Average Error: 29.0 → 17.4
Time: 7.6s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 20.327229482009734 \lor \neg \left(b \le 520.871493226619691\right) \land b \le 5060.8354328326695:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 20.327229482009734 \lor \neg \left(b \le 520.871493226619691\right) \land b \le 5060.8354328326695:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r100868 = b;
        double r100869 = -r100868;
        double r100870 = r100868 * r100868;
        double r100871 = 3.0;
        double r100872 = a;
        double r100873 = r100871 * r100872;
        double r100874 = c;
        double r100875 = r100873 * r100874;
        double r100876 = r100870 - r100875;
        double r100877 = sqrt(r100876);
        double r100878 = r100869 + r100877;
        double r100879 = r100878 / r100873;
        return r100879;
}
↓
double f(double a, double b, double c) {
        double r100880 = b;
        double r100881 = 20.327229482009734;
        bool r100882 = r100880 <= r100881;
        double r100883 = 520.8714932266197;
        bool r100884 = r100880 <= r100883;
        double r100885 = !r100884;
        double r100886 = 5060.8354328326695;
        bool r100887 = r100880 <= r100886;
        bool r100888 = r100885 && r100887;
        bool r100889 = r100882 || r100888;
        double r100890 = r100880 * r100880;
        double r100891 = 3.0;
        double r100892 = a;
        double r100893 = r100891 * r100892;
        double r100894 = c;
        double r100895 = r100893 * r100894;
        double r100896 = r100890 - r100895;
        double r100897 = 2.0;
        double r100898 = pow(r100880, r100897);
        double r100899 = r100896 - r100898;
        double r100900 = sqrt(r100896);
        double r100901 = r100900 + r100880;
        double r100902 = r100899 / r100901;
        double r100903 = r100902 / r100893;
        double r100904 = -0.5;
        double r100905 = r100894 / r100880;
        double r100906 = r100904 * r100905;
        double r100907 = r100889 ? r100903 : r100906;
        return r100907;
}

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 < 20.327229482009734 or 520.8714932266197 < b < 5060.8354328326695

    1. Initial program 18.2

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

    2. Simplified18.2

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

    4. Applied flip--18.2

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

    5. Simplified17.2

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

    if 20.327229482009734 < b < 520.8714932266197 or 5060.8354328326695 < b

    1. Initial program 34.7

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

    2. Simplified34.7

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

    3. Taylor expanded around inf 17.5

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

  4. Final simplification17.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 20.327229482009734 \lor \neg \left(b \le 520.871493226619691\right) \land b \le 5060.8354328326695:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right) - {b}^{2}}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :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 a) c)))) (* 3 a)))