Average Error: 34.0 → 9.6
Time: 6.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.5940112039867074 \cdot 10^{100}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.5940112039867074 \cdot 10^{100}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r171892 = b;
        double r171893 = -r171892;
        double r171894 = r171892 * r171892;
        double r171895 = 4.0;
        double r171896 = a;
        double r171897 = r171895 * r171896;
        double r171898 = c;
        double r171899 = r171897 * r171898;
        double r171900 = r171894 - r171899;
        double r171901 = sqrt(r171900);
        double r171902 = r171893 + r171901;
        double r171903 = 2.0;
        double r171904 = r171903 * r171896;
        double r171905 = r171902 / r171904;
        return r171905;
}
↓
double f(double a, double b, double c) {
        double r171906 = b;
        double r171907 = -3.5940112039867074e+100;
        bool r171908 = r171906 <= r171907;
        double r171909 = 1.0;
        double r171910 = c;
        double r171911 = r171910 / r171906;
        double r171912 = a;
        double r171913 = r171906 / r171912;
        double r171914 = r171911 - r171913;
        double r171915 = r171909 * r171914;
        double r171916 = 2.267195199467958e-82;
        bool r171917 = r171906 <= r171916;
        double r171918 = 1.0;
        double r171919 = 2.0;
        double r171920 = r171919 * r171912;
        double r171921 = -r171906;
        double r171922 = r171906 * r171906;
        double r171923 = 4.0;
        double r171924 = r171923 * r171912;
        double r171925 = r171924 * r171910;
        double r171926 = r171922 - r171925;
        double r171927 = sqrt(r171926);
        double r171928 = r171921 + r171927;
        double r171929 = r171920 / r171928;
        double r171930 = r171918 / r171929;
        double r171931 = -1.0;
        double r171932 = r171931 * r171911;
        double r171933 = r171917 ? r171930 : r171932;
        double r171934 = r171908 ? r171915 : r171933;
        return r171934;
}

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
Target20.7
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.5940112039867074e+100

    1. Initial program 47.3

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

    2. Taylor expanded around -inf 3.8

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

    3. Simplified3.8

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

    if -3.5940112039867074e+100 < b < 2.267195199467958e-82

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    if 2.267195199467958e-82 < b

    1. Initial program 52.9

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.5940112039867074 \cdot 10^{100}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))