Average Error: 34.4 → 14.2
Time: 22.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{\frac{1}{\frac{-1}{b}}}{a}\\ \mathbf{elif}\;b \le 5.860223638943180333955717619400031865396 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{b} - \frac{b}{c \cdot a} \cdot 1}}{a}\\ \end{array}\]
\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 -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;\frac{\frac{1}{\frac{-1}{b}}}{a}\\

\mathbf{elif}\;b \le 5.860223638943180333955717619400031865396 \cdot 10^{-17}:\\
\;\;\;\;\frac{\frac{1}{\frac{2}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1895957 = b;
        double r1895958 = -r1895957;
        double r1895959 = r1895957 * r1895957;
        double r1895960 = 4.0;
        double r1895961 = a;
        double r1895962 = r1895960 * r1895961;
        double r1895963 = c;
        double r1895964 = r1895962 * r1895963;
        double r1895965 = r1895959 - r1895964;
        double r1895966 = sqrt(r1895965);
        double r1895967 = r1895958 + r1895966;
        double r1895968 = 2.0;
        double r1895969 = r1895968 * r1895961;
        double r1895970 = r1895967 / r1895969;
        return r1895970;
}
↓
double f(double a, double b, double c) {
        double r1895971 = b;
        double r1895972 = -1.7633154797394035e+89;
        bool r1895973 = r1895971 <= r1895972;
        double r1895974 = 1.0;
        double r1895975 = -1.0;
        double r1895976 = r1895975 / r1895971;
        double r1895977 = r1895974 / r1895976;
        double r1895978 = a;
        double r1895979 = r1895977 / r1895978;
        double r1895980 = 5.86022363894318e-17;
        bool r1895981 = r1895971 <= r1895980;
        double r1895982 = 2.0;
        double r1895983 = r1895971 * r1895971;
        double r1895984 = 4.0;
        double r1895985 = c;
        double r1895986 = r1895985 * r1895978;
        double r1895987 = r1895984 * r1895986;
        double r1895988 = r1895983 - r1895987;
        double r1895989 = sqrt(r1895988);
        double r1895990 = r1895989 - r1895971;
        double r1895991 = r1895982 / r1895990;
        double r1895992 = r1895974 / r1895991;
        double r1895993 = r1895992 / r1895978;
        double r1895994 = 1.0;
        double r1895995 = r1895994 / r1895971;
        double r1895996 = r1895971 / r1895986;
        double r1895997 = r1895996 * r1895994;
        double r1895998 = r1895995 - r1895997;
        double r1895999 = r1895974 / r1895998;
        double r1896000 = r1895999 / r1895978;
        double r1896001 = r1895981 ? r1895993 : r1896000;
        double r1896002 = r1895973 ? r1895979 : r1896001;
        return r1896002;
}

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 3 regimes

  2. if b < -1.7633154797394035e+89

    1. Initial program 45.7

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

    2. Simplified45.7

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

    4. Applied clear-num45.8

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

    5. Taylor expanded around -inf 4.2

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

    if -1.7633154797394035e+89 < b < 5.86022363894318e-17

    1. Initial program 15.4

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

    2. Simplified15.4

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

    4. Applied clear-num15.4

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

    if 5.86022363894318e-17 < b

    1. Initial program 55.6

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

    2. Simplified55.6

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

    4. Applied clear-num55.6

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

    5. Taylor expanded around inf 17.3

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

    6. Simplified17.3

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

  4. Final simplification14.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{\frac{1}{\frac{-1}{b}}}{a}\\ \mathbf{elif}\;b \le 5.860223638943180333955717619400031865396 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{b} - \frac{b}{c \cdot a} \cdot 1}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))