- Split input into 3 regimes
if c < -9.900947129885563e+185
Initial program 43.9
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt43.9
\[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity43.9
\[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac43.9
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified43.9
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified30.8
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
Taylor expanded around -inf 11.6
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(-1 \cdot a\right)}\]
Simplified11.6
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(-a\right)}\]
if -9.900947129885563e+185 < c < 1.5638452564338392e+105
Initial program 20.1
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt20.1
\[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity20.1
\[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac20.1
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified20.1
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified12.2
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*l/12.1
\[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
Simplified12.1
\[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
if 1.5638452564338392e+105 < c
Initial program 39.5
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt39.5
\[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity39.5
\[\leadsto \frac{\color{blue}{1 \cdot \left(a \cdot c + b \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac39.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Simplified39.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Simplified25.6
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied associate-*l/25.5
\[\leadsto \color{blue}{\frac{1 \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}}\]
Simplified25.5
\[\leadsto \frac{\color{blue}{\frac{(d \cdot b + \left(c \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}}}{\sqrt{c^2 + d^2}^*}\]
Taylor expanded around 0 17.2
\[\leadsto \frac{\color{blue}{a}}{\sqrt{c^2 + d^2}^*}\]
- Recombined 3 regimes into one program.
Final simplification12.9
\[\leadsto \begin{array}{l}
\mathbf{if}\;c \le -9.900947129885563 \cdot 10^{+185}:\\
\;\;\;\;\frac{-1}{\sqrt{c^2 + d^2}^*} \cdot a\\
\mathbf{elif}\;c \le 1.5638452564338392 \cdot 10^{+105}:\\
\;\;\;\;\frac{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\
\mathbf{else}:\\
\;\;\;\;\frac{a}{\sqrt{c^2 + d^2}^*}\\
\end{array}\]
