Average Error: 34.0 → 9.9
Time: 5.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.4336717856540716 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.910803917041586 \cdot 10^{80}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.4336717856540716 \cdot 10^{-82}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.910803917041586 \cdot 10^{80}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r79832 = b;
        double r79833 = -r79832;
        double r79834 = r79832 * r79832;
        double r79835 = 4.0;
        double r79836 = a;
        double r79837 = c;
        double r79838 = r79836 * r79837;
        double r79839 = r79835 * r79838;
        double r79840 = r79834 - r79839;
        double r79841 = sqrt(r79840);
        double r79842 = r79833 - r79841;
        double r79843 = 2.0;
        double r79844 = r79843 * r79836;
        double r79845 = r79842 / r79844;
        return r79845;
}
↓
double f(double a, double b, double c) {
        double r79846 = b;
        double r79847 = -2.4336717856540716e-82;
        bool r79848 = r79846 <= r79847;
        double r79849 = -1.0;
        double r79850 = c;
        double r79851 = r79850 / r79846;
        double r79852 = r79849 * r79851;
        double r79853 = 2.910803917041586e+80;
        bool r79854 = r79846 <= r79853;
        double r79855 = 1.0;
        double r79856 = 2.0;
        double r79857 = a;
        double r79858 = r79856 * r79857;
        double r79859 = -r79846;
        double r79860 = r79846 * r79846;
        double r79861 = 4.0;
        double r79862 = r79857 * r79850;
        double r79863 = r79861 * r79862;
        double r79864 = r79860 - r79863;
        double r79865 = sqrt(r79864);
        double r79866 = r79859 - r79865;
        double r79867 = r79858 / r79866;
        double r79868 = r79855 / r79867;
        double r79869 = 1.0;
        double r79870 = r79846 / r79857;
        double r79871 = r79851 - r79870;
        double r79872 = r79869 * r79871;
        double r79873 = r79854 ? r79868 : r79872;
        double r79874 = r79848 ? r79852 : r79873;
        return r79874;
}

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

Original34.0
Target21.0
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.4336717856540716e-82

    1. Initial program 53.1

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

    2. Taylor expanded around -inf 9.4

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

    if -2.4336717856540716e-82 < b < 2.910803917041586e+80

    1. Initial program 12.8

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

    3. Applied clear-num12.9

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

    if 2.910803917041586e+80 < b

    1. Initial program 42.9

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

    2. Taylor expanded around inf 4.0

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

    3. Simplified4.0

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.4336717856540716 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.910803917041586 \cdot 10^{80}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(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)))