Average Error: 34.1 → 8.9
Time: 19.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.122942397323538653087473965252285768999 \cdot 10^{137}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.408354642852288642375909774932644593067 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{a \cdot c}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le -5.546621280225112292650318866994441138678 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.823335453796603439248590818149160856749 \cdot 10^{131}:\\ \;\;\;\;\frac{-\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.122942397323538653087473965252285768999 \cdot 10^{137}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.408354642852288642375909774932644593067 \cdot 10^{-45}:\\
\;\;\;\;\frac{\frac{a \cdot c}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le -5.546621280225112292650318866994441138678 \cdot 10^{-56}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.823335453796603439248590818149160856749 \cdot 10^{131}:\\
\;\;\;\;\frac{-\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r1463968 = b_2;
        double r1463969 = -r1463968;
        double r1463970 = r1463968 * r1463968;
        double r1463971 = a;
        double r1463972 = c;
        double r1463973 = r1463971 * r1463972;
        double r1463974 = r1463970 - r1463973;
        double r1463975 = sqrt(r1463974);
        double r1463976 = r1463969 - r1463975;
        double r1463977 = r1463976 / r1463971;
        return r1463977;
}
↓
double f(double a, double b_2, double c) {
        double r1463978 = b_2;
        double r1463979 = -2.1229423973235387e+137;
        bool r1463980 = r1463978 <= r1463979;
        double r1463981 = -0.5;
        double r1463982 = c;
        double r1463983 = r1463982 / r1463978;
        double r1463984 = r1463981 * r1463983;
        double r1463985 = -3.4083546428522886e-45;
        bool r1463986 = r1463978 <= r1463985;
        double r1463987 = a;
        double r1463988 = r1463987 * r1463982;
        double r1463989 = r1463988 / r1463987;
        double r1463990 = -r1463978;
        double r1463991 = r1463978 * r1463978;
        double r1463992 = r1463991 - r1463988;
        double r1463993 = sqrt(r1463992);
        double r1463994 = r1463990 + r1463993;
        double r1463995 = r1463989 / r1463994;
        double r1463996 = -5.546621280225112e-56;
        bool r1463997 = r1463978 <= r1463996;
        double r1463998 = 2.8233354537966034e+131;
        bool r1463999 = r1463978 <= r1463998;
        double r1464000 = r1463978 + r1463993;
        double r1464001 = -r1464000;
        double r1464002 = r1464001 / r1463987;
        double r1464003 = 0.5;
        double r1464004 = r1464003 * r1463983;
        double r1464005 = 2.0;
        double r1464006 = r1463978 / r1463987;
        double r1464007 = r1464005 * r1464006;
        double r1464008 = r1464004 - r1464007;
        double r1464009 = r1463999 ? r1464002 : r1464008;
        double r1464010 = r1463997 ? r1463984 : r1464009;
        double r1464011 = r1463986 ? r1463995 : r1464010;
        double r1464012 = r1463980 ? r1463984 : r1464011;
        return r1464012;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -2.1229423973235387e+137 or -3.4083546428522886e-45 < b_2 < -5.546621280225112e-56

    1. Initial program 61.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 2.3

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -2.1229423973235387e+137 < b_2 < -3.4083546428522886e-45

    1. Initial program 45.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv45.1

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
    4. Using strategy rm

    5. Applied flip--45.1

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{1}{a}\]

    6. Applied associate-*l/45.1

      \[\leadsto \color{blue}{\frac{\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    7. Simplified11.6

      \[\leadsto \frac{\color{blue}{\frac{0 + a \cdot c}{a}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    if -5.546621280225112e-56 < b_2 < 2.8233354537966034e+131

    1. Initial program 12.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv12.7

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
    4. Using strategy rm

    5. Applied associate-*r/12.5

      \[\leadsto \color{blue}{\frac{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot 1}{a}}\]

    6. Simplified12.5

      \[\leadsto \frac{\color{blue}{-\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]

    if 2.8233354537966034e+131 < b_2

    1. Initial program 56.4

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 2.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.122942397323538653087473965252285768999 \cdot 10^{137}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.408354642852288642375909774932644593067 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{a \cdot c}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le -5.546621280225112292650318866994441138678 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.823335453796603439248590818149160856749 \cdot 10^{131}:\\ \;\;\;\;\frac{-\left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))