Average Error: 33.1 → 8.0
Time: 35.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\
\;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\

\mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\
\;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1647681 = b_2;
        double r1647682 = -r1647681;
        double r1647683 = r1647681 * r1647681;
        double r1647684 = a;
        double r1647685 = c;
        double r1647686 = r1647684 * r1647685;
        double r1647687 = r1647683 - r1647686;
        double r1647688 = sqrt(r1647687);
        double r1647689 = r1647682 - r1647688;
        double r1647690 = r1647689 / r1647684;
        return r1647690;
}
↓
double f(double a, double b_2, double c) {
        double r1647691 = b_2;
        double r1647692 = -6.473972066548491e+100;
        bool r1647693 = r1647691 <= r1647692;
        double r1647694 = -0.5;
        double r1647695 = c;
        double r1647696 = r1647695 / r1647691;
        double r1647697 = r1647694 * r1647696;
        double r1647698 = -3.554031892664371e-133;
        bool r1647699 = r1647691 <= r1647698;
        double r1647700 = a;
        double r1647701 = cbrt(r1647700);
        double r1647702 = r1647695 / r1647701;
        double r1647703 = r1647701 * r1647701;
        double r1647704 = r1647700 / r1647703;
        double r1647705 = r1647702 * r1647704;
        double r1647706 = r1647691 * r1647691;
        double r1647707 = r1647695 * r1647700;
        double r1647708 = r1647706 - r1647707;
        double r1647709 = sqrt(r1647708);
        double r1647710 = -r1647691;
        double r1647711 = r1647709 + r1647710;
        double r1647712 = r1647705 / r1647711;
        double r1647713 = 1.983916337927056e+89;
        bool r1647714 = r1647691 <= r1647713;
        double r1647715 = r1647691 / r1647700;
        double r1647716 = -r1647715;
        double r1647717 = r1647709 / r1647700;
        double r1647718 = r1647716 - r1647717;
        double r1647719 = 0.5;
        double r1647720 = r1647719 * r1647696;
        double r1647721 = 2.0;
        double r1647722 = r1647715 * r1647721;
        double r1647723 = r1647720 - r1647722;
        double r1647724 = r1647714 ? r1647718 : r1647723;
        double r1647725 = r1647699 ? r1647712 : r1647724;
        double r1647726 = r1647693 ? r1647697 : r1647725;
        return r1647726;
}

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 < -6.473972066548491e+100

    1. Initial program 58.8

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

    2. Taylor expanded around -inf 2.3

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

    if -6.473972066548491e+100 < b_2 < -3.554031892664371e-133

    1. Initial program 39.0

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

    3. Applied div-inv39.1

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

    5. Applied flip--39.2

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

    6. Applied associate-*l/39.2

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

    7. Simplified13.1

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

    9. Applied add-cube-cbrt14.0

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    10. Applied times-frac10.6

      \[\leadsto \frac{\color{blue}{\frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{c}{\sqrt[3]{a}}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    if -3.554031892664371e-133 < b_2 < 1.983916337927056e+89

    1. Initial program 11.5

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

    3. Applied div-sub11.5

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

    if 1.983916337927056e+89 < b_2

    1. Initial program 42.0

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.473972066548491 \cdot 10^{+100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.554031892664371 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{c}{\sqrt[3]{a}} \cdot \frac{a}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.983916337927056 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))