if b < -6.636085511448982e+62

1. Started with
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   29.5
2. Applied taylor to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
   7.8
3. Taylor expanded around -inf to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases}\]
   7.8
4. Applied taylor to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
   7.8
5. Taylor expanded around inf to get
   \[\begin{cases} \frac{\left(-b\right) - \color{red}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
   7.8
6. Applied simplify to get
   \[\color{red}{\begin{cases} \frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{c}{1 \cdot b} - \frac{b - \left(-b\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases}}\]
   1.4
7. Using strategy rm
   1.4
8. Applied expm1-log1p-u to get
   \[\begin{cases} \color{red}{\frac{c}{1 \cdot b} - \frac{b - \left(-b\right)}{2 \cdot a}} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases} \leadsto \begin{cases} \color{blue}{(e^{\log_* (1 + \left(\frac{c}{1 \cdot b} - \frac{b - \left(-b\right)}{2 \cdot a}\right))} - 1)^*} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases}\]
   1.4
9. Applied simplify to get
   \[\begin{cases} (e^{\color{red}{\log_* (1 + \left(\frac{c}{1 \cdot b} - \frac{b - \left(-b\right)}{2 \cdot a}\right))}} - 1)^* & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases} \leadsto \begin{cases} (e^{\color{blue}{\log_* (1 + \left(\frac{c}{b} - \frac{b - \left(-b\right)}{2 \cdot a}\right))}} - 1)^* & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases}\]
   1.4
10. Applied taylor to get
    \[\begin{cases} (e^{\log_* (1 + \left(\frac{c}{b} - \frac{b - \left(-b\right)}{2 \cdot a}\right))} - 1)^* & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases} \leadsto \begin{cases} (e^{\log_* (1 + \left(\frac{b}{c} - \frac{a}{b}\right))} - 1)^* & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases}\]
    1.4
11. Taylor expanded around inf to get
    \[\begin{cases} (e^{\color{red}{\log_* (1 + \left(\frac{b}{c} - \frac{a}{b}\right))}} - 1)^* & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases} \leadsto \begin{cases} (e^{\color{blue}{\log_* (1 + \left(\frac{b}{c} - \frac{a}{b}\right))}} - 1)^* & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases}\]
    1.4
12. Applied simplify to get
    \[\begin{cases} (e^{\log_* (1 + \left(\frac{b}{c} - \frac{a}{b}\right))} - 1)^* & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{b}{c} - \frac{a}{b} & \text{when } b \ge 0 \\ \frac{c \cdot 2}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases}\]
    1.4

---

13. Applied final simplification

if -6.636085511448982e+62 < b < 2.0187374307498356e-297

1. Started with
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   9.6
2. Using strategy rm
   9.6
3. Applied flip-- to get
   \[\begin{cases} \frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   9.6
4. Applied simplify to get
   \[\begin{cases} \frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   9.6

if 2.0187374307498356e-297 < b < 2.7032921376893094e+83

1. Started with
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   8.4
2. Using strategy rm
   8.4
3. Applied flip-+ to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{red}{\frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \color{blue}{\frac{2 \cdot c}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}}} & \text{otherwise} \end{cases}\]
   8.4
4. Applied simplify to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{\color{red}{2 \cdot c}}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{\color{blue}{2 \cdot c}}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases}\]
   8.4
5. Applied taylor to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}} & \text{otherwise} \end{cases}\]
   8.4
6. Taylor expanded around -inf to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}}} & \text{otherwise} \end{cases}\]
   8.4
7. Applied simplify to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\frac{\left(4 \cdot a\right) \cdot c}{-2 \cdot \frac{c \cdot a}{b}}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - a \cdot \left(c \cdot 4\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \left(\frac{c}{4} \cdot \frac{-2}{a}\right) \cdot \frac{\frac{2}{1}}{\frac{b}{a}} & \text{otherwise} \end{cases}\]
   8.3

---

8. Applied final simplification
9. Applied simplify to get
   \[\color{red}{\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - a \cdot \left(c \cdot 4\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \left(\frac{c}{4} \cdot \frac{-2}{a}\right) \cdot \frac{\frac{2}{1}}{\frac{b}{a}} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - c \cdot \left(4 \cdot a\right)}}{a \cdot 2} & \text{when } b \ge 0 \\ \frac{\frac{-2}{a} \cdot \frac{c}{4}}{\frac{b}{a \cdot 2}} & \text{otherwise} \end{cases}}\]
   8.4

if 2.7032921376893094e+83 < b

1. Started with
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases}\]
   43.5
2. Applied taylor to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
   43.5
3. Taylor expanded around -inf to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{red}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}} & \text{otherwise} \end{cases}\]
   43.5
4. Applied taylor to get
   \[\begin{cases} \frac{\left(-b\right) - \sqrt{{b}^2 - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
   11.7
5. Taylor expanded around inf to get
   \[\begin{cases} \frac{\left(-b\right) - \color{red}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}\]
   11.7
6. Applied simplify to get
   \[\color{red}{\begin{cases} \frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)} & \text{otherwise} \end{cases}} \leadsto \color{blue}{\begin{cases} \frac{c}{1 \cdot b} - \frac{b - \left(-b\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases}}\]
   0.1
7. Using strategy rm
   0.1
8. Applied clear-num to get
   \[\begin{cases} \frac{c}{1 \cdot b} - \frac{b - \left(-b\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{2 \cdot c}{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*} & \text{otherwise} \end{cases} \leadsto \begin{cases} \frac{c}{1 \cdot b} - \frac{b - \left(-b\right)}{2 \cdot a} & \text{when } b \ge 0 \\ \frac{1}{\frac{(2 * \left(\frac{c}{b} \cdot a\right) + \left(\left(-b\right) - b\right))_*}{2 \cdot c}} & \text{otherwise} \end{cases}\]
   0.1