Average Error: 36.0 → 31.8
Time: 9.5s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 6.30056245054299891 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + -1 \cdot g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}}\\ \end{array}\]
\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 6.30056245054299891 \cdot 10^{-159}:\\
\;\;\;\;\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 \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}}\\

\end{array}
double f(double g, double h, double a) {
        double r121186 = 1.0;
        double r121187 = 2.0;
        double r121188 = a;
        double r121189 = r121187 * r121188;
        double r121190 = r121186 / r121189;
        double r121191 = g;
        double r121192 = -r121191;
        double r121193 = r121191 * r121191;
        double r121194 = h;
        double r121195 = r121194 * r121194;
        double r121196 = r121193 - r121195;
        double r121197 = sqrt(r121196);
        double r121198 = r121192 + r121197;
        double r121199 = r121190 * r121198;
        double r121200 = cbrt(r121199);
        double r121201 = r121192 - r121197;
        double r121202 = r121190 * r121201;
        double r121203 = cbrt(r121202);
        double r121204 = r121200 + r121203;
        return r121204;
}
↓
double f(double g, double h, double a) {
        double r121205 = g;
        double r121206 = 6.300562450542999e-159;
        bool r121207 = r121205 <= r121206;
        double r121208 = 1.0;
        double r121209 = -r121205;
        double r121210 = -1.0;
        double r121211 = r121210 * r121205;
        double r121212 = r121209 + r121211;
        double r121213 = r121208 * r121212;
        double r121214 = cbrt(r121213);
        double r121215 = 2.0;
        double r121216 = a;
        double r121217 = r121215 * r121216;
        double r121218 = cbrt(r121217);
        double r121219 = r121214 / r121218;
        double r121220 = r121208 / r121217;
        double r121221 = r121205 * r121205;
        double r121222 = h;
        double r121223 = r121222 * r121222;
        double r121224 = r121221 - r121223;
        double r121225 = sqrt(r121224);
        double r121226 = r121209 - r121225;
        double r121227 = r121220 * r121226;
        double r121228 = cbrt(r121227);
        double r121229 = r121219 + r121228;
        double r121230 = cbrt(r121220);
        double r121231 = r121209 + r121225;
        double r121232 = cbrt(r121231);
        double r121233 = r121230 * r121232;
        double r121234 = cbrt(r121225);
        double r121235 = r121234 * r121234;
        double r121236 = r121235 * r121234;
        double r121237 = r121209 - r121236;
        double r121238 = cbrt(r121237);
        double r121239 = r121230 * r121238;
        double r121240 = r121233 + r121239;
        double r121241 = r121207 ? r121229 : r121240;
        return r121241;
}

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 < 6.300562450542999e-159

    1. Initial program 37.1

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

    3. Applied associate-*l/37.1

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

    4. Applied cbrt-div33.2

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

    5. Taylor expanded around -inf 32.0

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

    if 6.300562450542999e-159 < g

    1. Initial program 34.8

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

    3. Applied cbrt-prod34.8

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

    5. Applied cbrt-prod31.6

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

    7. Applied add-cube-cbrt31.6

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

  4. Final simplification31.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 6.30056245054299891 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + -1 \cdot g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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))))))))