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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.595028172156282 \cdot 10^{+143}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.039671900760653 \cdot 10^{-263}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.4653326993720654 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

double f(double a, double b_2, double c) {
        double r7720477 = b_2;
        double r7720478 = -r7720477;
        double r7720479 = r7720477 * r7720477;
        double r7720480 = a;
        double r7720481 = c;
        double r7720482 = r7720480 * r7720481;
        double r7720483 = r7720479 - r7720482;
        double r7720484 = sqrt(r7720483);
        double r7720485 = r7720478 - r7720484;
        double r7720486 = r7720485 / r7720480;
        return r7720486;
}

↓

double f(double a, double b_2, double c) {
        double r7720487 = b_2;
        double r7720488 = -4.595028172156282e+143;
        bool r7720489 = r7720487 <= r7720488;
        double r7720490 = -0.5;
        double r7720491 = c;
        double r7720492 = r7720491 / r7720487;
        double r7720493 = r7720490 * r7720492;
        double r7720494 = -1.039671900760653e-263;
        bool r7720495 = r7720487 <= r7720494;
        double r7720496 = r7720487 * r7720487;
        double r7720497 = a;
        double r7720498 = r7720497 * r7720491;
        double r7720499 = r7720496 - r7720498;
        double r7720500 = sqrt(r7720499);
        double r7720501 = -r7720487;
        double r7720502 = r7720500 + r7720501;
        double r7720503 = r7720491 / r7720502;
        double r7720504 = 1.4653326993720654e+110;
        bool r7720505 = r7720487 <= r7720504;
        double r7720506 = r7720501 - r7720500;
        double r7720507 = r7720506 / r7720497;
        double r7720508 = -2.0;
        double r7720509 = r7720487 / r7720497;
        double r7720510 = r7720508 * r7720509;
        double r7720511 = r7720505 ? r7720507 : r7720510;
        double r7720512 = r7720495 ? r7720503 : r7720511;
        double r7720513 = r7720489 ? r7720493 : r7720512;
        return r7720513;
}

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

↓

\begin{array}{l}
\mathbf{if}\;b_2 \le -4.595028172156282 \cdot 10^{+143}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.039671900760653 \cdot 10^{-263}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\

\mathbf{elif}\;b_2 \le 1.4653326993720654 \cdot 10^{+110}:\\
\;\;\;\;\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}

Derivation

Split input into 4 regimes
if b_2 < -4.595028172156282e+143
1. Initial program 61.8
  \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
2. Using strategy rm
3. Applied div-inv61.8
  \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
4. Taylor expanded around -inf 1.7
  \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
if -4.595028172156282e+143 < b_2 < -1.039671900760653e-263
1. Initial program 34.9
  \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
2. Using strategy rm
3. Applied div-inv34.9
  \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
4. Using strategy rm
5. Applied flip--35.0
  \[\leadsto \color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{1}{a}\]
6. Applied associate-*l/35.0
  \[\leadsto \color{blue}{\frac{\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]
7. Simplified14.3
  \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{a}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]
8. Taylor expanded around inf 7.8
  \[\leadsto \frac{\color{blue}{c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]
if -1.039671900760653e-263 < b_2 < 1.4653326993720654e+110
1. Initial program 9.7
  \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
if 1.4653326993720654e+110 < b_2
1. Initial program 47.7
  \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
2. Using strategy rm
3. Applied div-inv47.7
  \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
4. Using strategy rm
5. Applied flip--61.6
  \[\leadsto \color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{1}{a}\]
6. Applied associate-*l/61.6
  \[\leadsto \color{blue}{\frac{\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]
7. Simplified61.8
  \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{a}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]
8. Taylor expanded around inf 61.8
  \[\leadsto \frac{\color{blue}{c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]
9. Taylor expanded around 0 3.4
  \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]
Recombined 4 regimes into one program.
Final simplification6.6
\[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.595028172156282 \cdot 10^{+143}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.039671900760653 \cdot 10^{-263}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le 1.4653326993720654 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Error

Derivation

`if b_2 < -4.595028172156282e+143`

`if -4.595028172156282e+143 < b_2 < -1.039671900760653e-263`

`if -1.039671900760653e-263 < b_2 < 1.4653326993720654e+110`

`if 1.4653326993720654e+110 < b_2`

Reproduce