Average Error: 34.0 → 10.4
Time: 6.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.71711085460076329 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 8.87069060473612544 \cdot 10^{-35}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.71711085460076329 \cdot 10^{118}:\\
\;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\

\mathbf{elif}\;b \le 8.87069060473612544 \cdot 10^{-35}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r135157 = b;
        double r135158 = -r135157;
        double r135159 = r135157 * r135157;
        double r135160 = 3.0;
        double r135161 = a;
        double r135162 = r135160 * r135161;
        double r135163 = c;
        double r135164 = r135162 * r135163;
        double r135165 = r135159 - r135164;
        double r135166 = sqrt(r135165);
        double r135167 = r135158 + r135166;
        double r135168 = r135167 / r135162;
        return r135168;
}
↓
double f(double a, double b, double c) {
        double r135169 = b;
        double r135170 = -3.7171108546007633e+118;
        bool r135171 = r135169 <= r135170;
        double r135172 = 0.5;
        double r135173 = c;
        double r135174 = r135173 / r135169;
        double r135175 = r135172 * r135174;
        double r135176 = 0.6666666666666666;
        double r135177 = a;
        double r135178 = r135169 / r135177;
        double r135179 = r135176 * r135178;
        double r135180 = r135175 - r135179;
        double r135181 = 8.870690604736125e-35;
        bool r135182 = r135169 <= r135181;
        double r135183 = -r135169;
        double r135184 = r135169 * r135169;
        double r135185 = 3.0;
        double r135186 = r135185 * r135177;
        double r135187 = r135186 * r135173;
        double r135188 = r135184 - r135187;
        double r135189 = sqrt(r135188);
        double r135190 = r135183 + r135189;
        double r135191 = 1.0;
        double r135192 = r135191 / r135186;
        double r135193 = r135190 * r135192;
        double r135194 = -0.5;
        double r135195 = r135194 * r135174;
        double r135196 = r135182 ? r135193 : r135195;
        double r135197 = r135171 ? r135180 : r135196;
        return r135197;
}

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 < -3.7171108546007633e+118

    1. Initial program 52.1

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

    2. Taylor expanded around -inf 3.3

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}}\]

    if -3.7171108546007633e+118 < b < 8.870690604736125e-35

    1. Initial program 14.2

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

    3. Applied div-inv14.2

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}}\]

    if 8.870690604736125e-35 < b

    1. Initial program 54.1

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

    2. Taylor expanded around inf 7.9

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.71711085460076329 \cdot 10^{118}:\\ \;\;\;\;0.5 \cdot \frac{c}{b} - 0.66666666666666663 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \le 8.87069060473612544 \cdot 10^{-35}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{1}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))