if c < -8.0443834f+10 or 2.6212168f+19 < c

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   20.9
2. Using strategy rm
   20.9
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   20.9
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   20.8
5. Applied taylor to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2}\]
   6.1
6. Taylor expanded around inf to get
   \[\color{red}{\left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2}} \leadsto \color{blue}{\left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2}}\]
   6.1
7. Applied taylor to get
   \[\left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2} \leadsto \left(\frac{b}{c} + 0\right) - \frac{d \cdot a}{{c}^2}\]
   6.0
8. Taylor expanded around inf to get
   \[\left(\frac{b}{c} + \color{red}{0}\right) - \frac{d \cdot a}{{c}^2} \leadsto \left(\frac{b}{c} + \color{blue}{0}\right) - \frac{d \cdot a}{{c}^2}\]
   6.0
9. Applied simplify to get
   \[\left(\frac{b}{c} + 0\right) - \frac{d \cdot a}{{c}^2} \leadsto \frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\]
   0.3

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10. Applied final simplification

if -8.0443834f+10 < c < 2.6212168f+19

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   8.8
2. Using strategy rm
   8.8
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   8.7
4. Applied add-sqr-sqrt to get
   \[\frac{\color{red}{b \cdot c - a \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
   20.3
5. Applied square-undiv to get
   \[\color{red}{\frac{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{b \cdot c - a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
   20.3
6. Applied simplify to get
   \[{\color{red}{\left(\frac{\sqrt{b \cdot c - a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto {\color{blue}{\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
   18.6
7. Applied taylor to get
   \[{\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto {\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
   18.6
8. Taylor expanded around 0 to get
   \[{\left(\frac{\sqrt{c \cdot b - \color{red}{d \cdot a}}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto {\left(\frac{\sqrt{c \cdot b - \color{blue}{d \cdot a}}}{\sqrt{c^2 + d^2}^*}\right)}^2\]
   18.6
9. Applied simplify to get
   \[{\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}^2 \leadsto \frac{b \cdot c - a \cdot d}{\sqrt{c^2 + d^2}^* \cdot \sqrt{c^2 + d^2}^*}\]
   5.4

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10. Applied final simplification