Average Error: 36.1 → 31.7
Time: 24.3s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -8.483083017653631513010138080609923608314 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]
\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 -8.483083017653631513010138080609923608314 \cdot 10^{-164}:\\
\;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\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)}\\

\mathbf{else}:\\
\;\;\;\;\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 \sqrt[3]{\left(-g\right) - g}\\

\end{array}
double f(double g, double h, double a) {
        double r127120 = 1.0;
        double r127121 = 2.0;
        double r127122 = a;
        double r127123 = r127121 * r127122;
        double r127124 = r127120 / r127123;
        double r127125 = g;
        double r127126 = -r127125;
        double r127127 = r127125 * r127125;
        double r127128 = h;
        double r127129 = r127128 * r127128;
        double r127130 = r127127 - r127129;
        double r127131 = sqrt(r127130);
        double r127132 = r127126 + r127131;
        double r127133 = r127124 * r127132;
        double r127134 = cbrt(r127133);
        double r127135 = r127126 - r127131;
        double r127136 = r127124 * r127135;
        double r127137 = cbrt(r127136);
        double r127138 = r127134 + r127137;
        return r127138;
}
↓
double f(double g, double h, double a) {
        double r127139 = g;
        double r127140 = -8.483083017653632e-164;
        bool r127141 = r127139 <= r127140;
        double r127142 = 1.0;
        double r127143 = r127139 * r127139;
        double r127144 = h;
        double r127145 = r127144 * r127144;
        double r127146 = r127143 - r127145;
        double r127147 = sqrt(r127146);
        double r127148 = r127147 - r127139;
        double r127149 = r127142 * r127148;
        double r127150 = cbrt(r127149);
        double r127151 = 2.0;
        double r127152 = a;
        double r127153 = r127151 * r127152;
        double r127154 = cbrt(r127153);
        double r127155 = r127150 / r127154;
        double r127156 = r127142 / r127153;
        double r127157 = -r127139;
        double r127158 = r127157 - r127147;
        double r127159 = r127156 * r127158;
        double r127160 = cbrt(r127159);
        double r127161 = r127155 + r127160;
        double r127162 = r127157 + r127147;
        double r127163 = r127156 * r127162;
        double r127164 = cbrt(r127163);
        double r127165 = cbrt(r127156);
        double r127166 = r127157 - r127139;
        double r127167 = cbrt(r127166);
        double r127168 = r127165 * r127167;
        double r127169 = r127164 + r127168;
        double r127170 = r127141 ? r127161 : r127169;
        return r127170;
}

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 < -8.483083017653632e-164

    1. Initial program 35.1

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

    3. Applied associate-*l/35.1

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

    4. Applied cbrt-div31.1

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

    5. Simplified31.1

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

    if -8.483083017653632e-164 < g

    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. Using strategy rm

    3. Applied cbrt-prod33.3

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

    4. Taylor expanded around inf 32.2

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

  4. Final simplification31.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -8.483083017653631513010138080609923608314 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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))))))))