

  5. Final simplification35.6

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018251 
(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))))))))