Average Error: 33.5 → 10.1
Time: 6.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.571370019832872361238263891019444684316 \cdot 10^{135}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 7.097044289113136495972982608457999659592 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -5.571370019832872361238263891019444684316 \cdot 10^{135}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 7.097044289113136495972982608457999659592 \cdot 10^{-99}:\\
\;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r95266 = b;
        double r95267 = -r95266;
        double r95268 = r95266 * r95266;
        double r95269 = 3.0;
        double r95270 = a;
        double r95271 = r95269 * r95270;
        double r95272 = c;
        double r95273 = r95271 * r95272;
        double r95274 = r95268 - r95273;
        double r95275 = sqrt(r95274);
        double r95276 = r95267 + r95275;
        double r95277 = r95276 / r95271;
        return r95277;
}
↓
double f(double a, double b, double c) {
        double r95278 = b;
        double r95279 = -5.5713700198328724e+135;
        bool r95280 = r95278 <= r95279;
        double r95281 = 0.5;
        double r95282 = c;
        double r95283 = r95282 / r95278;
        double r95284 = r95281 * r95283;
        double r95285 = 0.6666666666666666;
        double r95286 = a;
        double r95287 = r95278 / r95286;
        double r95288 = r95285 * r95287;
        double r95289 = r95284 - r95288;
        double r95290 = 7.0970442891131365e-99;
        bool r95291 = r95278 <= r95290;
        double r95292 = -r95278;
        double r95293 = r95278 * r95278;
        double r95294 = 3.0;
        double r95295 = r95294 * r95286;
        double r95296 = r95295 * r95282;
        double r95297 = r95293 - r95296;
        double r95298 = sqrt(r95297);
        double r95299 = r95292 + r95298;
        double r95300 = r95299 / r95294;
        double r95301 = r95300 / r95286;
        double r95302 = -0.5;
        double r95303 = r95302 * r95283;
        double r95304 = r95291 ? r95301 : r95303;
        double r95305 = r95280 ? r95289 : r95304;
        return r95305;
}

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 < -5.5713700198328724e+135

    1. Initial program 56.8

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

    2. Taylor expanded around -inf 3.0

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

    if -5.5713700198328724e+135 < b < 7.0970442891131365e-99

    1. Initial program 11.3

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

    3. Applied associate-/r*11.3

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

    if 7.0970442891131365e-99 < b

    1. Initial program 51.5

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

    2. Taylor expanded around inf 11.0

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.571370019832872361238263891019444684316 \cdot 10^{135}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.6666666666666666296592325124947819858789 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 7.097044289113136495972982608457999659592 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))