Average Error: 43.6 → 11.3
Time: 13.2s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.004017556086995931:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 0.004017556086995931:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r581837 = b;
        double r581838 = -r581837;
        double r581839 = r581837 * r581837;
        double r581840 = 4.0;
        double r581841 = a;
        double r581842 = r581840 * r581841;
        double r581843 = c;
        double r581844 = r581842 * r581843;
        double r581845 = r581839 - r581844;
        double r581846 = sqrt(r581845);
        double r581847 = r581838 + r581846;
        double r581848 = 2.0;
        double r581849 = r581848 * r581841;
        double r581850 = r581847 / r581849;
        return r581850;
}
↓
double f(double a, double b, double c) {
        double r581851 = b;
        double r581852 = 0.004017556086995931;
        bool r581853 = r581851 <= r581852;
        double r581854 = r581851 * r581851;
        double r581855 = 4.0;
        double r581856 = c;
        double r581857 = a;
        double r581858 = r581856 * r581857;
        double r581859 = r581855 * r581858;
        double r581860 = r581854 - r581859;
        double r581861 = r581860 - r581854;
        double r581862 = sqrt(r581860);
        double r581863 = r581851 + r581862;
        double r581864 = r581861 / r581863;
        double r581865 = r581864 / r581857;
        double r581866 = 2.0;
        double r581867 = r581865 / r581866;
        double r581868 = r581856 / r581851;
        double r581869 = -2.0;
        double r581870 = r581868 * r581869;
        double r581871 = r581870 / r581866;
        double r581872 = r581853 ? r581867 : r581871;
        return r581872;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if b < 0.004017556086995931

    1. Initial program 20.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified20.7

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}}\]
    3. Using strategy rm

    4. Applied flip--20.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}}{a}}{2}\]

    5. Simplified19.6

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(b \cdot b - \left(c \cdot a\right) \cdot 4\right) - b \cdot b}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}{a}}{2}\]

    if 0.004017556086995931 < b

    1. Initial program 45.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Simplified45.9

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}}\]

    3. Taylor expanded around inf 10.5

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.004017556086995931:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))