Average Error: 43.8 → 11.2
Time: 16.9s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.0283053499758227980476199547865689964965:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 0.0283053499758227980476199547865689964965:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3683597 = b;
        double r3683598 = -r3683597;
        double r3683599 = r3683597 * r3683597;
        double r3683600 = 3.0;
        double r3683601 = a;
        double r3683602 = r3683600 * r3683601;
        double r3683603 = c;
        double r3683604 = r3683602 * r3683603;
        double r3683605 = r3683599 - r3683604;
        double r3683606 = sqrt(r3683605);
        double r3683607 = r3683598 + r3683606;
        double r3683608 = r3683607 / r3683602;
        return r3683608;
}
↓
double f(double a, double b, double c) {
        double r3683609 = b;
        double r3683610 = 0.028305349975822798;
        bool r3683611 = r3683609 <= r3683610;
        double r3683612 = r3683609 * r3683609;
        double r3683613 = c;
        double r3683614 = 3.0;
        double r3683615 = a;
        double r3683616 = r3683614 * r3683615;
        double r3683617 = r3683613 * r3683616;
        double r3683618 = r3683612 - r3683617;
        double r3683619 = r3683618 - r3683612;
        double r3683620 = sqrt(r3683618);
        double r3683621 = r3683609 + r3683620;
        double r3683622 = r3683619 / r3683621;
        double r3683623 = r3683622 / r3683616;
        double r3683624 = -0.5;
        double r3683625 = r3683613 / r3683609;
        double r3683626 = r3683624 * r3683625;
        double r3683627 = r3683611 ? r3683623 : r3683626;
        return r3683627;
}

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 < 0.028305349975822798

    1. Initial program 22.1

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

    2. Simplified22.1

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

    4. Applied flip--22.0

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

    5. Simplified21.0

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

    if 0.028305349975822798 < b

    1. Initial program 46.8

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

    2. Simplified46.8

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

    3. Taylor expanded around inf 9.8

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.0283053499758227980476199547865689964965:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(3 \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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) (* (* 3.0 a) c)))) (* 3.0 a)))