Average Error: 35.6 → 35.7
Time: 34.4s
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\left(\sqrt[3]{\left(-g\right) + \left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}}} \cdot \left(\sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}\right)\right) \cdot \frac{1}{a \cdot 2}}\]

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 35.6

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

  3. Applied add-cube-cbrt35.7

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

  5. Applied add-cube-cbrt35.7

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

  6. Applied sqrt-prod35.7

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

  7. Simplified35.7

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

  8. Final simplification35.7

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

Runtime

Time bar (total: 34.4s)Debug logProfile

herbie shell --seed 2018249 
(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))))))))