if d < -3.6129147f-16 or 2.3232377f-12 < d < 1.5011559f+10

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   11.9
2. Using strategy rm
   11.9
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   11.9
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   10.2
5. Using strategy rm
   10.2
6. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   10.1
7. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   7.9

if -3.6129147f-16 < d < 2.3232377f-12

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   10.5
2. Using strategy rm
   10.5
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   10.5
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   6.7
5. Using strategy rm
   6.7
6. Applied add-sqr-sqrt to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}} \leadsto \color{blue}{{\left(\sqrt{\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}}\right)}^2}\]
   17.8
7. Applied taylor to get
   \[{\left(\sqrt{\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}}\right)}^2 \leadsto {\left(\sqrt{\frac{b \cdot c}{{\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|c\right|\right)}^2}}\right)}^2\]
   17.4
8. Taylor expanded around 0 to get
   \[{\left(\sqrt{\color{red}{\frac{b \cdot c}{{\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|c\right|\right)}^2}}}\right)}^2 \leadsto {\left(\sqrt{\color{blue}{\frac{b \cdot c}{{\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|c\right|\right)}^2}}}\right)}^2\]
   17.4
9. Applied simplify to get
   \[{\left(\sqrt{\frac{b \cdot c}{{\left(\left|c\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|c\right|\right)}^2}}\right)}^2 \leadsto \frac{b}{\left|c\right|} \cdot \frac{c}{\left|c\right|} - \frac{\frac{d \cdot a}{\left|c\right|}}{\left|c\right|}\]
   0.6

---

10. Applied final simplification

if 1.5011559f+10 < d

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   18.8
2. Using strategy rm
   18.8
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2} + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2}\right)}^2} + {d}^2}\]
   18.8
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2}\right)}}^2 + {d}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\left|c\right|\right)}}^2 + {d}^2}\]
   18.7
5. Using strategy rm
   18.7
6. Applied add-sqr-sqrt to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}} \leadsto \color{blue}{{\left(\sqrt{\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}}\right)}^2}\]
   21.3
7. Applied taylor to get
   \[{\left(\sqrt{\frac{b \cdot c - a \cdot d}{{\left(\left|c\right|\right)}^2 + {d}^2}}\right)}^2 \leadsto {\left(\sqrt{\frac{b \cdot c}{{d}^2} - \frac{a}{d}}\right)}^2\]
   15.9
8. Taylor expanded around inf to get
   \[{\left(\sqrt{\color{red}{\frac{b \cdot c}{{d}^2} - \frac{a}{d}}}\right)}^2 \leadsto {\left(\sqrt{\color{blue}{\frac{b \cdot c}{{d}^2} - \frac{a}{d}}}\right)}^2\]
   15.9
9. Applied simplify to get
   \[{\left(\sqrt{\frac{b \cdot c}{{d}^2} - \frac{a}{d}}\right)}^2 \leadsto \frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\]
   0.4

---

10. Applied final simplification