Initial program 1.1
\[\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{x}\right)}\right) - \left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}\right)\]
- Using strategy
rm Applied p16-flip--1.3
\[\leadsto \color{blue}{\frac{\left(\left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{x}\right)}\right) \cdot \left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{x}\right)}\right)\right) - \left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}\right) \cdot \left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}\right)\right)\right)}{\left(\frac{\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{x}\right)}\right)}{\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}\right)}\right)}}\]
- Using strategy
rm Applied *-commutative1.3
\[\leadsto \frac{\left(\color{blue}{\left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{x}\right)}\right) \cdot \left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{x}\right)}\right)\right)} - \left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}\right) \cdot \left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}\right)\right)\right)}{\left(\frac{\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{x}\right)}\right)}{\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}\right)}\right)}\]
Applied difference-of-squares1.2
\[\leadsto \frac{\color{blue}{\left(\left(\frac{\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{x}\right)}\right)}{\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}\right)}\right) \cdot \left(\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{x}\right)}\right) - \left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}\right)\right)\right)}}{\left(\frac{\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{x}\right)}\right)}{\left(\frac{\left(real->posit(1)\right)}{\left(\sqrt{\left(\frac{x}{\left(real->posit(1)\right)}\right)}\right)}\right)}\right)}\]
Final simplification1.2
\[\leadsto \frac{\left(\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}\right) \cdot \left(\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
