Average Error: 34.4 → 10.6
Time: 18.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\
\;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r4430899 = b;
        double r4430900 = -r4430899;
        double r4430901 = r4430899 * r4430899;
        double r4430902 = 4.0;
        double r4430903 = a;
        double r4430904 = r4430902 * r4430903;
        double r4430905 = c;
        double r4430906 = r4430904 * r4430905;
        double r4430907 = r4430901 - r4430906;
        double r4430908 = sqrt(r4430907);
        double r4430909 = r4430900 + r4430908;
        double r4430910 = 2.0;
        double r4430911 = r4430910 * r4430903;
        double r4430912 = r4430909 / r4430911;
        return r4430912;
}
↓
double f(double a, double b, double c) {
        double r4430913 = b;
        double r4430914 = -2.221067196710922e+149;
        bool r4430915 = r4430913 <= r4430914;
        double r4430916 = 2.0;
        double r4430917 = c;
        double r4430918 = r4430917 / r4430913;
        double r4430919 = r4430916 * r4430918;
        double r4430920 = a;
        double r4430921 = r4430913 / r4430920;
        double r4430922 = 2.0;
        double r4430923 = r4430921 * r4430922;
        double r4430924 = r4430919 - r4430923;
        double r4430925 = r4430924 / r4430916;
        double r4430926 = 2.8983489306952693e-35;
        bool r4430927 = r4430913 <= r4430926;
        double r4430928 = r4430913 * r4430913;
        double r4430929 = r4430917 * r4430920;
        double r4430930 = 4.0;
        double r4430931 = r4430929 * r4430930;
        double r4430932 = r4430928 - r4430931;
        double r4430933 = sqrt(r4430932);
        double r4430934 = r4430933 - r4430913;
        double r4430935 = r4430934 / r4430920;
        double r4430936 = r4430935 / r4430916;
        double r4430937 = -2.0;
        double r4430938 = r4430937 * r4430918;
        double r4430939 = r4430938 / r4430916;
        double r4430940 = r4430927 ? r4430936 : r4430939;
        double r4430941 = r4430915 ? r4430925 : r4430940;
        return r4430941;
}

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

Original34.4
Target21.5
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.221067196710922e+149

    1. Initial program 62.3

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

    2. Simplified62.3

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

    3. Taylor expanded around -inf 2.8

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    if -2.221067196710922e+149 < b < 2.8983489306952693e-35

    1. Initial program 14.6

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

    2. Simplified14.6

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

    if 2.8983489306952693e-35 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.3

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))