Average Error: 34.4 → 10.6
Time: 18.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\
\;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1838440 = b;
        double r1838441 = -r1838440;
        double r1838442 = r1838440 * r1838440;
        double r1838443 = 4.0;
        double r1838444 = a;
        double r1838445 = r1838443 * r1838444;
        double r1838446 = c;
        double r1838447 = r1838445 * r1838446;
        double r1838448 = r1838442 - r1838447;
        double r1838449 = sqrt(r1838448);
        double r1838450 = r1838441 + r1838449;
        double r1838451 = 2.0;
        double r1838452 = r1838451 * r1838444;
        double r1838453 = r1838450 / r1838452;
        return r1838453;
}
↓
double f(double a, double b, double c) {
        double r1838454 = b;
        double r1838455 = -2.221067196710922e+149;
        bool r1838456 = r1838454 <= r1838455;
        double r1838457 = 2.0;
        double r1838458 = c;
        double r1838459 = r1838458 / r1838454;
        double r1838460 = r1838457 * r1838459;
        double r1838461 = a;
        double r1838462 = r1838454 / r1838461;
        double r1838463 = 2.0;
        double r1838464 = r1838462 * r1838463;
        double r1838465 = r1838460 - r1838464;
        double r1838466 = r1838465 / r1838457;
        double r1838467 = 2.8983489306952693e-35;
        bool r1838468 = r1838454 <= r1838467;
        double r1838469 = r1838454 * r1838454;
        double r1838470 = r1838458 * r1838461;
        double r1838471 = 4.0;
        double r1838472 = r1838470 * r1838471;
        double r1838473 = r1838469 - r1838472;
        double r1838474 = sqrt(r1838473);
        double r1838475 = r1838474 - r1838454;
        double r1838476 = r1838475 / r1838461;
        double r1838477 = r1838476 / r1838457;
        double r1838478 = -2.0;
        double r1838479 = r1838478 * r1838459;
        double r1838480 = r1838479 / r1838457;
        double r1838481 = r1838468 ? r1838477 : r1838480;
        double r1838482 = r1838456 ? r1838466 : r1838481;
        return r1838482;
}

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 < -2.221067196710922e+149

    1. Initial program 62.3

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

    2. Simplified62.3

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

    3. Taylor expanded around -inf 2.8

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    if -2.221067196710922e+149 < b < 2.8983489306952693e-35

    1. Initial program 14.6

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

    2. Simplified14.6

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

    if 2.8983489306952693e-35 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.3

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))