Average Error: 34.8 → 10.7
Time: 5.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.60514141786167054 \cdot 10^{33}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.92049775718538 \cdot 10^{-66}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.60514141786167054 \cdot 10^{33}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.92049775718538 \cdot 10^{-66}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r117897 = b;
        double r117898 = -r117897;
        double r117899 = r117897 * r117897;
        double r117900 = 4.0;
        double r117901 = a;
        double r117902 = r117900 * r117901;
        double r117903 = c;
        double r117904 = r117902 * r117903;
        double r117905 = r117899 - r117904;
        double r117906 = sqrt(r117905);
        double r117907 = r117898 + r117906;
        double r117908 = 2.0;
        double r117909 = r117908 * r117901;
        double r117910 = r117907 / r117909;
        return r117910;
}
↓
double f(double a, double b, double c) {
        double r117911 = b;
        double r117912 = -4.6051414178616705e+33;
        bool r117913 = r117911 <= r117912;
        double r117914 = 1.0;
        double r117915 = c;
        double r117916 = r117915 / r117911;
        double r117917 = a;
        double r117918 = r117911 / r117917;
        double r117919 = r117916 - r117918;
        double r117920 = r117914 * r117919;
        double r117921 = 1.92049775718538e-66;
        bool r117922 = r117911 <= r117921;
        double r117923 = -r117911;
        double r117924 = r117911 * r117911;
        double r117925 = 4.0;
        double r117926 = r117925 * r117917;
        double r117927 = r117926 * r117915;
        double r117928 = r117924 - r117927;
        double r117929 = sqrt(r117928);
        double r117930 = r117923 + r117929;
        double r117931 = 1.0;
        double r117932 = 2.0;
        double r117933 = r117932 * r117917;
        double r117934 = r117931 / r117933;
        double r117935 = r117930 * r117934;
        double r117936 = -1.0;
        double r117937 = r117936 * r117916;
        double r117938 = r117922 ? r117935 : r117937;
        double r117939 = r117913 ? r117920 : r117938;
        return r117939;
}

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

Original34.8
Target21.2
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.6051414178616705e+33

    1. Initial program 36.3

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

    2. Taylor expanded around -inf 6.9

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

    3. Simplified6.9

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

    if -4.6051414178616705e+33 < b < 1.92049775718538e-66

    1. Initial program 15.2

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

    3. Applied div-inv15.3

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

    if 1.92049775718538e-66 < b

    1. Initial program 54.0

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

    2. Taylor expanded around inf 8.1

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.60514141786167054 \cdot 10^{33}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.92049775718538 \cdot 10^{-66}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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)))