Average Error: 43.9 → 12.6
Time: 14.6s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 58.70366934289741:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 58.70366934289741:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right)}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4124783 = b;
        double r4124784 = -r4124783;
        double r4124785 = r4124783 * r4124783;
        double r4124786 = 3.0;
        double r4124787 = a;
        double r4124788 = r4124786 * r4124787;
        double r4124789 = c;
        double r4124790 = r4124788 * r4124789;
        double r4124791 = r4124785 - r4124790;
        double r4124792 = sqrt(r4124791);
        double r4124793 = r4124784 + r4124792;
        double r4124794 = r4124793 / r4124788;
        return r4124794;
}
↓
double f(double a, double b, double c) {
        double r4124795 = b;
        double r4124796 = 58.70366934289741;
        bool r4124797 = r4124795 <= r4124796;
        double r4124798 = r4124795 * r4124795;
        double r4124799 = c;
        double r4124800 = 3.0;
        double r4124801 = a;
        double r4124802 = r4124800 * r4124801;
        double r4124803 = r4124799 * r4124802;
        double r4124804 = r4124798 - r4124803;
        double r4124805 = sqrt(r4124804);
        double r4124806 = r4124804 * r4124805;
        double r4124807 = r4124798 * r4124795;
        double r4124808 = r4124806 - r4124807;
        double r4124809 = r4124795 * r4124805;
        double r4124810 = r4124809 + r4124798;
        double r4124811 = r4124804 + r4124810;
        double r4124812 = r4124808 / r4124811;
        double r4124813 = r4124812 / r4124802;
        double r4124814 = -0.5;
        double r4124815 = r4124799 / r4124795;
        double r4124816 = r4124814 * r4124815;
        double r4124817 = r4124797 ? r4124813 : r4124816;
        return r4124817;
}

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 < 58.70366934289741

    1. Initial program 27.5

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

    2. Simplified27.5

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

    4. Applied flip3--27.6

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

    5. Simplified26.9

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

    6. Simplified26.9

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

    if 58.70366934289741 < b

    1. Initial program 48.6

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

    2. Simplified48.6

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

    3. Taylor expanded around inf 8.5

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

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 58.70366934289741:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) + \left(b \cdot \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} + b \cdot b\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))