Average Error: 28.7 → 16.1
Time: 17.0s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 315.484838761318258:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 315.484838761318258:\\
\;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r27959 = b;
        double r27960 = -r27959;
        double r27961 = r27959 * r27959;
        double r27962 = 4.0;
        double r27963 = a;
        double r27964 = r27962 * r27963;
        double r27965 = c;
        double r27966 = r27964 * r27965;
        double r27967 = r27961 - r27966;
        double r27968 = sqrt(r27967);
        double r27969 = r27960 + r27968;
        double r27970 = 2.0;
        double r27971 = r27970 * r27963;
        double r27972 = r27969 / r27971;
        return r27972;
}
↓
double f(double a, double b, double c) {
        double r27973 = b;
        double r27974 = 315.48483876131826;
        bool r27975 = r27973 <= r27974;
        double r27976 = r27973 * r27973;
        double r27977 = 4.0;
        double r27978 = a;
        double r27979 = r27977 * r27978;
        double r27980 = c;
        double r27981 = r27979 * r27980;
        double r27982 = r27981 + r27976;
        double r27983 = r27976 - r27982;
        double r27984 = r27976 - r27981;
        double r27985 = sqrt(r27984);
        double r27986 = r27985 + r27973;
        double r27987 = r27983 / r27986;
        double r27988 = 2.0;
        double r27989 = r27987 / r27988;
        double r27990 = r27989 / r27978;
        double r27991 = -1.0;
        double r27992 = r27980 / r27973;
        double r27993 = r27991 * r27992;
        double r27994 = r27975 ? r27990 : r27993;
        return r27994;
}

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 2 regimes

  2. if b < 315.48483876131826

    1. Initial program 16.3

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

    2. Simplified16.3

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

    4. Applied flip--16.3

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

    5. Simplified15.3

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

    if 315.48483876131826 < b

    1. Initial program 35.8

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

    2. Simplified35.8

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

    3. Taylor expanded around inf 16.6

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 315.484838761318258:\\ \;\;\;\;\frac{\frac{\frac{b \cdot b - \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))