Average Error: 34.0 → 10.5
Time: 19.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1147605 = b_2;
        double r1147606 = -r1147605;
        double r1147607 = r1147605 * r1147605;
        double r1147608 = a;
        double r1147609 = c;
        double r1147610 = r1147608 * r1147609;
        double r1147611 = r1147607 - r1147610;
        double r1147612 = sqrt(r1147611);
        double r1147613 = r1147606 - r1147612;
        double r1147614 = r1147613 / r1147608;
        return r1147614;
}
↓
double f(double a, double b_2, double c) {
        double r1147615 = b_2;
        double r1147616 = -9.332433396832084e-58;
        bool r1147617 = r1147615 <= r1147616;
        double r1147618 = -0.5;
        double r1147619 = c;
        double r1147620 = r1147619 / r1147615;
        double r1147621 = r1147618 * r1147620;
        double r1147622 = 3.038903409991338e+107;
        bool r1147623 = r1147615 <= r1147622;
        double r1147624 = -r1147615;
        double r1147625 = r1147615 * r1147615;
        double r1147626 = a;
        double r1147627 = r1147626 * r1147619;
        double r1147628 = r1147625 - r1147627;
        double r1147629 = sqrt(r1147628);
        double r1147630 = r1147624 - r1147629;
        double r1147631 = r1147630 / r1147626;
        double r1147632 = -2.0;
        double r1147633 = r1147615 * r1147632;
        double r1147634 = r1147633 / r1147626;
        double r1147635 = r1147623 ? r1147631 : r1147634;
        double r1147636 = r1147617 ? r1147621 : r1147635;
        return r1147636;
}

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 < -9.332433396832084e-58

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 8.7

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

    if -9.332433396832084e-58 < b_2 < 3.038903409991338e+107

    1. Initial program 14.0

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

    if 3.038903409991338e+107 < b_2

    1. Initial program 49.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--63.2

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

    4. Simplified62.2

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

    5. Simplified62.2

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

    6. Taylor expanded around 0 3.7

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.038903409991338138548211857189252856935 \cdot 10^{107}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))