Asymptote A

1. Initial program 79.7%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. frac-sub80.6%

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

   2. associate-/r*80.6%

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

   3. *-un-lft-identity80.6%

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

   4. *-rgt-identity80.6%

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

   5. associate--l-80.6%

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

   6. +-commutative80.6%

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

   7. +-commutative80.6%

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

   8. sub-neg80.6%

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

   9. metadata-eval80.6%

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

3. Applied egg-rr80.6%

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

   1. expm1-log1p-u79.4%

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

   2. expm1-udef77.8%

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

   3. associate-/l/77.8%

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

   4. associate-+r+77.7%

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

   5. metadata-eval77.7%

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

   6. +-commutative77.7%

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

   7. +-commutative77.7%

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

5. Applied egg-rr77.7%

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

   1. expm1-def79.4%

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

   2. expm1-log1p80.6%

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

   3. *-commutative80.6%

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

   4. associate-/r*80.6%

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

   5. associate--r+99.9%

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

   6. +-inverses99.9%

      \[\leadsto \frac{\frac{\color{blue}{0} - 2}{x + 1}}{x + -1} \]

   7. metadata-eval99.9%

      \[\leadsto \frac{\frac{\color{blue}{-2}}{x + 1}}{x + -1} \]

   8. metadata-eval99.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{-1}}}{x + 1}}{x + -1} \]

   9. associate-/r*99.9%

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

   10. neg-mul-199.9%

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

   11. +-commutative99.9%

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

   12. distribute-neg-in99.9%

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

   13. metadata-eval99.9%

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

   14. sub-neg99.9%

       \[\leadsto \frac{\frac{2}{\color{blue}{-1 - x}}}{x + -1} \]

7. Simplified99.9%

   \[\leadsto \color{blue}{\frac{\frac{2}{-1 - x}}{x + -1}} \]

8. Final simplification99.9%

   \[\leadsto \frac{\frac{2}{-1 - x}}{-1 + x} \]

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 1 < x

   1. Initial program 59.2%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around inf 97.5%

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

      1. unpow297.5%

         \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

   4. Simplified97.5%

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]
   5. Step-by-step derivation

      1. associate-/r*98.5%

         \[\leadsto \color{blue}{\frac{\frac{-2}{x}}{x}} \]

      2. div-inv98.3%

         \[\leadsto \color{blue}{\frac{-2}{x} \cdot \frac{1}{x}} \]

   6. Applied egg-rr98.3%

      \[\leadsto \color{blue}{\frac{-2}{x} \cdot \frac{1}{x}} \]
   7. Step-by-step derivation

      1. un-div-inv98.5%

         \[\leadsto \color{blue}{\frac{\frac{-2}{x}}{x}} \]

   8. Applied egg-rr98.5%

      \[\leadsto \color{blue}{\frac{\frac{-2}{x}}{x}} \]

   if -1.55000000000000004 < x < 1

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around 0 98.8%

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

      1. sub-neg98.8%

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

      2. metadata-eval98.8%

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

      3. +-commutative98.8%

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

      4. neg-mul-198.8%

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

      5. unsub-neg98.8%

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

   4. Simplified98.8%

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

4. Final simplification98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} + \left(x - -1\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 59.2%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around inf 97.5%

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

      1. unpow297.5%

         \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

   4. Simplified97.5%

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around 0 98.8%

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

      1. sub-neg98.8%

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

      2. metadata-eval98.8%

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

      3. +-commutative98.8%

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

      4. neg-mul-198.8%

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

      5. unsub-neg98.8%

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

   4. Simplified98.8%

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

   5. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{2 + {x}^{2}} \]
   6. Step-by-step derivation

      1. unpow298.8%

         \[\leadsto 2 + \color{blue}{x \cdot x} \]

   7. Simplified98.8%

      \[\leadsto \color{blue}{2 + x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;2 + x \cdot x\\ \end{array} \]

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 59.2%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around inf 97.5%

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

      1. unpow297.5%

         \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

   4. Simplified97.5%

      \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]
   5. Step-by-step derivation

      1. associate-/r*98.5%

         \[\leadsto \color{blue}{\frac{\frac{-2}{x}}{x}} \]

      2. div-inv98.3%

         \[\leadsto \color{blue}{\frac{-2}{x} \cdot \frac{1}{x}} \]

   6. Applied egg-rr98.3%

      \[\leadsto \color{blue}{\frac{-2}{x} \cdot \frac{1}{x}} \]
   7. Step-by-step derivation

      1. un-div-inv98.5%

         \[\leadsto \color{blue}{\frac{\frac{-2}{x}}{x}} \]

   8. Applied egg-rr98.5%

      \[\leadsto \color{blue}{\frac{\frac{-2}{x}}{x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

   2. Taylor expanded in x around 0 98.8%

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

      1. sub-neg98.8%

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

      2. metadata-eval98.8%

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

      3. +-commutative98.8%

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

      4. neg-mul-198.8%

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

      5. unsub-neg98.8%

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

   4. Simplified98.8%

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

   5. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{2 + {x}^{2}} \]
   6. Step-by-step derivation

      1. unpow298.8%

         \[\leadsto 2 + \color{blue}{x \cdot x} \]

   7. Simplified98.8%

      \[\leadsto \color{blue}{2 + x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{-2}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;2 + x \cdot x\\ \end{array} \]

1. Initial program 79.7%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. frac-sub80.6%

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

   2. associate-/r*80.6%

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

   3. *-un-lft-identity80.6%

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

   4. *-rgt-identity80.6%

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

   5. associate--l-80.6%

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

   6. +-commutative80.6%

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

   7. +-commutative80.6%

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

   8. sub-neg80.6%

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

   9. metadata-eval80.6%

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

3. Applied egg-rr80.6%

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

   1. clear-num80.6%

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

   2. inv-pow80.6%

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

   3. div-inv80.6%

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

   4. clear-num80.6%

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

   5. +-commutative80.6%

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

   6. associate-+r+80.6%

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

   7. metadata-eval80.6%

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

   8. +-commutative80.6%

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

5. Applied egg-rr80.6%

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

   1. unpow-180.6%

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

   2. associate-*r/80.6%

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

   3. associate--r+99.4%

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

   4. +-inverses99.4%

      \[\leadsto \frac{1}{\frac{\left(x + -1\right) \cdot \left(x + 1\right)}{\color{blue}{0} - 2}} \]

   5. metadata-eval99.4%

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

7. Simplified99.4%

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

   1. *-commutative99.4%

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

   2. metadata-eval99.4%

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

   3. sub-neg99.4%

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

   4. difference-of-sqr-199.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot x - 1}}{-2}} \]

9. Applied egg-rr99.4%

   \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot x - 1}}{-2}} \]

10. Final simplification99.4%

    \[\leadsto \frac{1}{\frac{-1 + x \cdot x}{-2}} \]

1. Initial program 79.7%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]

2. Taylor expanded in x around 0 51.1%

   \[\leadsto \color{blue}{2} \]

3. Final simplification51.1%

   \[\leadsto 2 \]

herbie shell --seed 2023166 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))