if b < -2.3005939f+20

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   30.9
2. Using strategy rm
   30.9
3. Applied flip-- to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   30.9
4. Applied simplify to get
   \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
   19.4
5. Using strategy rm
   19.4
6. Applied *-un-lft-identity to get
   \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{red}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]
   19.4
7. Applied times-frac to get
   \[\frac{\color{red}{\frac{c \cdot \left(4 \cdot a\right)}{1 \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   19.3
8. Applied times-frac to get
   \[\color{red}{\frac{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{c}{1}}{2} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}}\]
   19.3
9. Applied simplify to get
   \[\color{red}{\frac{\frac{c}{1}}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a} \leadsto \color{blue}{\frac{c}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}\]
   19.3
10. Applied simplify to get
    \[\frac{c}{2} \cdot \color{red}{\frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}} \leadsto \frac{c}{2} \cdot \color{blue}{\frac{4}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\]
    19.3
11. Using strategy rm
    19.3
12. Applied add-exp-log to get
    \[\frac{c}{2} \cdot \frac{4}{\color{red}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}} \leadsto \frac{c}{2} \cdot \frac{4}{\color{blue}{e^{\log \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}\right)}}}\]
    19.3
13. Applied simplify to get
    \[\frac{c}{2} \cdot \frac{4}{e^{\color{red}{\log \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}\right)}}} \leadsto \frac{c}{2} \cdot \frac{4}{e^{\color{blue}{\log \left(\left(-b\right) + \sqrt{{b}^2 - a \cdot \left(c \cdot 4\right)}\right)}}}\]
    19.3
14. Applied taylor to get
    \[\frac{c}{2} \cdot \frac{4}{e^{\log \left(\left(-b\right) + \sqrt{{b}^2 - a \cdot \left(c \cdot 4\right)}\right)}} \leadsto \frac{c}{2} \cdot \frac{4}{e^{\log \left(2 \cdot \frac{c \cdot a}{b} - 2 \cdot b\right)}}\]
    6.2
15. Taylor expanded around -inf to get
    \[\frac{c}{2} \cdot \frac{4}{e^{\log \color{red}{\left(2 \cdot \frac{c \cdot a}{b} - 2 \cdot b\right)}}} \leadsto \frac{c}{2} \cdot \frac{4}{e^{\log \color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - 2 \cdot b\right)}}}\]
    6.2
16. Applied simplify to get
    \[\frac{c}{2} \cdot \frac{4}{e^{\log \left(2 \cdot \frac{c \cdot a}{b} - 2 \cdot b\right)}} \leadsto \frac{\frac{4}{2} \cdot \frac{c}{2}}{\frac{a}{\frac{b}{c}} - b}\]
    0.2

---

17. Applied final simplification
18. Applied simplify to get
    \[\color{red}{\frac{\frac{4}{2} \cdot \frac{c}{2}}{\frac{a}{\frac{b}{c}} - b}} \leadsto \color{blue}{\frac{\frac{\frac{4}{2}}{2}}{\frac{a}{b} - \frac{b}{c}}}\]
    1.0

if -2.3005939f+20 < b < 1.0894418f-30

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   15.1
2. Using strategy rm
   15.1
3. Applied flip-- to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   16.7
4. Applied simplify to get
   \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
   6.9
5. Using strategy rm
   6.9
6. Applied *-un-lft-identity to get
   \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{red}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]
   6.9
7. Applied times-frac to get
   \[\frac{\color{red}{\frac{c \cdot \left(4 \cdot a\right)}{1 \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   4.7
8. Applied times-frac to get
   \[\color{red}{\frac{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{c}{1}}{2} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}}\]
   4.7
9. Applied simplify to get
   \[\color{red}{\frac{\frac{c}{1}}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a} \leadsto \color{blue}{\frac{c}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}\]
   4.7
10. Applied simplify to get
    \[\frac{c}{2} \cdot \color{red}{\frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}} \leadsto \frac{c}{2} \cdot \color{blue}{\frac{4}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\]
    4.7
11. Applied simplify to get
    \[\frac{c}{2} \cdot \frac{4}{\color{red}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}} \leadsto \frac{c}{2} \cdot \frac{4}{\color{blue}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4}}}\]
    4.7

if 1.0894418f-30 < b < 2.2299319f+17

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   3.4
2. Using strategy rm
   3.4
3. Applied *-un-lft-identity to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a}\]
   3.4
4. Applied times-frac to get
   \[\color{red}{\frac{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{2} \cdot \frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{a}}\]
   3.4

if 2.2299319f+17 < b

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   26.8
2. Applied taylor to get
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \leadsto \frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a}\]
   6.4
3. Taylor expanded around inf to get
   \[\frac{\color{red}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a} \leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]
   6.4
4. Applied simplify to get
   \[\color{red}{\frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{c}{b}}{1} - \frac{b}{a}}\]
   0.0
5. Applied simplify to get
   \[\color{red}{\frac{\frac{c}{b}}{1}} - \frac{b}{a} \leadsto \color{blue}{\frac{c}{b}} - \frac{b}{a}\]
   0.0