if b < -1.3686002f-10

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   28.0
2. Using strategy rm
   28.0
3. Applied flip-- to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   29.4
4. Applied simplify to get
   \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
   16.4
5. Applied taylor to get
   \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{2 \cdot a}\]
   7.5
6. Taylor expanded around -inf to get
   \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \color{red}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}}}{2 \cdot a}\]
   7.5
7. Applied simplify to get
   \[\color{red}{\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} - b\right)}}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{4 \cdot c}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)}}{2}}\]
   2.3
8. Applied taylor to get
   \[\frac{\frac{4 \cdot c}{\frac{2 \cdot a}{\frac{b}{c}} - \left(b - \left(-b\right)\right)}}{2} \leadsto \frac{\frac{4 \cdot c}{2 \cdot \frac{c \cdot a}{b} - \left(b - \left(-b\right)\right)}}{2}\]
   3.8
9. Taylor expanded around 0 to get
   \[\frac{\frac{4 \cdot c}{\color{red}{2 \cdot \frac{c \cdot a}{b}} - \left(b - \left(-b\right)\right)}}{2} \leadsto \frac{\frac{4 \cdot c}{\color{blue}{2 \cdot \frac{c \cdot a}{b}} - \left(b - \left(-b\right)\right)}}{2}\]
   3.8
10. Applied simplify to get
    \[\frac{\frac{4 \cdot c}{2 \cdot \frac{c \cdot a}{b} - \left(b - \left(-b\right)\right)}}{2} \leadsto \frac{\frac{c \cdot 4}{2}}{\frac{2 \cdot c}{\frac{b}{a}} - \left(b - \left(-b\right)\right)}\]
    2.1

---

11. Applied final simplification

if -1.3686002f-10 < b < 1.7973414f+17

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   5.7
2. Using strategy rm
   5.7
3. Applied *-un-lft-identity to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a}\]
   5.7
4. Applied times-frac to get
   \[\color{red}{\frac{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{a}}\]
   5.7

if 1.7973414f+17 < b

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   26.2
2. Using strategy rm
   26.2
3. Applied *-un-lft-identity to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a}\]
   26.2
4. Applied times-frac to get
   \[\color{red}{\frac{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{a}}\]
   26.2
5. Applied taylor to get
   \[\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{a} \leadsto \frac{1}{2} \cdot \frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{a}\]
   6.6
6. Taylor expanded around inf to get
   \[\frac{1}{2} \cdot \frac{\left(-b\right) - \color{red}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{a} \leadsto \frac{1}{2} \cdot \frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{a}\]
   6.6
7. Applied simplify to get
   \[\frac{1}{2} \cdot \frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{a} \leadsto \frac{\frac{1}{2}}{a} \cdot \left(\frac{c \cdot 2}{\frac{b}{a}} + \left(\left(-b\right) - b\right)\right)\]
   1.5

---

8. Applied final simplification
9. Applied simplify to get
   \[\color{red}{\frac{\frac{1}{2}}{a} \cdot \left(\frac{c \cdot 2}{\frac{b}{a}} + \left(\left(-b\right) - b\right)\right)} \leadsto \color{blue}{\frac{\frac{c}{b} \cdot \left(a \cdot 2\right) - \left(b - \left(-b\right)\right)}{a \cdot 2}}\]
   1.2