Average Error: 34.2 → 7.4
Time: 18.7s
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 r46908 = b;
        double r46909 = -r46908;
        double r46910 = r46908 * r46908;
        double r46911 = 4.0;
        double r46912 = a;
        double r46913 = r46911 * r46912;
        double r46914 = c;
        double r46915 = r46913 * r46914;
        double r46916 = r46910 - r46915;
        double r46917 = sqrt(r46916);
        double r46918 = r46909 + r46917;
        double r46919 = 2.0;
        double r46920 = r46919 * r46912;
        double r46921 = r46918 / r46920;
        return r46921;
}
↓
double f(double a, double b, double c) {
        double r46922 = b;
        double r46923 = -1.555632367828989e+101;
        bool r46924 = r46922 <= r46923;
        double r46925 = 1.0;
        double r46926 = c;
        double r46927 = r46926 / r46922;
        double r46928 = a;
        double r46929 = r46922 / r46928;
        double r46930 = r46927 - r46929;
        double r46931 = r46925 * r46930;
        double r46932 = -1.5885810260222291e-168;
        bool r46933 = r46922 <= r46932;
        double r46934 = -r46922;
        double r46935 = 2.0;
        double r46936 = pow(r46922, r46935);
        double r46937 = 4.0;
        double r46938 = r46928 * r46926;
        double r46939 = r46937 * r46938;
        double r46940 = r46936 - r46939;
        double r46941 = sqrt(r46940);
        double r46942 = r46934 + r46941;
        double r46943 = 2.0;
        double r46944 = r46943 * r46928;
        double r46945 = r46942 / r46944;
        double r46946 = 8.75372275402518e+40;
        bool r46947 = r46922 <= r46946;
        double r46948 = 1.0;
        double r46949 = r46948 / r46926;
        double r46950 = r46937 / r46949;
        double r46951 = r46922 * r46922;
        double r46952 = r46937 * r46928;
        double r46953 = r46952 * r46926;
        double r46954 = r46951 - r46953;
        double r46955 = sqrt(r46954);
        double r46956 = r46934 - r46955;
        double r46957 = r46950 / r46956;
        double r46958 = r46957 / r46943;
        double r46959 = -1.0;
        double r46960 = r46959 * r46927;
        double r46961 = r46947 ? r46958 : r46960;
        double r46962 = r46933 ? r46945 : r46961;
        double r46963 = r46924 ? r46931 : r46962;
        return r46963;
}

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)))