Average Error: 34.7 → 6.8
Time: 10.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.277091394223363542557795301452975127355 \cdot 10^{71}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le \frac{3416711217990983}{1.097224813758737736651187250237441854015 \cdot 10^{304}}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 7.49320603738817106440277485916837905827 \cdot 10^{90}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.277091394223363542557795301452975127355 \cdot 10^{71}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le \frac{3416711217990983}{1.097224813758737736651187250237441854015 \cdot 10^{304}}:\\
\;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\

\mathbf{elif}\;b_2 \le 7.49320603738817106440277485916837905827 \cdot 10^{90}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r25383 = b_2;
        double r25384 = -r25383;
        double r25385 = r25383 * r25383;
        double r25386 = a;
        double r25387 = c;
        double r25388 = r25386 * r25387;
        double r25389 = r25385 - r25388;
        double r25390 = sqrt(r25389);
        double r25391 = r25384 - r25390;
        double r25392 = r25391 / r25386;
        return r25392;
}
↓
double f(double a, double b_2, double c) {
        double r25393 = b_2;
        double r25394 = -1.2770913942233635e+71;
        bool r25395 = r25393 <= r25394;
        double r25396 = -0.5;
        double r25397 = c;
        double r25398 = r25397 / r25393;
        double r25399 = r25396 * r25398;
        double r25400 = 3416711217990983.0;
        double r25401 = 1.0972248137587377e+304;
        double r25402 = r25400 / r25401;
        bool r25403 = r25393 <= r25402;
        double r25404 = 1.0;
        double r25405 = r25393 * r25393;
        double r25406 = a;
        double r25407 = r25406 * r25397;
        double r25408 = r25405 - r25407;
        double r25409 = sqrt(r25408);
        double r25410 = r25409 - r25393;
        double r25411 = r25410 / r25397;
        double r25412 = r25404 / r25411;
        double r25413 = 7.493206037388171e+90;
        bool r25414 = r25393 <= r25413;
        double r25415 = -r25393;
        double r25416 = r25415 - r25409;
        double r25417 = r25416 / r25406;
        double r25418 = -2.0;
        double r25419 = r25393 / r25406;
        double r25420 = r25418 * r25419;
        double r25421 = r25414 ? r25417 : r25420;
        double r25422 = r25403 ? r25412 : r25421;
        double r25423 = r25395 ? r25399 : r25422;
        return r25423;
}

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 < -1.2770913942233635e+71

    1. Initial program 58.1

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

    2. Taylor expanded around -inf 3.5

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

    if -1.2770913942233635e+71 < b_2 < 3.113957299494927e-289

    1. Initial program 30.4

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

    3. Applied flip--30.4

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

    4. Simplified17.0

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

    5. Simplified17.0

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

    7. Applied clear-num17.1

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

    8. Simplified10.0

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

    if 3.113957299494927e-289 < b_2 < 7.493206037388171e+90

    1. Initial program 8.6

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

    if 7.493206037388171e+90 < b_2

    1. Initial program 45.9

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

    3. Applied flip--62.9

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

    4. Simplified62.0

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

    5. Simplified62.0

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

    6. Taylor expanded around 0 4.2

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.277091394223363542557795301452975127355 \cdot 10^{71}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le \frac{3416711217990983}{1.097224813758737736651187250237441854015 \cdot 10^{304}}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 7.49320603738817106440277485916837905827 \cdot 10^{90}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 197574269 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))