- Split input into 3 regimes
if c < -4.455259953183728e+101
Initial program 39.7
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt39.7
\[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity39.7
\[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac39.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified39.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified26.3
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*r/26.3
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot (b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
Taylor expanded around -inf 16.3
\[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{c^2 + d^2}^*}\]
Simplified16.3
\[\leadsto \frac{\color{blue}{-a}}{\sqrt{c^2 + d^2}^*}\]
if -4.455259953183728e+101 < c < 2.2124615645609753e+171
Initial program 19.1
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt19.1
\[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity19.1
\[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac19.1
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified19.1
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified11.9
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*r/11.9
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot (b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
if 2.2124615645609753e+171 < c
Initial program 43.7
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt43.7
\[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity43.7
\[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac43.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified43.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified28.9
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*r/28.9
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot (b \cdot d + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}\]
Taylor expanded around inf 13.0
\[\leadsto \frac{\color{blue}{a}}{\sqrt{c^2 + d^2}^*}\]
- Recombined 3 regimes into one program.
Final simplification12.8
\[\leadsto \begin{array}{l}
\mathbf{if}\;c \le -4.455259953183728 \cdot 10^{+101}:\\
\;\;\;\;\frac{-a}{\sqrt{c^2 + d^2}^*}\\
\mathbf{elif}\;c \le 2.2124615645609753 \cdot 10^{+171}:\\
\;\;\;\;\frac{(b \cdot d + \left(a \cdot c\right))_* \cdot \frac{1}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\
\mathbf{else}:\\
\;\;\;\;\frac{a}{\sqrt{c^2 + d^2}^*}\\
\end{array}\]
