Average Error: 35.9 → 31.8
Time: 8.4s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 4.54756140803492154 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(-\left(g + g\right)\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{1 \cdot \frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 \cdot \frac{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}{2 \cdot a}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < 4.54756140803492154e-163

    1. Initial program 37.0

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

    2. Simplified37.0

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

    4. Applied associate-*r/37.0

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

    5. Applied cbrt-div33.3

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

    6. Simplified33.3

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

    7. Taylor expanded around -inf 32.4

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

    8. Simplified32.4

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

    if 4.54756140803492154e-163 < g

    1. Initial program 34.8

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

    2. Simplified34.8

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

    4. Applied associate-*r/34.8

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

    5. Applied cbrt-div31.2

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

    7. Applied add-cube-cbrt31.2

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

    8. Applied sqrt-prod31.2

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

    9. Simplified31.2

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

  4. Final simplification31.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 4.54756140803492154 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(-\left(g + g\right)\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{1 \cdot \frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 \cdot \frac{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}{2 \cdot a}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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))))))))