Average Error: 36.4 → 31.9
Time: 37.4s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -2.422345052641111360634811561319161541182 \cdot 10^{-160}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}} - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]
\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 -2.422345052641111360634811561319161541182 \cdot 10^{-160}:\\
\;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\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}} - g} + \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(\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 r117209 = 1.0;
        double r117210 = 2.0;
        double r117211 = a;
        double r117212 = r117210 * r117211;
        double r117213 = r117209 / r117212;
        double r117214 = g;
        double r117215 = -r117214;
        double r117216 = r117214 * r117214;
        double r117217 = h;
        double r117218 = r117217 * r117217;
        double r117219 = r117216 - r117218;
        double r117220 = sqrt(r117219);
        double r117221 = r117215 + r117220;
        double r117222 = r117213 * r117221;
        double r117223 = cbrt(r117222);
        double r117224 = r117215 - r117220;
        double r117225 = r117213 * r117224;
        double r117226 = cbrt(r117225);
        double r117227 = r117223 + r117226;
        return r117227;
}
↓
double f(double g, double h, double a) {
        double r117228 = g;
        double r117229 = -2.4223450526411114e-160;
        bool r117230 = r117228 <= r117229;
        double r117231 = 1.0;
        double r117232 = 2.0;
        double r117233 = a;
        double r117234 = r117232 * r117233;
        double r117235 = r117231 / r117234;
        double r117236 = cbrt(r117235);
        double r117237 = r117228 * r117228;
        double r117238 = h;
        double r117239 = r117238 * r117238;
        double r117240 = r117237 - r117239;
        double r117241 = sqrt(r117240);
        double r117242 = cbrt(r117241);
        double r117243 = r117242 * r117242;
        double r117244 = r117243 * r117242;
        double r117245 = r117244 - r117228;
        double r117246 = cbrt(r117245);
        double r117247 = r117236 * r117246;
        double r117248 = -r117228;
        double r117249 = r117248 - r117241;
        double r117250 = r117235 * r117249;
        double r117251 = cbrt(r117250);
        double r117252 = r117247 + r117251;
        double r117253 = r117241 - r117228;
        double r117254 = r117235 * r117253;
        double r117255 = cbrt(r117254);
        double r117256 = r117248 - r117228;
        double r117257 = cbrt(r117256);
        double r117258 = r117236 * r117257;
        double r117259 = r117255 + r117258;
        double r117260 = r117230 ? r117252 : r117259;
        return r117260;
}

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 < -2.4223450526411114e-160

    1. Initial program 35.0

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

    2. Simplified35.0

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

    4. Applied cbrt-prod31.2

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

    6. Applied add-cube-cbrt31.2

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

    if -2.4223450526411114e-160 < g

    1. Initial program 37.6

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

    2. Simplified37.6

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

    4. Applied cbrt-prod33.8

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

    5. Taylor expanded around inf 32.5

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

  4. Final simplification31.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -2.422345052641111360634811561319161541182 \cdot 10^{-160}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}} - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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))))))))