Average Error: 33.1 → 6.6
Time: 58.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1884247920746475 \cdot 10^{+101}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -1.6086609448752587 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \frac{1}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{elif}\;b \le 9.34348145460108 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b} \cdot \left(c \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\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 -2.1884247920746475 \cdot 10^{+101}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le -1.6086609448752587 \cdot 10^{-299}:\\
\;\;\;\;\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \frac{1}{a \cdot 2} - \frac{b}{a \cdot 2}\\

\mathbf{elif}\;b \le 9.34348145460108 \cdot 10^{+88}:\\
\;\;\;\;\frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b} \cdot \left(c \cdot -2\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r4784977 = b;
        double r4784978 = -r4784977;
        double r4784979 = r4784977 * r4784977;
        double r4784980 = 4.0;
        double r4784981 = a;
        double r4784982 = r4784980 * r4784981;
        double r4784983 = c;
        double r4784984 = r4784982 * r4784983;
        double r4784985 = r4784979 - r4784984;
        double r4784986 = sqrt(r4784985);
        double r4784987 = r4784978 + r4784986;
        double r4784988 = 2.0;
        double r4784989 = r4784988 * r4784981;
        double r4784990 = r4784987 / r4784989;
        return r4784990;
}
↓
double f(double a, double b, double c) {
        double r4784991 = b;
        double r4784992 = -2.1884247920746475e+101;
        bool r4784993 = r4784991 <= r4784992;
        double r4784994 = c;
        double r4784995 = r4784994 / r4784991;
        double r4784996 = a;
        double r4784997 = r4784991 / r4784996;
        double r4784998 = r4784995 - r4784997;
        double r4784999 = -1.6086609448752587e-299;
        bool r4785000 = r4784991 <= r4784999;
        double r4785001 = r4784991 * r4784991;
        double r4785002 = r4784994 * r4784996;
        double r4785003 = 4.0;
        double r4785004 = r4785002 * r4785003;
        double r4785005 = r4785001 - r4785004;
        double r4785006 = sqrt(r4785005);
        double r4785007 = 1.0;
        double r4785008 = 2.0;
        double r4785009 = r4784996 * r4785008;
        double r4785010 = r4785007 / r4785009;
        double r4785011 = r4785006 * r4785010;
        double r4785012 = r4784991 / r4785009;
        double r4785013 = r4785011 - r4785012;
        double r4785014 = 9.34348145460108e+88;
        bool r4785015 = r4784991 <= r4785014;
        double r4785016 = r4785006 + r4784991;
        double r4785017 = r4785007 / r4785016;
        double r4785018 = -2.0;
        double r4785019 = r4784994 * r4785018;
        double r4785020 = r4785017 * r4785019;
        double r4785021 = -r4784994;
        double r4785022 = r4785021 / r4784991;
        double r4785023 = r4785015 ? r4785020 : r4785022;
        double r4785024 = r4785000 ? r4785013 : r4785023;
        double r4785025 = r4784993 ? r4784998 : r4785024;
        return r4785025;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -2.1884247920746475e+101

    1. Initial program 44.5

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

    2. Simplified44.5

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

    4. Applied div-sub44.5

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

    5. Taylor expanded around -inf 3.3

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

    if -2.1884247920746475e+101 < b < -1.6086609448752587e-299

    1. Initial program 8.7

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

    2. Simplified8.7

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

    4. Applied div-sub8.7

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

    6. Applied div-inv8.8

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

    if -1.6086609448752587e-299 < b < 9.34348145460108e+88

    1. Initial program 31.3

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

    2. Simplified31.3

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

    4. Applied *-un-lft-identity31.3

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

    5. Applied *-un-lft-identity31.3

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

    6. Applied distribute-lft-out--31.3

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

    7. Applied associate-/l*31.4

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}}\]
    8. Using strategy rm

    9. Applied flip--31.5

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

    10. Applied associate-/r/31.5

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

    11. Applied *-un-lft-identity31.5

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

    12. Applied times-frac31.5

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

    13. Simplified16.1

      \[\leadsto \color{blue}{\frac{0 - \left(c \cdot a\right) \cdot 4}{\frac{a}{\frac{1}{2}}}} \cdot \frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\]

    14. Taylor expanded around inf 9.2

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

    if 9.34348145460108e+88 < b

    1. Initial program 57.8

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

    2. Simplified57.8

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

    4. Applied div-sub58.8

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

    5. Taylor expanded around inf 2.8

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

    6. Simplified2.8

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1884247920746475 \cdot 10^{+101}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -1.6086609448752587 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \frac{1}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{elif}\;b \le 9.34348145460108 \cdot 10^{+88}:\\ \;\;\;\;\frac{1}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b} \cdot \left(c \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))