Average Error: 33.6 → 11.2
Time: 19.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.2890050783826923 \cdot 10^{-183}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.786204067849289 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}{a \cdot 2}\\ \end{array}\]
\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 -1.2890050783826923 \cdot 10^{-183}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.786204067849289 \cdot 10^{+100}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}{a \cdot 2}\\

\end{array}
double f(double a, double b, double c) {
        double r1621895 = b;
        double r1621896 = -r1621895;
        double r1621897 = r1621895 * r1621895;
        double r1621898 = 4.0;
        double r1621899 = a;
        double r1621900 = c;
        double r1621901 = r1621899 * r1621900;
        double r1621902 = r1621898 * r1621901;
        double r1621903 = r1621897 - r1621902;
        double r1621904 = sqrt(r1621903);
        double r1621905 = r1621896 - r1621904;
        double r1621906 = 2.0;
        double r1621907 = r1621906 * r1621899;
        double r1621908 = r1621905 / r1621907;
        return r1621908;
}
↓
double f(double a, double b, double c) {
        double r1621909 = b;
        double r1621910 = -1.2890050783826923e-183;
        bool r1621911 = r1621909 <= r1621910;
        double r1621912 = c;
        double r1621913 = r1621912 / r1621909;
        double r1621914 = -r1621913;
        double r1621915 = 1.786204067849289e+100;
        bool r1621916 = r1621909 <= r1621915;
        double r1621917 = -r1621909;
        double r1621918 = -4.0;
        double r1621919 = a;
        double r1621920 = r1621918 * r1621919;
        double r1621921 = r1621920 * r1621912;
        double r1621922 = r1621909 * r1621909;
        double r1621923 = r1621921 + r1621922;
        double r1621924 = sqrt(r1621923);
        double r1621925 = r1621917 - r1621924;
        double r1621926 = 2.0;
        double r1621927 = r1621919 * r1621926;
        double r1621928 = r1621925 / r1621927;
        double r1621929 = r1621919 / r1621909;
        double r1621930 = r1621929 * r1621912;
        double r1621931 = r1621930 - r1621909;
        double r1621932 = r1621926 * r1621931;
        double r1621933 = r1621932 / r1621927;
        double r1621934 = r1621916 ? r1621928 : r1621933;
        double r1621935 = r1621911 ? r1621914 : r1621934;
        return r1621935;
}

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

Original33.6
Target20.5
Herbie11.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2890050783826923e-183

    1. Initial program 48.2

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

    2. Taylor expanded around 0 48.2

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

    3. Simplified48.2

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

    5. Applied div-inv48.2

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

    6. Simplified48.2

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

    7. Taylor expanded around -inf 14.3

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

    8. Simplified14.3

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

    if -1.2890050783826923e-183 < b < 1.786204067849289e+100

    1. Initial program 10.5

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

    2. Taylor expanded around 0 10.5

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

    3. Simplified10.5

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

    if 1.786204067849289e+100 < b

    1. Initial program 44.2

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified3.5

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2890050783826923 \cdot 10^{-183}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.786204067849289 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a}{b} \cdot c - b\right)}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 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)))