if c < 1.5651492f+15

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
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   10.9
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\sqrt{c^2 + d^2}^*\right)}}^2}\]
   7.1
5. Applied taylor to get
   \[\frac{b \cdot c - a \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
   7.1
6. Taylor expanded around 0 to get
   \[\color{red}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
   7.1
7. Using strategy rm
   7.1
8. Applied *-un-lft-identity to get
   \[\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\color{red}{\left(\sqrt{c^2 + d^2}^*\right)}}^2} \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\color{blue}{\left(1 \cdot \sqrt{c^2 + d^2}^*\right)}}^2}\]
   7.1
9. Applied square-prod to get
   \[\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{\color{red}{{\left(1 \cdot \sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{\color{blue}{{1}^2 \cdot {\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
   7.1
10. Applied times-frac to get
    \[\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \color{red}{\frac{d \cdot a}{{1}^2 \cdot {\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \color{blue}{\frac{d}{{1}^2} \cdot \frac{a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
    3.0
11. Applied simplify to get
    \[\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \color{red}{\frac{d}{{1}^2}} \cdot \frac{a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \color{blue}{\frac{d}{1}} \cdot \frac{a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
    3.0

if 1.5651492f+15 < c

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   20.3
2. Using strategy rm
   20.3
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   20.2
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{\color{blue}{\left(\sqrt{c^2 + d^2}^*\right)}}^2}\]
   13.4
5. Applied taylor to get
   \[\frac{b \cdot c - a \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
   13.4
6. Taylor expanded around 0 to get
   \[\color{red}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
   13.4
7. Using strategy rm
   13.4
8. Applied *-un-lft-identity to get
   \[\frac{b \cdot c}{{\color{red}{\left(\sqrt{c^2 + d^2}^*\right)}}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{b \cdot c}{{\color{blue}{\left(1 \cdot \sqrt{c^2 + d^2}^*\right)}}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
   13.4
9. Applied square-prod to get
   \[\frac{b \cdot c}{\color{red}{{\left(1 \cdot \sqrt{c^2 + d^2}^*\right)}^2}} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{b \cdot c}{\color{blue}{{1}^2 \cdot {\left(\sqrt{c^2 + d^2}^*\right)}^2}} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
   13.4
10. Applied times-frac to get
    \[\color{red}{\frac{b \cdot c}{{1}^2 \cdot {\left(\sqrt{c^2 + d^2}^*\right)}^2}} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \color{blue}{\frac{b}{{1}^2} \cdot \frac{c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
    3.7
11. Applied simplify to get
    \[\color{red}{\frac{b}{{1}^2}} \cdot \frac{c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \color{blue}{\frac{b}{1}} \cdot \frac{c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} - \frac{d \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
    3.7