Average Error: 34.2 → 7.5
Time: 16.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.007820467288354043661462566796901658096 \cdot 10^{70}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -3.385925883276496138762890378525598892144 \cdot 10^{-167}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 87537227540251800037021545535125898395650:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.007820467288354043661462566796901658096 \cdot 10^{70}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -3.385925883276496138762890378525598892144 \cdot 10^{-167}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 87537227540251800037021545535125898395650:\\
\;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r18600 = b_2;
        double r18601 = -r18600;
        double r18602 = r18600 * r18600;
        double r18603 = a;
        double r18604 = c;
        double r18605 = r18603 * r18604;
        double r18606 = r18602 - r18605;
        double r18607 = sqrt(r18606);
        double r18608 = r18601 + r18607;
        double r18609 = r18608 / r18603;
        return r18609;
}
↓
double f(double a, double b_2, double c) {
        double r18610 = b_2;
        double r18611 = -2.007820467288354e+70;
        bool r18612 = r18610 <= r18611;
        double r18613 = 0.5;
        double r18614 = c;
        double r18615 = r18614 / r18610;
        double r18616 = r18613 * r18615;
        double r18617 = 2.0;
        double r18618 = a;
        double r18619 = r18610 / r18618;
        double r18620 = r18617 * r18619;
        double r18621 = r18616 - r18620;
        double r18622 = -3.385925883276496e-167;
        bool r18623 = r18610 <= r18622;
        double r18624 = -r18610;
        double r18625 = r18610 * r18610;
        double r18626 = r18618 * r18614;
        double r18627 = r18625 - r18626;
        double r18628 = sqrt(r18627);
        double r18629 = r18624 + r18628;
        double r18630 = 1.0;
        double r18631 = r18630 / r18618;
        double r18632 = r18629 * r18631;
        double r18633 = 8.75372275402518e+40;
        bool r18634 = r18610 <= r18633;
        double r18635 = r18624 - r18628;
        double r18636 = r18614 / r18635;
        double r18637 = -0.5;
        double r18638 = r18637 * r18615;
        double r18639 = r18634 ? r18636 : r18638;
        double r18640 = r18623 ? r18632 : r18639;
        double r18641 = r18612 ? r18621 : r18640;
        return r18641;
}

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 < -2.007820467288354e+70

    1. Initial program 41.4

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

    2. Taylor expanded around -inf 4.9

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

    if -2.007820467288354e+70 < b_2 < -3.385925883276496e-167

    1. Initial program 7.2

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

    3. Applied div-inv7.3

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

    if -3.385925883276496e-167 < b_2 < 8.75372275402518e+40

    1. Initial program 25.7

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

    3. Applied div-inv25.8

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

    5. Applied flip-+26.0

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

    6. Applied associate-*l/26.1

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

    7. Simplified16.7

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

    8. Taylor expanded around 0 11.5

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

    if 8.75372275402518e+40 < b_2

    1. Initial program 56.7

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

    2. Taylor expanded around inf 4.5

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

  4. Final simplification7.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.007820467288354043661462566796901658096 \cdot 10^{70}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -3.385925883276496138762890378525598892144 \cdot 10^{-167}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 87537227540251800037021545535125898395650:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))