Average Error: 35.2 → 31.7
Time: 26.8s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g}}{\frac{\sqrt[3]{a}}{\sqrt[3]{\frac{-1}{2}}}} + \frac{\sqrt[3]{\frac{1}{2} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right)}}{\sqrt[3]{a}}\]
\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)}
↓
\frac{\sqrt[3]{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g}}{\frac{\sqrt[3]{a}}{\sqrt[3]{\frac{-1}{2}}}} + \frac{\sqrt[3]{\frac{1}{2} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right)}}{\sqrt[3]{a}}
double f(double g, double h, double a) {
        double r5553938 = 1.0;
        double r5553939 = 2.0;
        double r5553940 = a;
        double r5553941 = r5553939 * r5553940;
        double r5553942 = r5553938 / r5553941;
        double r5553943 = g;
        double r5553944 = -r5553943;
        double r5553945 = r5553943 * r5553943;
        double r5553946 = h;
        double r5553947 = r5553946 * r5553946;
        double r5553948 = r5553945 - r5553947;
        double r5553949 = sqrt(r5553948);
        double r5553950 = r5553944 + r5553949;
        double r5553951 = r5553942 * r5553950;
        double r5553952 = cbrt(r5553951);
        double r5553953 = r5553944 - r5553949;
        double r5553954 = r5553942 * r5553953;
        double r5553955 = cbrt(r5553954);
        double r5553956 = r5553952 + r5553955;
        return r5553956;
}
↓
double f(double g, double h, double a) {
        double r5553957 = g;
        double r5553958 = h;
        double r5553959 = r5553957 - r5553958;
        double r5553960 = r5553958 + r5553957;
        double r5553961 = r5553959 * r5553960;
        double r5553962 = sqrt(r5553961);
        double r5553963 = r5553962 + r5553957;
        double r5553964 = cbrt(r5553963);
        double r5553965 = a;
        double r5553966 = cbrt(r5553965);
        double r5553967 = -0.5;
        double r5553968 = cbrt(r5553967);
        double r5553969 = r5553966 / r5553968;
        double r5553970 = r5553964 / r5553969;
        double r5553971 = 0.5;
        double r5553972 = r5553962 - r5553957;
        double r5553973 = r5553971 * r5553972;
        double r5553974 = cbrt(r5553973);
        double r5553975 = r5553974 / r5553966;
        double r5553976 = r5553970 + r5553975;
        return r5553976;
}

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. Initial program 35.2

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

  2. Simplified35.2

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

  4. Applied associate-*l/35.2

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

  5. Applied cbrt-div33.4

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

  7. Applied add-cbrt-cube33.4

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

  8. Simplified33.4

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

  10. Applied cbrt-div31.6

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

  12. Applied cbrt-div31.7

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

  13. Final simplification31.7

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

Reproduce

herbie shell --seed 2019164 +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))))))))