Average Error: 43.8 → 11.5
Time: 42.8s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 0.2404309694818497:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4.0 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{b \cdot \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)} + \left(b \cdot b + \left(b \cdot b - 4.0 \cdot \left(a \cdot c\right)\right)\right)}}{a}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2.0 \cdot \frac{c}{b}}{2.0}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le 0.2404309694818497:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4.0 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{b \cdot \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)} + \left(b \cdot b + \left(b \cdot b - 4.0 \cdot \left(a \cdot c\right)\right)\right)}}{a}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2.0 \cdot \frac{c}{b}}{2.0}\\

\end{array}
double f(double a, double b, double c) {
        double r1938034 = b;
        double r1938035 = -r1938034;
        double r1938036 = r1938034 * r1938034;
        double r1938037 = 4.0;
        double r1938038 = a;
        double r1938039 = r1938037 * r1938038;
        double r1938040 = c;
        double r1938041 = r1938039 * r1938040;
        double r1938042 = r1938036 - r1938041;
        double r1938043 = sqrt(r1938042);
        double r1938044 = r1938035 + r1938043;
        double r1938045 = 2.0;
        double r1938046 = r1938045 * r1938038;
        double r1938047 = r1938044 / r1938046;
        return r1938047;
}
↓
double f(double a, double b, double c) {
        double r1938048 = b;
        double r1938049 = 0.2404309694818497;
        bool r1938050 = r1938048 <= r1938049;
        double r1938051 = r1938048 * r1938048;
        double r1938052 = 4.0;
        double r1938053 = a;
        double r1938054 = c;
        double r1938055 = r1938053 * r1938054;
        double r1938056 = r1938052 * r1938055;
        double r1938057 = r1938051 - r1938056;
        double r1938058 = sqrt(r1938057);
        double r1938059 = r1938057 * r1938058;
        double r1938060 = r1938051 * r1938048;
        double r1938061 = r1938059 - r1938060;
        double r1938062 = r1938048 * r1938058;
        double r1938063 = r1938051 + r1938057;
        double r1938064 = r1938062 + r1938063;
        double r1938065 = r1938061 / r1938064;
        double r1938066 = r1938065 / r1938053;
        double r1938067 = 2.0;
        double r1938068 = r1938066 / r1938067;
        double r1938069 = -2.0;
        double r1938070 = r1938054 / r1938048;
        double r1938071 = r1938069 * r1938070;
        double r1938072 = r1938071 / r1938067;
        double r1938073 = r1938050 ? r1938068 : r1938072;
        return r1938073;
}

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 < 0.2404309694818497

    1. Initial program 24.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\]

    2. Simplified24.0

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4.0} - b}{a}}{2.0}}\]
    3. Using strategy rm

    4. Applied flip3--24.0

      \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4.0}\right)}^{3} - {b}^{3}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4.0} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4.0} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4.0} \cdot b\right)}}}{a}}{2.0}\]

    5. Simplified23.3

      \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} \cdot \left(b \cdot b - \left(a \cdot c\right) \cdot 4.0\right) - \left(b \cdot b\right) \cdot b}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4.0} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4.0} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4.0} \cdot b\right)}}{a}}{2.0}\]

    6. Simplified23.3

      \[\leadsto \frac{\frac{\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} \cdot \left(b \cdot b - \left(a \cdot c\right) \cdot 4.0\right) - \left(b \cdot b\right) \cdot b}{\color{blue}{b \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} + \left(\left(b \cdot b - \left(a \cdot c\right) \cdot 4.0\right) + b \cdot b\right)}}}{a}}{2.0}\]

    if 0.2404309694818497 < b

    1. Initial program 47.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4.0 \cdot a\right) \cdot c}}{2.0 \cdot a}\]

    2. Simplified47.1

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4.0} - b}{a}}{2.0}}\]

    3. Taylor expanded around inf 9.6

      \[\leadsto \frac{\color{blue}{-2.0 \cdot \frac{c}{b}}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 0.2404309694818497:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4.0 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{b \cdot \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)} + \left(b \cdot b + \left(b \cdot b - 4.0 \cdot \left(a \cdot c\right)\right)\right)}}{a}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2.0 \cdot \frac{c}{b}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))