Average Error: 35.2 → 31.2
Time: 2.9m
Precision: 64
Internal Precision: 128
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -6.028835061410279 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{\frac{-1}{2}}{a}} \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{2 \cdot g} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}} + \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a \cdot 2}}\\ \end{array}\]

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < -6.028835061410279e-172

    1. Initial program 34.1

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

    2. Simplified34.1

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

    4. Applied *-un-lft-identity34.1

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

    5. Applied *-un-lft-identity34.1

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

    6. Applied distribute-lft-out34.1

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

    7. Applied associate-*r*34.1

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

    8. Applied cbrt-prod34.1

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

    9. Simplified34.1

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

    11. Applied cbrt-div30.7

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

    12. Taylor expanded around -inf 30.5

      \[\leadsto \frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{\frac{-1}{2}}{a}} \cdot \sqrt[3]{\color{blue}{0}}\]

    if -6.028835061410279e-172 < g

    1. Initial program 36.3

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

    2. Simplified36.3

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

    4. Applied *-un-lft-identity36.3

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

    5. Applied *-un-lft-identity36.3

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

    6. Applied distribute-lft-out36.3

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

    7. Applied associate-*r*36.3

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

    8. Applied cbrt-prod32.5

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

    9. Simplified32.5

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

    10. Taylor expanded around inf 31.8

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

  4. Final simplification31.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -6.028835061410279 \cdot 10^{-172}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{a \cdot 2}} + \sqrt[3]{\frac{\frac{-1}{2}}{a}} \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{2 \cdot g} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}} + \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 +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))))))))