Jmat.Real.erfi, branch x greater than or equal to 5

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

4. Final simplification100.0%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in13.3%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)} \]

   2. +-commutative13.3%

      \[\leadsto \color{blue}{\left(\frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)} \]

3. Simplified99.6%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)} \]
4. Step-by-step derivation

   1. add-log-exp0.8%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\log \left(e^{\left|x\right| \cdot \sqrt{\pi}}\right)}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   2. *-un-lft-identity0.8%

      \[\leadsto \frac{e^{x \cdot x}}{\log \color{blue}{\left(1 \cdot e^{\left|x\right| \cdot \sqrt{\pi}}\right)}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   3. log-prod0.8%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\log 1 + \log \left(e^{\left|x\right| \cdot \sqrt{\pi}}\right)}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   4. metadata-eval0.8%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{0} + \log \left(e^{\left|x\right| \cdot \sqrt{\pi}}\right)} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   5. add-log-exp99.6%

      \[\leadsto \frac{e^{x \cdot x}}{0 + \color{blue}{\left|x\right| \cdot \sqrt{\pi}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   6. add-sqr-sqrt99.6%

      \[\leadsto \frac{e^{x \cdot x}}{0 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   7. fabs-sqr99.6%

      \[\leadsto \frac{e^{x \cdot x}}{0 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   8. add-sqr-sqrt99.6%

      \[\leadsto \frac{e^{x \cdot x}}{0 + \color{blue}{x} \cdot \sqrt{\pi}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

5. Applied egg-rr99.6%

   \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{0 + x \cdot \sqrt{\pi}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
6. Step-by-step derivation

   1. +-lft-identity99.6%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{x \cdot \sqrt{\pi}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

7. Simplified99.6%

   \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{x \cdot \sqrt{\pi}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

8. Final simplification99.6%

   \[\leadsto \frac{e^{x \cdot x}}{x \cdot \sqrt{\pi}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. distribute-rgt-in13.3%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)} \]

   2. +-commutative13.3%

      \[\leadsto \color{blue}{\left(\frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) + \left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)} \]

3. Simplified99.6%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)} \]
4. Step-by-step derivation

   1. add-log-exp0.8%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\log \left(e^{\left|x\right| \cdot \sqrt{\pi}}\right)}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   2. *-un-lft-identity0.8%

      \[\leadsto \frac{e^{x \cdot x}}{\log \color{blue}{\left(1 \cdot e^{\left|x\right| \cdot \sqrt{\pi}}\right)}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   3. log-prod0.8%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\log 1 + \log \left(e^{\left|x\right| \cdot \sqrt{\pi}}\right)}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   4. metadata-eval0.8%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{0} + \log \left(e^{\left|x\right| \cdot \sqrt{\pi}}\right)} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   5. add-log-exp99.6%

      \[\leadsto \frac{e^{x \cdot x}}{0 + \color{blue}{\left|x\right| \cdot \sqrt{\pi}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   6. add-sqr-sqrt99.6%

      \[\leadsto \frac{e^{x \cdot x}}{0 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   7. fabs-sqr99.6%

      \[\leadsto \frac{e^{x \cdot x}}{0 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   8. add-sqr-sqrt99.6%

      \[\leadsto \frac{e^{x \cdot x}}{0 + \color{blue}{x} \cdot \sqrt{\pi}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

5. Applied egg-rr99.6%

   \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{0 + x \cdot \sqrt{\pi}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
6. Step-by-step derivation

   1. +-lft-identity99.6%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{x \cdot \sqrt{\pi}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

7. Simplified99.6%

   \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{x \cdot \sqrt{\pi}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

8. Taylor expanded in x around inf 99.0%

   \[\leadsto \frac{e^{x \cdot x}}{x \cdot \sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.5}{{x}^{2}}} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
9. Step-by-step derivation

   1. unpow299.4%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \]

10. Simplified99.0%

    \[\leadsto \frac{e^{x \cdot x}}{x \cdot \sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.5}{x \cdot x}} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

11. Final simplification99.0%

    \[\leadsto \frac{e^{x \cdot x}}{x \cdot \sqrt{\pi}} \cdot \left(\left(1 + \frac{1.875}{{x}^{6}}\right) + \frac{0.5}{x \cdot x}\right) \]

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

4. Taylor expanded in x around inf 99.4%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{{x}^{2}}}\right)\right) \]
5. Step-by-step derivation

   1. unpow299.4%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \]

6. Simplified99.4%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{x \cdot x}}\right)\right) \]

7. Taylor expanded in x around 0 52.0%

   \[\leadsto \frac{\frac{\color{blue}{1 + {x}^{2}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. unpow252.0%

      \[\leadsto \frac{\frac{1 + \color{blue}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

9. Simplified52.0%

   \[\leadsto \frac{\frac{\color{blue}{1 + x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

10. Taylor expanded in x around inf 52.0%

    \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{\left|x\right|}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
11. Step-by-step derivation

    1. unpow252.0%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. associate-/l*5.6%

       \[\leadsto \frac{\color{blue}{\frac{x}{\frac{\left|x\right|}{x}}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    3. associate-/r/5.6%

       \[\leadsto \frac{\color{blue}{\frac{x}{\left|x\right|} \cdot x}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    4. unpow15.6%

       \[\leadsto \frac{\frac{x}{\left|\color{blue}{{x}^{1}}\right|} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    5. sqr-pow5.6%

       \[\leadsto \frac{\frac{x}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    6. metadata-eval5.6%

       \[\leadsto \frac{\frac{x}{\left|{x}^{\color{blue}{0.5}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    7. unpow1/25.6%

       \[\leadsto \frac{\frac{x}{\left|\color{blue}{\sqrt{x}} \cdot {x}^{\left(\frac{1}{2}\right)}\right|} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    8. metadata-eval5.6%

       \[\leadsto \frac{\frac{x}{\left|\sqrt{x} \cdot {x}^{\color{blue}{0.5}}\right|} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    9. unpow1/25.6%

       \[\leadsto \frac{\frac{x}{\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    10. fabs-sqr5.6%

        \[\leadsto \frac{\frac{x}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    11. unpow1/25.6%

        \[\leadsto \frac{\frac{x}{\color{blue}{{x}^{0.5}} \cdot \sqrt{x}} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    12. metadata-eval5.6%

        \[\leadsto \frac{\frac{x}{{x}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \sqrt{x}} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    13. unpow1/25.6%

        \[\leadsto \frac{\frac{x}{{x}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{x}^{0.5}}} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    14. metadata-eval5.6%

        \[\leadsto \frac{\frac{x}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\color{blue}{\left(\frac{1}{2}\right)}}} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    15. sqr-pow5.6%

        \[\leadsto \frac{\frac{x}{\color{blue}{{x}^{1}}} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    16. unpow15.6%

        \[\leadsto \frac{\frac{x}{\color{blue}{x}} \cdot x}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

12. Simplified5.6%

    \[\leadsto \frac{\color{blue}{\frac{x}{x} \cdot x}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
13. Step-by-step derivation

    1. add-sqr-sqrt5.6%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{x}{x} \cdot x}{\sqrt{\pi}}} \cdot \sqrt{\frac{\frac{x}{x} \cdot x}{\sqrt{\pi}}}\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. sqrt-unprod52.0%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{x}{x} \cdot x}{\sqrt{\pi}} \cdot \frac{\frac{x}{x} \cdot x}{\sqrt{\pi}}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    3. *-inverses52.0%

       \[\leadsto \sqrt{\frac{\color{blue}{1} \cdot x}{\sqrt{\pi}} \cdot \frac{\frac{x}{x} \cdot x}{\sqrt{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    4. *-un-lft-identity52.0%

       \[\leadsto \sqrt{\frac{\color{blue}{x}}{\sqrt{\pi}} \cdot \frac{\frac{x}{x} \cdot x}{\sqrt{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    5. *-inverses52.0%

       \[\leadsto \sqrt{\frac{x}{\sqrt{\pi}} \cdot \frac{\color{blue}{1} \cdot x}{\sqrt{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    6. *-un-lft-identity52.0%

       \[\leadsto \sqrt{\frac{x}{\sqrt{\pi}} \cdot \frac{\color{blue}{x}}{\sqrt{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    7. frac-times52.0%

       \[\leadsto \sqrt{\color{blue}{\frac{x \cdot x}{\sqrt{\pi} \cdot \sqrt{\pi}}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    8. add-sqr-sqrt52.0%

       \[\leadsto \sqrt{\frac{x \cdot x}{\color{blue}{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

14. Applied egg-rr52.0%

    \[\leadsto \color{blue}{\sqrt{\frac{x \cdot x}{\pi}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

15. Final simplification52.0%

    \[\leadsto \sqrt{\frac{x \cdot x}{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Step-by-step derivation

   1. associate-+l+100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \left(\frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Simplified100.0%

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]

4. Taylor expanded in x around inf 99.4%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{{x}^{2}}}\right)\right) \]
5. Step-by-step derivation

   1. unpow299.4%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right) \]

6. Simplified99.4%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \color{blue}{\frac{0.5}{x \cdot x}}\right)\right) \]

7. Taylor expanded in x around 0 52.0%

   \[\leadsto \frac{\frac{\color{blue}{1 + {x}^{2}}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. unpow252.0%

      \[\leadsto \frac{\frac{1 + \color{blue}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

9. Simplified52.0%

   \[\leadsto \frac{\frac{\color{blue}{1 + x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
10. Step-by-step derivation

    1. add-log-exp99.3%

       \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{1 + x \cdot x}{\left|x\right|}}\right)}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    2. *-un-lft-identity99.3%

       \[\leadsto \frac{\log \color{blue}{\left(1 \cdot e^{\frac{1 + x \cdot x}{\left|x\right|}}\right)}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    3. log-prod99.3%

       \[\leadsto \frac{\color{blue}{\log 1 + \log \left(e^{\frac{1 + x \cdot x}{\left|x\right|}}\right)}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    4. metadata-eval99.3%

       \[\leadsto \frac{\color{blue}{0} + \log \left(e^{\frac{1 + x \cdot x}{\left|x\right|}}\right)}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    5. add-log-exp52.0%

       \[\leadsto \frac{0 + \color{blue}{\frac{1 + x \cdot x}{\left|x\right|}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    6. add-sqr-sqrt52.0%

       \[\leadsto \frac{0 + \frac{1 + x \cdot x}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    7. fabs-sqr52.0%

       \[\leadsto \frac{0 + \frac{1 + x \cdot x}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    8. add-sqr-sqrt52.0%

       \[\leadsto \frac{0 + \frac{1 + x \cdot x}{\color{blue}{x}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    9. +-commutative52.0%

       \[\leadsto \frac{0 + \frac{\color{blue}{x \cdot x + 1}}{x}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

    10. fma-def52.0%

        \[\leadsto \frac{0 + \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{x}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

11. Applied egg-rr52.0%

    \[\leadsto \frac{\color{blue}{0 + \frac{\mathsf{fma}\left(x, x, 1\right)}{x}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]
12. Step-by-step derivation

    1. +-lft-identity52.0%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{x}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

13. Simplified52.0%

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, x, 1\right)}{x}}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

14. Taylor expanded in x around inf 5.6%

    \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \]

15. Final simplification5.6%

    \[\leadsto \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5}{x \cdot x}\right)\right) \cdot \frac{x}{\sqrt{\pi}} \]

herbie shell --seed 2023166 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))