Logistic distribution

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. *-lft-identity99.8%

      \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   2. associate-*r/99.8%

      \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   3. associate-/l*99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

   4. distribute-frac-neg99.8%

      \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

   5. exp-neg99.8%

      \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   6. associate-/r/99.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

   7. /-rgt-identity99.8%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

   8. associate-*l*99.8%

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

3. Simplified99.9%

   \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

4. Final simplification99.9%

   \[\leadsto \frac{1}{\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)} \]

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. *-lft-identity99.8%

      \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   2. associate-*r/99.8%

      \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   3. associate-*l*99.9%

      \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   4. times-frac99.0%

      \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   5. associate-*r/99.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   6. associate-/l*99.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

   7. distribute-frac-neg99.0%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

   8. exp-neg99.0%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

3. Simplified99.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]

4. Final simplification99.1%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. *-lft-identity99.8%

      \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   2. associate-*r/99.8%

      \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   3. associate-*l*99.9%

      \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   4. times-frac99.0%

      \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   5. associate-*r/99.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   6. associate-/l*99.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

   7. distribute-frac-neg99.0%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

   8. exp-neg99.0%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

3. Simplified99.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}} \]

4. Final simplification99.1%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)} \]

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. *-lft-identity99.8%

      \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   2. associate-*r/99.8%

      \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   3. associate-*l*99.9%

      \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   4. times-frac99.0%

      \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   5. associate-*r/99.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   6. associate-/l*99.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

   7. distribute-frac-neg99.0%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

   8. exp-neg99.0%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

3. Simplified99.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]
4. Step-by-step derivation

   1. expm1-log1p-u99.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}} + 2\right)\right)}} \]

   2. expm1-udef99.0%

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

   3. add-sqr-sqrt-0.0%

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

   4. sqrt-unprod92.4%

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

   5. sqr-neg92.4%

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

   6. sqrt-unprod95.4%

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

   7. add-sqr-sqrt95.4%

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

   8. add-sqr-sqrt47.5%

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

   9. fabs-sqr47.5%

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

   10. add-sqr-sqrt97.4%

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

5. Applied egg-rr97.4%

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

   1. expm1-def97.4%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 2\right)\right)}} \]

   2. expm1-log1p97.4%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(e^{\frac{x}{s}} + 2\right)}} \]

   3. +-commutative97.4%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]

7. Simplified97.4%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt97.4%

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

   2. sqrt-unprod97.4%

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

   3. sqr-neg97.4%

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

   4. distribute-frac-neg97.4%

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

   5. distribute-frac-neg97.4%

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

   6. sqrt-unprod-0.0%

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

   7. add-sqr-sqrt63.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   8. distribute-frac-neg63.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{-\frac{\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   9. exp-neg63.0%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   10. add-sqr-sqrt49.2%

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

   11. fabs-sqr49.2%

       \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   12. add-sqr-sqrt99.1%

       \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{x}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

9. Applied egg-rr99.1%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{x}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]
10. Step-by-step derivation

    1. rec-exp99.1%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{-\frac{x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

    2. distribute-neg-frac99.1%

       \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

11. Simplified99.1%

    \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

12. Final simplification99.1%

    \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \left(2 + e^{\frac{x}{s}}\right)} \]

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. *-lft-identity99.8%

      \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   2. associate-*r/99.8%

      \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   3. associate-*l*99.9%

      \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   4. times-frac99.0%

      \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   5. associate-*r/99.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   6. associate-/l*99.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

   7. distribute-frac-neg99.0%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

   8. exp-neg99.0%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

3. Simplified99.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]

4. Taylor expanded in s around inf 95.9%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]

5. Final simplification95.9%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3} \]

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. associate-*l*99.9%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   2. +-commutative99.9%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]

   3. +-commutative99.9%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

3. Simplified99.9%

   \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]

4. Taylor expanded in s around inf 95.9%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \color{blue}{4}} \]

5. Final simplification95.9%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4} \]

1. Split input into 4 regimes

2. if x < -1.8e9

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 1.1%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+1.1%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative1.1%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+1.1%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified1.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 95.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow295.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot x}}{s}} \]

   9. Simplified95.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s}}} \]

   if -1.8e9 < x < -5.00000016e-23

   1. Initial program 99.7%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.7%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.9%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac99.9%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/99.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*99.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg99.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg99.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified99.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u99.8%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}} + 2\right)\right)}} \]

      2. expm1-udef99.8%

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

      3. add-sqr-sqrt-0.0%

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

      4. sqrt-unprod94.2%

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

      5. sqr-neg94.2%

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

      6. sqrt-unprod94.2%

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

      7. add-sqr-sqrt94.2%

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

      8. add-sqr-sqrt-0.0%

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

      9. fabs-sqr-0.0%

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

      10. add-sqr-sqrt99.8%

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

   5. Applied egg-rr99.8%

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

      1. expm1-def99.8%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 2\right)\right)}} \]

      2. expm1-log1p99.8%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(e^{\frac{x}{s}} + 2\right)}} \]

      3. +-commutative99.8%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]

   7. Simplified99.8%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt99.7%

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

      2. sqrt-unprod99.8%

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

      3. sqr-neg99.8%

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

      4. distribute-frac-neg99.8%

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

      5. distribute-frac-neg99.8%

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

      6. sqrt-unprod-0.0%

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

      7. add-sqr-sqrt19.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      8. distribute-frac-neg19.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{-\frac{\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      9. exp-neg19.9%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      10. add-sqr-sqrt-0.0%

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

      11. fabs-sqr-0.0%

          \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      12. add-sqr-sqrt99.8%

          \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{x}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   9. Applied egg-rr99.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{x}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]
   10. Step-by-step derivation

       1. rec-exp99.8%

          \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{-\frac{x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

       2. distribute-neg-frac99.8%

          \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   11. Simplified99.8%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   12. Taylor expanded in x around 0 97.5%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   13. Step-by-step derivation

       1. associate-+r+97.5%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{\left(\left(3 + \frac{x}{s}\right) + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]

       2. +-commutative97.5%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \left(\color{blue}{\left(\frac{x}{s} + 3\right)} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

       3. unpow297.5%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)} \]

       4. unpow297.5%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)} \]

   14. Simplified97.5%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{\left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}} \]

   15. Taylor expanded in x around 0 69.6%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)\right)} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]
   16. Step-by-step derivation

       1. mul-1-neg69.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + \color{blue}{\left(-\frac{x}{s}\right)}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       2. unsub-neg69.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} - \frac{x}{s}\right)}\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       3. associate-*r/69.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       4. unpow269.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       5. unpow269.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       6. associate-*r*69.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{\color{blue}{\left(0.5 \cdot x\right) \cdot x}}{s \cdot s} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

   17. Simplified69.6%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(\frac{\left(0.5 \cdot x\right) \cdot x}{s \cdot s} - \frac{x}{s}\right)\right)} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

   if -5.00000016e-23 < x < 6.5000002e-21

   1. Initial program 99.5%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.5%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.5%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg99.6%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg99.5%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/99.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity99.4%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*99.6%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified99.8%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 81.2%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+81.2%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative81.2%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+81.2%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified81.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around 0 81.2%

      \[\leadsto \frac{1}{\color{blue}{4 \cdot s + \frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. *-commutative81.2%

         \[\leadsto \frac{1}{\color{blue}{s \cdot 4} + \frac{{x}^{2}}{s}} \]

      2. fma-def81.2%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(s, 4, \frac{{x}^{2}}{s}\right)}} \]

      3. unpow281.2%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, 4, \frac{\color{blue}{x \cdot x}}{s}\right)} \]

      4. associate-*l/83.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, 4, \color{blue}{\frac{x}{s} \cdot x}\right)} \]

      5. *-commutative83.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, 4, \color{blue}{x \cdot \frac{x}{s}}\right)} \]

   9. Simplified83.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(s, 4, x \cdot \frac{x}{s}\right)}} \]

   if 6.5000002e-21 < x

   1. Initial program 99.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.9%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.9%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.9%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac98.9%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/98.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*98.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg98.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg98.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified98.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]

   4. Taylor expanded in s around inf 98.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt98.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}} + 3} \]

      2. sqrt-unprod97.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s \cdot s}}}} + 3} \]

      3. sqr-neg97.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}} + 3} \]

      4. sqrt-unprod-0.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}} + 3} \]

      5. add-sqr-sqrt7.3%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 3} \]

      6. expm1-log1p-u7.3%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)\right)} + 3} \]

      7. expm1-udef7.3%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)} - 1\right)} + 3} \]

   6. Applied egg-rr98.0%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\left(e^{\frac{x}{s}} + 1\right) - 1\right)} + 3} \]
   7. Step-by-step derivation

      1. associate--l+98.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\frac{x}{s}} + \left(1 - 1\right)\right)} + 3} \]

      2. metadata-eval98.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 3} \]

      3. +-rgt-identity98.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   8. Simplified98.0%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   9. Taylor expanded in x around 0 81.5%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} + 3} \]
   10. Step-by-step derivation

       1. unpow281.5%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)\right) + 3} \]

       2. unpow281.5%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)\right) + 3} \]

   11. Simplified81.5%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)} + 3} \]
3. Recombined 4 regimes into one program.

4. Final simplification82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1800000000:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{elif}\;x \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(1 + \left(\frac{x \cdot \left(x \cdot 0.5\right)}{s \cdot s} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}\\ \mathbf{elif}\;x \leq 6.50000019762268 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(s, 4, x \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{3 + \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \]

1. Split input into 3 regimes

2. if x < -1.8e9

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 1.1%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+1.1%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative1.1%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+1.1%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified1.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 95.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow295.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot x}}{s}} \]

   9. Simplified95.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s}}} \]

   if -1.8e9 < x < -5.99999992e-24

   1. Initial program 99.7%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.7%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.9%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac99.9%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/99.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*99.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg99.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg99.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified99.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u99.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}} + 2\right)\right)}} \]

      2. expm1-udef99.7%

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

      3. add-sqr-sqrt-0.0%

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

      4. sqrt-unprod93.3%

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

      5. sqr-neg93.3%

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

      6. sqrt-unprod93.3%

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

      7. add-sqr-sqrt93.3%

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

      8. add-sqr-sqrt-0.0%

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

      9. fabs-sqr-0.0%

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

      10. add-sqr-sqrt99.7%

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

   5. Applied egg-rr99.7%

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

      1. expm1-def99.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 2\right)\right)}} \]

      2. expm1-log1p99.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(e^{\frac{x}{s}} + 2\right)}} \]

      3. +-commutative99.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]

   7. Simplified99.7%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt99.7%

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

      2. sqrt-unprod99.7%

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

      3. sqr-neg99.7%

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

      4. distribute-frac-neg99.7%

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

      5. distribute-frac-neg99.7%

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

      6. sqrt-unprod-0.0%

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

      7. add-sqr-sqrt21.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      8. distribute-frac-neg21.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{-\frac{\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      9. exp-neg21.9%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      10. add-sqr-sqrt-0.0%

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

      11. fabs-sqr-0.0%

          \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      12. add-sqr-sqrt99.7%

          \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{x}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   9. Applied egg-rr99.7%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{x}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]
   10. Step-by-step derivation

       1. rec-exp99.7%

          \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{-\frac{x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

       2. distribute-neg-frac99.7%

          \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   11. Simplified99.7%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   12. Taylor expanded in x around 0 96.9%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   13. Step-by-step derivation

       1. associate-+r+96.9%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{\left(\left(3 + \frac{x}{s}\right) + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]

       2. +-commutative96.9%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \left(\color{blue}{\left(\frac{x}{s} + 3\right)} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

       3. unpow296.9%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)} \]

       4. unpow296.9%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)} \]

   14. Simplified96.9%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{\left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}} \]

   15. Taylor expanded in x around 0 70.1%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)\right)} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]
   16. Step-by-step derivation

       1. mul-1-neg70.1%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + \color{blue}{\left(-\frac{x}{s}\right)}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       2. unsub-neg70.1%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} - \frac{x}{s}\right)}\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       3. associate-*r/70.0%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       4. unpow270.0%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       5. unpow270.0%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       6. associate-*r*70.1%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{\color{blue}{\left(0.5 \cdot x\right) \cdot x}}{s \cdot s} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

   17. Simplified70.1%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(\frac{\left(0.5 \cdot x\right) \cdot x}{s \cdot s} - \frac{x}{s}\right)\right)} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

   if -5.99999992e-24 < x

   1. Initial program 99.8%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.8%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.8%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac99.1%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/99.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*99.1%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg99.1%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg99.1%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified99.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]

   4. Taylor expanded in s around inf 95.9%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt95.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}} + 3} \]

      2. sqrt-unprod90.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s \cdot s}}}} + 3} \]

      3. sqr-neg90.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}} + 3} \]

      4. sqrt-unprod-0.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}} + 3} \]

      5. add-sqr-sqrt30.4%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 3} \]

      6. expm1-log1p-u30.4%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)\right)} + 3} \]

      7. expm1-udef30.4%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)} - 1\right)} + 3} \]

   6. Applied egg-rr92.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\left(e^{\frac{x}{s}} + 1\right) - 1\right)} + 3} \]
   7. Step-by-step derivation

      1. associate--l+92.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\frac{x}{s}} + \left(1 - 1\right)\right)} + 3} \]

      2. metadata-eval92.8%

         \[\leadsto \frac{\frac{1}{s}}{\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 3} \]

      3. +-rgt-identity92.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   8. Simplified92.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]
3. Recombined 3 regimes into one program.

4. Final simplification88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1800000000:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{elif}\;x \leq -5.999999920033662 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(1 + \left(\frac{x \cdot \left(x \cdot 0.5\right)}{s \cdot s} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < 1e-28

   1. Initial program 99.8%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.8%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.9%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac99.1%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/99.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*99.1%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg99.1%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg99.1%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified99.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u99.1%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}} + 2\right)\right)}} \]

      2. expm1-udef99.1%

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

      3. add-sqr-sqrt-0.0%

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

      4. sqrt-unprod89.2%

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

      5. sqr-neg89.2%

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

      6. sqrt-unprod94.4%

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

      7. add-sqr-sqrt94.4%

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

      8. add-sqr-sqrt7.5%

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

      9. fabs-sqr7.5%

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

      10. add-sqr-sqrt98.2%

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

   5. Applied egg-rr98.2%

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

      1. expm1-def98.2%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 2\right)\right)}} \]

      2. expm1-log1p98.2%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(e^{\frac{x}{s}} + 2\right)}} \]

      3. +-commutative98.2%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]

   7. Simplified98.2%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt98.2%

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

      2. sqrt-unprod98.2%

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

      3. sqr-neg98.2%

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

      4. distribute-frac-neg98.2%

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

      5. distribute-frac-neg98.2%

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

      6. sqrt-unprod-0.0%

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

      7. add-sqr-sqrt33.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      8. distribute-frac-neg33.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{-\frac{\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      9. exp-neg33.7%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      10. add-sqr-sqrt8.5%

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

      11. fabs-sqr8.5%

          \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      12. add-sqr-sqrt99.1%

          \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{x}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   9. Applied egg-rr99.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{x}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]
   10. Step-by-step derivation

       1. rec-exp99.1%

          \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{-\frac{x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

       2. distribute-neg-frac99.1%

          \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   11. Simplified99.1%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   12. Taylor expanded in x around 0 94.7%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{3}} \]

   if 1e-28 < x

   1. Initial program 99.8%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.8%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.8%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac98.9%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/99.0%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*98.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg98.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg99.0%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified99.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt99.0%

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

      2. sqrt-unprod98.0%

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

      3. sqr-neg98.0%

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

      4. sqrt-unprod-0.0%

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

      5. add-sqr-sqrt12.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{-s}}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]

      6. add-sqr-sqrt-0.0%

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

      7. sqrt-unprod98.0%

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

      8. sqr-neg98.0%

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

      9. sqrt-unprod99.0%

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

      10. add-sqr-sqrt99.0%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\frac{\left|x\right|}{\color{blue}{s}}} + 2\right)} \]

      11. expm1-log1p-u98.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}\right)\right)} \]

      12. expm1-udef98.4%

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

   5. Applied egg-rr96.1%

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

      1. expm1-def96.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{\left(e^{\frac{x}{s}} + e^{\frac{x}{s}}\right) + 2}\right)\right)} \]

      2. expm1-log1p96.5%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\left(e^{\frac{x}{s}} + e^{\frac{x}{s}}\right) + 2}} \]

      3. associate-/r*97.4%

         \[\leadsto \color{blue}{\frac{1}{s \cdot \left(\left(e^{\frac{x}{s}} + e^{\frac{x}{s}}\right) + 2\right)}} \]

      4. +-commutative97.4%

         \[\leadsto \frac{1}{s \cdot \color{blue}{\left(2 + \left(e^{\frac{x}{s}} + e^{\frac{x}{s}}\right)\right)}} \]

      5. count-297.4%

         \[\leadsto \frac{1}{s \cdot \left(2 + \color{blue}{2 \cdot e^{\frac{x}{s}}}\right)} \]

   7. Simplified97.4%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{-x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < -1.00000002e-33

   1. Initial program 99.8%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.8%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.9%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac99.0%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/99.0%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*98.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg98.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg98.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified99.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u98.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}} + 2\right)\right)}} \]

      2. expm1-udef98.9%

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

      3. add-sqr-sqrt-0.0%

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

      4. sqrt-unprod92.7%

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

      5. sqr-neg92.7%

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

      6. sqrt-unprod94.4%

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

      7. add-sqr-sqrt94.4%

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

      8. add-sqr-sqrt-0.0%

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

      9. fabs-sqr-0.0%

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

      10. add-sqr-sqrt98.9%

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

   5. Applied egg-rr98.9%

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

      1. expm1-def98.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 2\right)\right)}} \]

      2. expm1-log1p99.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(e^{\frac{x}{s}} + 2\right)}} \]

      3. +-commutative99.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]

   7. Simplified99.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt98.9%

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

      2. sqrt-unprod99.0%

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

      3. sqr-neg99.0%

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

      4. distribute-frac-neg99.0%

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

      5. distribute-frac-neg99.0%

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

      6. sqrt-unprod-0.0%

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

      7. add-sqr-sqrt18.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      8. distribute-frac-neg18.7%

         \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{-\frac{\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      9. exp-neg18.7%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      10. add-sqr-sqrt-0.0%

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

      11. fabs-sqr-0.0%

          \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      12. add-sqr-sqrt99.0%

          \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{x}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   9. Applied egg-rr99.0%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{x}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]
   10. Step-by-step derivation

       1. rec-exp99.0%

          \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{-\frac{x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

       2. distribute-neg-frac99.0%

          \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   11. Simplified99.0%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   12. Taylor expanded in x around 0 94.7%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{3}} \]

   if -1.00000002e-33 < x

   1. Initial program 99.8%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.8%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.8%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.9%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac99.1%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/99.2%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*99.1%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg99.1%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg99.1%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified99.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]

   4. Taylor expanded in s around inf 96.8%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt96.8%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}} + 3} \]

      2. sqrt-unprod92.6%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s \cdot s}}}} + 3} \]

      3. sqr-neg92.6%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}} + 3} \]

      4. sqrt-unprod-0.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}} + 3} \]

      5. add-sqr-sqrt27.8%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 3} \]

      6. expm1-log1p-u27.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)\right)} + 3} \]

      7. expm1-udef27.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)} - 1\right)} + 3} \]

   6. Applied egg-rr96.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\left(e^{\frac{x}{s}} + 1\right) - 1\right)} + 3} \]
   7. Step-by-step derivation

      1. associate--l+96.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\frac{x}{s}} + \left(1 - 1\right)\right)} + 3} \]

      2. metadata-eval96.8%

         \[\leadsto \frac{\frac{1}{s}}{\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 3} \]

      3. +-rgt-identity96.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   8. Simplified96.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.000000023742228 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{-x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]

1. Split input into 4 regimes

2. if x < -1.8e9

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 1.1%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+1.1%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative1.1%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+1.1%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified1.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 95.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow295.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot x}}{s}} \]

   9. Simplified95.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s}}} \]

   if -1.8e9 < x < -5.00000016e-23

   1. Initial program 99.7%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.7%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.9%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac99.9%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/99.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*99.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg99.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg99.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified99.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u99.8%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}} + 2\right)\right)}} \]

      2. expm1-udef99.8%

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

      3. add-sqr-sqrt-0.0%

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

      4. sqrt-unprod94.2%

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

      5. sqr-neg94.2%

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

      6. sqrt-unprod94.2%

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

      7. add-sqr-sqrt94.2%

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

      8. add-sqr-sqrt-0.0%

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

      9. fabs-sqr-0.0%

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

      10. add-sqr-sqrt99.8%

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

   5. Applied egg-rr99.8%

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

      1. expm1-def99.8%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 2\right)\right)}} \]

      2. expm1-log1p99.8%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(e^{\frac{x}{s}} + 2\right)}} \]

      3. +-commutative99.8%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]

   7. Simplified99.8%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(2 + e^{\frac{x}{s}}\right)}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt99.7%

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

      2. sqrt-unprod99.8%

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

      3. sqr-neg99.8%

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

      4. distribute-frac-neg99.8%

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

      5. distribute-frac-neg99.8%

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

      6. sqrt-unprod-0.0%

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

      7. add-sqr-sqrt19.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      8. distribute-frac-neg19.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{-\frac{\left|x\right|}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      9. exp-neg19.9%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      10. add-sqr-sqrt-0.0%

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

      11. fabs-sqr-0.0%

          \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

      12. add-sqr-sqrt99.8%

          \[\leadsto \frac{\frac{1}{s}}{\frac{1}{e^{\frac{\color{blue}{x}}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   9. Applied egg-rr99.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{1}{e^{\frac{x}{s}}}} + \left(2 + e^{\frac{x}{s}}\right)} \]
   10. Step-by-step derivation

       1. rec-exp99.8%

          \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{-\frac{x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

       2. distribute-neg-frac99.8%

          \[\leadsto \frac{\frac{1}{s}}{e^{\color{blue}{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   11. Simplified99.8%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{-x}{s}}} + \left(2 + e^{\frac{x}{s}}\right)} \]

   12. Taylor expanded in x around 0 97.5%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   13. Step-by-step derivation

       1. associate-+r+97.5%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{\left(\left(3 + \frac{x}{s}\right) + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]

       2. +-commutative97.5%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \left(\color{blue}{\left(\frac{x}{s} + 3\right)} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)} \]

       3. unpow297.5%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)} \]

       4. unpow297.5%

          \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)} \]

   14. Simplified97.5%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{-x}{s}} + \color{blue}{\left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}} \]

   15. Taylor expanded in x around 0 69.6%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + -1 \cdot \frac{x}{s}\right)\right)} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]
   16. Step-by-step derivation

       1. mul-1-neg69.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} + \color{blue}{\left(-\frac{x}{s}\right)}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       2. unsub-neg69.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{{s}^{2}} - \frac{x}{s}\right)}\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       3. associate-*r/69.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       4. unpow269.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       5. unpow269.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

       6. associate-*r*69.6%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{\color{blue}{\left(0.5 \cdot x\right) \cdot x}}{s \cdot s} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

   17. Simplified69.6%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(\frac{\left(0.5 \cdot x\right) \cdot x}{s \cdot s} - \frac{x}{s}\right)\right)} + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)} \]

   if -5.00000016e-23 < x < 6.5000002e-21

   1. Initial program 99.5%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.5%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.5%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg99.6%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg99.5%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/99.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity99.4%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*99.6%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified99.8%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 81.2%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+81.2%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative81.2%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+81.2%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified81.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]
   7. Step-by-step derivation

      1. fma-udef81.2%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]

      2. associate-/l*81.4%

         \[\leadsto \frac{1}{\left(2 \cdot \color{blue}{\frac{x}{\frac{s}{x}}} + s \cdot 4\right) - \frac{x \cdot x}{s}} \]

   8. Applied egg-rr81.4%

      \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x}{\frac{s}{x}} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]
   9. Step-by-step derivation

      1. associate--l+81.4%

         \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\frac{s}{x}} + \left(s \cdot 4 - \frac{x \cdot x}{s}\right)}} \]

      2. associate-*r/81.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot x}{\frac{s}{x}}} + \left(s \cdot 4 - \frac{x \cdot x}{s}\right)} \]

      3. *-commutative81.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot 2}}{\frac{s}{x}} + \left(s \cdot 4 - \frac{x \cdot x}{s}\right)} \]

      4. associate-/l*83.7%

         \[\leadsto \frac{1}{\frac{x \cdot 2}{\frac{s}{x}} + \left(s \cdot 4 - \color{blue}{\frac{x}{\frac{s}{x}}}\right)} \]

      5. div-inv83.7%

         \[\leadsto \frac{1}{\frac{x \cdot 2}{\frac{s}{x}} + \left(s \cdot 4 - \color{blue}{x \cdot \frac{1}{\frac{s}{x}}}\right)} \]

      6. clear-num83.7%

         \[\leadsto \frac{1}{\frac{x \cdot 2}{\frac{s}{x}} + \left(s \cdot 4 - x \cdot \color{blue}{\frac{x}{s}}\right)} \]

   10. Applied egg-rr83.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot 2}{\frac{s}{x}} + \left(s \cdot 4 - x \cdot \frac{x}{s}\right)}} \]

   if 6.5000002e-21 < x

   1. Initial program 99.9%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.9%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.9%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.9%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac98.9%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/98.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*98.9%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg98.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg98.9%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified98.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]

   4. Taylor expanded in s around inf 98.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt98.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}} + 3} \]

      2. sqrt-unprod97.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s \cdot s}}}} + 3} \]

      3. sqr-neg97.9%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}} + 3} \]

      4. sqrt-unprod-0.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}} + 3} \]

      5. add-sqr-sqrt7.3%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 3} \]

      6. expm1-log1p-u7.3%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)\right)} + 3} \]

      7. expm1-udef7.3%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)} - 1\right)} + 3} \]

   6. Applied egg-rr98.0%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\left(e^{\frac{x}{s}} + 1\right) - 1\right)} + 3} \]
   7. Step-by-step derivation

      1. associate--l+98.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\frac{x}{s}} + \left(1 - 1\right)\right)} + 3} \]

      2. metadata-eval98.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 3} \]

      3. +-rgt-identity98.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   8. Simplified98.0%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   9. Taylor expanded in x around 0 81.5%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} + 3} \]
   10. Step-by-step derivation

       1. unpow281.5%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)\right) + 3} \]

       2. unpow281.5%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)\right) + 3} \]

   11. Simplified81.5%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)} + 3} \]
3. Recombined 4 regimes into one program.

4. Final simplification82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1800000000:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{elif}\;x \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(1 + \left(\frac{x \cdot \left(x \cdot 0.5\right)}{s \cdot s} - \frac{x}{s}\right)\right) + \left(\left(\frac{x}{s} + 3\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}\\ \mathbf{elif}\;x \leq 6.50000019762268 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\frac{x \cdot 2}{\frac{s}{x}} + \left(s \cdot 4 - x \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{3 + \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < -1e9

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 1.3%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+1.3%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative1.3%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+1.3%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified1.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 93.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow293.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot x}}{s}} \]

   9. Simplified93.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s}}} \]

   if -1e9 < x

   1. Initial program 99.7%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.7%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.8%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac99.3%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/99.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*99.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg99.3%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg99.3%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified99.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]

   4. Taylor expanded in s around inf 95.4%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt95.4%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}} + 3} \]

      2. sqrt-unprod91.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s \cdot s}}}} + 3} \]

      3. sqr-neg91.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}} + 3} \]

      4. sqrt-unprod-0.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}} + 3} \]

      5. add-sqr-sqrt28.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 3} \]

      6. expm1-log1p-u28.5%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)\right)} + 3} \]

      7. expm1-udef28.5%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)} - 1\right)} + 3} \]

   6. Applied egg-rr75.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\left(e^{\frac{x}{s}} + 1\right) - 1\right)} + 3} \]
   7. Step-by-step derivation

      1. associate--l+75.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\frac{x}{s}} + \left(1 - 1\right)\right)} + 3} \]

      2. metadata-eval75.8%

         \[\leadsto \frac{\frac{1}{s}}{\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 3} \]

      3. +-rgt-identity75.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   8. Simplified75.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   9. Taylor expanded in x around 0 74.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} + 3} \]
   10. Step-by-step derivation

       1. unpow274.1%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)\right) + 3} \]

       2. unpow274.1%

          \[\leadsto \frac{\frac{1}{s}}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)\right) + 3} \]

   11. Simplified74.1%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)} + 3} \]
3. Recombined 2 regimes into one program.

4. Final simplification77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000000:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{3 + \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < -1e9

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 1.3%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+1.3%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative1.3%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+1.3%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified1.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 93.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow293.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot x}}{s}} \]

   9. Simplified93.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s}}} \]

   if -1e9 < x

   1. Initial program 99.7%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity99.7%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/99.7%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-*l*99.8%

         \[\leadsto \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      4. times-frac99.3%

         \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      5. associate-*r/99.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      6. associate-/l*99.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      7. distribute-frac-neg99.3%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      8. exp-neg99.3%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   3. Simplified99.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]

   4. Taylor expanded in s around inf 95.4%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{3}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt95.4%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}} + 3} \]

      2. sqrt-unprod91.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{s \cdot s}}}} + 3} \]

      3. sqr-neg91.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}} + 3} \]

      4. sqrt-unprod-0.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}} + 3} \]

      5. add-sqr-sqrt28.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{\color{blue}{-s}}} + 3} \]

      6. expm1-log1p-u28.5%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)\right)} + 3} \]

      7. expm1-udef28.5%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{\left|x\right|}{-s}}\right)} - 1\right)} + 3} \]

   6. Applied egg-rr75.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\left(e^{\frac{x}{s}} + 1\right) - 1\right)} + 3} \]
   7. Step-by-step derivation

      1. associate--l+75.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(e^{\frac{x}{s}} + \left(1 - 1\right)\right)} + 3} \]

      2. metadata-eval75.8%

         \[\leadsto \frac{\frac{1}{s}}{\left(e^{\frac{x}{s}} + \color{blue}{0}\right) + 3} \]

      3. +-rgt-identity75.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   8. Simplified75.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}}} + 3} \]

   9. Taylor expanded in x around 0 74.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{4 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   10. Step-by-step derivation

       1. +-commutative74.1%

          \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right) + 4}} \]

       2. associate-*r/74.1%

          \[\leadsto \frac{\frac{1}{s}}{\left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right) + 4} \]

       3. unpow274.1%

          \[\leadsto \frac{\frac{1}{s}}{\left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right) + 4} \]

       4. times-frac70.9%

          \[\leadsto \frac{\frac{1}{s}}{\left(\frac{x}{s} + \color{blue}{\frac{0.5}{s} \cdot \frac{{x}^{2}}{s}}\right) + 4} \]

       5. unpow270.9%

          \[\leadsto \frac{\frac{1}{s}}{\left(\frac{x}{s} + \frac{0.5}{s} \cdot \frac{\color{blue}{x \cdot x}}{s}\right) + 4} \]

       6. associate-*r/70.9%

          \[\leadsto \frac{\frac{1}{s}}{\left(\frac{x}{s} + \frac{0.5}{s} \cdot \color{blue}{\left(x \cdot \frac{x}{s}\right)}\right) + 4} \]

   11. Simplified70.9%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{x}{s} + \frac{0.5}{s} \cdot \left(x \cdot \frac{x}{s}\right)\right) + 4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000000000:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \left(\frac{x}{s} + \left(x \cdot \frac{x}{s}\right) \cdot \frac{0.5}{s}\right)}\\ \end{array} \]

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. *-lft-identity99.8%

      \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   2. associate-*r/99.8%

      \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

   3. associate-/l*99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

   4. distribute-frac-neg99.8%

      \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

   5. exp-neg99.8%

      \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

   6. associate-/r/99.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

   7. /-rgt-identity99.8%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

   8. associate-*l*99.8%

      \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

3. Simplified99.9%

   \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

4. Taylor expanded in s around inf 25.8%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
5. Step-by-step derivation

   1. associate-+r+25.8%

      \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

   2. +-commutative25.8%

      \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

   3. associate-+r+25.8%

      \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

6. Simplified25.8%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]
7. Step-by-step derivation

   1. fma-udef25.8%

      \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x \cdot x}{s} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]

   2. associate-/l*25.9%

      \[\leadsto \frac{1}{\left(2 \cdot \color{blue}{\frac{x}{\frac{s}{x}}} + s \cdot 4\right) - \frac{x \cdot x}{s}} \]

8. Applied egg-rr25.9%

   \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \frac{x}{\frac{s}{x}} + s \cdot 4\right)} - \frac{x \cdot x}{s}} \]
9. Step-by-step derivation

   1. *-un-lft-identity25.9%

      \[\leadsto \color{blue}{1 \cdot \frac{1}{\left(2 \cdot \frac{x}{\frac{s}{x}} + s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   2. fma-def25.9%

      \[\leadsto 1 \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x}{\frac{s}{x}}, s \cdot 4\right)} - \frac{x \cdot x}{s}} \]

   3. div-inv25.9%

      \[\leadsto 1 \cdot \frac{1}{\mathsf{fma}\left(2, \color{blue}{x \cdot \frac{1}{\frac{s}{x}}}, s \cdot 4\right) - \frac{x \cdot x}{s}} \]

   4. clear-num25.9%

      \[\leadsto 1 \cdot \frac{1}{\mathsf{fma}\left(2, x \cdot \color{blue}{\frac{x}{s}}, s \cdot 4\right) - \frac{x \cdot x}{s}} \]

   5. associate-/l*26.4%

      \[\leadsto 1 \cdot \frac{1}{\mathsf{fma}\left(2, x \cdot \frac{x}{s}, s \cdot 4\right) - \color{blue}{\frac{x}{\frac{s}{x}}}} \]

   6. div-inv26.4%

      \[\leadsto 1 \cdot \frac{1}{\mathsf{fma}\left(2, x \cdot \frac{x}{s}, s \cdot 4\right) - \color{blue}{x \cdot \frac{1}{\frac{s}{x}}}} \]

   7. clear-num26.4%

      \[\leadsto 1 \cdot \frac{1}{\mathsf{fma}\left(2, x \cdot \frac{x}{s}, s \cdot 4\right) - x \cdot \color{blue}{\frac{x}{s}}} \]

10. Applied egg-rr26.4%

    \[\leadsto \color{blue}{1 \cdot \frac{1}{\mathsf{fma}\left(2, x \cdot \frac{x}{s}, s \cdot 4\right) - x \cdot \frac{x}{s}}} \]
11. Step-by-step derivation

    1. *-lft-identity26.4%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(2, x \cdot \frac{x}{s}, s \cdot 4\right) - x \cdot \frac{x}{s}}} \]

    2. fma-udef26.4%

       \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left(x \cdot \frac{x}{s}\right) + s \cdot 4\right)} - x \cdot \frac{x}{s}} \]

    3. associate-*r/25.7%

       \[\leadsto \frac{1}{\left(2 \cdot \color{blue}{\frac{x \cdot x}{s}} + s \cdot 4\right) - x \cdot \frac{x}{s}} \]

    4. unpow225.7%

       \[\leadsto \frac{1}{\left(2 \cdot \frac{\color{blue}{{x}^{2}}}{s} + s \cdot 4\right) - x \cdot \frac{x}{s}} \]

    5. *-commutative25.7%

       \[\leadsto \frac{1}{\left(2 \cdot \frac{{x}^{2}}{s} + \color{blue}{4 \cdot s}\right) - x \cdot \frac{x}{s}} \]

    6. +-commutative25.7%

       \[\leadsto \frac{1}{\color{blue}{\left(4 \cdot s + 2 \cdot \frac{{x}^{2}}{s}\right)} - x \cdot \frac{x}{s}} \]

    7. associate-+r-25.7%

       \[\leadsto \frac{1}{\color{blue}{4 \cdot s + \left(2 \cdot \frac{{x}^{2}}{s} - x \cdot \frac{x}{s}\right)}} \]

    8. *-commutative25.7%

       \[\leadsto \frac{1}{\color{blue}{s \cdot 4} + \left(2 \cdot \frac{{x}^{2}}{s} - x \cdot \frac{x}{s}\right)} \]

    9. associate-*r/25.7%

       \[\leadsto \frac{1}{s \cdot 4 + \left(\color{blue}{\frac{2 \cdot {x}^{2}}{s}} - x \cdot \frac{x}{s}\right)} \]

    10. *-commutative25.7%

        \[\leadsto \frac{1}{s \cdot 4 + \left(\frac{\color{blue}{{x}^{2} \cdot 2}}{s} - x \cdot \frac{x}{s}\right)} \]

    11. unpow225.7%

        \[\leadsto \frac{1}{s \cdot 4 + \left(\frac{\color{blue}{\left(x \cdot x\right)} \cdot 2}{s} - x \cdot \frac{x}{s}\right)} \]

    12. associate-*r*25.8%

        \[\leadsto \frac{1}{s \cdot 4 + \left(\frac{\color{blue}{x \cdot \left(x \cdot 2\right)}}{s} - x \cdot \frac{x}{s}\right)} \]

    13. associate-*r/25.8%

        \[\leadsto \frac{1}{s \cdot 4 + \left(\frac{x \cdot \left(x \cdot 2\right)}{s} - \color{blue}{\frac{x \cdot x}{s}}\right)} \]

    14. unpow225.8%

        \[\leadsto \frac{1}{s \cdot 4 + \left(\frac{x \cdot \left(x \cdot 2\right)}{s} - \frac{\color{blue}{{x}^{2}}}{s}\right)} \]

    15. div-sub38.1%

        \[\leadsto \frac{1}{s \cdot 4 + \color{blue}{\frac{x \cdot \left(x \cdot 2\right) - {x}^{2}}{s}}} \]

    16. unpow238.1%

        \[\leadsto \frac{1}{s \cdot 4 + \frac{x \cdot \left(x \cdot 2\right) - \color{blue}{x \cdot x}}{s}} \]

12. Simplified65.8%

    \[\leadsto \color{blue}{\frac{1}{s \cdot 4 + \frac{x \cdot \left(x \cdot 2 - x\right)}{s}}} \]

13. Final simplification65.8%

    \[\leadsto \frac{1}{s \cdot 4 + \frac{x \cdot \left(x \cdot 2 - x\right)}{s}} \]

1. Split input into 2 regimes

2. if x < -0.0199999996 or 4.99999987e-5 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 4.1%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+4.1%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative4.1%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+4.1%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified4.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 76.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow276.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot x}}{s}} \]

      2. associate-*l/76.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{x}{s} \cdot x}} \]

      3. *-commutative76.2%

         \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{x}{s}}} \]

   9. Simplified76.2%

      \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{x}{s}}} \]

   if -0.0199999996 < x < 4.99999987e-5

   1. Initial program 99.5%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. associate-*l*99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      2. +-commutative99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]

      3. +-commutative99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

   3. Simplified99.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]

   4. Taylor expanded in s around inf 50.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.019999999552965164 \lor \neg \left(x \leq 4.999999873689376 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < -0.0199999996 or 4.99999987e-5 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 4.1%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+4.1%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative4.1%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+4.1%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified4.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 76.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   8. Step-by-step derivation

      1. unpow276.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot x}}{s}} \]

   9. Simplified76.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s}}} \]

   if -0.0199999996 < x < 4.99999987e-5

   1. Initial program 99.5%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. associate-*l*99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      2. +-commutative99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]

      3. +-commutative99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

   3. Simplified99.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]

   4. Taylor expanded in s around inf 50.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.019999999552965164 \lor \neg \left(x \leq 4.999999873689376 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < -0.0199999996 or 4.99999987e-5 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 4.1%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+4.1%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative4.1%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+4.1%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified4.1%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 73.8%

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow273.8%

         \[\leadsto \frac{s}{\color{blue}{x \cdot x}} \]

   9. Simplified73.8%

      \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]

   if -0.0199999996 < x < 4.99999987e-5

   1. Initial program 99.5%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. associate-*l*99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      2. +-commutative99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]

      3. +-commutative99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

   3. Simplified99.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]

   4. Taylor expanded in s around inf 50.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.019999999552965164 \lor \neg \left(x \leq 4.999999873689376 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

1. Split input into 3 regimes

2. if x < -0.0199999996

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 4.3%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+4.3%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative4.3%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+4.3%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified4.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 71.7%

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow271.7%

         \[\leadsto \frac{s}{\color{blue}{x \cdot x}} \]

      2. associate-/r*71.7%

         \[\leadsto \color{blue}{\frac{\frac{s}{x}}{x}} \]

   9. Simplified71.7%

      \[\leadsto \color{blue}{\frac{\frac{s}{x}}{x}} \]

   if -0.0199999996 < x < 4.99999987e-5

   1. Initial program 99.5%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. associate-*l*99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      2. +-commutative99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]

      3. +-commutative99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

   3. Simplified99.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]

   4. Taylor expanded in s around inf 50.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   if 4.99999987e-5 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 3.9%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+3.9%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative3.9%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+3.9%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified3.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 75.4%

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow275.4%

         \[\leadsto \frac{s}{\color{blue}{x \cdot x}} \]

   9. Simplified75.4%

      \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.019999999552965164:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \mathbf{elif}\;x \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

1. Split input into 3 regimes

2. if x < -0.0199999996

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 4.3%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+4.3%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative4.3%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+4.3%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified4.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 71.7%

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow271.7%

         \[\leadsto \frac{s}{\color{blue}{x \cdot x}} \]

   9. Simplified71.7%

      \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   10. Step-by-step derivation

       1. clear-num75.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{s}}} \]

       2. inv-pow75.1%

          \[\leadsto \color{blue}{{\left(\frac{x \cdot x}{s}\right)}^{-1}} \]

   11. Applied egg-rr75.1%

       \[\leadsto \color{blue}{{\left(\frac{x \cdot x}{s}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-175.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{s}}} \]

       2. associate-/l*75.1%

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{s}{x}}}} \]

       3. associate-/r/71.7%

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{s}{x}} \]

   13. Simplified71.7%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{s}{x}} \]

   if -0.0199999996 < x < 4.99999987e-5

   1. Initial program 99.5%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. associate-*l*99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

      2. +-commutative99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]

      3. +-commutative99.7%

         \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

   3. Simplified99.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]

   4. Taylor expanded in s around inf 50.1%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]

   if 4.99999987e-5 < x

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. *-lft-identity100.0%

         \[\leadsto \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      2. associate-*r/100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

      3. associate-/l*100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

      4. distribute-frac-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

      5. exp-neg100.0%

         \[\leadsto \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

      6. associate-/r/100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

      7. /-rgt-identity100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

      8. associate-*l*100.0%

         \[\leadsto \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]

   4. Taylor expanded in s around inf 3.9%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \left|x\right| + \left(-2 \cdot \left|x\right| + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-+r+3.9%

         \[\leadsto \frac{1}{\color{blue}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}} \]

      2. +-commutative3.9%

         \[\leadsto \frac{1}{\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \color{blue}{\left(\left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      3. associate-+r+3.9%

         \[\leadsto \frac{1}{\color{blue}{\left(\left(2 \cdot \left|x\right| + -2 \cdot \left|x\right|\right) + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right) + 2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified3.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in x around inf 75.4%

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow275.4%

         \[\leadsto \frac{s}{\color{blue}{x \cdot x}} \]

   9. Simplified75.4%

      \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.019999999552965164:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

1. Initial program 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. associate-*l*99.9%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   2. +-commutative99.9%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]

   3. +-commutative99.9%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

3. Simplified99.9%

   \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]

4. Taylor expanded in s around inf 24.9%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]

5. Final simplification24.9%

   \[\leadsto \frac{0.25}{s} \]

herbie shell --seed 2023166 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))