Average Error: 33.2 → 9.8
Time: 44.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.303853124735619 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.2295616480632551 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.303853124735619 \cdot 10^{+50}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.2295616480632551 \cdot 10^{-79}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1396838 = b_2;
        double r1396839 = -r1396838;
        double r1396840 = r1396838 * r1396838;
        double r1396841 = a;
        double r1396842 = c;
        double r1396843 = r1396841 * r1396842;
        double r1396844 = r1396840 - r1396843;
        double r1396845 = sqrt(r1396844);
        double r1396846 = r1396839 + r1396845;
        double r1396847 = r1396846 / r1396841;
        return r1396847;
}
↓
double f(double a, double b_2, double c) {
        double r1396848 = b_2;
        double r1396849 = -3.303853124735619e+50;
        bool r1396850 = r1396848 <= r1396849;
        double r1396851 = 0.5;
        double r1396852 = c;
        double r1396853 = r1396852 / r1396848;
        double r1396854 = r1396851 * r1396853;
        double r1396855 = a;
        double r1396856 = r1396848 / r1396855;
        double r1396857 = 2.0;
        double r1396858 = r1396856 * r1396857;
        double r1396859 = r1396854 - r1396858;
        double r1396860 = 1.2295616480632551e-79;
        bool r1396861 = r1396848 <= r1396860;
        double r1396862 = r1396848 * r1396848;
        double r1396863 = r1396852 * r1396855;
        double r1396864 = r1396862 - r1396863;
        double r1396865 = sqrt(r1396864);
        double r1396866 = r1396865 / r1396855;
        double r1396867 = r1396866 - r1396856;
        double r1396868 = -0.5;
        double r1396869 = r1396853 * r1396868;
        double r1396870 = r1396861 ? r1396867 : r1396869;
        double r1396871 = r1396850 ? r1396859 : r1396870;
        return r1396871;
}

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 < -3.303853124735619e+50

    1. Initial program 35.2

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

    2. Simplified35.2

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

    3. Taylor expanded around -inf 5.1

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

    if -3.303853124735619e+50 < b_2 < 1.2295616480632551e-79

    1. Initial program 12.6

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

    2. Simplified12.6

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

    4. Applied div-sub12.6

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

    if 1.2295616480632551e-79 < b_2

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 9.5

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.303853124735619 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.2295616480632551 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))