Average Error: 34.2 → 7.4
Time: 11.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -84351863079940134392111797632084475904000:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.779221184338331637638800633144967588071 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 2.730494439370032074747470763239053019705 \cdot 10^{75}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -84351863079940134392111797632084475904000:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.779221184338331637638800633144967588071 \cdot 10^{-301}:\\
\;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\

\mathbf{elif}\;b_2 \le 2.730494439370032074747470763239053019705 \cdot 10^{75}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r26434 = b_2;
        double r26435 = -r26434;
        double r26436 = r26434 * r26434;
        double r26437 = a;
        double r26438 = c;
        double r26439 = r26437 * r26438;
        double r26440 = r26436 - r26439;
        double r26441 = sqrt(r26440);
        double r26442 = r26435 - r26441;
        double r26443 = r26442 / r26437;
        return r26443;
}
↓
double f(double a, double b_2, double c) {
        double r26444 = b_2;
        double r26445 = -8.435186307994013e+40;
        bool r26446 = r26444 <= r26445;
        double r26447 = -0.5;
        double r26448 = c;
        double r26449 = r26448 / r26444;
        double r26450 = r26447 * r26449;
        double r26451 = 2.7792211843383316e-301;
        bool r26452 = r26444 <= r26451;
        double r26453 = 1.0;
        double r26454 = r26444 * r26444;
        double r26455 = a;
        double r26456 = r26455 * r26448;
        double r26457 = r26454 - r26456;
        double r26458 = sqrt(r26457);
        double r26459 = r26458 - r26444;
        double r26460 = r26459 / r26448;
        double r26461 = r26453 / r26460;
        double r26462 = 2.730494439370032e+75;
        bool r26463 = r26444 <= r26462;
        double r26464 = -r26444;
        double r26465 = r26464 / r26455;
        double r26466 = r26458 / r26455;
        double r26467 = r26465 - r26466;
        double r26468 = -2.0;
        double r26469 = r26468 * r26444;
        double r26470 = r26469 / r26455;
        double r26471 = r26463 ? r26467 : r26470;
        double r26472 = r26452 ? r26461 : r26471;
        double r26473 = r26446 ? r26450 : r26472;
        return r26473;
}

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 < -8.435186307994013e+40

    1. Initial program 56.9

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

    2. Taylor expanded around -inf 4.4

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

    if -8.435186307994013e+40 < b_2 < 2.7792211843383316e-301

    1. Initial program 28.4

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

    3. Applied flip--28.4

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

    4. Simplified16.9

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

    5. Simplified16.9

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

    7. Applied clear-num17.0

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

    8. Simplified10.1

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

    if 2.7792211843383316e-301 < b_2 < 2.730494439370032e+75

    1. Initial program 9.5

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

    3. Applied div-sub9.5

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

    if 2.730494439370032e+75 < b_2

    1. Initial program 43.1

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

    3. Applied flip--62.3

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

    4. Simplified61.5

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

    5. Simplified61.5

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

    6. Taylor expanded around 0 5.2

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -84351863079940134392111797632084475904000:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.779221184338331637638800633144967588071 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 2.730494439370032074747470763239053019705 \cdot 10^{75}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))