Jmat.Real.erfi, branch x less than or equal to 0.5

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

2. Final simplification99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right| \]

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. *-rgt-identity99.9%

      \[\leadsto \left|\color{blue}{\left(\frac{1}{\sqrt{\pi}} \cdot 1\right)} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified99.9%

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right|} \]

4. Final simplification99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \]

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-*l/99.4%

      \[\leadsto \left|\color{blue}{\frac{1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}}\right| \]

3. Simplified99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|} \]
4. Step-by-step derivation

   1. expm1-log1p-u98.9%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   2. expm1-udef39.0%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   3. add-sqr-sqrt2.9%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   4. fabs-sqr2.9%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   5. add-sqr-sqrt5.1%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

5. Applied egg-rr5.1%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
6. Step-by-step derivation

   1. expm1-def65.0%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   2. expm1-log1p99.1%

      \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

7. Simplified99.4%

   \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

8. Taylor expanded in x around 0 99.4%

   \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right| \]

9. Final simplification99.4%

   \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-+l+99.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

   2. +-commutative99.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

3. Simplified99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around inf 99.2%

   \[\leadsto \left|\frac{\mathsf{fma}\left(2, x, \color{blue}{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}\right)}{\sqrt{\pi}}\right| \]

5. Final simplification99.2%

   \[\leadsto \left|\frac{\mathsf{fma}\left(2, x, 0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}\right| \]

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-*l/99.4%

      \[\leadsto \left|\color{blue}{\frac{1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}}\right| \]

3. Simplified99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|} \]

4. Taylor expanded in x around inf 99.2%

   \[\leadsto \left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right| \]

5. Taylor expanded in x around 0 99.1%

   \[\leadsto \left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\color{blue}{2} + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]
6. Step-by-step derivation

   1. associate-*l/99.1%

      \[\leadsto \left|\color{blue}{\frac{\left|x\right| \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}}\right| \]

   2. clear-num99.2%

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{\sqrt{\pi}}{\left|x\right| \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)}}}\right| \]

   3. add-sqr-sqrt35.5%

      \[\leadsto \left|\frac{1}{\frac{\sqrt{\pi}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)}}\right| \]

   4. fabs-sqr35.5%

      \[\leadsto \left|\frac{1}{\frac{\sqrt{\pi}}{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)}}\right| \]

   5. add-sqr-sqrt99.2%

      \[\leadsto \left|\frac{1}{\frac{\sqrt{\pi}}{\color{blue}{x} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)}}\right| \]

   6. +-commutative99.2%

      \[\leadsto \left|\frac{1}{\frac{\sqrt{\pi}}{x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)}}}\right| \]

   7. fma-def99.2%

      \[\leadsto \left|\frac{1}{\frac{\sqrt{\pi}}{x \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}}\right| \]

7. Applied egg-rr99.2%

   \[\leadsto \left|\color{blue}{\frac{1}{\frac{\sqrt{\pi}}{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}}\right| \]

8. Final simplification99.2%

   \[\leadsto \left|\frac{1}{\frac{\sqrt{\pi}}{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}\right| \]

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-*l/99.4%

      \[\leadsto \left|\color{blue}{\frac{1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}}\right| \]

3. Simplified99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|} \]

4. Taylor expanded in x around inf 99.2%

   \[\leadsto \left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right| \]

5. Taylor expanded in x around 0 99.1%

   \[\leadsto \left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\color{blue}{2} + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]
6. Step-by-step derivation

   1. expm1-log1p-u98.9%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   2. expm1-udef39.0%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   3. add-sqr-sqrt2.9%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   4. fabs-sqr2.9%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   5. add-sqr-sqrt5.1%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

7. Applied egg-rr5.1%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]
8. Step-by-step derivation

   1. expm1-def65.0%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   2. expm1-log1p99.1%

      \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

9. Simplified99.1%

   \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]
10. Step-by-step derivation

    1. distribute-lft-in99.1%

       \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}} \cdot 2 + \frac{x}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)}\right| \]

11. Applied egg-rr99.1%

    \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}} \cdot 2 + \frac{x}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)}\right| \]
12. Step-by-step derivation

    1. distribute-lft-out99.1%

       \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)}\right| \]

    2. +-commutative99.1%

       \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)}\right| \]

    3. fma-udef99.1%

       \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}\right| \]

    4. associate-/r/99.1%

       \[\leadsto \left|\color{blue}{\frac{x}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}}\right| \]

13. Simplified99.1%

    \[\leadsto \left|\color{blue}{\frac{x}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}}\right| \]

14. Final simplification99.1%

    \[\leadsto \left|\frac{x}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}\right| \]

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-*l/99.4%

      \[\leadsto \left|\color{blue}{\frac{1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}}\right| \]

3. Simplified99.4%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|} \]

4. Taylor expanded in x around inf 99.2%

   \[\leadsto \left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right| \]

5. Taylor expanded in x around 0 99.1%

   \[\leadsto \left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\color{blue}{2} + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]
6. Step-by-step derivation

   1. expm1-log1p-u98.9%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   2. expm1-udef39.0%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   3. add-sqr-sqrt2.9%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   4. fabs-sqr2.9%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   5. add-sqr-sqrt5.1%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

7. Applied egg-rr5.1%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]
8. Step-by-step derivation

   1. expm1-def65.0%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   2. expm1-log1p99.1%

      \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

9. Simplified99.1%

   \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

10. Final simplification99.1%

    \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

1. Split input into 2 regimes

2. if x < -2.14999999999999991

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

      2. +-commutative99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

   3. Simplified99.9%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 99.1%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. associate-*r*99.1%

         \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

   6. Simplified99.1%

      \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]
   7. Step-by-step derivation

      1. add-sqr-sqrt0.0%

         \[\leadsto \left|\color{blue}{\sqrt{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \cdot \sqrt{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}}\right| \]

      2. sqrt-unprod92.8%

         \[\leadsto \left|\color{blue}{\sqrt{\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}}\right| \]

      3. *-commutative92.8%

         \[\leadsto \left|\sqrt{\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)} \cdot \left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      4. *-commutative92.8%

         \[\leadsto \left|\sqrt{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)}}\right| \]

      5. swap-sqr92.8%

         \[\leadsto \left|\sqrt{\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)}}\right| \]

      6. add-sqr-sqrt92.8%

         \[\leadsto \left|\sqrt{\color{blue}{\frac{1}{\pi}} \cdot \left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

      7. *-commutative92.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi} \cdot \left(\color{blue}{\left({x}^{7} \cdot 0.047619047619047616\right)} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

      8. *-commutative92.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi} \cdot \left(\left({x}^{7} \cdot 0.047619047619047616\right) \cdot \color{blue}{\left({x}^{7} \cdot 0.047619047619047616\right)}\right)}\right| \]

      9. swap-sqr92.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi} \cdot \color{blue}{\left(\left({x}^{7} \cdot {x}^{7}\right) \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)\right)}}\right| \]

      10. pow-prod-up92.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi} \cdot \left(\color{blue}{{x}^{\left(7 + 7\right)}} \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)\right)}\right| \]

      11. metadata-eval92.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi} \cdot \left({x}^{\color{blue}{14}} \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)\right)}\right| \]

      12. metadata-eval92.9%

          \[\leadsto \left|\sqrt{\frac{1}{\pi} \cdot \left({x}^{14} \cdot \color{blue}{0.0022675736961451248}\right)}\right| \]

   8. Applied egg-rr92.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi} \cdot \left({x}^{14} \cdot 0.0022675736961451248\right)}}\right| \]
   9. Step-by-step derivation

      1. associate-*l/92.9%

         \[\leadsto \left|\sqrt{\color{blue}{\frac{1 \cdot \left({x}^{14} \cdot 0.0022675736961451248\right)}{\pi}}}\right| \]

      2. *-lft-identity92.9%

         \[\leadsto \left|\sqrt{\frac{\color{blue}{{x}^{14} \cdot 0.0022675736961451248}}{\pi}}\right| \]

   10. Simplified92.9%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{{x}^{14} \cdot 0.0022675736961451248}{\pi}}}\right| \]

   if -2.14999999999999991 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

      2. +-commutative99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

   3. Simplified99.2%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 99.9%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. +-commutative99.9%

         \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. associate-*r*99.9%

         \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. associate-*r*99.9%

         \[\leadsto \left|\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      4. distribute-rgt-out99.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

      5. *-commutative99.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x \cdot 2} + 0.6666666666666666 \cdot {x}^{3}\right)\right| \]

   6. Simplified99.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]
   7. Step-by-step derivation

      1. unpow399.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right| \]

   8. Applied egg-rr99.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14} \cdot 0.0022675736961451248}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \]

1. Split input into 2 regimes

2. if x < -2.14999999999999991

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

      2. +-commutative99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

   3. Simplified99.9%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 99.1%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. associate-*r*99.1%

         \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

   6. Simplified99.1%

      \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]
   7. Step-by-step derivation

      1. expm1-log1p-u0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef0.0%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1}\right| \]

      3. sqrt-div0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right| \]

      4. metadata-eval0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right| \]

      5. un-div-inv0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}}\right)} - 1\right| \]

   8. Applied egg-rr0.0%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right)\right)}\right| \]

      2. expm1-log1p99.2%

         \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}}\right| \]

      3. associate-/l*99.2%

         \[\leadsto \left|\color{blue}{\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}}\right| \]

   10. Simplified99.2%

       \[\leadsto \left|\color{blue}{\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}}\right| \]
   11. Step-by-step derivation

       1. expm1-log1p-u0.0%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\right)\right)}\right| \]

       2. expm1-udef0.0%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\right)} - 1}\right| \]

       3. associate-/r/0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{7}}\right)} - 1\right| \]

   12. Applied egg-rr0.0%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{7}\right)} - 1}\right| \]
   13. Step-by-step derivation

       1. expm1-def0.0%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{7}\right)\right)}\right| \]

       2. expm1-log1p99.2%

          \[\leadsto \left|\color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{7}}\right| \]

       3. associate-*l/99.2%

          \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}}\right| \]

       4. associate-*r/99.2%

          \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}}\right| \]

   14. Simplified99.2%

       \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}}\right| \]

   if -2.14999999999999991 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

      2. +-commutative99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

   3. Simplified99.2%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 99.9%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. +-commutative99.9%

         \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. associate-*r*99.9%

         \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. associate-*r*99.9%

         \[\leadsto \left|\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      4. distribute-rgt-out99.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

      5. *-commutative99.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x \cdot 2} + 0.6666666666666666 \cdot {x}^{3}\right)\right| \]

   6. Simplified99.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]
   7. Step-by-step derivation

      1. unpow399.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right| \]

   8. Applied egg-rr99.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \]

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-+l+99.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

   2. +-commutative99.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

3. Simplified99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around 0 88.2%

   \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
5. Step-by-step derivation

   1. +-commutative88.2%

      \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   2. associate-*r*88.2%

      \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   3. associate-*r*88.2%

      \[\leadsto \left|\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

   4. distribute-rgt-out88.2%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

   5. *-commutative88.2%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x \cdot 2} + 0.6666666666666666 \cdot {x}^{3}\right)\right| \]

6. Simplified88.2%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]
7. Step-by-step derivation

   1. unpow388.2%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right| \]

8. Applied egg-rr88.2%

   \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right)\right| \]

9. Final simplification88.2%

   \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right| \]

1. Split input into 2 regimes

2. if x < -1e14

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

      2. +-commutative99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 5.4%

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

      1. *-commutative5.4%

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

      2. associate-*l*5.4%

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

   6. Simplified5.4%

      \[\leadsto \left|\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]
   7. Step-by-step derivation

      1. add-sqr-sqrt0.0%

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

      2. sqrt-unprod53.1%

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

      3. associate-*r*53.1%

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

      4. sqrt-div53.1%

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

      5. metadata-eval53.1%

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

      6. div-inv53.1%

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

      7. associate-*r*53.1%

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

      8. sqrt-div53.1%

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

      9. metadata-eval53.1%

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

      10. div-inv53.1%

          \[\leadsto \left|\sqrt{\left(\frac{x}{\sqrt{\pi}} \cdot 2\right) \cdot \left(\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot 2\right)}\right| \]

      11. swap-sqr53.1%

          \[\leadsto \left|\sqrt{\color{blue}{\left(\frac{x}{\sqrt{\pi}} \cdot \frac{x}{\sqrt{\pi}}\right) \cdot \left(2 \cdot 2\right)}}\right| \]

      12. frac-times53.1%

          \[\leadsto \left|\sqrt{\color{blue}{\frac{x \cdot x}{\sqrt{\pi} \cdot \sqrt{\pi}}} \cdot \left(2 \cdot 2\right)}\right| \]

      13. add-sqr-sqrt53.1%

          \[\leadsto \left|\sqrt{\frac{x \cdot x}{\color{blue}{\pi}} \cdot \left(2 \cdot 2\right)}\right| \]

      14. metadata-eval53.1%

          \[\leadsto \left|\sqrt{\frac{x \cdot x}{\pi} \cdot \color{blue}{4}}\right| \]

   8. Applied egg-rr53.1%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{x \cdot x}{\pi} \cdot 4}}\right| \]

   if -1e14 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

      2. +-commutative99.9%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

   3. Simplified99.2%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 98.8%

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

      1. *-commutative98.8%

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

      2. associate-*l*98.8%

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

   6. Simplified98.8%

      \[\leadsto \left|\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-log1p-u98.7%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)\right)\right)}\right| \]

      2. expm1-udef7.8%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)\right)} - 1}\right| \]

      3. associate-*r*7.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right)} - 1\right| \]

      4. sqrt-div7.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right) \cdot 2\right)} - 1\right| \]

      5. metadata-eval7.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right) \cdot 2\right)} - 1\right| \]

      6. div-inv7.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot 2\right)} - 1\right| \]

   8. Applied egg-rr7.8%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot 2\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def98.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot 2\right)\right)}\right| \]

      2. expm1-log1p98.1%

         \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}} \cdot 2}\right| \]

      3. associate-*l/98.1%

         \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]

   10. Simplified98.1%

       \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]
   11. Step-by-step derivation

       1. expm1-log1p-u98.0%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-udef7.8%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)} - 1}\right| \]

   12. Applied egg-rr7.8%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)} - 1}\right| \]
   13. Step-by-step derivation

       1. expm1-def98.0%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-log1p98.1%

          \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]

       3. associate-*r/98.8%

          \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

   14. Simplified98.8%

       \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\left|\sqrt{4 \cdot \frac{x \cdot x}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-+l+99.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

   2. +-commutative99.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)}\right| \]

3. Simplified99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around 0 67.4%

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

   1. *-commutative67.4%

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

   2. associate-*l*67.4%

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

6. Simplified67.4%

   \[\leadsto \left|\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)}\right| \]
7. Step-by-step derivation

   1. expm1-log1p-u65.5%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)\right)\right)}\right| \]

   2. expm1-udef5.1%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 2\right)\right)} - 1}\right| \]

   3. associate-*r*5.1%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 2}\right)} - 1\right| \]

   4. sqrt-div5.1%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right) \cdot 2\right)} - 1\right| \]

   5. metadata-eval5.1%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right) \cdot 2\right)} - 1\right| \]

   6. div-inv5.1%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot 2\right)} - 1\right| \]

8. Applied egg-rr5.1%

   \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot 2\right)} - 1}\right| \]
9. Step-by-step derivation

   1. expm1-def65.0%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}} \cdot 2\right)\right)}\right| \]

   2. expm1-log1p66.9%

      \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}} \cdot 2}\right| \]

   3. associate-*l/66.9%

      \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]

10. Simplified66.9%

    \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]
11. Step-by-step derivation

    1. expm1-log1p-u65.0%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)\right)}\right| \]

    2. expm1-udef5.1%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)} - 1}\right| \]

12. Applied egg-rr5.1%

    \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)} - 1}\right| \]
13. Step-by-step derivation

    1. expm1-def65.0%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)\right)}\right| \]

    2. expm1-log1p66.9%

       \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]

    3. associate-*r/67.4%

       \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

14. Simplified67.4%

    \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

15. Final simplification67.4%

    \[\leadsto \left|x \cdot \frac{2}{\sqrt{\pi}}\right| \]

herbie shell --seed 2023166 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))