Average Error: 33.9 → 10.4
Time: 20.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.5961406266953245 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.1115579814291686 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.5961406266953245 \cdot 10^{-58}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.1115579814291686 \cdot 10^{+29}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r690448 = b_2;
        double r690449 = -r690448;
        double r690450 = r690448 * r690448;
        double r690451 = a;
        double r690452 = c;
        double r690453 = r690451 * r690452;
        double r690454 = r690450 - r690453;
        double r690455 = sqrt(r690454);
        double r690456 = r690449 - r690455;
        double r690457 = r690456 / r690451;
        return r690457;
}
↓
double f(double a, double b_2, double c) {
        double r690458 = b_2;
        double r690459 = -1.5961406266953245e-58;
        bool r690460 = r690458 <= r690459;
        double r690461 = -0.5;
        double r690462 = c;
        double r690463 = r690462 / r690458;
        double r690464 = r690461 * r690463;
        double r690465 = 3.1115579814291686e+29;
        bool r690466 = r690458 <= r690465;
        double r690467 = -r690458;
        double r690468 = r690458 * r690458;
        double r690469 = a;
        double r690470 = r690469 * r690462;
        double r690471 = r690468 - r690470;
        double r690472 = sqrt(r690471);
        double r690473 = r690467 - r690472;
        double r690474 = r690473 / r690469;
        double r690475 = r690458 / r690469;
        double r690476 = -2.0;
        double r690477 = r690475 * r690476;
        double r690478 = r690466 ? r690474 : r690477;
        double r690479 = r690460 ? r690464 : r690478;
        return r690479;
}

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 3 regimes

  2. if b_2 < -1.5961406266953245e-58

    1. Initial program 53.4

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

    2. Taylor expanded around -inf 53.4

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified53.4

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

    4. Taylor expanded around -inf 8.1

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

    if -1.5961406266953245e-58 < b_2 < 3.1115579814291686e+29

    1. Initial program 14.7

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

    3. Applied div-inv14.8

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

    5. Applied associate-*r/14.7

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

    6. Simplified14.7

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

    if 3.1115579814291686e+29 < b_2

    1. Initial program 34.4

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

    2. Taylor expanded around -inf 34.4

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified34.4

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

    5. Applied *-un-lft-identity34.4

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

    6. Applied associate-/l*34.5

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

    7. Taylor expanded around 0 6.5

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.5961406266953245 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.1115579814291686 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))