Average Error: 28.5 → 16.2
Time: 17.8s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2495.5039318207096:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 2495.5039318207096:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r680983 = b;
        double r680984 = -r680983;
        double r680985 = r680983 * r680983;
        double r680986 = 4.0;
        double r680987 = a;
        double r680988 = r680986 * r680987;
        double r680989 = c;
        double r680990 = r680988 * r680989;
        double r680991 = r680985 - r680990;
        double r680992 = sqrt(r680991);
        double r680993 = r680984 + r680992;
        double r680994 = 2.0;
        double r680995 = r680994 * r680987;
        double r680996 = r680993 / r680995;
        return r680996;
}
↓
double f(double a, double b, double c) {
        double r680997 = b;
        double r680998 = 2495.5039318207096;
        bool r680999 = r680997 <= r680998;
        double r681000 = r680997 * r680997;
        double r681001 = a;
        double r681002 = c;
        double r681003 = r681001 * r681002;
        double r681004 = 4.0;
        double r681005 = r681003 * r681004;
        double r681006 = r681000 - r681005;
        double r681007 = sqrt(r681006);
        double r681008 = r681006 * r681007;
        double r681009 = r681000 * r680997;
        double r681010 = r681008 - r681009;
        double r681011 = r680997 + r681007;
        double r681012 = r680997 * r681011;
        double r681013 = r681006 + r681012;
        double r681014 = r681010 / r681013;
        double r681015 = r681014 / r681001;
        double r681016 = 2.0;
        double r681017 = r681015 / r681016;
        double r681018 = -2.0;
        double r681019 = r681002 / r680997;
        double r681020 = r681018 * r681019;
        double r681021 = r681020 / r681016;
        double r681022 = r680999 ? r681017 : r681021;
        return r681022;
}

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 < 2495.5039318207096

    1. Initial program 17.8

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

    2. Simplified17.8

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

    4. Applied flip3--17.9

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

    5. Simplified17.2

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

    6. Simplified17.2

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

    if 2495.5039318207096 < b

    1. Initial program 37.3

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

    2. Simplified37.3

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

    3. Taylor expanded around inf 15.5

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2495.5039318207096:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot c\right) \cdot 4\right) + b \cdot \left(b + \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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)))