- Split input into 3 regimes
if c < -2.1110781218601205e+158
Initial program 46.0
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Simplified46.0
\[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
- Using strategy
rm Applied add-sqr-sqrt46.0
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied associate-/r*46.0
\[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
- Using strategy
rm Applied fma-udef46.0
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied hypot-def46.0
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{d^2 + c^2}^*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
- Using strategy
rm Applied fma-udef46.0
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
Applied hypot-def29.1
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
Taylor expanded around -inf 14.1
\[\leadsto \frac{\color{blue}{-1 \cdot a}}{\sqrt{d^2 + c^2}^*}\]
Simplified14.1
\[\leadsto \frac{\color{blue}{-a}}{\sqrt{d^2 + c^2}^*}\]
if -2.1110781218601205e+158 < c < 5.614617575965483e+189
Initial program 20.5
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Simplified20.5
\[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
- Using strategy
rm Applied add-sqr-sqrt20.5
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied associate-/r*20.4
\[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
- Using strategy
rm Applied fma-udef20.4
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied hypot-def20.4
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{d^2 + c^2}^*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
- Using strategy
rm Applied fma-udef20.4
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
Applied hypot-def12.5
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
if 5.614617575965483e+189 < c
Initial program 42.9
\[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]
Simplified42.9
\[\leadsto \color{blue}{\frac{(a \cdot c + \left(b \cdot d\right))_*}{(d \cdot d + \left(c \cdot c\right))_*}}\]
- Using strategy
rm Applied add-sqr-sqrt42.9
\[\leadsto \frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*} \cdot \sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
Applied associate-/r*42.9
\[\leadsto \color{blue}{\frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}}\]
- Using strategy
rm Applied fma-udef42.9
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
Applied hypot-def42.9
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\color{blue}{\sqrt{d^2 + c^2}^*}}}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]
- Using strategy
rm Applied fma-udef42.9
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{\color{blue}{d \cdot d + c \cdot c}}}\]
Applied hypot-def30.4
\[\leadsto \frac{\frac{(a \cdot c + \left(b \cdot d\right))_*}{\sqrt{d^2 + c^2}^*}}{\color{blue}{\sqrt{d^2 + c^2}^*}}\]
Taylor expanded around inf 11.8
\[\leadsto \frac{\color{blue}{a}}{\sqrt{d^2 + c^2}^*}\]
- Recombined 3 regimes into one program.
Final simplification12.7
\[\leadsto \begin{array}{l}
\mathbf{if}\;c \le -2.1110781218601205 \cdot 10^{+158}:\\
\;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\
\mathbf{elif}\;c \le 5.614617575965483 \cdot 10^{+189}:\\
\;\;\;\;\frac{\frac{(a \cdot c + \left(d \cdot b\right))_*}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\
\mathbf{else}:\\
\;\;\;\;\frac{a}{\sqrt{d^2 + c^2}^*}\\
\end{array}\]
