Average Error: 33.6 → 10.2
Time: 14.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r40210 = b;
        double r40211 = -r40210;
        double r40212 = r40210 * r40210;
        double r40213 = 4.0;
        double r40214 = a;
        double r40215 = r40213 * r40214;
        double r40216 = c;
        double r40217 = r40215 * r40216;
        double r40218 = r40212 - r40217;
        double r40219 = sqrt(r40218);
        double r40220 = r40211 + r40219;
        double r40221 = 2.0;
        double r40222 = r40221 * r40214;
        double r40223 = r40220 / r40222;
        return r40223;
}
↓
double f(double a, double b, double c) {
        double r40224 = b;
        double r40225 = -1.2609617020890706e+118;
        bool r40226 = r40224 <= r40225;
        double r40227 = 1.0;
        double r40228 = c;
        double r40229 = r40228 / r40224;
        double r40230 = a;
        double r40231 = r40224 / r40230;
        double r40232 = r40229 - r40231;
        double r40233 = r40227 * r40232;
        double r40234 = 5.81843322574321e-115;
        bool r40235 = r40224 <= r40234;
        double r40236 = -r40224;
        double r40237 = r40224 * r40224;
        double r40238 = 4.0;
        double r40239 = r40238 * r40230;
        double r40240 = r40239 * r40228;
        double r40241 = r40237 - r40240;
        double r40242 = sqrt(r40241);
        double r40243 = r40236 + r40242;
        double r40244 = 2.0;
        double r40245 = r40244 * r40230;
        double r40246 = r40243 / r40245;
        double r40247 = -1.0;
        double r40248 = r40247 * r40229;
        double r40249 = r40235 ? r40246 : r40248;
        double r40250 = r40226 ? r40233 : r40249;
        return r40250;
}

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 < -1.2609617020890706e+118

    1. Initial program 51.6

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

    2. Taylor expanded around -inf 2.7

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

    3. Simplified2.7

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

    if -1.2609617020890706e+118 < b < 5.81843322574321e-115

    1. Initial program 11.5

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

    if 5.81843322574321e-115 < b

    1. Initial program 51.3

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

    2. Taylor expanded around inf 11.3

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.260961702089070630848300788408824469286 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.818433225743210113099557178165353186607 \cdot 10^{-115}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))