Average Error: 34.5 → 10.4
Time: 21.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - \frac{b}{a} \cdot 0.6666666666666666296592325124947819858789\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{3} \cdot \left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]
\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 -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - \frac{b}{a} \cdot 0.6666666666666666296592325124947819858789\\

\mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\
\;\;\;\;\frac{\frac{1}{3} \cdot \left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -0.5\\

\end{array}
double f(double a, double b, double c) {
        double r3967912 = b;
        double r3967913 = -r3967912;
        double r3967914 = r3967912 * r3967912;
        double r3967915 = 3.0;
        double r3967916 = a;
        double r3967917 = r3967915 * r3967916;
        double r3967918 = c;
        double r3967919 = r3967917 * r3967918;
        double r3967920 = r3967914 - r3967919;
        double r3967921 = sqrt(r3967920);
        double r3967922 = r3967913 + r3967921;
        double r3967923 = r3967922 / r3967917;
        return r3967923;
}
↓
double f(double a, double b, double c) {
        double r3967924 = b;
        double r3967925 = -1.7633154797394035e+89;
        bool r3967926 = r3967924 <= r3967925;
        double r3967927 = 0.5;
        double r3967928 = c;
        double r3967929 = r3967928 / r3967924;
        double r3967930 = r3967927 * r3967929;
        double r3967931 = a;
        double r3967932 = r3967924 / r3967931;
        double r3967933 = 0.6666666666666666;
        double r3967934 = r3967932 * r3967933;
        double r3967935 = r3967930 - r3967934;
        double r3967936 = 9.136492990928292e-23;
        bool r3967937 = r3967924 <= r3967936;
        double r3967938 = 1.0;
        double r3967939 = 3.0;
        double r3967940 = r3967938 / r3967939;
        double r3967941 = r3967924 * r3967924;
        double r3967942 = r3967928 * r3967939;
        double r3967943 = r3967931 * r3967942;
        double r3967944 = r3967941 - r3967943;
        double r3967945 = sqrt(r3967944);
        double r3967946 = r3967945 - r3967924;
        double r3967947 = r3967940 * r3967946;
        double r3967948 = r3967947 / r3967931;
        double r3967949 = -0.5;
        double r3967950 = r3967929 * r3967949;
        double r3967951 = r3967937 ? r3967948 : r3967950;
        double r3967952 = r3967926 ? r3967935 : r3967951;
        return r3967952;
}

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 3 regimes

  2. if b < -1.7633154797394035e+89

    1. Initial program 45.8

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

    2. Taylor expanded around -inf 4.2

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}}\]

    if -1.7633154797394035e+89 < b < 9.136492990928292e-23

    1. Initial program 15.1

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

    3. Applied associate-/r*15.2

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

    4. Simplified15.2

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

    6. Applied div-inv15.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right) \cdot \frac{1}{3}}}{a}\]

    if 9.136492990928292e-23 < b

    1. Initial program 55.4

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

    2. Taylor expanded around inf 6.7

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - \frac{b}{a} \cdot 0.6666666666666666296592325124947819858789\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{3} \cdot \left(\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))