Average Error: 34.5 → 11.0
Time: 7.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.388070047225937856958905133202240499626 \cdot 10^{-143}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.388070047225937856958905133202240499626 \cdot 10^{-143}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r123895 = b;
        double r123896 = -r123895;
        double r123897 = r123895 * r123895;
        double r123898 = 4.0;
        double r123899 = a;
        double r123900 = r123898 * r123899;
        double r123901 = c;
        double r123902 = r123900 * r123901;
        double r123903 = r123897 - r123902;
        double r123904 = sqrt(r123903);
        double r123905 = r123896 + r123904;
        double r123906 = 2.0;
        double r123907 = r123906 * r123899;
        double r123908 = r123905 / r123907;
        return r123908;
}
↓
double f(double a, double b, double c) {
        double r123909 = b;
        double r123910 = -2.6152573735422387e+153;
        bool r123911 = r123909 <= r123910;
        double r123912 = 1.0;
        double r123913 = c;
        double r123914 = r123913 / r123909;
        double r123915 = a;
        double r123916 = r123909 / r123915;
        double r123917 = r123914 - r123916;
        double r123918 = r123912 * r123917;
        double r123919 = 1.3880700472259379e-143;
        bool r123920 = r123909 <= r123919;
        double r123921 = -r123909;
        double r123922 = r123909 * r123909;
        double r123923 = 4.0;
        double r123924 = r123923 * r123915;
        double r123925 = r123924 * r123913;
        double r123926 = r123922 - r123925;
        double r123927 = sqrt(r123926);
        double r123928 = r123921 + r123927;
        double r123929 = 2.0;
        double r123930 = r123929 * r123915;
        double r123931 = r123928 / r123930;
        double r123932 = -1.0;
        double r123933 = r123932 * r123914;
        double r123934 = r123920 ? r123931 : r123933;
        double r123935 = r123911 ? r123918 : r123934;
        return r123935;
}

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

Original34.5
Target21.5
Herbie11.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.6152573735422387e+153

    1. Initial program 63.8

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

    2. Taylor expanded around -inf 2.1

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

    3. Simplified2.1

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

    if -2.6152573735422387e+153 < b < 1.3880700472259379e-143

    1. Initial program 11.5

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

    if 1.3880700472259379e-143 < b

    1. Initial program 50.3

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

    2. Taylor expanded around inf 12.6

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.388070047225937856958905133202240499626 \cdot 10^{-143}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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)))