Average Error: 33.8 → 9.5
Time: 19.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.7064129236785206 \cdot 10^{+142}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.1804820682342164 \cdot 10^{-93}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.7064129236785206 \cdot 10^{+142}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.1804820682342164 \cdot 10^{-93}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r562425 = b_2;
        double r562426 = -r562425;
        double r562427 = r562425 * r562425;
        double r562428 = a;
        double r562429 = c;
        double r562430 = r562428 * r562429;
        double r562431 = r562427 - r562430;
        double r562432 = sqrt(r562431);
        double r562433 = r562426 + r562432;
        double r562434 = r562433 / r562428;
        return r562434;
}
↓
double f(double a, double b_2, double c) {
        double r562435 = b_2;
        double r562436 = -3.7064129236785206e+142;
        bool r562437 = r562435 <= r562436;
        double r562438 = 0.5;
        double r562439 = c;
        double r562440 = r562439 / r562435;
        double r562441 = r562438 * r562440;
        double r562442 = a;
        double r562443 = r562435 / r562442;
        double r562444 = 2.0;
        double r562445 = r562443 * r562444;
        double r562446 = r562441 - r562445;
        double r562447 = 1.1804820682342164e-93;
        bool r562448 = r562435 <= r562447;
        double r562449 = 1.0;
        double r562450 = r562449 / r562442;
        double r562451 = r562435 * r562435;
        double r562452 = r562439 * r562442;
        double r562453 = r562451 - r562452;
        double r562454 = sqrt(r562453);
        double r562455 = r562454 - r562435;
        double r562456 = r562450 * r562455;
        double r562457 = -0.5;
        double r562458 = r562457 * r562440;
        double r562459 = r562448 ? r562456 : r562458;
        double r562460 = r562437 ? r562446 : r562459;
        return r562460;
}

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 < -3.7064129236785206e+142

    1. Initial program 56.4

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

    2. Simplified56.4

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around -inf 2.7

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

    if -3.7064129236785206e+142 < b_2 < 1.1804820682342164e-93

    1. Initial program 11.5

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

    2. Simplified11.5

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-inv11.7

      \[\leadsto \color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}}\]

    if 1.1804820682342164e-93 < b_2

    1. Initial program 52.4

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

    2. Simplified52.4

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf 9.1

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.7064129236785206 \cdot 10^{+142}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.1804820682342164 \cdot 10^{-93}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))