ENA, Section 1.4, Mentioned, B

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]
2. Step-by-step derivation

   1. sub-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

   2. +-commutative87.8%

      \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

   3. neg-sub087.8%

      \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

   4. associate-+l-87.8%

      \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

   5. sub0-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

   6. neg-mul-187.8%

      \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

   7. associate-/r*87.8%

      \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

   8. metadata-eval87.8%

      \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

   9. fma-neg99.5%

      \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

   10. metadata-eval99.5%

       \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

3. Simplified99.5%

   \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]

4. Final simplification99.5%

   \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \]

1. Split input into 2 regimes

2. if (*.f64 x x) < 1

   1. Initial program 88.4%

      \[\frac{10}{1 - x \cdot x} \]
   2. Step-by-step derivation

      1. sub-neg88.4%

         \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

      2. +-commutative88.4%

         \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

      3. neg-sub088.4%

         \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

      4. associate-+l-88.4%

         \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

      5. sub0-neg88.4%

         \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

      6. neg-mul-188.4%

         \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

      7. associate-/r*88.4%

         \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

      8. metadata-eval88.4%

         \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

      9. fma-neg99.5%

         \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

      10. metadata-eval99.5%

          \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

   3. Simplified99.5%

      \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]
   4. Step-by-step derivation

      1. frac-2neg99.5%

         \[\leadsto \color{blue}{\frac{--10}{-\mathsf{fma}\left(x, x, -1\right)}} \]

      2. metadata-eval99.5%

         \[\leadsto \frac{\color{blue}{10}}{-\mathsf{fma}\left(x, x, -1\right)} \]

      3. fma-udef88.4%

         \[\leadsto \frac{10}{-\color{blue}{\left(x \cdot x + -1\right)}} \]

      4. distribute-neg-in88.4%

         \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + \left(--1\right)}} \]

      5. metadata-eval88.4%

         \[\leadsto \frac{10}{\left(-x \cdot x\right) + \color{blue}{1}} \]

      6. +-commutative88.4%

         \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

      7. sub-neg88.4%

         \[\leadsto \frac{10}{\color{blue}{1 - x \cdot x}} \]

      8. add-sqr-sqrt88.4%

         \[\leadsto \color{blue}{\sqrt{\frac{10}{1 - x \cdot x}} \cdot \sqrt{\frac{10}{1 - x \cdot x}}} \]

      9. sqrt-unprod88.4%

         \[\leadsto \color{blue}{\sqrt{\frac{10}{1 - x \cdot x} \cdot \frac{10}{1 - x \cdot x}}} \]

      10. pow1/288.4%

          \[\leadsto \color{blue}{{\left(\frac{10}{1 - x \cdot x} \cdot \frac{10}{1 - x \cdot x}\right)}^{0.5}} \]

      11. pow288.4%

          \[\leadsto {\color{blue}{\left({\left(\frac{10}{1 - x \cdot x}\right)}^{2}\right)}}^{0.5} \]

      12. metadata-eval88.4%

          \[\leadsto {\left({\left(\frac{\color{blue}{--10}}{1 - x \cdot x}\right)}^{2}\right)}^{0.5} \]

      13. sub-neg88.4%

          \[\leadsto {\left({\left(\frac{--10}{\color{blue}{1 + \left(-x \cdot x\right)}}\right)}^{2}\right)}^{0.5} \]

      14. +-commutative88.4%

          \[\leadsto {\left({\left(\frac{--10}{\color{blue}{\left(-x \cdot x\right) + 1}}\right)}^{2}\right)}^{0.5} \]

      15. metadata-eval88.4%

          \[\leadsto {\left({\left(\frac{--10}{\left(-x \cdot x\right) + \color{blue}{\left(--1\right)}}\right)}^{2}\right)}^{0.5} \]

      16. distribute-neg-in88.4%

          \[\leadsto {\left({\left(\frac{--10}{\color{blue}{-\left(x \cdot x + -1\right)}}\right)}^{2}\right)}^{0.5} \]

      17. fma-udef99.5%

          \[\leadsto {\left({\left(\frac{--10}{-\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{2}\right)}^{0.5} \]

      18. frac-2neg99.5%

          \[\leadsto {\left({\color{blue}{\left(\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{2}\right)}^{0.5} \]

      19. pow299.5%

          \[\leadsto {\color{blue}{\left(\frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{-10}{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{0.5} \]

      20. frac-times99.4%

          \[\leadsto {\color{blue}{\left(\frac{-10 \cdot -10}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}}^{0.5} \]

      21. metadata-eval99.4%

          \[\leadsto {\left(\frac{\color{blue}{100}}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}^{0.5} \]

      22. pow299.4%

          \[\leadsto {\left(\frac{100}{\color{blue}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}\right)}^{0.5} \]

   5. Applied egg-rr99.4%

      \[\leadsto \color{blue}{{\left(\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. unpow1/299.4%

         \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

   7. Simplified99.4%

      \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

   8. Taylor expanded in x around inf 13.5%

      \[\leadsto \color{blue}{\frac{10}{{x}^{2}}} \]
   9. Step-by-step derivation

      1. unpow213.5%

         \[\leadsto \frac{10}{\color{blue}{x \cdot x}} \]

   10. Simplified13.5%

       \[\leadsto \color{blue}{\frac{10}{x \cdot x}} \]

   if 1 < (*.f64 x x)

   1. Initial program 86.8%

      \[\frac{10}{1 - x \cdot x} \]
   2. Step-by-step derivation

      1. sub-neg86.8%

         \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

      2. +-commutative86.8%

         \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

      3. neg-sub086.8%

         \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

      4. associate-+l-86.8%

         \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

      5. sub0-neg86.8%

         \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

      6. neg-mul-186.8%

         \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

      7. associate-/r*86.8%

         \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

      8. metadata-eval86.8%

         \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

      9. fma-neg99.6%

         \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

      10. metadata-eval99.6%

          \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

   3. Simplified99.6%

      \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]
   4. Step-by-step derivation

      1. frac-2neg99.6%

         \[\leadsto \color{blue}{\frac{--10}{-\mathsf{fma}\left(x, x, -1\right)}} \]

      2. metadata-eval99.6%

         \[\leadsto \frac{\color{blue}{10}}{-\mathsf{fma}\left(x, x, -1\right)} \]

      3. fma-udef86.8%

         \[\leadsto \frac{10}{-\color{blue}{\left(x \cdot x + -1\right)}} \]

      4. distribute-neg-in86.8%

         \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + \left(--1\right)}} \]

      5. metadata-eval86.8%

         \[\leadsto \frac{10}{\left(-x \cdot x\right) + \color{blue}{1}} \]

      6. +-commutative86.8%

         \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

      7. sub-neg86.8%

         \[\leadsto \frac{10}{\color{blue}{1 - x \cdot x}} \]

      8. add-sqr-sqrt0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{10}{1 - x \cdot x}} \cdot \sqrt{\frac{10}{1 - x \cdot x}}} \]

      9. sqrt-unprod1.5%

         \[\leadsto \color{blue}{\sqrt{\frac{10}{1 - x \cdot x} \cdot \frac{10}{1 - x \cdot x}}} \]

      10. pow1/21.5%

          \[\leadsto \color{blue}{{\left(\frac{10}{1 - x \cdot x} \cdot \frac{10}{1 - x \cdot x}\right)}^{0.5}} \]

      11. pow21.5%

          \[\leadsto {\color{blue}{\left({\left(\frac{10}{1 - x \cdot x}\right)}^{2}\right)}}^{0.5} \]

      12. metadata-eval1.5%

          \[\leadsto {\left({\left(\frac{\color{blue}{--10}}{1 - x \cdot x}\right)}^{2}\right)}^{0.5} \]

      13. sub-neg1.5%

          \[\leadsto {\left({\left(\frac{--10}{\color{blue}{1 + \left(-x \cdot x\right)}}\right)}^{2}\right)}^{0.5} \]

      14. +-commutative1.5%

          \[\leadsto {\left({\left(\frac{--10}{\color{blue}{\left(-x \cdot x\right) + 1}}\right)}^{2}\right)}^{0.5} \]

      15. metadata-eval1.5%

          \[\leadsto {\left({\left(\frac{--10}{\left(-x \cdot x\right) + \color{blue}{\left(--1\right)}}\right)}^{2}\right)}^{0.5} \]

      16. distribute-neg-in1.5%

          \[\leadsto {\left({\left(\frac{--10}{\color{blue}{-\left(x \cdot x + -1\right)}}\right)}^{2}\right)}^{0.5} \]

      17. fma-udef1.5%

          \[\leadsto {\left({\left(\frac{--10}{-\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{2}\right)}^{0.5} \]

      18. frac-2neg1.5%

          \[\leadsto {\left({\color{blue}{\left(\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{2}\right)}^{0.5} \]

      19. pow21.5%

          \[\leadsto {\color{blue}{\left(\frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{-10}{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{0.5} \]

      20. frac-times1.5%

          \[\leadsto {\color{blue}{\left(\frac{-10 \cdot -10}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}}^{0.5} \]

      21. metadata-eval1.5%

          \[\leadsto {\left(\frac{\color{blue}{100}}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}^{0.5} \]

      22. pow21.5%

          \[\leadsto {\left(\frac{100}{\color{blue}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}\right)}^{0.5} \]

   5. Applied egg-rr1.5%

      \[\leadsto \color{blue}{{\left(\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. unpow1/21.5%

         \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

   7. Simplified1.5%

      \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

   8. Taylor expanded in x around 0 13.5%

      \[\leadsto \color{blue}{-10} \]
3. Recombined 2 regimes into one program.

4. Final simplification13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;\frac{10}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-10\\ \end{array} \]

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]
2. Step-by-step derivation

   1. sub-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

   2. +-commutative87.8%

      \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

   3. neg-sub087.8%

      \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

   4. associate-+l-87.8%

      \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

   5. sub0-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

   6. neg-mul-187.8%

      \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

   7. associate-/r*87.8%

      \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

   8. metadata-eval87.8%

      \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

   9. fma-neg99.5%

      \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

   10. metadata-eval99.5%

       \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

3. Simplified99.5%

   \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]
4. Step-by-step derivation

   1. frac-2neg99.5%

      \[\leadsto \color{blue}{\frac{--10}{-\mathsf{fma}\left(x, x, -1\right)}} \]

   2. metadata-eval99.5%

      \[\leadsto \frac{\color{blue}{10}}{-\mathsf{fma}\left(x, x, -1\right)} \]

   3. fma-udef87.8%

      \[\leadsto \frac{10}{-\color{blue}{\left(x \cdot x + -1\right)}} \]

   4. distribute-neg-in87.8%

      \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + \left(--1\right)}} \]

   5. metadata-eval87.8%

      \[\leadsto \frac{10}{\left(-x \cdot x\right) + \color{blue}{1}} \]

   6. +-commutative87.8%

      \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

   7. sub-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{1 - x \cdot x}} \]

   8. div-inv87.8%

      \[\leadsto \color{blue}{10 \cdot \frac{1}{1 - x \cdot x}} \]

   9. *-commutative87.8%

      \[\leadsto \color{blue}{\frac{1}{1 - x \cdot x} \cdot 10} \]

5. Applied egg-rr87.8%

   \[\leadsto \color{blue}{\frac{1}{1 - x \cdot x} \cdot 10} \]

6. Final simplification87.8%

   \[\leadsto \frac{1}{1 - x \cdot x} \cdot 10 \]

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]

2. Final simplification87.8%

   \[\leadsto \frac{10}{1 - x \cdot x} \]

1. Split input into 2 regimes

2. if (*.f64 x x) < 1

   1. Initial program 88.4%

      \[\frac{10}{1 - x \cdot x} \]
   2. Step-by-step derivation

      1. sub-neg88.4%

         \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

      2. +-commutative88.4%

         \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

      3. neg-sub088.4%

         \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

      4. associate-+l-88.4%

         \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

      5. sub0-neg88.4%

         \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

      6. neg-mul-188.4%

         \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

      7. associate-/r*88.4%

         \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

      8. metadata-eval88.4%

         \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

      9. fma-neg99.5%

         \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

      10. metadata-eval99.5%

          \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

   3. Simplified99.5%

      \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]

   4. Taylor expanded in x around 0 13.5%

      \[\leadsto \color{blue}{10} \]

   if 1 < (*.f64 x x)

   1. Initial program 86.8%

      \[\frac{10}{1 - x \cdot x} \]
   2. Step-by-step derivation

      1. sub-neg86.8%

         \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

      2. +-commutative86.8%

         \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

      3. neg-sub086.8%

         \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

      4. associate-+l-86.8%

         \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

      5. sub0-neg86.8%

         \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

      6. neg-mul-186.8%

         \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

      7. associate-/r*86.8%

         \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

      8. metadata-eval86.8%

         \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

      9. fma-neg99.6%

         \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

      10. metadata-eval99.6%

          \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

   3. Simplified99.6%

      \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]
   4. Step-by-step derivation

      1. frac-2neg99.6%

         \[\leadsto \color{blue}{\frac{--10}{-\mathsf{fma}\left(x, x, -1\right)}} \]

      2. metadata-eval99.6%

         \[\leadsto \frac{\color{blue}{10}}{-\mathsf{fma}\left(x, x, -1\right)} \]

      3. fma-udef86.8%

         \[\leadsto \frac{10}{-\color{blue}{\left(x \cdot x + -1\right)}} \]

      4. distribute-neg-in86.8%

         \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + \left(--1\right)}} \]

      5. metadata-eval86.8%

         \[\leadsto \frac{10}{\left(-x \cdot x\right) + \color{blue}{1}} \]

      6. +-commutative86.8%

         \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

      7. sub-neg86.8%

         \[\leadsto \frac{10}{\color{blue}{1 - x \cdot x}} \]

      8. add-sqr-sqrt0.0%

         \[\leadsto \color{blue}{\sqrt{\frac{10}{1 - x \cdot x}} \cdot \sqrt{\frac{10}{1 - x \cdot x}}} \]

      9. sqrt-unprod1.5%

         \[\leadsto \color{blue}{\sqrt{\frac{10}{1 - x \cdot x} \cdot \frac{10}{1 - x \cdot x}}} \]

      10. pow1/21.5%

          \[\leadsto \color{blue}{{\left(\frac{10}{1 - x \cdot x} \cdot \frac{10}{1 - x \cdot x}\right)}^{0.5}} \]

      11. pow21.5%

          \[\leadsto {\color{blue}{\left({\left(\frac{10}{1 - x \cdot x}\right)}^{2}\right)}}^{0.5} \]

      12. metadata-eval1.5%

          \[\leadsto {\left({\left(\frac{\color{blue}{--10}}{1 - x \cdot x}\right)}^{2}\right)}^{0.5} \]

      13. sub-neg1.5%

          \[\leadsto {\left({\left(\frac{--10}{\color{blue}{1 + \left(-x \cdot x\right)}}\right)}^{2}\right)}^{0.5} \]

      14. +-commutative1.5%

          \[\leadsto {\left({\left(\frac{--10}{\color{blue}{\left(-x \cdot x\right) + 1}}\right)}^{2}\right)}^{0.5} \]

      15. metadata-eval1.5%

          \[\leadsto {\left({\left(\frac{--10}{\left(-x \cdot x\right) + \color{blue}{\left(--1\right)}}\right)}^{2}\right)}^{0.5} \]

      16. distribute-neg-in1.5%

          \[\leadsto {\left({\left(\frac{--10}{\color{blue}{-\left(x \cdot x + -1\right)}}\right)}^{2}\right)}^{0.5} \]

      17. fma-udef1.5%

          \[\leadsto {\left({\left(\frac{--10}{-\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{2}\right)}^{0.5} \]

      18. frac-2neg1.5%

          \[\leadsto {\left({\color{blue}{\left(\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{2}\right)}^{0.5} \]

      19. pow21.5%

          \[\leadsto {\color{blue}{\left(\frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{-10}{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{0.5} \]

      20. frac-times1.5%

          \[\leadsto {\color{blue}{\left(\frac{-10 \cdot -10}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}}^{0.5} \]

      21. metadata-eval1.5%

          \[\leadsto {\left(\frac{\color{blue}{100}}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}^{0.5} \]

      22. pow21.5%

          \[\leadsto {\left(\frac{100}{\color{blue}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}\right)}^{0.5} \]

   5. Applied egg-rr1.5%

      \[\leadsto \color{blue}{{\left(\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}\right)}^{0.5}} \]
   6. Step-by-step derivation

      1. unpow1/21.5%

         \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

   7. Simplified1.5%

      \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

   8. Taylor expanded in x around 0 13.5%

      \[\leadsto \color{blue}{-10} \]
3. Recombined 2 regimes into one program.

4. Final simplification13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;10\\ \mathbf{else}:\\ \;\;\;\;-10\\ \end{array} \]

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]
2. Step-by-step derivation

   1. sub-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

   2. +-commutative87.8%

      \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + 1}} \]

   3. neg-sub087.8%

      \[\leadsto \frac{10}{\color{blue}{\left(0 - x \cdot x\right)} + 1} \]

   4. associate-+l-87.8%

      \[\leadsto \frac{10}{\color{blue}{0 - \left(x \cdot x - 1\right)}} \]

   5. sub0-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{-\left(x \cdot x - 1\right)}} \]

   6. neg-mul-187.8%

      \[\leadsto \frac{10}{\color{blue}{-1 \cdot \left(x \cdot x - 1\right)}} \]

   7. associate-/r*87.8%

      \[\leadsto \color{blue}{\frac{\frac{10}{-1}}{x \cdot x - 1}} \]

   8. metadata-eval87.8%

      \[\leadsto \frac{\color{blue}{-10}}{x \cdot x - 1} \]

   9. fma-neg99.5%

      \[\leadsto \frac{-10}{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \]

   10. metadata-eval99.5%

       \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \]

3. Simplified99.5%

   \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]
4. Step-by-step derivation

   1. frac-2neg99.5%

      \[\leadsto \color{blue}{\frac{--10}{-\mathsf{fma}\left(x, x, -1\right)}} \]

   2. metadata-eval99.5%

      \[\leadsto \frac{\color{blue}{10}}{-\mathsf{fma}\left(x, x, -1\right)} \]

   3. fma-udef87.8%

      \[\leadsto \frac{10}{-\color{blue}{\left(x \cdot x + -1\right)}} \]

   4. distribute-neg-in87.8%

      \[\leadsto \frac{10}{\color{blue}{\left(-x \cdot x\right) + \left(--1\right)}} \]

   5. metadata-eval87.8%

      \[\leadsto \frac{10}{\left(-x \cdot x\right) + \color{blue}{1}} \]

   6. +-commutative87.8%

      \[\leadsto \frac{10}{\color{blue}{1 + \left(-x \cdot x\right)}} \]

   7. sub-neg87.8%

      \[\leadsto \frac{10}{\color{blue}{1 - x \cdot x}} \]

   8. add-sqr-sqrt54.9%

      \[\leadsto \color{blue}{\sqrt{\frac{10}{1 - x \cdot x}} \cdot \sqrt{\frac{10}{1 - x \cdot x}}} \]

   9. sqrt-unprod55.5%

      \[\leadsto \color{blue}{\sqrt{\frac{10}{1 - x \cdot x} \cdot \frac{10}{1 - x \cdot x}}} \]

   10. pow1/255.5%

       \[\leadsto \color{blue}{{\left(\frac{10}{1 - x \cdot x} \cdot \frac{10}{1 - x \cdot x}\right)}^{0.5}} \]

   11. pow255.5%

       \[\leadsto {\color{blue}{\left({\left(\frac{10}{1 - x \cdot x}\right)}^{2}\right)}}^{0.5} \]

   12. metadata-eval55.5%

       \[\leadsto {\left({\left(\frac{\color{blue}{--10}}{1 - x \cdot x}\right)}^{2}\right)}^{0.5} \]

   13. sub-neg55.5%

       \[\leadsto {\left({\left(\frac{--10}{\color{blue}{1 + \left(-x \cdot x\right)}}\right)}^{2}\right)}^{0.5} \]

   14. +-commutative55.5%

       \[\leadsto {\left({\left(\frac{--10}{\color{blue}{\left(-x \cdot x\right) + 1}}\right)}^{2}\right)}^{0.5} \]

   15. metadata-eval55.5%

       \[\leadsto {\left({\left(\frac{--10}{\left(-x \cdot x\right) + \color{blue}{\left(--1\right)}}\right)}^{2}\right)}^{0.5} \]

   16. distribute-neg-in55.5%

       \[\leadsto {\left({\left(\frac{--10}{\color{blue}{-\left(x \cdot x + -1\right)}}\right)}^{2}\right)}^{0.5} \]

   17. fma-udef62.4%

       \[\leadsto {\left({\left(\frac{--10}{-\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)}^{2}\right)}^{0.5} \]

   18. frac-2neg62.4%

       \[\leadsto {\left({\color{blue}{\left(\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{2}\right)}^{0.5} \]

   19. pow262.4%

       \[\leadsto {\color{blue}{\left(\frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{-10}{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{0.5} \]

   20. frac-times62.3%

       \[\leadsto {\color{blue}{\left(\frac{-10 \cdot -10}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}}^{0.5} \]

   21. metadata-eval62.3%

       \[\leadsto {\left(\frac{\color{blue}{100}}{\mathsf{fma}\left(x, x, -1\right) \cdot \mathsf{fma}\left(x, x, -1\right)}\right)}^{0.5} \]

   22. pow262.3%

       \[\leadsto {\left(\frac{100}{\color{blue}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}\right)}^{0.5} \]

5. Applied egg-rr62.3%

   \[\leadsto \color{blue}{{\left(\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}\right)}^{0.5}} \]
6. Step-by-step derivation

   1. unpow1/262.3%

      \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

7. Simplified62.3%

   \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

8. Taylor expanded in x around 0 6.1%

   \[\leadsto \color{blue}{-10} \]

9. Final simplification6.1%

   \[\leadsto -10 \]

herbie shell --seed 2023166 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, B"
  :precision binary64
  :pre (and (<= 0.999 x) (<= x 1.001))
  (/ 10.0 (- 1.0 (* x x))))