if d < -1.4521623f+15

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   20.5
2. Using strategy rm
   20.5
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   20.5
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   20.4
5. Using strategy rm
   20.4
6. Applied flip-- to get
   \[\frac{\color{red}{b \cdot c - a \cdot d}}{{\left(\left|c\right|\right)}^2 + {d}^2} \leadsto \frac{\color{blue}{\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{b \cdot c + a \cdot d}}}{{\left(\left|c\right|\right)}^2 + {d}^2}\]
   23.6
7. Applied associate-/l/ to get
   \[\color{red}{\frac{\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{b \cdot c + a \cdot d}}{{\left(\left|c\right|\right)}^2 + {d}^2}} \leadsto \color{blue}{\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{\left({\left(\left|c\right|\right)}^2 + {d}^2\right) \cdot \left(b \cdot c + a \cdot d\right)}}\]
   23.6
8. Applied taylor to get
   \[\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{\left({\left(\left|c\right|\right)}^2 + {d}^2\right) \cdot \left(b \cdot c + a \cdot d\right)} \leadsto \frac{b \cdot c}{{d}^2} - \frac{a}{d}\]
   6.0
9. Taylor expanded around inf to get
   \[\color{red}{\frac{b \cdot c}{{d}^2} - \frac{a}{d}} \leadsto \color{blue}{\frac{b \cdot c}{{d}^2} - \frac{a}{d}}\]
   6.0
10. Applied simplify to get
    \[\frac{b \cdot c}{{d}^2} - \frac{a}{d} \leadsto \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\]
    0.3

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11. Applied final simplification

if -1.4521623f+15 < d < -8.914957f-21 or 1.3204445f-22 < d

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   11.0
2. Using strategy rm
   11.0
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   11.0
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   9.2
5. Using strategy rm
   9.2
6. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   9.2
7. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   7.4

if -8.914957f-21 < d < 1.3204445f-22

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   10.4
2. Using strategy rm
   10.4
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   10.4
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   5.9
5. Using strategy rm
   5.9
6. Applied flip-- to get
   \[\frac{\color{red}{b \cdot c - a \cdot d}}{{\left(\left|c\right|\right)}^2 + {d}^2} \leadsto \frac{\color{blue}{\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{b \cdot c + a \cdot d}}}{{\left(\left|c\right|\right)}^2 + {d}^2}\]
   14.8
7. Applied associate-/l/ to get
   \[\color{red}{\frac{\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{b \cdot c + a \cdot d}}{{\left(\left|c\right|\right)}^2 + {d}^2}} \leadsto \color{blue}{\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{\left({\left(\left|c\right|\right)}^2 + {d}^2\right) \cdot \left(b \cdot c + a \cdot d\right)}}\]
   14.8
8. Applied taylor to get
   \[\frac{{\left(b \cdot c\right)}^2 - {\left(a \cdot d\right)}^2}{\left({\left(\left|c\right|\right)}^2 + {d}^2\right) \cdot \left(b \cdot c + a \cdot d\right)} \leadsto \frac{b \cdot c}{{\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|c\right|\right)}^2}\]
   5.8
9. Taylor expanded around 0 to get
   \[\color{red}{\frac{b \cdot c}{{\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|c\right|\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|c\right|\right)}^2}}\]
   5.8
10. Applied simplify to get
    \[\frac{b \cdot c}{{\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|c\right|\right)}^2} \leadsto \frac{b}{\left|c\right|} \cdot \frac{c}{\left|c\right|} - \frac{a}{\left|c\right|} \cdot \frac{d}{\left|c\right|}\]
    0

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11. Applied final simplification