Average Error: 34.7 → 31.1
Time: 2.7m
Precision: 64
Internal Precision: 128
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\frac{\sqrt{\frac{1}{2}}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \left(\sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\frac{\sqrt[3]{\sqrt[3]{a}}}{\sqrt{\frac{1}{2}}}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}}}\right) + \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{\frac{a}{\frac{1}{2}}}}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Initial program 34.7

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

  2. Simplified34.7

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

  4. Applied cbrt-div32.7

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

  6. Applied add-sqr-sqrt32.8

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

  7. Applied add-cube-cbrt32.8

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

  8. Applied times-frac32.8

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

  9. Applied *-un-lft-identity32.8

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

  10. Applied *-un-lft-identity32.8

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

  11. Applied distribute-lft-out--32.8

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

  12. Applied times-frac32.8

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

  13. Applied cbrt-prod31.0

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

  14. Simplified31.0

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

  16. Applied *-un-lft-identity31.0

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

  17. Applied add-cube-cbrt31.0

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

  18. Applied times-frac31.0

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

  19. Applied *-un-lft-identity31.0

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

  20. Applied *-un-lft-identity31.0

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

  21. Applied distribute-rgt-neg-in31.0

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

  22. Applied distribute-lft-out--31.0

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

  23. Applied times-frac31.0

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

  24. Applied cbrt-prod31.1

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

  25. Simplified31.1

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

  26. Final simplification31.1

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

Reproduce

herbie shell --seed 2019072 
(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))))))))