Average Error: 34.0 → 10.2
Time: 16.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.4367588244629113 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 1.744031351412433 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.4367588244629113 \cdot 10^{+152}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\

\mathbf{elif}\;b \le 1.744031351412433 \cdot 10^{-142}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4222256 = b;
        double r4222257 = -r4222256;
        double r4222258 = r4222256 * r4222256;
        double r4222259 = 3.0;
        double r4222260 = a;
        double r4222261 = r4222259 * r4222260;
        double r4222262 = c;
        double r4222263 = r4222261 * r4222262;
        double r4222264 = r4222258 - r4222263;
        double r4222265 = sqrt(r4222264);
        double r4222266 = r4222257 + r4222265;
        double r4222267 = r4222266 / r4222261;
        return r4222267;
}
↓
double f(double a, double b, double c) {
        double r4222268 = b;
        double r4222269 = -1.4367588244629113e+152;
        bool r4222270 = r4222268 <= r4222269;
        double r4222271 = 0.5;
        double r4222272 = c;
        double r4222273 = r4222272 / r4222268;
        double r4222274 = r4222271 * r4222273;
        double r4222275 = a;
        double r4222276 = r4222268 / r4222275;
        double r4222277 = 0.6666666666666666;
        double r4222278 = r4222276 * r4222277;
        double r4222279 = r4222274 - r4222278;
        double r4222280 = 1.744031351412433e-142;
        bool r4222281 = r4222268 <= r4222280;
        double r4222282 = r4222268 * r4222268;
        double r4222283 = 3.0;
        double r4222284 = r4222272 * r4222283;
        double r4222285 = r4222275 * r4222284;
        double r4222286 = r4222282 - r4222285;
        double r4222287 = sqrt(r4222286);
        double r4222288 = r4222287 - r4222268;
        double r4222289 = r4222283 * r4222275;
        double r4222290 = r4222288 / r4222289;
        double r4222291 = -0.5;
        double r4222292 = r4222291 * r4222273;
        double r4222293 = r4222281 ? r4222290 : r4222292;
        double r4222294 = r4222270 ? r4222279 : r4222293;
        return r4222294;
}

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 3 regimes

  2. if b < -1.4367588244629113e+152

    1. Initial program 60.0

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

    2. Simplified60.0

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

    3. Taylor expanded around -inf 2.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}}\]

    if -1.4367588244629113e+152 < b < 1.744031351412433e-142

    1. Initial program 10.3

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

    2. Simplified10.3

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

    3. Taylor expanded around 0 10.4

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

    4. Simplified10.4

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

    if 1.744031351412433e-142 < b

    1. Initial program 50.1

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

    2. Simplified50.1

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

    3. Taylor expanded around inf 12.0

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.4367588244629113 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b} - \frac{b}{a} \cdot \frac{2}{3}\\ \mathbf{elif}\;b \le 1.744031351412433 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))