if c < -1.1288553016514488e+169 or 7.979289197910273e+111 < c

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   40.3
2. Using strategy rm
   40.3
3. Applied clear-num to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}}\]
   40.3
4. Using strategy rm
   40.3
5. Applied add-sqr-sqrt to get
   \[\frac{1}{\frac{{c}^2 + {d}^2}{\color{red}{b \cdot c - a \cdot d}}} \leadsto \frac{1}{\frac{{c}^2 + {d}^2}{\color{blue}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}}\]
   52.4
6. Applied add-sqr-sqrt to get
   \[\frac{1}{\frac{\color{red}{{c}^2 + {d}^2}}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}} \leadsto \frac{1}{\frac{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}\]
   52.4
7. Applied square-undiv to get
   \[\frac{1}{\color{red}{\frac{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}} \leadsto \frac{1}{\color{blue}{{\left(\frac{\sqrt{{c}^2 + {d}^2}}{\sqrt{b \cdot c - a \cdot d}}\right)}^2}}\]
   52.4
8. Applied simplify to get
   \[\frac{1}{{\color{red}{\left(\frac{\sqrt{{c}^2 + {d}^2}}{\sqrt{b \cdot c - a \cdot d}}\right)}}^2} \leadsto \frac{1}{{\color{blue}{\left(\frac{\sqrt{c^2 + d^2}^*}{\sqrt{c \cdot b - a \cdot d}}\right)}}^2}\]
   49.2
9. Applied taylor to get
   \[\frac{1}{{\left(\frac{\sqrt{c^2 + d^2}^*}{\sqrt{c \cdot b - a \cdot d}}\right)}^2} \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
   35.9
10. Taylor expanded around 0 to get
    \[\color{red}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
    35.9
11. Applied simplify to get
    \[\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{\frac{c \cdot b}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\]
    23.3

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12. Applied final simplification

if -1.1288553016514488e+169 < c < 7.979289197910273e+111

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   19.3
2. Using strategy rm
   19.3
3. Applied clear-num to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}}\]
   19.6