Average Error: 35.0 → 31.6
Time: 54.8s
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g}}{\sqrt[3]{a \cdot 2}}\]

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 35.0

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

  3. Applied associate-*l/35.0

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

  4. Applied cbrt-div33.2

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

  5. Simplified33.2

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

  7. Applied associate-*l/33.2

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

  8. Applied cbrt-div31.6

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

  9. Simplified31.6

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

  10. Final simplification31.6

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

Runtime

Time bar (total: 54.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.631.629.42.20%
herbie shell --seed 2018263 
(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))))))))