ENA, Section 1.4, Exercise 4b, n=2

1. Initial program 71.1%

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

   1. unpow271.1%

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

3. Simplified71.1%

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

   1. unpow271.1%

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

   2. +-commutative71.1%

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

   3. +-commutative71.1%

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

5. Applied egg-rr71.1%

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

   1. distribute-rgt-in71.2%

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

   2. *-commutative71.2%

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

   3. associate--l+74.6%

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

   4. +-commutative74.6%

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

   5. distribute-lft-out--74.6%

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

   6. +-commutative74.6%

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

   7. *-commutative74.6%

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

   8. +-commutative74.6%

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

7. Applied egg-rr74.6%

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

8. Taylor expanded in x around 0 99.9%

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

   1. +-commutative99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(x + \varepsilon\right) + x \cdot \varepsilon} \]

   2. *-commutative99.9%

      \[\leadsto \varepsilon \cdot \left(x + \varepsilon\right) + \color{blue}{\varepsilon \cdot x} \]

   3. distribute-lft-out100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(x + \varepsilon\right) + x\right)} \]

10. Applied egg-rr100.0%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(x + \varepsilon\right) + x\right)} \]

11. Final simplification100.0%

    \[\leadsto \varepsilon \cdot \left(x + \left(\varepsilon + x\right)\right) \]

1. Split input into 2 regimes

2. if x < -3.8000000000000001e-96 or 1.1499999999999999e-99 < x

   1. Initial program 35.9%

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

      1. unpow235.9%

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

      2. unpow235.9%

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

      3. difference-of-squares35.8%

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

      4. *-commutative35.8%

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

      5. +-commutative35.8%

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

      6. associate--l+99.9%

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

      7. +-inverses99.9%

         \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      8. +-rgt-identity99.9%

         \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      9. +-commutative99.9%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

      10. associate-+r+99.9%

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

      11. count-299.9%

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

      12. fma-def99.9%

          \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

   3. Simplified99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]

   4. Taylor expanded in eps around 0 86.4%

      \[\leadsto \color{blue}{2 \cdot \left(\varepsilon \cdot x\right)} \]

   if -3.8000000000000001e-96 < x < 1.1499999999999999e-99

   1. Initial program 96.9%

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

      1. unpow296.9%

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

      2. unpow296.9%

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

      3. difference-of-squares96.9%

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

      4. *-commutative96.9%

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

      5. +-commutative96.9%

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

      6. associate--l+100.0%

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

      7. +-inverses100.0%

         \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      8. +-rgt-identity100.0%

         \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      9. +-commutative100.0%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

      10. associate-+r+100.0%

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

      11. count-2100.0%

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

      12. fma-def100.0%

          \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]

   4. Taylor expanded in eps around inf 96.2%

      \[\leadsto \color{blue}{{\varepsilon}^{2}} \]
   5. Step-by-step derivation

      1. unpow296.2%

         \[\leadsto \color{blue}{\varepsilon \cdot \varepsilon} \]

   6. Simplified96.2%

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

4. Final simplification92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-96} \lor \neg \left(x \leq 1.15 \cdot 10^{-99}\right):\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

1. Split input into 3 regimes

2. if x < -3.9999999999999996e-96

   1. Initial program 29.8%

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

      1. unpow229.8%

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

      2. unpow229.8%

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

      3. difference-of-squares29.8%

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

      4. *-commutative29.8%

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

      5. +-commutative29.8%

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

      6. associate--l+100.0%

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

      7. +-inverses100.0%

         \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      8. +-rgt-identity100.0%

         \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      9. +-commutative100.0%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

      10. associate-+r+99.9%

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

      11. count-299.9%

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

      12. fma-def99.9%

          \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

   3. Simplified99.9%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]

   4. Taylor expanded in eps around 0 88.5%

      \[\leadsto \color{blue}{2 \cdot \left(\varepsilon \cdot x\right)} \]

   if -3.9999999999999996e-96 < x < 1.7999999999999999e-100

   1. Initial program 96.9%

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

      1. unpow296.9%

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

      2. unpow296.9%

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

      3. difference-of-squares96.9%

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

      4. *-commutative96.9%

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

      5. +-commutative96.9%

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

      6. associate--l+100.0%

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

      7. +-inverses100.0%

         \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      8. +-rgt-identity100.0%

         \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      9. +-commutative100.0%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

      10. associate-+r+100.0%

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

      11. count-2100.0%

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

      12. fma-def100.0%

          \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

   3. Simplified100.0%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]

   4. Taylor expanded in eps around inf 96.2%

      \[\leadsto \color{blue}{{\varepsilon}^{2}} \]
   5. Step-by-step derivation

      1. unpow296.2%

         \[\leadsto \color{blue}{\varepsilon \cdot \varepsilon} \]

   6. Simplified96.2%

      \[\leadsto \color{blue}{\varepsilon \cdot \varepsilon} \]

   if 1.7999999999999999e-100 < x

   1. Initial program 41.7%

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

      1. unpow241.7%

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

      2. unpow241.7%

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

      3. difference-of-squares41.7%

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

      4. *-commutative41.7%

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

      5. +-commutative41.7%

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

      6. associate--l+99.9%

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

      7. +-inverses99.9%

         \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      8. +-rgt-identity99.9%

         \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

      9. +-commutative99.9%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

      10. associate-+r+99.9%

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

      11. count-299.9%

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

      12. fma-def99.9%

          \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

   3. Simplified99.9%

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

      1. fma-udef99.9%

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

      2. distribute-rgt-in99.9%

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

   5. Applied egg-rr99.9%

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

   6. Taylor expanded in x around inf 84.3%

      \[\leadsto \color{blue}{2 \cdot \left(\varepsilon \cdot x\right)} \]
   7. Step-by-step derivation

      1. associate-*r*84.5%

         \[\leadsto \color{blue}{\left(2 \cdot \varepsilon\right) \cdot x} \]

      2. *-commutative84.5%

         \[\leadsto \color{blue}{\left(\varepsilon \cdot 2\right)} \cdot x \]

      3. associate-*r*84.5%

         \[\leadsto \color{blue}{\varepsilon \cdot \left(2 \cdot x\right)} \]

      4. count-284.5%

         \[\leadsto \varepsilon \cdot \color{blue}{\left(x + x\right)} \]

   8. Simplified84.5%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(x + x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-96}:\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-100}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \end{array} \]

1. Initial program 71.1%

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

   1. unpow271.1%

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

3. Simplified71.1%

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

   1. unpow271.1%

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

   2. +-commutative71.1%

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

   3. +-commutative71.1%

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

5. Applied egg-rr71.1%

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

   1. distribute-rgt-in71.2%

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

   2. *-commutative71.2%

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

   3. associate--l+74.6%

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

   4. +-commutative74.6%

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

   5. distribute-lft-out--74.6%

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

   6. +-commutative74.6%

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

   7. *-commutative74.6%

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

   8. +-commutative74.6%

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

7. Applied egg-rr74.6%

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

8. Taylor expanded in x around 0 99.9%

   \[\leadsto x \cdot \color{blue}{\varepsilon} + \varepsilon \cdot \left(x + \varepsilon\right) \]

9. Taylor expanded in x around 0 99.9%

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

    1. unpow299.9%

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

    2. *-commutative99.9%

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

    3. associate-*r*100.0%

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

    4. count-2100.0%

       \[\leadsto \varepsilon \cdot \varepsilon + \color{blue}{\left(x + x\right)} \cdot \varepsilon \]

    5. distribute-rgt-in100.0%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon + \left(x + x\right)\right)} \]

11. Simplified100.0%

    \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon + \left(x + x\right)\right)} \]

12. Final simplification100.0%

    \[\leadsto \varepsilon \cdot \left(\varepsilon + \left(x + x\right)\right) \]

1. Initial program 71.1%

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

   1. unpow271.1%

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

   2. unpow271.1%

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

   3. difference-of-squares71.1%

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

   4. *-commutative71.1%

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

   5. +-commutative71.1%

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

   6. associate--l+100.0%

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

   7. +-inverses100.0%

      \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   8. +-rgt-identity100.0%

      \[\leadsto \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]

   9. +-commutative100.0%

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]

   10. associate-+r+100.0%

       \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]

   11. count-2100.0%

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

   12. fma-def100.0%

       \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

3. Simplified100.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)} \]

4. Taylor expanded in eps around inf 68.7%

   \[\leadsto \color{blue}{{\varepsilon}^{2}} \]
5. Step-by-step derivation

   1. unpow268.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \varepsilon} \]

6. Simplified68.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \varepsilon} \]

7. Final simplification68.7%

   \[\leadsto \varepsilon \cdot \varepsilon \]

herbie shell --seed 2023167 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=2"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 2.0) (pow x 2.0)))