Average Error: 34.1 → 8.2
Time: 5.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.4834941205945284 \cdot 10^{67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.329606077024855 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{a}{\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{elif}\;b_2 \le 4.8331091207749691 \cdot 10^{125}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.4834941205945284 \cdot 10^{67}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.329606077024855 \cdot 10^{-296}:\\
\;\;\;\;\frac{1}{a} \cdot \frac{a}{\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\

\mathbf{elif}\;b_2 \le 4.8331091207749691 \cdot 10^{125}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{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 r14404 = b_2;
        double r14405 = -r14404;
        double r14406 = r14404 * r14404;
        double r14407 = a;
        double r14408 = c;
        double r14409 = r14407 * r14408;
        double r14410 = r14406 - r14409;
        double r14411 = sqrt(r14410);
        double r14412 = r14405 - r14411;
        double r14413 = r14412 / r14407;
        return r14413;
}
↓
double f(double a, double b_2, double c) {
        double r14414 = b_2;
        double r14415 = -2.4834941205945284e+67;
        bool r14416 = r14414 <= r14415;
        double r14417 = -0.5;
        double r14418 = c;
        double r14419 = r14418 / r14414;
        double r14420 = r14417 * r14419;
        double r14421 = 2.329606077024855e-296;
        bool r14422 = r14414 <= r14421;
        double r14423 = 1.0;
        double r14424 = a;
        double r14425 = r14423 / r14424;
        double r14426 = -r14414;
        double r14427 = r14414 * r14414;
        double r14428 = r14424 * r14418;
        double r14429 = r14427 - r14428;
        double r14430 = sqrt(r14429);
        double r14431 = r14426 + r14430;
        double r14432 = r14431 / r14418;
        double r14433 = r14424 / r14432;
        double r14434 = r14425 * r14433;
        double r14435 = 4.833109120774969e+125;
        bool r14436 = r14414 <= r14435;
        double r14437 = r14426 - r14430;
        double r14438 = r14437 / r14424;
        double r14439 = 0.5;
        double r14440 = r14439 * r14419;
        double r14441 = 2.0;
        double r14442 = r14414 / r14424;
        double r14443 = r14441 * r14442;
        double r14444 = r14440 - r14443;
        double r14445 = r14436 ? r14438 : r14444;
        double r14446 = r14422 ? r14434 : r14445;
        double r14447 = r14416 ? r14420 : r14446;
        return r14447;
}

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 < -2.4834941205945284e+67

    1. Initial program 57.7

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

    2. Taylor expanded around -inf 3.7

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

    if -2.4834941205945284e+67 < b_2 < 2.329606077024855e-296

    1. Initial program 30.5

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

    3. Applied clear-num30.6

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

    5. Applied div-inv30.6

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

    6. Applied add-cube-cbrt30.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{a \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    7. Applied times-frac30.6

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{a} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}\]

    8. Simplified30.6

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    9. Simplified30.6

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

    11. Applied flip--30.6

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

    12. Applied frac-times35.6

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

    13. Simplified22.4

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

    15. Applied *-un-lft-identity22.4

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

    16. Applied times-frac17.5

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

    17. Simplified15.1

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

    if 2.329606077024855e-296 < b_2 < 4.833109120774969e+125

    1. Initial program 8.5

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

    if 4.833109120774969e+125 < b_2

    1. Initial program 53.6

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

    2. Taylor expanded around inf 2.5

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.4834941205945284 \cdot 10^{67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.329606077024855 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{a}{\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{elif}\;b_2 \le 4.8331091207749691 \cdot 10^{125}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))