1. Initial program 85.9%

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

   1. frac-sub57.9%

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

   2. div-inv57.4%

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

   3. /-rgt-identity57.4%

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

   4. *-un-lft-identity57.4%

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

   5. /-rgt-identity57.4%

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

   6. +-commutative57.4%

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

   7. *-commutative57.4%

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

   8. +-commutative57.4%

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

3. Applied egg-rr57.4%

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

   1. *-commutative57.4%

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

   2. +-commutative57.4%

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

   3. associate-/r*59.0%

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

   4. +-commutative59.0%

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

5. Simplified59.0%

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

   1. +-commutative59.0%

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

   2. frac-2neg59.0%

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

   3. metadata-eval59.0%

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

   4. associate-*r/85.6%

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

   5. frac-add84.5%

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

7. Applied egg-rr85.6%

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

   1. neg-mul-185.6%

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

   2. metadata-eval85.6%

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

   3. distribute-lft-in85.6%

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

   4. +-commutative85.6%

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

   5. *-commutative85.6%

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

   6. *-commutative85.6%

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

   7. +-commutative85.6%

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

   8. *-commutative85.6%

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

   9. +-commutative85.6%

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

   10. distribute-rgt1-in85.6%

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

   11. *-commutative85.6%

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

   12. neg-mul-185.6%

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

   13. unsub-neg85.6%

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

9. Simplified85.6%

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

10. Taylor expanded in x around 0 99.9%

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

11. Taylor expanded in x around 0 99.9%

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

    1. mul-1-neg99.9%

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

    2. unsub-neg99.9%

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

    3. unpow299.9%

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

13. Simplified99.9%

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

14. Final simplification99.9%

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

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 71.7%

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

      1. frac-sub15.1%

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

      2. div-inv14.2%

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

      3. /-rgt-identity14.2%

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

      4. *-un-lft-identity14.2%

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

      5. /-rgt-identity14.2%

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

      6. +-commutative14.2%

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

      7. *-commutative14.2%

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

      8. +-commutative14.2%

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

   3. Applied egg-rr14.2%

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

      1. *-commutative14.2%

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

      2. +-commutative14.2%

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

      3. associate-/r*17.3%

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

      4. +-commutative17.3%

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

   5. Simplified17.3%

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

      1. +-commutative17.3%

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

      2. frac-2neg17.3%

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

      3. metadata-eval17.3%

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

      4. associate-*r/71.0%

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

      5. frac-add68.8%

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

   7. Applied egg-rr70.9%

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

      1. neg-mul-170.9%

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

      2. metadata-eval70.9%

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

      3. distribute-lft-in70.9%

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

      4. +-commutative70.9%

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

      5. *-commutative70.9%

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

      6. *-commutative70.9%

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

      7. +-commutative70.9%

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

      8. *-commutative70.9%

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

      9. +-commutative70.9%

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

      10. distribute-rgt1-in70.9%

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

      11. *-commutative70.9%

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

      12. neg-mul-170.9%

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

      13. unsub-neg70.9%

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

   9. Simplified70.9%

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

   10. Taylor expanded in x around 0 99.9%

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

   11. Taylor expanded in x around inf 98.5%

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

       1. mul-1-neg98.5%

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

       2. unpow298.5%

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

   13. Simplified98.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.5%

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

      1. associate-*r/99.5%

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

      2. metadata-eval99.5%

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

   4. Simplified99.5%

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

4. Final simplification99.0%

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

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 71.7%

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

   2. Taylor expanded in x around inf 69.9%

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

   3. Taylor expanded in x around inf 60.9%

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

      1. unpow260.9%

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

   5. Simplified60.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.3%

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

   3. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-199.3%

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

      2. associate-*r/99.3%

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

      3. metadata-eval99.3%

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

   5. Simplified99.3%

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

4. Final simplification80.2%

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

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 71.7%

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

   2. Taylor expanded in x around inf 69.9%

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

   3. Taylor expanded in x around inf 60.9%

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

      1. unpow260.9%

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

   5. Simplified60.9%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.2%

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

4. Final simplification80.2%

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

1. Initial program 85.9%

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

2. Taylor expanded in x around 0 51.7%

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

3. Taylor expanded in x around 0 84.6%

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

4. Final simplification84.6%

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

1. Initial program 85.9%

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

2. Taylor expanded in x around 0 52.6%

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

3. Final simplification52.6%

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

1. Initial program 85.9%

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

2. Taylor expanded in x around 0 51.7%

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

3. Taylor expanded in x around inf 3.3%

   \[\leadsto \color{blue}{1} \]

4. Final simplification3.3%

   \[\leadsto 1 \]

herbie shell --seed 2023166 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))