Average Error: 28.5 → 16.7
Time: 16.3s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2892.1913455639924:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b + \left(c \cdot a\right) \cdot -4\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - \left(b \cdot b\right) \cdot b}{\left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} + b \cdot b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 2892.1913455639924:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b + \left(c \cdot a\right) \cdot -4\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - \left(b \cdot b\right) \cdot b}{\left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} + b \cdot b}}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1061885 = b;
        double r1061886 = -r1061885;
        double r1061887 = r1061885 * r1061885;
        double r1061888 = 4.0;
        double r1061889 = a;
        double r1061890 = r1061888 * r1061889;
        double r1061891 = c;
        double r1061892 = r1061890 * r1061891;
        double r1061893 = r1061887 - r1061892;
        double r1061894 = sqrt(r1061893);
        double r1061895 = r1061886 + r1061894;
        double r1061896 = 2.0;
        double r1061897 = r1061896 * r1061889;
        double r1061898 = r1061895 / r1061897;
        return r1061898;
}
↓
double f(double a, double b, double c) {
        double r1061899 = b;
        double r1061900 = 2892.1913455639924;
        bool r1061901 = r1061899 <= r1061900;
        double r1061902 = r1061899 * r1061899;
        double r1061903 = c;
        double r1061904 = a;
        double r1061905 = r1061903 * r1061904;
        double r1061906 = -4.0;
        double r1061907 = r1061905 * r1061906;
        double r1061908 = r1061902 + r1061907;
        double r1061909 = sqrt(r1061908);
        double r1061910 = r1061908 * r1061909;
        double r1061911 = r1061902 * r1061899;
        double r1061912 = r1061910 - r1061911;
        double r1061913 = r1061899 + r1061909;
        double r1061914 = r1061913 * r1061909;
        double r1061915 = r1061914 + r1061902;
        double r1061916 = r1061912 / r1061915;
        double r1061917 = r1061916 / r1061904;
        double r1061918 = 2.0;
        double r1061919 = r1061917 / r1061918;
        double r1061920 = -2.0;
        double r1061921 = r1061903 / r1061899;
        double r1061922 = r1061920 * r1061921;
        double r1061923 = r1061922 / r1061918;
        double r1061924 = r1061901 ? r1061919 : r1061923;
        return r1061924;
}

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 < 2892.1913455639924

    1. Initial program 18.5

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

    2. Simplified18.5

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

    4. Applied flip3--18.6

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}^{3} - {b}^{3}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)}}}{a}}{2}\]

    5. Simplified17.9

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

    6. Simplified17.9

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

    if 2892.1913455639924 < b

    1. Initial program 36.9

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

    2. Simplified36.9

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

    3. Taylor expanded around inf 15.7

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

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2892.1913455639924:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b + \left(c \cdot a\right) \cdot -4\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - \left(b \cdot b\right) \cdot b}{\left(b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4}\right) \cdot \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} + b \cdot b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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 a) c)))) (* 2 a)))