Average Error: 36.2 → 31.6
Time: 37.3s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -3.421984391530689600469067551996753796002 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h\right)}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}} + \frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]
\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 -3.421984391530689600469067551996753796002 \cdot 10^{-165}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h\right)}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}} + \frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\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 r110210 = 1.0;
        double r110211 = 2.0;
        double r110212 = a;
        double r110213 = r110211 * r110212;
        double r110214 = r110210 / r110213;
        double r110215 = g;
        double r110216 = -r110215;
        double r110217 = r110215 * r110215;
        double r110218 = h;
        double r110219 = r110218 * r110218;
        double r110220 = r110217 - r110219;
        double r110221 = sqrt(r110220);
        double r110222 = r110216 + r110221;
        double r110223 = r110214 * r110222;
        double r110224 = cbrt(r110223);
        double r110225 = r110216 - r110221;
        double r110226 = r110214 * r110225;
        double r110227 = cbrt(r110226);
        double r110228 = r110224 + r110227;
        return r110228;
}
↓
double f(double g, double h, double a) {
        double r110229 = g;
        double r110230 = -3.4219843915306896e-165;
        bool r110231 = r110229 <= r110230;
        double r110232 = 1.0;
        double r110233 = 2.0;
        double r110234 = a;
        double r110235 = r110233 * r110234;
        double r110236 = r110232 / r110235;
        double r110237 = h;
        double r110238 = r110237 * r110237;
        double r110239 = r110236 * r110238;
        double r110240 = cbrt(r110239);
        double r110241 = r110229 * r110229;
        double r110242 = r110241 - r110238;
        double r110243 = sqrt(r110242);
        double r110244 = r110243 - r110229;
        double r110245 = cbrt(r110244);
        double r110246 = r110240 / r110245;
        double r110247 = r110232 * r110244;
        double r110248 = cbrt(r110247);
        double r110249 = cbrt(r110235);
        double r110250 = r110248 / r110249;
        double r110251 = r110246 + r110250;
        double r110252 = r110236 * r110244;
        double r110253 = cbrt(r110252);
        double r110254 = cbrt(r110236);
        double r110255 = -r110229;
        double r110256 = r110255 - r110229;
        double r110257 = cbrt(r110256);
        double r110258 = r110254 * r110257;
        double r110259 = r110253 + r110258;
        double r110260 = r110231 ? r110251 : r110259;
        return r110260;
}

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 < -3.4219843915306896e-165

    1. Initial program 35.3

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

    2. Simplified35.3

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

    4. Applied associate-*l/35.3

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

    5. Applied cbrt-div31.4

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

    7. Applied flip--31.3

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

    8. Applied associate-*r/31.4

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

    9. Applied cbrt-div31.4

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

    10. Simplified31.6

      \[\leadsto \frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}} + \frac{\color{blue}{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h + 0\right)}}}{\sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}}\]

    11. Simplified31.6

      \[\leadsto \frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h + 0\right)}}{\color{blue}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}}\]

    if -3.4219843915306896e-165 < g

    1. Initial program 36.9

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

    2. Simplified36.9

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

    4. Applied cbrt-prod33.0

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

    5. Taylor expanded around inf 31.7

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

  4. Final simplification31.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -3.421984391530689600469067551996753796002 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h\right)}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}} + \frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))))))))