Average Error: 35.8 → 32.0
Time: 32.6s
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\frac{\frac{-1}{2}}{a}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} + g} \cdot \sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} + g}\right) \cdot \sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} + g}} + \frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{2 \cdot a}}\]

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 35.8

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

  2. Initial simplification35.7

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

  4. Applied cbrt-div33.7

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

  6. Applied cbrt-prod32.0

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

  8. Applied add-cube-cbrt32.0

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

  9. Final simplification32.0

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

Runtime

Time bar (total: 32.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes32.032.030.31.70%
herbie shell --seed 2018285 
(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))))))))