Average Error: 34.0 → 7.2
Time: 5.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.71711085460076329 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.9300475349170912 \cdot 10^{-278}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3461964491124549:\\ \;\;\;\;{\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.71711085460076329 \cdot 10^{118}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -2.9300475349170912 \cdot 10^{-278}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 3461964491124549:\\
\;\;\;\;{\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}\\

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

\end{array}
double f(double a, double b, double c) {
        double r65929 = b;
        double r65930 = -r65929;
        double r65931 = r65929 * r65929;
        double r65932 = 4.0;
        double r65933 = a;
        double r65934 = r65932 * r65933;
        double r65935 = c;
        double r65936 = r65934 * r65935;
        double r65937 = r65931 - r65936;
        double r65938 = sqrt(r65937);
        double r65939 = r65930 + r65938;
        double r65940 = 2.0;
        double r65941 = r65940 * r65933;
        double r65942 = r65939 / r65941;
        return r65942;
}
↓
double f(double a, double b, double c) {
        double r65943 = b;
        double r65944 = -3.7171108546007633e+118;
        bool r65945 = r65943 <= r65944;
        double r65946 = 1.0;
        double r65947 = c;
        double r65948 = r65947 / r65943;
        double r65949 = a;
        double r65950 = r65943 / r65949;
        double r65951 = r65948 - r65950;
        double r65952 = r65946 * r65951;
        double r65953 = -2.930047534917091e-278;
        bool r65954 = r65943 <= r65953;
        double r65955 = -r65943;
        double r65956 = r65943 * r65943;
        double r65957 = 4.0;
        double r65958 = r65957 * r65949;
        double r65959 = r65958 * r65947;
        double r65960 = r65956 - r65959;
        double r65961 = sqrt(r65960);
        double r65962 = r65955 + r65961;
        double r65963 = 2.0;
        double r65964 = r65963 * r65949;
        double r65965 = r65962 / r65964;
        double r65966 = 3461964491124549.0;
        bool r65967 = r65943 <= r65966;
        double r65968 = r65963 * r65947;
        double r65969 = r65955 - r65961;
        double r65970 = r65968 / r65969;
        double r65971 = 1.0;
        double r65972 = pow(r65970, r65971);
        double r65973 = -1.0;
        double r65974 = r65973 * r65948;
        double r65975 = r65967 ? r65972 : r65974;
        double r65976 = r65954 ? r65965 : r65975;
        double r65977 = r65945 ? r65952 : r65976;
        return r65977;
}

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 < -3.7171108546007633e+118

    1. Initial program 52.0

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

    2. Taylor expanded around -inf 2.9

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

    3. Simplified2.9

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

    if -3.7171108546007633e+118 < b < -2.930047534917091e-278

    1. Initial program 8.5

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

    if -2.930047534917091e-278 < b < 3461964491124549.0

    1. Initial program 26.5

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

    3. Applied flip-+26.6

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

    4. Simplified16.3

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

    6. Applied div-inv16.4

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

    8. Applied pow116.4

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

    9. Applied pow116.4

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

    10. Applied pow-prod-down16.4

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

    11. Simplified16.2

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

    12. Taylor expanded around 0 10.0

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

    if 3461964491124549.0 < b

    1. Initial program 55.9

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

    2. Taylor expanded around inf 5.5

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.71711085460076329 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.9300475349170912 \cdot 10^{-278}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3461964491124549:\\ \;\;\;\;{\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))