Average Error: 35.0 → 31.0
Time: 1.4m
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -3.1207087438865974 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \left(-g\right)}}{\sqrt[3]{\frac{a}{\frac{1}{2}}}} + \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{\frac{a}{\frac{1}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{\left(-g\right) - g}{\frac{\sqrt[3]{a}}{\sqrt[3]{\frac{1}{2}}}}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < -3.1207087438865974e-164

    1. Initial program 33.7

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

    2. Simplified33.7

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

    4. Applied cbrt-div33.6

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

    6. Applied cbrt-div30.3

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

    7. Taylor expanded around -inf 30.2

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

    8. Simplified30.2

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

    if -3.1207087438865974e-164 < g

    1. Initial program 36.3

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

    2. Simplified36.3

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

    4. Applied add-cube-cbrt36.3

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

    5. Applied add-cube-cbrt36.4

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

    6. Applied times-frac36.4

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

    7. Applied *-un-lft-identity36.4

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

    8. Applied *-un-lft-identity36.4

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

    9. Applied distribute-lft-out--36.4

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

    10. Applied times-frac36.4

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

    11. Applied cbrt-prod32.7

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

    12. Simplified32.7

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

    13. Taylor expanded around inf 31.8

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{\left(-g\right) - \color{blue}{g}}{\frac{\sqrt[3]{a}}{\sqrt[3]{\frac{1}{2}}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification31.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -3.1207087438865974 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \left(-g\right)}}{\sqrt[3]{\frac{a}{\frac{1}{2}}}} + \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{\frac{a}{\frac{1}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{\left(-g\right) - g}{\frac{\sqrt[3]{a}}{\sqrt[3]{\frac{1}{2}}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(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))))))))