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

↓

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

double f(double a, double b_2, double c) {
        double r1048086 = b_2;
        double r1048087 = -r1048086;
        double r1048088 = r1048086 * r1048086;
        double r1048089 = a;
        double r1048090 = c;
        double r1048091 = r1048089 * r1048090;
        double r1048092 = r1048088 - r1048091;
        double r1048093 = sqrt(r1048092);
        double r1048094 = r1048087 - r1048093;
        double r1048095 = r1048094 / r1048089;
        return r1048095;
}

↓

double f(double a, double b_2, double c) {
        double r1048096 = b_2;
        double r1048097 = -4.595028172156282e+143;
        bool r1048098 = r1048096 <= r1048097;
        double r1048099 = -0.5;
        double r1048100 = c;
        double r1048101 = r1048100 / r1048096;
        double r1048102 = r1048099 * r1048101;
        double r1048103 = -1.039671900760653e-263;
        bool r1048104 = r1048096 <= r1048103;
        double r1048105 = r1048096 * r1048096;
        double r1048106 = a;
        double r1048107 = r1048106 * r1048100;
        double r1048108 = r1048105 - r1048107;
        double r1048109 = sqrt(r1048108);
        double r1048110 = -r1048096;
        double r1048111 = r1048109 + r1048110;
        double r1048112 = r1048100 / r1048111;
        double r1048113 = 1.4653326993720654e+110;
        bool r1048114 = r1048096 <= r1048113;
        double r1048115 = r1048110 - r1048109;
        double r1048116 = r1048115 / r1048106;
        double r1048117 = -2.0;
        double r1048118 = r1048096 / r1048106;
        double r1048119 = r1048117 * r1048118;
        double r1048120 = r1048114 ? r1048116 : r1048119;
        double r1048121 = r1048104 ? r1048112 : r1048120;
        double r1048122 = r1048098 ? r1048102 : r1048121;
        return r1048122;
}

\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