Average Error: 32.8 → 7.0
Time: 24.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.812499047218918 \cdot 10^{+50}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.8716662261062222 \cdot 10^{-300}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 9.19242293018462 \cdot 10^{+63}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.812499047218918 \cdot 10^{+50}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.8716662261062222 \cdot 10^{-300}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 9.19242293018462 \cdot 10^{+63}:\\
\;\;\;\;\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} - \frac{b_2}{a} \cdot 2\\

\end{array}
double f(double a, double b_2, double c) {
        double r929503 = b_2;
        double r929504 = -r929503;
        double r929505 = r929503 * r929503;
        double r929506 = a;
        double r929507 = c;
        double r929508 = r929506 * r929507;
        double r929509 = r929505 - r929508;
        double r929510 = sqrt(r929509);
        double r929511 = r929504 - r929510;
        double r929512 = r929511 / r929506;
        return r929512;
}
↓
double f(double a, double b_2, double c) {
        double r929513 = b_2;
        double r929514 = -3.812499047218918e+50;
        bool r929515 = r929513 <= r929514;
        double r929516 = -0.5;
        double r929517 = c;
        double r929518 = r929517 / r929513;
        double r929519 = r929516 * r929518;
        double r929520 = 1.8716662261062222e-300;
        bool r929521 = r929513 <= r929520;
        double r929522 = r929513 * r929513;
        double r929523 = a;
        double r929524 = r929523 * r929517;
        double r929525 = r929522 - r929524;
        double r929526 = sqrt(r929525);
        double r929527 = r929526 - r929513;
        double r929528 = r929517 / r929527;
        double r929529 = 9.19242293018462e+63;
        bool r929530 = r929513 <= r929529;
        double r929531 = -r929513;
        double r929532 = r929531 - r929526;
        double r929533 = 1.0;
        double r929534 = r929533 / r929523;
        double r929535 = r929532 * r929534;
        double r929536 = 0.5;
        double r929537 = r929536 * r929518;
        double r929538 = r929513 / r929523;
        double r929539 = 2.0;
        double r929540 = r929538 * r929539;
        double r929541 = r929537 - r929540;
        double r929542 = r929530 ? r929535 : r929541;
        double r929543 = r929521 ? r929528 : r929542;
        double r929544 = r929515 ? r929519 : r929543;
        return r929544;
}

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 < -3.812499047218918e+50

    1. Initial program 55.8

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

    2. Taylor expanded around -inf 4.0

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

    if -3.812499047218918e+50 < b_2 < 1.8716662261062222e-300

    1. Initial program 28.9

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

    3. Applied flip--29.0

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

    4. Simplified16.2

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

    5. Simplified16.2

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

    7. Applied *-un-lft-identity16.2

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

    8. Applied *-un-lft-identity16.2

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

    9. Applied *-un-lft-identity16.2

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

    10. Applied times-frac16.2

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

    11. Applied times-frac16.2

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

    12. Simplified16.2

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

    13. Simplified15.9

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

    14. Taylor expanded around 0 9.2

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

    if 1.8716662261062222e-300 < b_2 < 9.19242293018462e+63

    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-inv9.7

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

    if 9.19242293018462e+63 < b_2

    1. Initial program 38.1

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

    2. Taylor expanded around inf 4.5

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

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.812499047218918 \cdot 10^{+50}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.8716662261062222 \cdot 10^{-300}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 9.19242293018462 \cdot 10^{+63}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))