Average Error: 34.3 → 8.7
Time: 4.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.8746290509448952 \cdot 10^{74}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.1885507100476909 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 1.1800329617120703 \cdot 10^{123}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -1.8746290509448952 \cdot 10^{74}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.1885507100476909 \cdot 10^{-281}:\\
\;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 1.1800329617120703 \cdot 10^{123}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r79857 = b;
        double r79858 = -r79857;
        double r79859 = r79857 * r79857;
        double r79860 = 4.0;
        double r79861 = a;
        double r79862 = c;
        double r79863 = r79861 * r79862;
        double r79864 = r79860 * r79863;
        double r79865 = r79859 - r79864;
        double r79866 = sqrt(r79865);
        double r79867 = r79858 - r79866;
        double r79868 = 2.0;
        double r79869 = r79868 * r79861;
        double r79870 = r79867 / r79869;
        return r79870;
}
↓
double f(double a, double b, double c) {
        double r79871 = b;
        double r79872 = -1.8746290509448952e+74;
        bool r79873 = r79871 <= r79872;
        double r79874 = -1.0;
        double r79875 = c;
        double r79876 = r79875 / r79871;
        double r79877 = r79874 * r79876;
        double r79878 = -1.1885507100476909e-281;
        bool r79879 = r79871 <= r79878;
        double r79880 = 1.0;
        double r79881 = 2.0;
        double r79882 = r79880 / r79881;
        double r79883 = 2.0;
        double r79884 = pow(r79871, r79883);
        double r79885 = r79884 - r79884;
        double r79886 = 4.0;
        double r79887 = a;
        double r79888 = r79887 * r79875;
        double r79889 = r79886 * r79888;
        double r79890 = r79885 + r79889;
        double r79891 = r79882 * r79890;
        double r79892 = r79891 / r79887;
        double r79893 = -r79871;
        double r79894 = r79871 * r79871;
        double r79895 = r79894 - r79889;
        double r79896 = sqrt(r79895);
        double r79897 = r79893 + r79896;
        double r79898 = r79892 / r79897;
        double r79899 = 1.1800329617120703e+123;
        bool r79900 = r79871 <= r79899;
        double r79901 = r79881 * r79887;
        double r79902 = r79893 - r79896;
        double r79903 = r79901 / r79902;
        double r79904 = r79880 / r79903;
        double r79905 = r79871 / r79887;
        double r79906 = r79874 * r79905;
        double r79907 = r79900 ? r79904 : r79906;
        double r79908 = r79879 ? r79898 : r79907;
        double r79909 = r79873 ? r79877 : r79908;
        return r79909;
}

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

Original34.3
Target21.3
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8746290509448952e+74

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 3.3

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

    if -1.8746290509448952e+74 < b < -1.1885507100476909e-281

    1. Initial program 31.9

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

    3. Applied clear-num32.0

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

    5. Applied flip--32.0

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

    6. Applied associate-/r/32.0

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

    7. Applied associate-/r*32.1

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

    8. Simplified15.9

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

    if -1.1885507100476909e-281 < b < 1.1800329617120703e+123

    1. Initial program 9.3

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

    3. Applied clear-num9.4

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

    if 1.1800329617120703e+123 < b

    1. Initial program 53.2

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

    3. Applied clear-num53.3

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

    4. Taylor expanded around 0 3.2

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.8746290509448952 \cdot 10^{74}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.1885507100476909 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 1.1800329617120703 \cdot 10^{123}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

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