Average Error: 32.9 → 10.6
Time: 21.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.1701110130378705 \cdot 10^{+68}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.4352467544377554 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -6.1701110130378705 \cdot 10^{+68}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.4352467544377554 \cdot 10^{-114}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} - \frac{b}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2833865 = b;
        double r2833866 = -r2833865;
        double r2833867 = r2833865 * r2833865;
        double r2833868 = 4.0;
        double r2833869 = a;
        double r2833870 = c;
        double r2833871 = r2833869 * r2833870;
        double r2833872 = r2833868 * r2833871;
        double r2833873 = r2833867 - r2833872;
        double r2833874 = sqrt(r2833873);
        double r2833875 = r2833866 + r2833874;
        double r2833876 = 2.0;
        double r2833877 = r2833876 * r2833869;
        double r2833878 = r2833875 / r2833877;
        return r2833878;
}
↓
double f(double a, double b, double c) {
        double r2833879 = b;
        double r2833880 = -6.1701110130378705e+68;
        bool r2833881 = r2833879 <= r2833880;
        double r2833882 = c;
        double r2833883 = r2833882 / r2833879;
        double r2833884 = a;
        double r2833885 = r2833879 / r2833884;
        double r2833886 = r2833883 - r2833885;
        double r2833887 = 1.4352467544377554e-114;
        bool r2833888 = r2833879 <= r2833887;
        double r2833889 = 1.0;
        double r2833890 = 2.0;
        double r2833891 = r2833884 * r2833890;
        double r2833892 = r2833879 * r2833879;
        double r2833893 = 4.0;
        double r2833894 = r2833893 * r2833884;
        double r2833895 = r2833894 * r2833882;
        double r2833896 = r2833892 - r2833895;
        double r2833897 = sqrt(r2833896);
        double r2833898 = r2833891 / r2833897;
        double r2833899 = r2833889 / r2833898;
        double r2833900 = r2833879 / r2833891;
        double r2833901 = r2833899 - r2833900;
        double r2833902 = -r2833883;
        double r2833903 = r2833888 ? r2833901 : r2833902;
        double r2833904 = r2833881 ? r2833886 : r2833903;
        return r2833904;
}

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

Target

Original32.9
Target20.2
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.1701110130378705e+68

    1. Initial program 38.1

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

    2. Simplified38.1

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

    4. Applied div-sub38.1

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

    5. Taylor expanded around -inf 4.7

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

    if -6.1701110130378705e+68 < b < 1.4352467544377554e-114

    1. Initial program 12.0

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

    2. Simplified12.0

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

    4. Applied div-sub12.1

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

    6. Applied clear-num12.1

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

    if 1.4352467544377554e-114 < b

    1. Initial program 50.8

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

    2. Simplified50.9

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

    3. Taylor expanded around inf 11.5

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

    4. Simplified11.5

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.1701110130378705 \cdot 10^{+68}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.4352467544377554 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))