Average Error: 35.6 → 31.6
Time: 22.5s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 2.321285034107555607621474765303676741173 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + -1 \cdot g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]
\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 2.321285034107555607621474765303676741173 \cdot 10^{-166}:\\
\;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + -1 \cdot 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)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}\\

\end{array}
double f(double g, double h, double a) {
        double r113223 = 1.0;
        double r113224 = 2.0;
        double r113225 = a;
        double r113226 = r113224 * r113225;
        double r113227 = r113223 / r113226;
        double r113228 = g;
        double r113229 = -r113228;
        double r113230 = r113228 * r113228;
        double r113231 = h;
        double r113232 = r113231 * r113231;
        double r113233 = r113230 - r113232;
        double r113234 = sqrt(r113233);
        double r113235 = r113229 + r113234;
        double r113236 = r113227 * r113235;
        double r113237 = cbrt(r113236);
        double r113238 = r113229 - r113234;
        double r113239 = r113227 * r113238;
        double r113240 = cbrt(r113239);
        double r113241 = r113237 + r113240;
        return r113241;
}
↓
double f(double g, double h, double a) {
        double r113242 = g;
        double r113243 = 2.3212850341075556e-166;
        bool r113244 = r113242 <= r113243;
        double r113245 = 1.0;
        double r113246 = -r113242;
        double r113247 = -1.0;
        double r113248 = r113247 * r113242;
        double r113249 = r113246 + r113248;
        double r113250 = r113245 * r113249;
        double r113251 = cbrt(r113250);
        double r113252 = 2.0;
        double r113253 = a;
        double r113254 = r113252 * r113253;
        double r113255 = cbrt(r113254);
        double r113256 = r113251 / r113255;
        double r113257 = r113245 / r113254;
        double r113258 = r113242 * r113242;
        double r113259 = h;
        double r113260 = r113259 * r113259;
        double r113261 = r113258 - r113260;
        double r113262 = sqrt(r113261);
        double r113263 = r113246 - r113262;
        double r113264 = r113257 * r113263;
        double r113265 = cbrt(r113264);
        double r113266 = r113256 + r113265;
        double r113267 = r113246 + r113262;
        double r113268 = r113257 * r113267;
        double r113269 = cbrt(r113268);
        double r113270 = r113245 * r113263;
        double r113271 = cbrt(r113270);
        double r113272 = r113271 / r113255;
        double r113273 = r113269 + r113272;
        double r113274 = r113244 ? r113266 : r113273;
        return r113274;
}

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 < 2.3212850341075556e-166

    1. Initial program 36.6

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

    3. Applied associate-*l/36.6

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

    4. Applied cbrt-div33.3

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

    5. Taylor expanded around -inf 32.4

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

    if 2.3212850341075556e-166 < g

    1. Initial program 34.5

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

    3. Applied associate-*l/34.5

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

    4. Applied cbrt-div30.7

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

  4. Final simplification31.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 2.321285034107555607621474765303676741173 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + -1 \cdot g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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))))))))