Average Error: 44.7 → 11.0
Time: 24.6s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.09946845057046652:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 0.09946845057046652:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r1918920 = b;
        double r1918921 = -r1918920;
        double r1918922 = r1918920 * r1918920;
        double r1918923 = 3.0;
        double r1918924 = a;
        double r1918925 = r1918923 * r1918924;
        double r1918926 = c;
        double r1918927 = r1918925 * r1918926;
        double r1918928 = r1918922 - r1918927;
        double r1918929 = sqrt(r1918928);
        double r1918930 = r1918921 + r1918929;
        double r1918931 = r1918930 / r1918925;
        return r1918931;
}
↓
double f(double a, double b, double c) {
        double r1918932 = b;
        double r1918933 = 0.09946845057046652;
        bool r1918934 = r1918932 <= r1918933;
        double r1918935 = r1918932 * r1918932;
        double r1918936 = a;
        double r1918937 = 3.0;
        double r1918938 = c;
        double r1918939 = r1918937 * r1918938;
        double r1918940 = r1918936 * r1918939;
        double r1918941 = r1918935 - r1918940;
        double r1918942 = sqrt(r1918941);
        double r1918943 = r1918941 * r1918942;
        double r1918944 = r1918935 * r1918932;
        double r1918945 = r1918943 - r1918944;
        double r1918946 = r1918932 * r1918942;
        double r1918947 = r1918946 + r1918935;
        double r1918948 = r1918941 + r1918947;
        double r1918949 = r1918945 / r1918948;
        double r1918950 = r1918936 * r1918937;
        double r1918951 = r1918949 / r1918950;
        double r1918952 = -0.5;
        double r1918953 = r1918938 / r1918932;
        double r1918954 = r1918952 * r1918953;
        double r1918955 = r1918934 ? r1918951 : r1918954;
        return r1918955;
}

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 < 0.09946845057046652

    1. Initial program 24.0

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

    2. Simplified24.0

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

    4. Applied flip3--24.1

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

    5. Simplified23.5

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

    6. Simplified23.5

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

    if 0.09946845057046652 < b

    1. Initial program 47.8

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

    2. Simplified47.8

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

    3. Taylor expanded around inf 9.2

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.09946845057046652:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - a \cdot \left(3 \cdot c\right)\right) + \left(b \cdot \sqrt{b \cdot b - a \cdot \left(3 \cdot c\right)} + b \cdot b\right)}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))