Average Error: 35.7 → 31.8
Time: 9.5s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -2.3907293248482188 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]
\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 -2.3907293248482188 \cdot 10^{-163}:\\
\;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\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}}\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) - g\right)}}{\sqrt[3]{2 \cdot a}}\\

\end{array}
double f(double g, double h, double a) {
        double r168022 = 1.0;
        double r168023 = 2.0;
        double r168024 = a;
        double r168025 = r168023 * r168024;
        double r168026 = r168022 / r168025;
        double r168027 = g;
        double r168028 = -r168027;
        double r168029 = r168027 * r168027;
        double r168030 = h;
        double r168031 = r168030 * r168030;
        double r168032 = r168029 - r168031;
        double r168033 = sqrt(r168032);
        double r168034 = r168028 + r168033;
        double r168035 = r168026 * r168034;
        double r168036 = cbrt(r168035);
        double r168037 = r168028 - r168033;
        double r168038 = r168026 * r168037;
        double r168039 = cbrt(r168038);
        double r168040 = r168036 + r168039;
        return r168040;
}
↓
double f(double g, double h, double a) {
        double r168041 = g;
        double r168042 = -2.3907293248482188e-163;
        bool r168043 = r168041 <= r168042;
        double r168044 = 1.0;
        double r168045 = -r168041;
        double r168046 = r168041 * r168041;
        double r168047 = h;
        double r168048 = r168047 * r168047;
        double r168049 = r168046 - r168048;
        double r168050 = sqrt(r168049);
        double r168051 = cbrt(r168050);
        double r168052 = r168051 * r168051;
        double r168053 = r168052 * r168051;
        double r168054 = r168045 + r168053;
        double r168055 = r168044 * r168054;
        double r168056 = cbrt(r168055);
        double r168057 = 2.0;
        double r168058 = a;
        double r168059 = r168057 * r168058;
        double r168060 = cbrt(r168059);
        double r168061 = r168056 / r168060;
        double r168062 = r168044 / r168059;
        double r168063 = r168045 - r168050;
        double r168064 = r168062 * r168063;
        double r168065 = cbrt(r168064);
        double r168066 = r168061 + r168065;
        double r168067 = r168045 + r168050;
        double r168068 = r168062 * r168067;
        double r168069 = cbrt(r168068);
        double r168070 = r168045 - r168041;
        double r168071 = r168044 * r168070;
        double r168072 = cbrt(r168071);
        double r168073 = r168072 / r168060;
        double r168074 = r168069 + r168073;
        double r168075 = r168043 ? r168066 : r168074;
        return r168075;
}

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 < -2.3907293248482188e-163

    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. Using strategy rm

    3. Applied associate-*l/35.0

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

    4. Applied cbrt-div31.6

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

    6. Applied add-cube-cbrt31.6

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

    if -2.3907293248482188e-163 < g

    1. Initial program 36.3

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

    3. Applied associate-*l/36.3

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

    4. Applied cbrt-div33.2

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

    5. Taylor expanded around inf 32.0

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

  4. Final simplification31.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -2.3907293248482188 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +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))))))))