Average Error: 35.6 → 31.3
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 4.0812065297008384 \cdot 10^{-202}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot 0\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if g < 4.0812065297008384e-202

    1. Initial program 35.7

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

    3. Applied cbrt-prod31.7

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

    4. Simplified31.7

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

    5. Taylor expanded around -inf 31.2

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

    6. Simplified31.2

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

    if 4.0812065297008384e-202 < g

    1. Initial program 35.6

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

    3. Applied cbrt-prod35.5

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

    4. Simplified35.5

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

    6. Applied associate-*l/35.5

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

    7. Applied cbrt-div31.6

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

    8. Simplified31.6

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

    9. Taylor expanded around inf 31.3

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

  4. Final simplification31.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 4.0812065297008384 \cdot 10^{-202}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{\frac{1}{2}}{a}} \cdot 0\\ \end{array}\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018249 +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))))))))