Average Error: 34.5 → 10.3
Time: 20.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - \frac{b}{a} \cdot 0.6666666666666666296592325124947819858789\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - \frac{b}{a} \cdot 0.6666666666666666296592325124947819858789\\

\mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4442269 = b;
        double r4442270 = -r4442269;
        double r4442271 = r4442269 * r4442269;
        double r4442272 = 3.0;
        double r4442273 = a;
        double r4442274 = r4442272 * r4442273;
        double r4442275 = c;
        double r4442276 = r4442274 * r4442275;
        double r4442277 = r4442271 - r4442276;
        double r4442278 = sqrt(r4442277);
        double r4442279 = r4442270 + r4442278;
        double r4442280 = r4442279 / r4442274;
        return r4442280;
}
↓
double f(double a, double b, double c) {
        double r4442281 = b;
        double r4442282 = -1.7633154797394035e+89;
        bool r4442283 = r4442281 <= r4442282;
        double r4442284 = 0.5;
        double r4442285 = c;
        double r4442286 = r4442285 / r4442281;
        double r4442287 = r4442284 * r4442286;
        double r4442288 = a;
        double r4442289 = r4442281 / r4442288;
        double r4442290 = 0.6666666666666666;
        double r4442291 = r4442289 * r4442290;
        double r4442292 = r4442287 - r4442291;
        double r4442293 = 9.136492990928292e-23;
        bool r4442294 = r4442281 <= r4442293;
        double r4442295 = r4442281 * r4442281;
        double r4442296 = 3.0;
        double r4442297 = r4442285 * r4442288;
        double r4442298 = r4442296 * r4442297;
        double r4442299 = r4442295 - r4442298;
        double r4442300 = sqrt(r4442299);
        double r4442301 = r4442300 - r4442281;
        double r4442302 = r4442301 / r4442296;
        double r4442303 = r4442302 / r4442288;
        double r4442304 = -0.5;
        double r4442305 = r4442304 * r4442286;
        double r4442306 = r4442294 ? r4442303 : r4442305;
        double r4442307 = r4442283 ? r4442292 : r4442306;
        return r4442307;
}

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 < -1.7633154797394035e+89

    1. Initial program 45.8

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

    2. Taylor expanded around -inf 4.2

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}}\]

    if -1.7633154797394035e+89 < b < 9.136492990928292e-23

    1. Initial program 15.1

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

    3. Applied associate-/r*15.2

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

    4. Simplified15.1

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

    if 9.136492990928292e-23 < b

    1. Initial program 55.4

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

    2. Taylor expanded around inf 6.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - \frac{b}{a} \cdot 0.6666666666666666296592325124947819858789\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))