Average Error: 36.0 → 32.3
Time: 47.6s
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\frac{\sqrt[3]{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g} \cdot \left(\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g} \cdot \sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\]

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 36.0

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

  3. Applied cbrt-prod34.0

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

  4. Simplified34.0

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

  6. Applied associate-*l/34.0

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

  7. Applied cbrt-div32.3

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

  8. Simplified32.3

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

  10. Applied add-cube-cbrt32.3

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

  11. Final simplification32.3

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

Runtime

Time bar (total: 47.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes32.332.330.12.20%
herbie shell --seed 2018286 +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))))))))