Average Error: 33.7 → 10.2
Time: 20.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.3248915655872564 \cdot 10^{+79}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.2796532586596585 \cdot 10^{-91}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -5.3248915655872564 \cdot 10^{+79}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 4.2796532586596585 \cdot 10^{-91}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1190903 = b;
        double r1190904 = -r1190903;
        double r1190905 = r1190903 * r1190903;
        double r1190906 = 4.0;
        double r1190907 = a;
        double r1190908 = r1190906 * r1190907;
        double r1190909 = c;
        double r1190910 = r1190908 * r1190909;
        double r1190911 = r1190905 - r1190910;
        double r1190912 = sqrt(r1190911);
        double r1190913 = r1190904 + r1190912;
        double r1190914 = 2.0;
        double r1190915 = r1190914 * r1190907;
        double r1190916 = r1190913 / r1190915;
        return r1190916;
}
↓
double f(double a, double b, double c) {
        double r1190917 = b;
        double r1190918 = -5.3248915655872564e+79;
        bool r1190919 = r1190917 <= r1190918;
        double r1190920 = c;
        double r1190921 = r1190920 / r1190917;
        double r1190922 = a;
        double r1190923 = r1190917 / r1190922;
        double r1190924 = r1190921 - r1190923;
        double r1190925 = 4.2796532586596585e-91;
        bool r1190926 = r1190917 <= r1190925;
        double r1190927 = -r1190917;
        double r1190928 = r1190917 * r1190917;
        double r1190929 = 4.0;
        double r1190930 = r1190929 * r1190922;
        double r1190931 = r1190920 * r1190930;
        double r1190932 = r1190928 - r1190931;
        double r1190933 = sqrt(r1190932);
        double r1190934 = r1190927 + r1190933;
        double r1190935 = 0.5;
        double r1190936 = r1190935 / r1190922;
        double r1190937 = r1190934 * r1190936;
        double r1190938 = -r1190921;
        double r1190939 = r1190926 ? r1190937 : r1190938;
        double r1190940 = r1190919 ? r1190924 : r1190939;
        return r1190940;
}

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 3 regimes

  2. if b < -5.3248915655872564e+79

    1. Initial program 41.1

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

    2. Taylor expanded around -inf 4.6

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]

    if -5.3248915655872564e+79 < b < 4.2796532586596585e-91

    1. Initial program 13.0

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

    3. Applied div-inv13.1

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}}\]

    4. Simplified13.1

      \[\leadsto \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\]

    if 4.2796532586596585e-91 < b

    1. Initial program 52.0

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

    2. Taylor expanded around inf 9.6

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

    3. Simplified9.6

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.3248915655872564 \cdot 10^{+79}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.2796532586596585 \cdot 10^{-91}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))