Average Error: 34.4 → 8.2
Time: 21.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 4.744536474227931203452738058362502791987 \cdot 10^{-289}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2 \cdot 2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 4.744536474227931203452738058362502791987 \cdot 10^{-289}:\\
\;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1089720 = b_2;
        double r1089721 = -r1089720;
        double r1089722 = r1089720 * r1089720;
        double r1089723 = a;
        double r1089724 = c;
        double r1089725 = r1089723 * r1089724;
        double r1089726 = r1089722 - r1089725;
        double r1089727 = sqrt(r1089726);
        double r1089728 = r1089721 - r1089727;
        double r1089729 = r1089728 / r1089723;
        return r1089729;
}
↓
double f(double a, double b_2, double c) {
        double r1089730 = b_2;
        double r1089731 = -7.741777288939024e+81;
        bool r1089732 = r1089730 <= r1089731;
        double r1089733 = -0.5;
        double r1089734 = c;
        double r1089735 = r1089734 / r1089730;
        double r1089736 = r1089733 * r1089735;
        double r1089737 = 4.744536474227931e-289;
        bool r1089738 = r1089730 <= r1089737;
        double r1089739 = a;
        double r1089740 = r1089730 * r1089730;
        double r1089741 = r1089734 * r1089739;
        double r1089742 = r1089740 - r1089741;
        double r1089743 = sqrt(r1089742);
        double r1089744 = r1089743 - r1089730;
        double r1089745 = r1089734 / r1089744;
        double r1089746 = r1089739 * r1089745;
        double r1089747 = r1089746 / r1089739;
        double r1089748 = 3.355858625783055e+101;
        bool r1089749 = r1089730 <= r1089748;
        double r1089750 = 1.0;
        double r1089751 = r1089750 / r1089739;
        double r1089752 = -r1089730;
        double r1089753 = r1089752 - r1089743;
        double r1089754 = r1089751 * r1089753;
        double r1089755 = 2.0;
        double r1089756 = r1089735 / r1089755;
        double r1089757 = r1089730 * r1089755;
        double r1089758 = r1089757 / r1089739;
        double r1089759 = r1089756 - r1089758;
        double r1089760 = r1089749 ? r1089754 : r1089759;
        double r1089761 = r1089738 ? r1089747 : r1089760;
        double r1089762 = r1089732 ? r1089736 : r1089761;
        return r1089762;
}

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 < -7.741777288939024e+81

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 2.9

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

    if -7.741777288939024e+81 < b_2 < 4.744536474227931e-289

    1. Initial program 30.9

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

    3. Applied flip--30.9

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

    4. Simplified16.5

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

    5. Simplified16.5

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity16.5

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

    8. Applied times-frac13.7

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

    9. Simplified13.7

      \[\leadsto \frac{\color{blue}{a} \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]

    if 4.744536474227931e-289 < b_2 < 3.355858625783055e+101

    1. Initial program 9.1

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

    3. Applied div-inv9.2

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

    if 3.355858625783055e+101 < b_2

    1. Initial program 46.6

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

    2. Taylor expanded around inf 4.5

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

    3. Simplified4.6

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.741777288939024183924384840560245543701 \cdot 10^{81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 4.744536474227931203452738058362502791987 \cdot 10^{-289}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 3.355858625783055094237525774982320834143 \cdot 10^{101}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2 \cdot 2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))