Average Error: 34.7 → 10.8
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r52262 = b;
        double r52263 = -r52262;
        double r52264 = r52262 * r52262;
        double r52265 = 4.0;
        double r52266 = a;
        double r52267 = r52265 * r52266;
        double r52268 = c;
        double r52269 = r52267 * r52268;
        double r52270 = r52264 - r52269;
        double r52271 = sqrt(r52270);
        double r52272 = r52263 + r52271;
        double r52273 = 2.0;
        double r52274 = r52273 * r52266;
        double r52275 = r52272 / r52274;
        return r52275;
}
↓
double f(double a, double b, double c) {
        double r52276 = b;
        double r52277 = -6.371698442415157e+150;
        bool r52278 = r52276 <= r52277;
        double r52279 = 1.0;
        double r52280 = c;
        double r52281 = r52280 / r52276;
        double r52282 = a;
        double r52283 = r52276 / r52282;
        double r52284 = r52281 - r52283;
        double r52285 = r52279 * r52284;
        double r52286 = 2.3065444773801163e-129;
        bool r52287 = r52276 <= r52286;
        double r52288 = -r52276;
        double r52289 = r52276 * r52276;
        double r52290 = 4.0;
        double r52291 = r52290 * r52282;
        double r52292 = r52291 * r52280;
        double r52293 = r52289 - r52292;
        double r52294 = sqrt(r52293);
        double r52295 = r52288 + r52294;
        double r52296 = 1.0;
        double r52297 = 2.0;
        double r52298 = r52297 * r52282;
        double r52299 = r52296 / r52298;
        double r52300 = r52295 * r52299;
        double r52301 = -1.0;
        double r52302 = r52301 * r52281;
        double r52303 = r52287 ? r52300 : r52302;
        double r52304 = r52278 ? r52285 : r52303;
        return r52304;
}

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 < -6.371698442415157e+150

    1. Initial program 63.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around -inf 2.5

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

    3. Simplified2.5

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

    if -6.371698442415157e+150 < b < 2.3065444773801163e-129

    1. Initial program 11.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv11.5

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}}\]

    if 2.3065444773801163e-129 < b

    1. Initial program 51.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around inf 12.3

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))