Average Error: 35.4 → 31.3
Time: 7.6s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -2.9415560676817554 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{1 \cdot \frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 \cdot \frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < -2.9415560676817554e-173

    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. Simplified34.6

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

    4. Applied associate-*r/34.6

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

    5. Applied cbrt-div30.7

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

    7. Applied add-cube-cbrt30.7

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

    if -2.9415560676817554e-173 < g

    1. Initial program 36.2

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

    2. Simplified36.2

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

    4. Applied associate-*r/36.2

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

    5. Applied cbrt-div32.7

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

    6. Taylor expanded around inf 32.0

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

  4. Final simplification31.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -2.9415560676817554 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{1 \cdot \frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 \cdot \frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (neg g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (neg g) (sqrt (- (* g g) (* h h))))))))