Average Error: 34.4 → 8.2
Time: 7.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.2038615298990798 \cdot 10^{25}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 9.57911224209110995 \cdot 10^{-234}:\\ \;\;\;\;\frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 4.06577896741184288 \cdot 10^{116}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.2038615298990798 \cdot 10^{25}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 9.57911224209110995 \cdot 10^{-234}:\\
\;\;\;\;\frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{a}\\

\mathbf{elif}\;b_2 \le 4.06577896741184288 \cdot 10^{116}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r23429 = b_2;
        double r23430 = -r23429;
        double r23431 = r23429 * r23429;
        double r23432 = a;
        double r23433 = c;
        double r23434 = r23432 * r23433;
        double r23435 = r23431 - r23434;
        double r23436 = sqrt(r23435);
        double r23437 = r23430 - r23436;
        double r23438 = r23437 / r23432;
        return r23438;
}
↓
double f(double a, double b_2, double c) {
        double r23439 = b_2;
        double r23440 = -9.20386152989908e+25;
        bool r23441 = r23439 <= r23440;
        double r23442 = -0.5;
        double r23443 = c;
        double r23444 = r23443 / r23439;
        double r23445 = r23442 * r23444;
        double r23446 = 9.57911224209111e-234;
        bool r23447 = r23439 <= r23446;
        double r23448 = a;
        double r23449 = r23439 * r23439;
        double r23450 = r23448 * r23443;
        double r23451 = r23449 - r23450;
        double r23452 = sqrt(r23451);
        double r23453 = r23452 - r23439;
        double r23454 = r23453 / r23443;
        double r23455 = r23448 / r23454;
        double r23456 = r23455 / r23448;
        double r23457 = 4.065778967411843e+116;
        bool r23458 = r23439 <= r23457;
        double r23459 = -r23439;
        double r23460 = r23459 - r23452;
        double r23461 = 1.0;
        double r23462 = r23461 / r23448;
        double r23463 = r23460 * r23462;
        double r23464 = 0.5;
        double r23465 = r23464 * r23444;
        double r23466 = 2.0;
        double r23467 = r23439 / r23448;
        double r23468 = r23466 * r23467;
        double r23469 = r23465 - r23468;
        double r23470 = r23458 ? r23463 : r23469;
        double r23471 = r23447 ? r23456 : r23470;
        double r23472 = r23441 ? r23445 : r23471;
        return r23472;
}

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 < -9.20386152989908e+25

    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 5.0

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

    if -9.20386152989908e+25 < b_2 < 9.57911224209111e-234

    1. Initial program 25.9

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

    3. Applied flip--26.0

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

    4. Simplified17.1

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

    5. Simplified17.1

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity17.1

      \[\leadsto \frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

    8. Applied associate-/r*17.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{1}}{a}}\]

    9. Simplified14.3

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

    if 9.57911224209111e-234 < b_2 < 4.065778967411843e+116

    1. Initial program 7.8

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

    3. Applied div-inv7.9

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

    if 4.065778967411843e+116 < b_2

    1. Initial program 51.7

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.2038615298990798 \cdot 10^{25}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 9.57911224209110995 \cdot 10^{-234}:\\ \;\;\;\;\frac{\frac{a}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 4.06577896741184288 \cdot 10^{116}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))