Average Error: 35.1 → 30.3
Time: 2.5m
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -6.249865151781854 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{\frac{g \cdot g + \left(h + g\right) \cdot \left(h - g\right)}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{2}} \cdot \sqrt[3]{\frac{1}{a}}\\ \mathbf{if}\;g \le 1.945276223793963 \cdot 10^{-134}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{\left(-g\right) - g}{2}} + \sqrt[3]{\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\left(-h\right) \cdot h}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} + g}}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}{2}}\\ \end{array}\]

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 < -6.249865151781854e-161

    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. Applied simplify34.4

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

    4. Applied *-un-lft-identity34.4

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

    5. Applied times-frac34.4

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

    6. Applied cbrt-prod34.4

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

    8. Applied cbrt-div30.8

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

    10. Applied flip--30.7

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

    11. Applied simplify30.6

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

    12. Applied simplify30.6

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

    if -6.249865151781854e-161 < g < 1.945276223793963e-134

    1. Initial program 50.6

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

    2. Applied simplify50.6

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

    4. Applied *-un-lft-identity50.6

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

    5. Applied times-frac50.6

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

    6. Applied cbrt-prod45.1

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

    7. Taylor expanded around inf 33.2

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

    if 1.945276223793963e-134 < g

    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. Applied simplify33.7

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

    4. Applied *-un-lft-identity33.7

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

    5. Applied times-frac33.7

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

    6. Applied cbrt-prod30.5

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

    8. Applied cbrt-div30.5

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

    10. Applied flip--30.5

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

    11. Applied simplify29.5

      \[\leadsto \frac{\sqrt[3]{\frac{\color{blue}{h \cdot \left(0 - h\right) + 0}}{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g}}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(g - h\right) \cdot \left(h + g\right)}}{2}}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify30.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;g \le -6.249865151781854 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{\frac{g \cdot g + \left(h + g\right) \cdot \left(h - g\right)}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}}{2}} \cdot \sqrt[3]{\frac{1}{a}}\\ \mathbf{if}\;g \le 1.945276223793963 \cdot 10^{-134}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{\left(-g\right) - g}{2}} + \sqrt[3]{\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{\left(-h\right) \cdot h}{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} + g}}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}{2}}\\ \end{array}}\]

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2020178 
(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))))))))