Average Error: 43.9 → 11.2
Time: 12.4s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.07798087007627403643983399206263129599392:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 0.07798087007627403643983399206263129599392:\\
\;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - b \cdot b}{\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 r88539 = b;
        double r88540 = -r88539;
        double r88541 = r88539 * r88539;
        double r88542 = 3.0;
        double r88543 = a;
        double r88544 = r88542 * r88543;
        double r88545 = c;
        double r88546 = r88544 * r88545;
        double r88547 = r88541 - r88546;
        double r88548 = sqrt(r88547);
        double r88549 = r88540 + r88548;
        double r88550 = r88549 / r88544;
        return r88550;
}
↓
double f(double a, double b, double c) {
        double r88551 = b;
        double r88552 = 0.07798087007627404;
        bool r88553 = r88551 <= r88552;
        double r88554 = 2.0;
        double r88555 = pow(r88551, r88554);
        double r88556 = 3.0;
        double r88557 = a;
        double r88558 = c;
        double r88559 = r88557 * r88558;
        double r88560 = r88556 * r88559;
        double r88561 = r88555 - r88560;
        double r88562 = r88551 * r88551;
        double r88563 = r88561 - r88562;
        double r88564 = r88556 * r88557;
        double r88565 = r88564 * r88558;
        double r88566 = r88562 - r88565;
        double r88567 = sqrt(r88566);
        double r88568 = r88567 + r88551;
        double r88569 = r88563 / r88568;
        double r88570 = r88569 / r88556;
        double r88571 = r88570 / r88557;
        double r88572 = -0.5;
        double r88573 = r88558 / r88551;
        double r88574 = r88572 * r88573;
        double r88575 = r88553 ? r88571 : r88574;
        return r88575;
}

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 < 0.07798087007627404

    1. Initial program 22.9

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

    2. Simplified22.9

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

    4. Applied flip--22.9

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

    5. Simplified21.9

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

    if 0.07798087007627404 < b

    1. Initial program 47.1

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

    2. Simplified47.1

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

    3. Taylor expanded around inf 9.5

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.07798087007627403643983399206263129599392:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 3 \cdot \left(a \cdot c\right)\right) - b \cdot b}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))