Average Error: 35.1 → 30.9
Time: 39.9s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -5.110014296461794 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}} - g}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{elif}\;g \le 6.420654691865451 \cdot 10^{-40}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot g - \left(g \cdot g - h \cdot h\right)}}{\sqrt[3]{\left(a \cdot 2\right) \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]
\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 -5.110014296461794 \cdot 10^{-199}:\\
\;\;\;\;\frac{\sqrt[3]{\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}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\\

\mathbf{elif}\;g \le 6.420654691865451 \cdot 10^{-40}:\\
\;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{g \cdot g - \left(g \cdot g - h \cdot h\right)}}{\sqrt[3]{\left(a \cdot 2\right) \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\\

\end{array}
double f(double g, double h, double a) {
        double r5478058 = 1.0;
        double r5478059 = 2.0;
        double r5478060 = a;
        double r5478061 = r5478059 * r5478060;
        double r5478062 = r5478058 / r5478061;
        double r5478063 = g;
        double r5478064 = -r5478063;
        double r5478065 = r5478063 * r5478063;
        double r5478066 = h;
        double r5478067 = r5478066 * r5478066;
        double r5478068 = r5478065 - r5478067;
        double r5478069 = sqrt(r5478068);
        double r5478070 = r5478064 + r5478069;
        double r5478071 = r5478062 * r5478070;
        double r5478072 = cbrt(r5478071);
        double r5478073 = r5478064 - r5478069;
        double r5478074 = r5478062 * r5478073;
        double r5478075 = cbrt(r5478074);
        double r5478076 = r5478072 + r5478075;
        return r5478076;
}
↓
double f(double g, double h, double a) {
        double r5478077 = g;
        double r5478078 = -5.110014296461794e-199;
        bool r5478079 = r5478077 <= r5478078;
        double r5478080 = r5478077 * r5478077;
        double r5478081 = h;
        double r5478082 = r5478081 * r5478081;
        double r5478083 = r5478080 - r5478082;
        double r5478084 = sqrt(r5478083);
        double r5478085 = cbrt(r5478084);
        double r5478086 = r5478085 * r5478085;
        double r5478087 = r5478086 * r5478085;
        double r5478088 = r5478087 - r5478077;
        double r5478089 = cbrt(r5478088);
        double r5478090 = a;
        double r5478091 = 2.0;
        double r5478092 = r5478090 * r5478091;
        double r5478093 = cbrt(r5478092);
        double r5478094 = r5478089 / r5478093;
        double r5478095 = -r5478077;
        double r5478096 = r5478095 - r5478084;
        double r5478097 = cbrt(r5478096);
        double r5478098 = r5478097 / r5478093;
        double r5478099 = r5478094 + r5478098;
        double r5478100 = 6.420654691865451e-40;
        bool r5478101 = r5478077 <= r5478100;
        double r5478102 = 1.0;
        double r5478103 = r5478102 / r5478092;
        double r5478104 = r5478084 + r5478095;
        double r5478105 = r5478103 * r5478104;
        double r5478106 = cbrt(r5478105);
        double r5478107 = r5478095 - r5478077;
        double r5478108 = cbrt(r5478107);
        double r5478109 = r5478108 / r5478093;
        double r5478110 = r5478106 + r5478109;
        double r5478111 = r5478080 - r5478083;
        double r5478112 = cbrt(r5478111);
        double r5478113 = r5478092 * r5478096;
        double r5478114 = cbrt(r5478113);
        double r5478115 = r5478112 / r5478114;
        double r5478116 = r5478115 + r5478098;
        double r5478117 = r5478101 ? r5478110 : r5478116;
        double r5478118 = r5478079 ? r5478099 : r5478117;
        return r5478118;
}

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. Split input into 3 regimes

  2. if g < -5.110014296461794e-199

    1. Initial program 33.9

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

    3. Applied associate-*l/33.9

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

    4. Applied cbrt-div33.8

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

    6. Applied associate-*l/33.8

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

    7. Applied cbrt-div30.6

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

    8. Simplified30.6

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

    10. Applied add-cube-cbrt30.6

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

    if -5.110014296461794e-199 < g < 6.420654691865451e-40

    1. Initial program 28.8

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

    3. Applied associate-*l/28.8

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

    4. Applied cbrt-div23.1

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

    5. Taylor expanded around inf 18.0

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

    if 6.420654691865451e-40 < g

    1. Initial program 38.2

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

    3. Applied associate-*l/38.2

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

    4. Applied cbrt-div34.9

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

    6. Applied flip-+34.9

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

    7. Applied frac-times34.9

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

    8. Applied cbrt-div34.8

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

    9. Simplified34.8

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

  4. Final simplification30.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -5.110014296461794 \cdot 10^{-199}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}} - g}}{\sqrt[3]{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{elif}\;g \le 6.420654691865451 \cdot 10^{-40}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot g - \left(g \cdot g - h \cdot h\right)}}{\sqrt[3]{\left(a \cdot 2\right) \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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))))))))