Average Error: 35.2 → 30.8
Time: 1.0m
Precision: 64
Internal Precision: 128
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -2.7316583182296233 \cdot 10^{-215}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(\sqrt{\sqrt{\left(h + g\right) \cdot \left(g - h\right)}} + \sqrt{-g}\right) \cdot \left(\sqrt{-g} - \sqrt{\sqrt{\left(h + g\right) \cdot \left(g - h\right)}}\right)}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{elif}\;g \le 4.502549453381009 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{h \cdot h}}{\sqrt[3]{\left(2 \cdot a\right) \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}\\ \end{array}\]

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. Split input into 3 regimes

  2. if g < -2.7316583182296233e-215

    1. Initial program 34.4

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

    3. Applied associate-*l/34.4

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

    4. Applied cbrt-div34.3

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

    5. Simplified34.3

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

    7. Applied associate-*l/34.3

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

    8. Applied cbrt-div30.4

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

    9. Simplified30.4

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

    11. Applied add-sqr-sqrt38.3

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

    12. Applied add-sqr-sqrt30.4

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

    13. Applied difference-of-squares30.4

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

    if -2.7316583182296233e-215 < g < 4.502549453381009e-58

    1. Initial program 32.8

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

    3. Applied associate-*l/32.8

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

    4. Applied cbrt-div26.8

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

    5. Simplified26.8

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

    6. Taylor expanded around inf 21.0

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

    if 4.502549453381009e-58 < g

    1. Initial program 36.6

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

    3. Applied associate-*l/36.7

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

    4. Applied cbrt-div33.7

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

    5. Simplified33.7

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

    7. Applied flip-+33.7

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

    8. Applied frac-times33.8

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

    9. Applied cbrt-div33.7

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

    10. Simplified33.3

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

  4. Final simplification30.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -2.7316583182296233 \cdot 10^{-215}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(\sqrt{\sqrt{\left(h + g\right) \cdot \left(g - h\right)}} + \sqrt{-g}\right) \cdot \left(\sqrt{-g} - \sqrt{\sqrt{\left(h + g\right) \cdot \left(g - h\right)}}\right)}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{elif}\;g \le 4.502549453381009 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{h \cdot h}}{\sqrt[3]{\left(2 \cdot a\right) \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019030 +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))))))))