Average Error: 28.7 → 16.2
Time: 22.7s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2134.20029068235863:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 2134.20029068235863:\\
\;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r66872 = b;
        double r66873 = -r66872;
        double r66874 = r66872 * r66872;
        double r66875 = 3.0;
        double r66876 = a;
        double r66877 = r66875 * r66876;
        double r66878 = c;
        double r66879 = r66877 * r66878;
        double r66880 = r66874 - r66879;
        double r66881 = sqrt(r66880);
        double r66882 = r66873 + r66881;
        double r66883 = r66882 / r66877;
        return r66883;
}
↓
double f(double a, double b, double c) {
        double r66884 = b;
        double r66885 = 2134.2002906823586;
        bool r66886 = r66884 <= r66885;
        double r66887 = r66884 * r66884;
        double r66888 = 3.0;
        double r66889 = a;
        double r66890 = r66888 * r66889;
        double r66891 = c;
        double r66892 = r66890 * r66891;
        double r66893 = r66892 + r66887;
        double r66894 = r66887 - r66893;
        double r66895 = r66887 - r66892;
        double r66896 = sqrt(r66895);
        double r66897 = r66896 + r66884;
        double r66898 = r66894 / r66897;
        double r66899 = r66898 / r66888;
        double r66900 = r66899 / r66889;
        double r66901 = -0.5;
        double r66902 = r66891 / r66884;
        double r66903 = r66901 * r66902;
        double r66904 = r66886 ? r66900 : r66903;
        return r66904;
}

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 2 regimes

  2. if b < 2134.2002906823586

    1. Initial program 18.0

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

    2. Simplified18.0

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

    4. Applied flip--18.0

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

    5. Simplified16.9

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

    if 2134.2002906823586 < b

    1. Initial program 37.0

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

    2. Simplified37.0

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

    3. Taylor expanded around inf 15.7

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

    4. Taylor expanded around 0 15.6

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2134.20029068235863:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))