Average Error: 33.5 → 10.4
Time: 59.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.691277786452672 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.502350718288979 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.691277786452672 \cdot 10^{-38}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.502350718288979 \cdot 10^{+75}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r2793371 = b_2;
        double r2793372 = -r2793371;
        double r2793373 = r2793371 * r2793371;
        double r2793374 = a;
        double r2793375 = c;
        double r2793376 = r2793374 * r2793375;
        double r2793377 = r2793373 - r2793376;
        double r2793378 = sqrt(r2793377);
        double r2793379 = r2793372 - r2793378;
        double r2793380 = r2793379 / r2793374;
        return r2793380;
}
↓
double f(double a, double b_2, double c) {
        double r2793381 = b_2;
        double r2793382 = -5.691277786452672e-38;
        bool r2793383 = r2793381 <= r2793382;
        double r2793384 = -0.5;
        double r2793385 = c;
        double r2793386 = r2793385 / r2793381;
        double r2793387 = r2793384 * r2793386;
        double r2793388 = 1.502350718288979e+75;
        bool r2793389 = r2793381 <= r2793388;
        double r2793390 = -r2793381;
        double r2793391 = r2793381 * r2793381;
        double r2793392 = a;
        double r2793393 = r2793392 * r2793385;
        double r2793394 = r2793391 - r2793393;
        double r2793395 = sqrt(r2793394);
        double r2793396 = r2793390 - r2793395;
        double r2793397 = r2793396 / r2793392;
        double r2793398 = 0.5;
        double r2793399 = r2793386 * r2793398;
        double r2793400 = 2.0;
        double r2793401 = r2793381 / r2793392;
        double r2793402 = r2793400 * r2793401;
        double r2793403 = r2793399 - r2793402;
        double r2793404 = r2793389 ? r2793397 : r2793403;
        double r2793405 = r2793383 ? r2793387 : r2793404;
        return r2793405;
}

Error

Bits error versus a

Bits error versus b_2

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 < -5.691277786452672e-38

    1. Initial program 54.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 7.9

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -5.691277786452672e-38 < b_2 < 1.502350718288979e+75

    1. Initial program 14.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around 0 14.7

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified14.7

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    if 1.502350718288979e+75 < b_2

    1. Initial program 40.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 4.5

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.691277786452672 \cdot 10^{-38}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.502350718288979 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))