if d < -4.083719811311209e+189 or 4.999857972009831e+168 < d

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   40.1
2. Using strategy rm
   40.1
3. Applied div-sub to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
   40.1
4. Using strategy rm
   40.1
5. Applied *-un-lft-identity to get
   \[\frac{b \cdot c}{\color{red}{{c}^2 + {d}^2}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b \cdot c}{\color{blue}{1 \cdot \left({c}^2 + {d}^2\right)}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
   40.1
6. Applied times-frac to get
   \[\color{red}{\frac{b \cdot c}{1 \cdot \left({c}^2 + {d}^2\right)}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \color{blue}{\frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
   36.9
7. Using strategy rm
   36.9
8. Applied add-sqr-sqrt to get
   \[\frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \frac{a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \frac{a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   36.9
9. Applied add-sqr-sqrt to get
   \[\frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \frac{\color{red}{a \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \frac{\color{blue}{{\left(\sqrt{a \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
   50.8
10. Applied square-undiv to get
    \[\frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \color{red}{\frac{{\left(\sqrt{a \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - \color{blue}{{\left(\frac{\sqrt{a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
    50.8
11. Applied simplify to get
    \[\frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - {\color{red}{\left(\frac{\sqrt{a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto \frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - {\color{blue}{\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
    44.6
12. Applied taylor to get
    \[\frac{b}{1} \cdot \frac{c}{{c}^2 + {d}^2} - {\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto \frac{b}{1} \cdot 0 - {\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
    44.6
13. Taylor expanded around 0 to get
    \[\frac{b}{1} \cdot \color{red}{0} - {\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto \frac{b}{1} \cdot \color{blue}{0} - {\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
    44.6
14. Applied simplify to get
    \[\frac{b}{1} \cdot 0 - {\left(\frac{\sqrt{d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto -\frac{d}{\sqrt{c^2 + d^2}^*} \cdot \frac{a}{\sqrt{c^2 + d^2}^*}\]
    0.0

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15. Applied final simplification

if -4.083719811311209e+189 < d < 4.999857972009831e+168

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   20.3
2. Using strategy rm
   20.3
3. Applied div-sub to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{c}^2 + {d}^2} - \frac{a \cdot d}{{c}^2 + {d}^2}}\]
   20.3
4. Using strategy rm
   20.3
5. Applied associate-/l* to get
   \[\color{red}{\frac{b \cdot c}{{c}^2 + {d}^2}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \color{blue}{\frac{b}{\frac{{c}^2 + {d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
   18.9
6. Applied taylor to get
   \[\frac{b}{\frac{{c}^2 + {d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{c + \frac{{d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
   10.1
7. Taylor expanded around 0 to get
   \[\frac{b}{\color{red}{c + \frac{{d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{\color{blue}{c + \frac{{d}^2}{c}}} - \frac{a \cdot d}{{c}^2 + {d}^2}\]
   10.1
8. Applied simplify to get
   \[\frac{b}{c + \frac{{d}^2}{c}} - \frac{a \cdot d}{{c}^2 + {d}^2} \leadsto \frac{b}{\frac{d}{\frac{c}{d}} + c} - \frac{a \cdot d}{(c * c + \left(d \cdot d\right))_*}\]
   9.6

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9. Applied final simplification
10. Applied simplify to get
    \[\color{red}{\frac{b}{\frac{d}{\frac{c}{d}} + c} - \frac{a \cdot d}{(c * c + \left(d \cdot d\right))_*}} \leadsto \color{blue}{\frac{b}{(\left(\frac{d}{c}\right) * d + c)_*} - \frac{d \cdot a}{(c * c + \left(d \cdot d\right))_*}}\]
    9.6