Average Error: 28.4 → 16.3
Time: 18.9s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 835.2343651472419878700748085975646972656:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 835.2343651472419878700748085975646972656:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1670034 = b;
        double r1670035 = -r1670034;
        double r1670036 = r1670034 * r1670034;
        double r1670037 = 4.0;
        double r1670038 = a;
        double r1670039 = r1670037 * r1670038;
        double r1670040 = c;
        double r1670041 = r1670039 * r1670040;
        double r1670042 = r1670036 - r1670041;
        double r1670043 = sqrt(r1670042);
        double r1670044 = r1670035 + r1670043;
        double r1670045 = 2.0;
        double r1670046 = r1670045 * r1670038;
        double r1670047 = r1670044 / r1670046;
        return r1670047;
}
↓
double f(double a, double b, double c) {
        double r1670048 = b;
        double r1670049 = 835.234365147242;
        bool r1670050 = r1670048 <= r1670049;
        double r1670051 = r1670048 * r1670048;
        double r1670052 = a;
        double r1670053 = 4.0;
        double r1670054 = r1670052 * r1670053;
        double r1670055 = c;
        double r1670056 = r1670054 * r1670055;
        double r1670057 = r1670051 - r1670056;
        double r1670058 = sqrt(r1670057);
        double r1670059 = r1670057 * r1670058;
        double r1670060 = r1670051 * r1670048;
        double r1670061 = r1670059 - r1670060;
        double r1670062 = r1670048 * r1670058;
        double r1670063 = r1670062 + r1670051;
        double r1670064 = r1670057 + r1670063;
        double r1670065 = r1670061 / r1670064;
        double r1670066 = r1670065 / r1670052;
        double r1670067 = 2.0;
        double r1670068 = r1670066 / r1670067;
        double r1670069 = -2.0;
        double r1670070 = r1670055 / r1670048;
        double r1670071 = r1670069 * r1670070;
        double r1670072 = r1670071 / r1670067;
        double r1670073 = r1670050 ? r1670068 : r1670072;
        return r1670073;
}

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 < 835.234365147242

    1. Initial program 16.7

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

    2. Simplified16.7

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

    4. Applied flip3--16.8

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

    5. Simplified16.1

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

    6. Simplified16.1

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

    if 835.234365147242 < b

    1. Initial program 36.1

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

    2. Simplified36.1

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

    3. Taylor expanded around inf 16.4

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 835.2343651472419878700748085975646972656:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + \left(b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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)))