- Split input into 3 regimes
if d < -6.632121896129618e+89
Initial program 38.6
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification38.6
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt38.6
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity38.6
\[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac38.6
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified38.6
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified24.7
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
Taylor expanded around -inf 16.4
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(-1 \cdot b\right)}\]
Simplified16.4
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\left(-b\right)}\]
if -6.632121896129618e+89 < d < 1.5750457472641093e+91
Initial program 18.2
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification18.2
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt18.3
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity18.3
\[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac18.3
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified18.3
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified11.3
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*l/11.2
\[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
Simplified11.2
\[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
- Using strategy
rm Applied add-sqr-sqrt11.4
\[\leadsto \frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\color{blue}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}}\]
if 1.5750457472641093e+91 < d
Initial program 37.6
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Initial simplification37.6
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt37.6
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied *-un-lft-identity37.6
\[\leadsto \frac{\color{blue}{1 \cdot (a \cdot c + \left(b \cdot d\right))_*}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied times-frac37.6
\[\leadsto \color{blue}{\frac{1}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Simplified37.6
\[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Simplified25.8
\[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}\]
- Using strategy
rm Applied associate-*l/25.8
\[\leadsto \color{blue}{\frac{1 \cdot \frac{(d \cdot b + \left(a \cdot c\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]
Simplified25.8
\[\leadsto \frac{\color{blue}{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
Taylor expanded around 0 15.6
\[\leadsto \frac{\color{blue}{b}}{\sqrt{d^2 + c^2}^*}\]
- Recombined 3 regimes into one program.
Final simplification13.1
\[\leadsto \begin{array}{l}
\mathbf{if}\;d \le -6.632121896129618 \cdot 10^{+89}:\\
\;\;\;\;\frac{-1}{\sqrt{d^2 + c^2}^*} \cdot b\\
\mathbf{elif}\;d \le 1.5750457472641093 \cdot 10^{+91}:\\
\;\;\;\;\frac{\frac{(c \cdot a + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{\sqrt{d^2 + c^2}^*} \cdot \sqrt{\sqrt{d^2 + c^2}^*}}\\
\mathbf{else}:\\
\;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\
\end{array}\]
