Average Error: 35.4 → 31.5
Time: 9.1s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 3.0797478610904441 \cdot 10^{-234}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) \cdot \left(-g\right) - \sqrt{g \cdot g - h \cdot h} \cdot \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g + h} \cdot \sqrt{g - h}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]
\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 3.0797478610904441 \cdot 10^{-234}:\\
\;\;\;\;\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]{\frac{1}{2 \cdot a} \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(-g\right) + \sqrt{g \cdot g - h \cdot h}}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r176004 = 1.0;
        double r176005 = 2.0;
        double r176006 = a;
        double r176007 = r176005 * r176006;
        double r176008 = r176004 / r176007;
        double r176009 = g;
        double r176010 = -r176009;
        double r176011 = r176009 * r176009;
        double r176012 = h;
        double r176013 = r176012 * r176012;
        double r176014 = r176011 - r176013;
        double r176015 = sqrt(r176014);
        double r176016 = r176010 + r176015;
        double r176017 = r176008 * r176016;
        double r176018 = cbrt(r176017);
        double r176019 = r176010 - r176015;
        double r176020 = r176008 * r176019;
        double r176021 = cbrt(r176020);
        double r176022 = r176018 + r176021;
        return r176022;
}
↓
double f(double g, double h, double a) {
        double r176023 = g;
        double r176024 = 3.079747861090444e-234;
        bool r176025 = r176023 <= r176024;
        double r176026 = 1.0;
        double r176027 = -r176023;
        double r176028 = r176023 * r176023;
        double r176029 = h;
        double r176030 = r176029 * r176029;
        double r176031 = r176028 - r176030;
        double r176032 = sqrt(r176031);
        double r176033 = r176027 + r176032;
        double r176034 = r176026 * r176033;
        double r176035 = cbrt(r176034);
        double r176036 = 2.0;
        double r176037 = a;
        double r176038 = r176036 * r176037;
        double r176039 = cbrt(r176038);
        double r176040 = r176035 / r176039;
        double r176041 = r176026 / r176038;
        double r176042 = r176027 * r176027;
        double r176043 = r176032 * r176032;
        double r176044 = r176042 - r176043;
        double r176045 = r176041 * r176044;
        double r176046 = cbrt(r176045);
        double r176047 = cbrt(r176033);
        double r176048 = r176046 / r176047;
        double r176049 = r176040 + r176048;
        double r176050 = r176041 * r176033;
        double r176051 = cbrt(r176050);
        double r176052 = r176023 + r176029;
        double r176053 = sqrt(r176052);
        double r176054 = r176023 - r176029;
        double r176055 = sqrt(r176054);
        double r176056 = r176053 * r176055;
        double r176057 = r176027 - r176056;
        double r176058 = r176026 * r176057;
        double r176059 = cbrt(r176058);
        double r176060 = r176059 / r176039;
        double r176061 = r176051 + r176060;
        double r176062 = r176025 ? r176049 : r176061;
        return r176062;
}

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 2 regimes

  2. if g < 3.079747861090444e-234

    1. Initial program 35.9

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

    3. Applied associate-*l/35.9

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

    4. Applied cbrt-div32.3

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

    6. Applied flip--32.1

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

    7. Applied associate-*r/32.2

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

    8. Applied cbrt-div32.2

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

    if 3.079747861090444e-234 < g

    1. Initial program 34.9

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

    3. Applied associate-*l/34.9

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

    4. Applied cbrt-div31.4

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

    6. Applied difference-of-squares31.4

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

    7. Applied sqrt-prod30.8

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

  4. Final simplification31.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 3.0797478610904441 \cdot 10^{-234}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) \cdot \left(-g\right) - \sqrt{g \cdot g - h \cdot h} \cdot \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g + h} \cdot \sqrt{g - h}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))