if (/ (- (* b c) (* a d)) (+ (sqr c) (sqr d))) < -1.4069939366766694e-241

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   9.9
2. Using strategy rm
   9.9
3. Applied clear-num to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}}\]
   10.0

if -1.4069939366766694e-241 < (/ (- (* b c) (* a d)) (+ (sqr c) (sqr d))) < 0.0

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   26.1
2. Using strategy rm
   26.1
3. Applied clear-num to get
   \[\color{red}{\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}} \leadsto \color{blue}{\frac{1}{\frac{{c}^2 + {d}^2}{b \cdot c - a \cdot d}}}\]
   26.1
4. Using strategy rm
   26.1
5. Applied add-sqr-sqrt to get
   \[\frac{1}{\frac{{c}^2 + {d}^2}{\color{red}{b \cdot c - a \cdot d}}} \leadsto \frac{1}{\frac{{c}^2 + {d}^2}{\color{blue}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}}\]
   47.5
6. Applied add-sqr-sqrt to get
   \[\frac{1}{\frac{\color{red}{{c}^2 + {d}^2}}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}} \leadsto \frac{1}{\frac{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}\]
   47.5
7. Applied square-undiv to get
   \[\frac{1}{\color{red}{\frac{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}} \leadsto \frac{1}{\color{blue}{{\left(\frac{\sqrt{{c}^2 + {d}^2}}{\sqrt{b \cdot c - a \cdot d}}\right)}^2}}\]
   47.5
8. Applied simplify to get
   \[\frac{1}{{\color{red}{\left(\frac{\sqrt{{c}^2 + {d}^2}}{\sqrt{b \cdot c - a \cdot d}}\right)}}^2} \leadsto \frac{1}{{\color{blue}{\left(\frac{\sqrt{c^2 + d^2}^*}{\sqrt{c \cdot b - a \cdot d}}\right)}}^2}\]
   35.4
9. Applied taylor to get
   \[\frac{1}{{\left(\frac{\sqrt{c^2 + d^2}^*}{\sqrt{c \cdot b - a \cdot d}}\right)}^2} \leadsto \frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
   24.4
10. Taylor expanded around 0 to get
    \[\color{red}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
    24.4
11. Applied simplify to get
    \[\frac{b \cdot c}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{\frac{c \cdot b}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\]
    11.6

---

12. Applied final simplification

if 0.0 < (/ (- (* b c) (* a d)) (+ (sqr c) (sqr d)))

1. Started with
   \[\frac{b \cdot c - a \cdot d}{{c}^2 + {d}^2}\]
   33.1
2. Using strategy rm
   33.1
3. Applied add-sqr-sqrt to get
   \[\frac{b \cdot c - a \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
   33.1
4. Applied add-sqr-sqrt to get
   \[\frac{\color{red}{b \cdot c - a \cdot d}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}\]
   33.2
5. Applied square-undiv to get
   \[\color{red}{\frac{{\left(\sqrt{b \cdot c - a \cdot d}\right)}^2}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{b \cdot c - a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}^2}\]
   33.2
6. Applied simplify to get
   \[{\color{red}{\left(\frac{\sqrt{b \cdot c - a \cdot d}}{\sqrt{{c}^2 + {d}^2}}\right)}}^2 \leadsto {\color{blue}{\left(\frac{\sqrt{c \cdot b - d \cdot a}}{\sqrt{c^2 + d^2}^*}\right)}}^2\]
   31.7