- Split input into 3 regimes
if d < -1.306144681609094e+168
Initial program 43.5
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification43.5
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt43.5
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity43.5
\[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac43.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified43.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified27.4
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*r/27.4
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot (d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
Taylor expanded around -inf 13.3
\[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{d^2 + c^2}^*}\]
Simplified13.3
\[\leadsto \frac{\color{blue}{-b}}{\sqrt{d^2 + c^2}^*}\]
if -1.306144681609094e+168 < d < 4.777330353203238e+199
Initial program 20.9
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification20.9
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt21.0
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity21.0
\[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac21.0
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified21.0
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified12.7
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*r/12.7
\[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{d^2 + c^2}^*} \cdot (d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
if 4.777330353203238e+199 < d
Initial program 38.8
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification38.8
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt38.8
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity38.8
\[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac38.8
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified38.8
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified27.0
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*l/27.0
\[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
Simplified27.0
\[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
Taylor expanded around 0 10.8
\[\leadsto \frac{\color{blue}{b}}{\sqrt{d^2 + c^2}^*}\]
- Recombined 3 regimes into one program.
Final simplification12.6
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \le -1.306144681609094 \cdot 10^{+168}:\\
\;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\
\mathbf{elif}\;d \le 4.777330353203238 \cdot 10^{+199}:\\
\;\;\;\;\frac{(d \cdot b + \left(a \cdot c\right))_* \cdot \frac{1}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\
\mathbf{else}:\\
\;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\
\end{array}\]
