if b < -8.642142288968947e+88

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   58.9
2. Using strategy rm
   58.9
3. Applied flip-- to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   59.0
4. Applied simplify to get
   \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
   34.8
5. Using strategy rm
   34.8
6. Applied clear-num to get
   \[\color{red}{\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}}\]
   34.9
7. Applied simplify to get
   \[\frac{1}{\color{red}{\frac{2 \cdot a}{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}} \leadsto \frac{1}{\color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)}}\]
   33.0
8. Applied taylor to get
   \[\frac{1}{\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)} \leadsto \frac{1}{\left(2 \cdot \frac{c \cdot a}{b} - 2 \cdot b\right) \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)}\]
   8.7
9. Taylor expanded around -inf to get
   \[\frac{1}{\color{red}{\left(2 \cdot \frac{c \cdot a}{b} - 2 \cdot b\right)} \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)} \leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - 2 \cdot b\right)} \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)}\]
   8.7
10. Applied simplify to get
    \[\color{red}{\frac{1}{\left(2 \cdot \frac{c \cdot a}{b} - 2 \cdot b\right) \cdot \left(\frac{1}{c} \cdot \frac{2}{4}\right)}} \leadsto \color{blue}{\frac{1}{\frac{2}{4}} \cdot \frac{\frac{c}{2}}{\frac{a}{b} \cdot c - b}}\]
    1.0
11. Applied simplify to get
    \[\color{red}{\frac{1}{\frac{2}{4}}} \cdot \frac{\frac{c}{2}}{\frac{a}{b} \cdot c - b} \leadsto \color{blue}{\frac{4}{2}} \cdot \frac{\frac{c}{2}}{\frac{a}{b} \cdot c - b}\]
    1.0

if -8.642142288968947e+88 < b < -3.247958591749237e-290

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   32.7
2. Using strategy rm
   32.7
3. Applied flip-- to get
   \[\frac{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   32.8
4. Applied simplify to get
   \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
   17.0
5. Using strategy rm
   17.0
6. Applied *-un-lft-identity to get
   \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{red}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{blue}{1 \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]
   17.0
7. Applied times-frac to get
   \[\frac{\color{red}{\frac{c \cdot \left(4 \cdot a\right)}{1 \cdot \left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   15.2
8. Applied times-frac to get
   \[\color{red}{\frac{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{c}{1}}{2} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}}\]
   10.1
9. Applied simplify to get
   \[\color{red}{\frac{\frac{c}{1}}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a} \leadsto \color{blue}{\frac{c}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}\]
   10.1
10. Applied simplify to get
    \[\frac{c}{2} \cdot \color{red}{\frac{\frac{4 \cdot a}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}} \leadsto \frac{c}{2} \cdot \color{blue}{\frac{4}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\]
    8.9
11. Applied simplify to get
    \[\frac{c}{2} \cdot \frac{4}{\color{red}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}} \leadsto \frac{c}{2} \cdot \frac{4}{\color{blue}{\left(-b\right) + \sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4}}}\]
    8.8

if -3.247958591749237e-290 < b < 2.2736054037680634e+124

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   9.2

if 2.2736054037680634e+124 < b

1. Started with
   \[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   51.8
2. Using strategy rm
   51.8
3. Applied add-sqr-sqrt to get
   \[\color{red}{\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \leadsto \color{blue}{{\left(\sqrt{\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\right)}^2}\]
   58.3
4. Applied taylor to get
   \[{\left(\sqrt{\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\right)}^2 \leadsto {\left(\sqrt{\frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a}}\right)}^2\]
   38.5
5. Taylor expanded around inf to get
   \[{\left(\sqrt{\frac{\left(-b\right) - \color{red}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a}}\right)}^2 \leadsto {\left(\sqrt{\frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{c \cdot a}{b}\right)}}{2 \cdot a}}\right)}^2\]
   38.5
6. Applied simplify to get
   \[{\left(\sqrt{\frac{\left(-b\right) - \left(b - 2 \cdot \frac{c \cdot a}{b}\right)}{2 \cdot a}}\right)}^2 \leadsto \frac{\left(\left(-b\right) - b\right) + \frac{2 \cdot c}{\frac{b}{a}}}{a \cdot 2}\]
   1.8

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7. Applied final simplification