Average Error: 34.2 → 7.4
Time: 19.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.555632367828988861043913196266489993904 \cdot 10^{101}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.588581026022229142935221773282266391902 \cdot 10^{-168}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 87537227540251800037021545535125898395650:\\ \;\;\;\;\frac{\frac{\frac{4}{\frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.555632367828988861043913196266489993904 \cdot 10^{101}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.588581026022229142935221773282266391902 \cdot 10^{-168}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le 87537227540251800037021545535125898395650:\\
\;\;\;\;\frac{\frac{\frac{4}{\frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r46955 = b;
        double r46956 = -r46955;
        double r46957 = r46955 * r46955;
        double r46958 = 4.0;
        double r46959 = a;
        double r46960 = r46958 * r46959;
        double r46961 = c;
        double r46962 = r46960 * r46961;
        double r46963 = r46957 - r46962;
        double r46964 = sqrt(r46963);
        double r46965 = r46956 + r46964;
        double r46966 = 2.0;
        double r46967 = r46966 * r46959;
        double r46968 = r46965 / r46967;
        return r46968;
}
↓
double f(double a, double b, double c) {
        double r46969 = b;
        double r46970 = -1.555632367828989e+101;
        bool r46971 = r46969 <= r46970;
        double r46972 = 1.0;
        double r46973 = c;
        double r46974 = r46973 / r46969;
        double r46975 = a;
        double r46976 = r46969 / r46975;
        double r46977 = r46974 - r46976;
        double r46978 = r46972 * r46977;
        double r46979 = -1.5885810260222291e-168;
        bool r46980 = r46969 <= r46979;
        double r46981 = -r46969;
        double r46982 = 2.0;
        double r46983 = pow(r46969, r46982);
        double r46984 = 4.0;
        double r46985 = r46975 * r46973;
        double r46986 = r46984 * r46985;
        double r46987 = r46983 - r46986;
        double r46988 = sqrt(r46987);
        double r46989 = r46981 + r46988;
        double r46990 = 2.0;
        double r46991 = r46990 * r46975;
        double r46992 = r46989 / r46991;
        double r46993 = 8.75372275402518e+40;
        bool r46994 = r46969 <= r46993;
        double r46995 = 1.0;
        double r46996 = r46995 / r46973;
        double r46997 = r46984 / r46996;
        double r46998 = r46969 * r46969;
        double r46999 = r46984 * r46975;
        double r47000 = r46999 * r46973;
        double r47001 = r46998 - r47000;
        double r47002 = sqrt(r47001);
        double r47003 = r46981 - r47002;
        double r47004 = r46997 / r47003;
        double r47005 = r47004 / r46990;
        double r47006 = -1.0;
        double r47007 = r47006 * r46974;
        double r47008 = r46994 ? r47005 : r47007;
        double r47009 = r46980 ? r46992 : r47008;
        double r47010 = r46971 ? r46978 : r47009;
        return r47010;
}

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 < -1.555632367828989e+101

    1. Initial program 47.4

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -1.555632367828989e+101 < b < -1.5885810260222291e-168

    1. Initial program 7.2

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

    2. Taylor expanded around 0 7.2

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

    if -1.5885810260222291e-168 < b < 8.75372275402518e+40

    1. Initial program 25.8

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

    3. Applied flip-+26.0

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

    4. Simplified16.9

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(c \cdot 4\right) \cdot a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied div-inv17.0

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

    7. Applied times-frac23.1

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

    8. Simplified23.1

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

    10. Applied associate-*l/23.1

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

    11. Simplified22.9

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

    13. Applied associate-/r*16.8

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

    14. Simplified11.6

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

    if 8.75372275402518e+40 < b

    1. Initial program 56.7

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

    2. Taylor expanded around inf 4.5

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.555632367828988861043913196266489993904 \cdot 10^{101}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.588581026022229142935221773282266391902 \cdot 10^{-168}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 87537227540251800037021545535125898395650:\\ \;\;\;\;\frac{\frac{\frac{4}{\frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))