Average Error: 36.5 → 32.0
Time: 37.7s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -8.764066339061834934650093734982364279507 \cdot 10^{-157}:\\ \;\;\;\;\sqrt[3]{\left(-g\right) + \sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}} + \frac{\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot 1}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right) \cdot \frac{1}{2 \cdot a}}\\ \end{array}\]
\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 -8.764066339061834934650093734982364279507 \cdot 10^{-157}:\\
\;\;\;\;\sqrt[3]{\left(-g\right) + \sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}} + \frac{\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot 1}}{\sqrt[3]{2 \cdot a}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r6170115 = 1.0;
        double r6170116 = 2.0;
        double r6170117 = a;
        double r6170118 = r6170116 * r6170117;
        double r6170119 = r6170115 / r6170118;
        double r6170120 = g;
        double r6170121 = -r6170120;
        double r6170122 = r6170120 * r6170120;
        double r6170123 = h;
        double r6170124 = r6170123 * r6170123;
        double r6170125 = r6170122 - r6170124;
        double r6170126 = sqrt(r6170125);
        double r6170127 = r6170121 + r6170126;
        double r6170128 = r6170119 * r6170127;
        double r6170129 = cbrt(r6170128);
        double r6170130 = r6170121 - r6170126;
        double r6170131 = r6170119 * r6170130;
        double r6170132 = cbrt(r6170131);
        double r6170133 = r6170129 + r6170132;
        return r6170133;
}
↓
double f(double g, double h, double a) {
        double r6170134 = g;
        double r6170135 = -8.764066339061835e-157;
        bool r6170136 = r6170134 <= r6170135;
        double r6170137 = -r6170134;
        double r6170138 = r6170134 * r6170134;
        double r6170139 = h;
        double r6170140 = r6170139 * r6170139;
        double r6170141 = r6170138 - r6170140;
        double r6170142 = sqrt(r6170141);
        double r6170143 = sqrt(r6170142);
        double r6170144 = r6170143 * r6170143;
        double r6170145 = r6170137 + r6170144;
        double r6170146 = cbrt(r6170145);
        double r6170147 = 1.0;
        double r6170148 = 2.0;
        double r6170149 = a;
        double r6170150 = r6170148 * r6170149;
        double r6170151 = r6170147 / r6170150;
        double r6170152 = cbrt(r6170151);
        double r6170153 = r6170146 * r6170152;
        double r6170154 = r6170137 - r6170142;
        double r6170155 = r6170154 * r6170147;
        double r6170156 = cbrt(r6170155);
        double r6170157 = cbrt(r6170150);
        double r6170158 = r6170156 / r6170157;
        double r6170159 = r6170153 + r6170158;
        double r6170160 = r6170137 - r6170134;
        double r6170161 = r6170147 * r6170160;
        double r6170162 = cbrt(r6170161);
        double r6170163 = r6170162 / r6170157;
        double r6170164 = r6170142 + r6170137;
        double r6170165 = r6170164 * r6170151;
        double r6170166 = cbrt(r6170165);
        double r6170167 = r6170163 + r6170166;
        double r6170168 = r6170136 ? r6170159 : r6170167;
        return r6170168;
}

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 < -8.764066339061835e-157

    1. Initial program 35.5

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

    3. Applied associate-*l/35.5

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

    4. Applied cbrt-div35.5

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

    6. Applied cbrt-prod32.1

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

    8. Applied add-sqr-sqrt32.1

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

    9. Applied sqrt-prod32.1

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

    if -8.764066339061835e-157 < g

    1. Initial program 37.3

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

    3. Applied associate-*l/37.3

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

    4. Applied cbrt-div33.3

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

    5. Taylor expanded around inf 31.9

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

  4. Final simplification32.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -8.764066339061834934650093734982364279507 \cdot 10^{-157}:\\ \;\;\;\;\sqrt[3]{\left(-g\right) + \sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}} + \frac{\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot 1}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right) \cdot \frac{1}{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))