Average Error: 33.7 → 13.1
Time: 4.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\ \;\;\;\;\frac{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \le -5.98152550694987672 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(\sqrt[3]{\left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.0331115085790278 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{-\left(b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\
\;\;\;\;\frac{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{3 \cdot a}\\

\mathbf{elif}\;b \le -5.98152550694987672 \cdot 10^{-298}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(\sqrt[3]{\left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\

\mathbf{elif}\;b \le 1.0331115085790278 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{-\left(b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r109895 = b;
        double r109896 = -r109895;
        double r109897 = r109895 * r109895;
        double r109898 = 3.0;
        double r109899 = a;
        double r109900 = r109898 * r109899;
        double r109901 = c;
        double r109902 = r109900 * r109901;
        double r109903 = r109897 - r109902;
        double r109904 = sqrt(r109903);
        double r109905 = r109896 + r109904;
        double r109906 = r109905 / r109900;
        return r109906;
}
↓
double f(double a, double b, double c) {
        double r109907 = b;
        double r109908 = -2.272454186637281e+165;
        bool r109909 = r109907 <= r109908;
        double r109910 = 1.5;
        double r109911 = a;
        double r109912 = c;
        double r109913 = r109911 * r109912;
        double r109914 = r109913 / r109907;
        double r109915 = r109910 * r109914;
        double r109916 = 2.0;
        double r109917 = r109916 * r109907;
        double r109918 = r109915 - r109917;
        double r109919 = 3.0;
        double r109920 = r109919 * r109911;
        double r109921 = r109918 / r109920;
        double r109922 = -5.981525506949877e-298;
        bool r109923 = r109907 <= r109922;
        double r109924 = -r109907;
        double r109925 = r109907 * r109907;
        double r109926 = r109920 * r109912;
        double r109927 = cbrt(r109926);
        double r109928 = r109927 * r109927;
        double r109929 = r109928 * r109927;
        double r109930 = r109925 - r109929;
        double r109931 = sqrt(r109930);
        double r109932 = r109924 + r109931;
        double r109933 = r109932 / r109920;
        double r109934 = 1.0331115085790278e+154;
        bool r109935 = r109907 <= r109934;
        double r109936 = pow(r109907, r109916);
        double r109937 = r109936 - r109936;
        double r109938 = r109937 + r109926;
        double r109939 = r109925 - r109926;
        double r109940 = sqrt(r109939);
        double r109941 = r109907 + r109940;
        double r109942 = -r109941;
        double r109943 = r109938 / r109942;
        double r109944 = r109943 / r109920;
        double r109945 = -1.5;
        double r109946 = r109945 * r109914;
        double r109947 = r109946 / r109920;
        double r109948 = r109935 ? r109944 : r109947;
        double r109949 = r109923 ? r109933 : r109948;
        double r109950 = r109909 ? r109921 : r109949;
        return r109950;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b < -2.272454186637281e+165

    1. Initial program 64.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around -inf 10.5

      \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3 \cdot a}\]

    if -2.272454186637281e+165 < b < -5.981525506949877e-298

    1. Initial program 9.5

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

    3. Applied add-cube-cbrt9.7

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{\left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

    if -5.981525506949877e-298 < b < 1.0331115085790278e+154

    1. Initial program 33.9

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

    3. Applied add-cube-cbrt37.4

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

    5. Applied flip-+37.4

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \left(\left(\sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right)}{\left(-b\right) - \left(\sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}{3 \cdot a}\]

    6. Simplified16.8

      \[\leadsto \frac{\frac{\color{blue}{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \left(\sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

    7. Simplified16.3

      \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{\color{blue}{-\left(b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a}\]

    if 1.0331115085790278e+154 < b

    1. Initial program 64.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

    2. Taylor expanded around inf 14.5

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2724541866372811 \cdot 10^{165}:\\ \;\;\;\;\frac{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{3 \cdot a}\\ \mathbf{elif}\;b \le -5.98152550694987672 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(\sqrt[3]{\left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{elif}\;b \le 1.0331115085790278 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{\left({b}^{2} - {b}^{2}\right) + \left(3 \cdot a\right) \cdot c}{-\left(b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))