Average Error: 34.5 → 8.7
Time: 7.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.95460202615535102 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.82564652270824723 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 2.039304743620259 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.95460202615535102 \cdot 10^{-16}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.82564652270824723 \cdot 10^{-308}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}}}{a}\\

\mathbf{elif}\;b_2 \le 2.039304743620259 \cdot 10^{111}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r98409 = b_2;
        double r98410 = -r98409;
        double r98411 = r98409 * r98409;
        double r98412 = a;
        double r98413 = c;
        double r98414 = r98412 * r98413;
        double r98415 = r98411 - r98414;
        double r98416 = sqrt(r98415);
        double r98417 = r98410 - r98416;
        double r98418 = r98417 / r98412;
        return r98418;
}
↓
double f(double a, double b_2, double c) {
        double r98419 = b_2;
        double r98420 = -1.954602026155351e-16;
        bool r98421 = r98419 <= r98420;
        double r98422 = -0.5;
        double r98423 = c;
        double r98424 = r98423 / r98419;
        double r98425 = r98422 * r98424;
        double r98426 = -3.825646522708247e-308;
        bool r98427 = r98419 <= r98426;
        double r98428 = 1.0;
        double r98429 = r98419 * r98419;
        double r98430 = a;
        double r98431 = r98430 * r98423;
        double r98432 = r98429 - r98431;
        double r98433 = sqrt(r98432);
        double r98434 = r98433 - r98419;
        double r98435 = r98434 / r98430;
        double r98436 = r98435 / r98423;
        double r98437 = r98428 / r98436;
        double r98438 = r98437 / r98430;
        double r98439 = 2.0393047436202585e+111;
        bool r98440 = r98419 <= r98439;
        double r98441 = -r98419;
        double r98442 = r98441 - r98433;
        double r98443 = r98428 / r98430;
        double r98444 = r98442 * r98443;
        double r98445 = -2.0;
        double r98446 = r98445 * r98419;
        double r98447 = r98446 / r98430;
        double r98448 = r98440 ? r98444 : r98447;
        double r98449 = r98427 ? r98438 : r98448;
        double r98450 = r98421 ? r98425 : r98449;
        return r98450;
}

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 < -1.954602026155351e-16

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 6.7

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

    if -1.954602026155351e-16 < b_2 < -3.825646522708247e-308

    1. Initial program 25.3

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

    3. Applied flip--25.3

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

    4. Simplified17.6

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

    5. Simplified17.6

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

    7. Applied clear-num17.6

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

    8. Simplified14.2

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

    if -3.825646522708247e-308 < b_2 < 2.0393047436202585e+111

    1. Initial program 9.7

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

    3. Applied div-inv9.9

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

    if 2.0393047436202585e+111 < b_2

    1. Initial program 49.6

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

    3. Applied flip--63.2

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

    4. Simplified62.3

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

    5. Simplified62.3

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

    6. Taylor expanded around 0 3.7

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.95460202615535102 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.82564652270824723 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 2.039304743620259 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))