Average Error: 35.0 → 31.1
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 2.302731106712237 \cdot 10^{-203}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{\frac{-1}{2}}{a}} + \sqrt[3]{\frac{\left(-g\right) - g}{2}} \cdot \sqrt[3]{\frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{-1}{2} \cdot \left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{a \cdot 2}}\\ \end{array}\]

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 2 regimes

  2. if g < 2.302731106712237e-203

    1. Initial program 35.3

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

    2. Initial simplification35.3

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

    4. Applied *-un-lft-identity35.3

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

    5. Applied times-frac35.3

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

    6. Applied cbrt-prod31.6

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

    7. Taylor expanded around -inf 31.5

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

    8. Simplified31.5

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

    if 2.302731106712237e-203 < g

    1. Initial program 34.6

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

    2. Initial simplification34.6

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

    4. Applied associate-*l/34.6

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

    5. Applied cbrt-div30.7

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

  4. Final simplification31.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 2.302731106712237 \cdot 10^{-203}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{\frac{-1}{2}}{a}} + \sqrt[3]{\frac{\left(-g\right) - g}{2}} \cdot \sqrt[3]{\frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{-1}{2} \cdot \left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{a \cdot 2}}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018336 
(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))))))))