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

1. Initial program 72.9%

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

   1. unpow272.9%

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

   2. unpow272.9%

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

   3. difference-of-squares72.8%

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

   4. *-commutative72.8%

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

   5. +-commutative72.8%

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

   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+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. Final simplification100.0%

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

1. Split input into 2 regimes

2. if x < -1.8999999999999999e-126 or 1.8499999999999999e-113 < x

   1. Initial program 31.3%

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

      1. unpow231.3%

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

      2. unpow231.3%

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

      3. difference-of-squares31.2%

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

      4. *-commutative31.2%

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

      5. +-commutative31.2%

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

      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+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 0 91.1%

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

   if -1.8999999999999999e-126 < x < 1.8499999999999999e-113

   1. Initial program 99.1%

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

      1. unpow299.1%

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

      2. unpow299.1%

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

      3. difference-of-squares99.1%

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

      4. *-commutative99.1%

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

      5. +-commutative99.1%

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

      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 98.4%

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

      1. unpow298.4%

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

   6. Simplified98.4%

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

4. Final simplification95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-126} \lor \neg \left(x \leq 1.85 \cdot 10^{-113}\right):\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

1. Split input into 2 regimes

2. if x < -3.5e-126 or 9.50000000000000004e-110 < x

   1. Initial program 31.3%

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

      1. unpow231.3%

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

      2. unpow231.3%

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

      3. difference-of-squares31.2%

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

      4. *-commutative31.2%

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

      5. +-commutative31.2%

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

      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+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. Step-by-step derivation

      1. fma-udef100.0%

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

      2. distribute-rgt-in100.0%

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

   5. Applied egg-rr100.0%

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

   6. Taylor expanded in x around inf 91.1%

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

      1. count-291.1%

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

      2. distribute-lft-out91.1%

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

   8. Simplified91.1%

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

   if -3.5e-126 < x < 9.50000000000000004e-110

   1. Initial program 99.1%

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

      1. unpow299.1%

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

      2. unpow299.1%

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

      3. difference-of-squares99.1%

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

      4. *-commutative99.1%

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

      5. +-commutative99.1%

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

      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 98.4%

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

      1. unpow298.4%

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

   6. Simplified98.4%

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

4. Final simplification95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-126} \lor \neg \left(x \leq 9.5 \cdot 10^{-110}\right):\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

1. Initial program 72.9%

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

   1. unpow272.9%

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

   2. unpow272.9%

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

   3. difference-of-squares72.8%

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

   4. *-commutative72.8%

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

   5. +-commutative72.8%

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

   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+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. Step-by-step derivation

   1. fma-udef100.0%

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

   2. distribute-rgt-in100.0%

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

5. Applied egg-rr100.0%

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

6. Final simplification100.0%

   \[\leadsto \varepsilon \cdot \left(2 \cdot x\right) + \varepsilon \cdot \varepsilon \]

1. Initial program 72.9%

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

   1. unpow272.9%

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

   2. unpow272.9%

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

   3. difference-of-squares72.8%

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

   4. *-commutative72.8%

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

   5. +-commutative72.8%

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

   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+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 70.8%

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

   1. unpow270.8%

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

6. Simplified70.8%

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

7. Final simplification70.8%

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

herbie shell --seed 2023166 
(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)))