Average Error: 33.6 → 10.8
Time: 22.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.0410715251838527 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.136683434005781 \cdot 10^{-32}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.0410715251838527 \cdot 10^{+49}:\\
\;\;\;\;\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 r788884 = b_2;
        double r788885 = -r788884;
        double r788886 = r788884 * r788884;
        double r788887 = a;
        double r788888 = c;
        double r788889 = r788887 * r788888;
        double r788890 = r788886 - r788889;
        double r788891 = sqrt(r788890);
        double r788892 = r788885 - r788891;
        double r788893 = r788892 / r788887;
        return r788893;
}
↓
double f(double a, double b_2, double c) {
        double r788894 = b_2;
        double r788895 = -3.136683434005781e-32;
        bool r788896 = r788894 <= r788895;
        double r788897 = -0.5;
        double r788898 = c;
        double r788899 = r788898 / r788894;
        double r788900 = r788897 * r788899;
        double r788901 = 2.0410715251838527e+49;
        bool r788902 = r788894 <= r788901;
        double r788903 = -r788894;
        double r788904 = r788894 * r788894;
        double r788905 = a;
        double r788906 = r788905 * r788898;
        double r788907 = r788904 - r788906;
        double r788908 = sqrt(r788907);
        double r788909 = r788903 - r788908;
        double r788910 = r788909 / r788905;
        double r788911 = 0.5;
        double r788912 = r788899 * r788911;
        double r788913 = 2.0;
        double r788914 = r788894 / r788905;
        double r788915 = r788913 * r788914;
        double r788916 = r788912 - r788915;
        double r788917 = r788902 ? r788910 : r788916;
        double r788918 = r788896 ? r788900 : r788917;
        return r788918;
}

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 < -3.136683434005781e-32

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 7.3

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

    if -3.136683434005781e-32 < b_2 < 2.0410715251838527e+49

    1. Initial program 15.8

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

    if 2.0410715251838527e+49 < b_2

    1. Initial program 36.2

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

    2. Taylor expanded around inf 6.2

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.0410715251838527 \cdot 10^{+49}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))