if d < -2.958337f+19 or 4.5961277f+15 < d

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   21.9
2. Using strategy rm
   21.9
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   21.9
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   15.9
5. Applied taylor to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c}{{\left(\left|d\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2}\]
   13.8
6. Taylor expanded around 0 to get
   \[\color{red}{\frac{b \cdot c}{{\left(\left|d\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\left|d\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2}}\]
   13.8
7. Applied simplify to get
   \[\frac{b \cdot c}{{\left(\left|d\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2} \leadsto \frac{b}{\left|d\right|} \cdot \frac{c}{\left|d\right|} - \frac{a}{\left|d\right|} \cdot \frac{d}{\left|d\right|}\]
   0

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8. Applied final simplification

if -2.958337f+19 < d < 4.5961277f+15

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   9.0
2. Using strategy rm
   9.0
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   9.0
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   6.1
5. Using strategy rm
   6.1
6. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   6.1
7. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   5.7
8. Applied taylor to get
   \[\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {\left(\left|d\right|\right)}^2} \leadsto \frac{b \cdot c}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}\]
   5.7
9. Taylor expanded around 0 to get
   \[\color{red}{\frac{b \cdot c}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2 + {\left(\left|c\right|\right)}^2}}\]
   5.7