Average Error: 35.0 → 30.9
Time: 36.7s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 4.0607805810600363 \cdot 10^{-188}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]
\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 4.0607805810600363 \cdot 10^{-188}:\\
\;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \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 r18953181 = 1.0;
        double r18953182 = 2.0;
        double r18953183 = a;
        double r18953184 = r18953182 * r18953183;
        double r18953185 = r18953181 / r18953184;
        double r18953186 = g;
        double r18953187 = -r18953186;
        double r18953188 = r18953186 * r18953186;
        double r18953189 = h;
        double r18953190 = r18953189 * r18953189;
        double r18953191 = r18953188 - r18953190;
        double r18953192 = sqrt(r18953191);
        double r18953193 = r18953187 + r18953192;
        double r18953194 = r18953185 * r18953193;
        double r18953195 = cbrt(r18953194);
        double r18953196 = r18953187 - r18953192;
        double r18953197 = r18953185 * r18953196;
        double r18953198 = cbrt(r18953197);
        double r18953199 = r18953195 + r18953198;
        return r18953199;
}
↓
double f(double g, double h, double a) {
        double r18953200 = g;
        double r18953201 = 4.0607805810600363e-188;
        bool r18953202 = r18953200 <= r18953201;
        double r18953203 = -r18953200;
        double r18953204 = r18953200 * r18953200;
        double r18953205 = h;
        double r18953206 = r18953205 * r18953205;
        double r18953207 = r18953204 - r18953206;
        double r18953208 = sqrt(r18953207);
        double r18953209 = r18953203 - r18953208;
        double r18953210 = 1.0;
        double r18953211 = a;
        double r18953212 = 2.0;
        double r18953213 = r18953211 * r18953212;
        double r18953214 = r18953210 / r18953213;
        double r18953215 = r18953209 * r18953214;
        double r18953216 = cbrt(r18953215);
        double r18953217 = r18953203 - r18953200;
        double r18953218 = cbrt(r18953217);
        double r18953219 = cbrt(r18953213);
        double r18953220 = r18953218 / r18953219;
        double r18953221 = r18953216 + r18953220;
        double r18953222 = r18953203 + r18953208;
        double r18953223 = r18953214 * r18953222;
        double r18953224 = cbrt(r18953223);
        double r18953225 = cbrt(r18953209);
        double r18953226 = r18953225 / r18953219;
        double r18953227 = r18953224 + r18953226;
        double r18953228 = r18953202 ? r18953221 : r18953227;
        return r18953228;
}

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 < 4.0607805810600363e-188

    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

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

    4. Applied cbrt-div32.0

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

    5. Simplified32.0

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

    6. Taylor expanded around -inf 31.3

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

    7. Simplified31.3

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

    if 4.0607805810600363e-188 < g

    1. Initial program 34.3

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

    3. Applied associate-*l/34.3

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

    4. Applied cbrt-div30.5

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

    5. Simplified30.5

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

  4. Final simplification30.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 4.0607805810600363 \cdot 10^{-188}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a \cdot 2}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(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))))))))