if d < -133854424.0f0 or 1.6446285f+15 < d

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   20.5
2. Using strategy rm
   20.5
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   20.4
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   15.4
5. Using strategy rm
   15.4
6. Applied add-exp-log to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{e^{\log \left(\frac{b \cdot c - a \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}\right)}}\]
   21.6
7. Applied taylor to get
   \[e^{\log \left(\frac{b \cdot c - a \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}\right)} \leadsto \frac{b \cdot c}{{\left(\left|d\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2}\]
   12.5
8. Taylor expanded around 0 to get
   \[\color{red}{\frac{b \cdot c}{{\left(\left|d\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\left|d\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2}}\]
   12.5
9. Applied simplify to get
   \[\frac{b \cdot c}{{\left(\left|d\right|\right)}^2} - \frac{d \cdot a}{{\left(\left|d\right|\right)}^2} \leadsto \frac{b}{\left|d\right|} \cdot \frac{c}{\left|d\right|} - \frac{a}{\left|d\right|} \cdot \frac{d}{\left|d\right|}\]
   0

---

10. Applied final simplification

if -133854424.0f0 < d < -9.938622f-17

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   16.2
2. Using strategy rm
   16.2
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   16.2
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   16.2
5. Applied taylor to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2}\]
   10.7
6. Taylor expanded around inf to get
   \[\color{red}{\left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2}} \leadsto \color{blue}{\left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2}}\]
   10.7
7. Applied taylor to get
   \[\left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2} \leadsto \left(\frac{b}{c} + 0\right) - \frac{d \cdot a}{{c}^2}\]
   2.7
8. Taylor expanded around inf to get
   \[\left(\frac{b}{c} + \color{red}{0}\right) - \frac{d \cdot a}{{c}^2} \leadsto \left(\frac{b}{c} + \color{blue}{0}\right) - \frac{d \cdot a}{{c}^2}\]
   2.7
9. Applied simplify to get
   \[\left(\frac{b}{c} + 0\right) - \frac{d \cdot a}{{c}^2} \leadsto \frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\]
   0.1

---

10. Applied final simplification

if -9.938622f-17 < d < 3.2039805f-25

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   10.2
2. Using strategy rm
   10.2
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   10.2
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   10.2
5. Applied taylor to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2} \leadsto \left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2}\]
   15.0
6. Taylor expanded around inf to get
   \[\color{red}{\left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2}} \leadsto \color{blue}{\left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2}}\]
   15.0
7. Applied taylor to get
   \[\left(\frac{b}{c} + \frac{d \cdot \left(a \cdot {\left(\left|\frac{1}{d}\right|\right)}^2\right)}{{c}^{4}}\right) - \frac{d \cdot a}{{c}^2} \leadsto \left(\frac{b}{c} + 0\right) - \frac{d \cdot a}{{c}^2}\]
   3.0
8. Taylor expanded around inf to get
   \[\left(\frac{b}{c} + \color{red}{0}\right) - \frac{d \cdot a}{{c}^2} \leadsto \left(\frac{b}{c} + \color{blue}{0}\right) - \frac{d \cdot a}{{c}^2}\]
   3.0
9. Applied simplify to get
   \[\left(\frac{b}{c} + 0\right) - \frac{d \cdot a}{{c}^2} \leadsto \frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\]
   1.5

---

10. Applied final simplification

if 3.2039805f-25 < d < 1.6446285f+15

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   7.6
2. Using strategy rm
   7.6
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + \color{red}{{d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + \color{blue}{{\left(\sqrt{{d}^2}\right)}^2}}\]
   7.6
4. Applied simplify to get
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {\color{red}{\left(\sqrt{{d}^2}\right)}}^2} \leadsto \frac{b \cdot c - a \cdot d}{{c}^2 + {\color{blue}{\left(\left|d\right|\right)}}^2}\]
   7.4
5. Using strategy rm
   7.4
6. Applied add-cube-cbrt to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{b \cdot c - a \cdot d}{{c}^2 + {\left(\left|d\right|\right)}^2}}\right)}^3}\]
   7.7