1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000001e-9

   1. Initial program 39.2%

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

      1. frac-sub39.3%

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

      2. div-inv39.3%

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

      3. *-un-lft-identity39.3%

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

      4. +-commutative39.3%

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

      5. *-rgt-identity39.3%

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

      6. metadata-eval39.3%

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

      7. frac-times39.3%

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

      8. un-div-inv39.3%

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

      9. pow1/239.3%

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

      10. pow-flip39.3%

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

      11. metadata-eval39.3%

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

      12. +-commutative39.3%

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

   3. Applied egg-rr39.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   4. Step-by-step derivation

      1. associate-*r/39.3%

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

      2. *-rgt-identity39.3%

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

      3. times-frac39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      4. div-sub39.3%

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

      5. *-inverses39.3%

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

      6. /-rgt-identity39.3%

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

   5. Simplified39.3%

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

   6. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\left(\left(0.5 \cdot \frac{1}{x} + 0.3125 \cdot \frac{1}{{x}^{3}}\right) - 0.375 \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate--l+99.6%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot {x}^{-0.5} \]

      2. associate-*r/99.6%

         \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      3. metadata-eval99.6%

         \[\leadsto \left(\frac{\color{blue}{0.5}}{x} + \left(0.3125 \cdot \frac{1}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      4. associate-*r/99.6%

         \[\leadsto \left(\frac{0.5}{x} + \left(\color{blue}{\frac{0.3125 \cdot 1}{{x}^{3}}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      5. metadata-eval99.6%

         \[\leadsto \left(\frac{0.5}{x} + \left(\frac{\color{blue}{0.3125}}{{x}^{3}} - 0.375 \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      6. associate-*r/99.6%

         \[\leadsto \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right)\right) \cdot {x}^{-0.5} \]

      7. metadata-eval99.6%

         \[\leadsto \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{\color{blue}{0.375}}{{x}^{2}}\right)\right) \cdot {x}^{-0.5} \]

      8. unpow299.6%

         \[\leadsto \left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{\color{blue}{x \cdot x}}\right)\right) \cdot {x}^{-0.5} \]

   8. Simplified99.6%

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right)} \cdot {x}^{-0.5} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. frac-sub99.3%

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

      2. div-inv99.3%

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

      3. *-un-lft-identity99.3%

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

      4. +-commutative99.3%

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

      5. *-rgt-identity99.3%

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

      6. metadata-eval99.3%

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

      7. frac-times99.3%

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

      8. un-div-inv99.3%

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

      9. pow1/299.3%

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

      10. pow-flip99.8%

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

      11. metadata-eval99.8%

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

      12. +-commutative99.8%

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

   3. Applied egg-rr99.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   4. Step-by-step derivation

      1. associate-*r/99.8%

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

      2. *-rgt-identity99.8%

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

      3. times-frac99.8%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      4. div-sub99.8%

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

      5. *-inverses99.8%

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

      6. /-rgt-identity99.8%

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

   5. Simplified99.8%

      \[\leadsto \color{blue}{\left(1 - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot {x}^{-0.5}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity99.8%

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

      2. sqrt-undiv99.8%

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

   7. Applied egg-rr99.8%

      \[\leadsto \left(1 - \color{blue}{1 \cdot \sqrt{\frac{x}{1 + x}}}\right) \cdot {x}^{-0.5} \]
   8. Step-by-step derivation

      1. *-lft-identity99.8%

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

   9. Simplified99.8%

      \[\leadsto \left(1 - \color{blue}{\sqrt{\frac{x}{1 + x}}}\right) \cdot {x}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left(\frac{0.5}{x} + \left(\frac{0.3125}{{x}^{3}} - \frac{0.375}{x \cdot x}\right)\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000001e-9

   1. Initial program 39.2%

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

      1. frac-sub39.3%

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

      2. div-inv39.3%

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

      3. *-un-lft-identity39.3%

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

      4. +-commutative39.3%

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

      5. *-rgt-identity39.3%

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

      6. metadata-eval39.3%

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

      7. frac-times39.3%

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

      8. un-div-inv39.3%

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

      9. pow1/239.3%

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

      10. pow-flip39.3%

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

      11. metadata-eval39.3%

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

      12. +-commutative39.3%

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

   3. Applied egg-rr39.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   4. Step-by-step derivation

      1. associate-*r/39.3%

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

      2. *-rgt-identity39.3%

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

      3. times-frac39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      4. div-sub39.3%

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

      5. *-inverses39.3%

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

      6. /-rgt-identity39.3%

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

   5. Simplified39.3%

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

   6. Taylor expanded in x around inf 99.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*r/99.4%

         \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      2. metadata-eval99.4%

         \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      3. associate-*r/99.4%

         \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right) \cdot {x}^{-0.5} \]

      4. metadata-eval99.4%

         \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      5. unpow299.4%

         \[\leadsto \left(\frac{0.5}{x} - \frac{0.375}{\color{blue}{x \cdot x}}\right) \cdot {x}^{-0.5} \]

   8. Simplified99.4%

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)} \cdot {x}^{-0.5} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. frac-sub99.3%

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

      2. div-inv99.3%

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

      3. *-un-lft-identity99.3%

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

      4. +-commutative99.3%

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

      5. *-rgt-identity99.3%

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

      6. metadata-eval99.3%

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

      7. frac-times99.3%

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

      8. un-div-inv99.3%

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

      9. pow1/299.3%

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

      10. pow-flip99.8%

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

      11. metadata-eval99.8%

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

      12. +-commutative99.8%

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

   3. Applied egg-rr99.8%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   4. Step-by-step derivation

      1. associate-*r/99.8%

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

      2. *-rgt-identity99.8%

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

      3. times-frac99.8%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      4. div-sub99.8%

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

      5. *-inverses99.8%

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

      6. /-rgt-identity99.8%

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

   5. Simplified99.8%

      \[\leadsto \color{blue}{\left(1 - \frac{\sqrt{x}}{\sqrt{1 + x}}\right) \cdot {x}^{-0.5}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity99.8%

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

      2. sqrt-undiv99.8%

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

   7. Applied egg-rr99.8%

      \[\leadsto \left(1 - \color{blue}{1 \cdot \sqrt{\frac{x}{1 + x}}}\right) \cdot {x}^{-0.5} \]
   8. Step-by-step derivation

      1. *-lft-identity99.8%

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

   9. Simplified99.8%

      \[\leadsto \left(1 - \color{blue}{\sqrt{\frac{x}{1 + x}}}\right) \cdot {x}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \sqrt{\frac{x}{1 + x}}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000001e-9

   1. Initial program 39.2%

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

      1. frac-sub39.3%

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

      2. div-inv39.3%

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

      3. *-un-lft-identity39.3%

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

      4. +-commutative39.3%

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

      5. *-rgt-identity39.3%

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

      6. metadata-eval39.3%

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

      7. frac-times39.3%

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

      8. un-div-inv39.3%

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

      9. pow1/239.3%

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

      10. pow-flip39.3%

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

      11. metadata-eval39.3%

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

      12. +-commutative39.3%

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

   3. Applied egg-rr39.3%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   4. Step-by-step derivation

      1. associate-*r/39.3%

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

      2. *-rgt-identity39.3%

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

      3. times-frac39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      4. div-sub39.3%

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

      5. *-inverses39.3%

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

      6. /-rgt-identity39.3%

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

   5. Simplified39.3%

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

   6. Taylor expanded in x around inf 99.4%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*r/99.4%

         \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      2. metadata-eval99.4%

         \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      3. associate-*r/99.4%

         \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right) \cdot {x}^{-0.5} \]

      4. metadata-eval99.4%

         \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      5. unpow299.4%

         \[\leadsto \left(\frac{0.5}{x} - \frac{0.375}{\color{blue}{x \cdot x}}\right) \cdot {x}^{-0.5} \]

   8. Simplified99.4%

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)} \cdot {x}^{-0.5} \]

   if 5.0000000000000001e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. *-un-lft-identity99.3%

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

      2. clear-num99.3%

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

      3. associate-/r/99.3%

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

      4. prod-diff99.3%

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

      5. *-un-lft-identity99.3%

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

      6. fma-neg99.3%

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

      7. *-un-lft-identity99.3%

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

      8. inv-pow99.3%

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

      9. sqrt-pow299.7%

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

      10. metadata-eval99.7%

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

      11. pow1/299.7%

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

      12. pow-flip99.8%

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

      13. +-commutative99.8%

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

      14. metadata-eval99.8%

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

   3. Applied egg-rr99.8%

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

      1. fma-udef99.8%

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

      2. distribute-lft1-in99.8%

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

      3. metadata-eval99.8%

         \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} \]

      4. mul0-lft99.8%

         \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \]

      5. +-rgt-identity99.8%

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

   5. Simplified99.8%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 98.5%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{0.5 \cdot x + 1}} \]
   3. Step-by-step derivation

      1. add-log-exp5.1%

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

      2. *-un-lft-identity5.1%

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

      3. log-prod5.1%

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

      4. metadata-eval5.1%

         \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right) - \frac{1}{0.5 \cdot x + 1} \]

      5. add-log-exp98.5%

         \[\leadsto \left(0 + \color{blue}{\frac{1}{\sqrt{x}}}\right) - \frac{1}{0.5 \cdot x + 1} \]

      6. pow1/298.5%

         \[\leadsto \left(0 + \frac{1}{\color{blue}{{x}^{0.5}}}\right) - \frac{1}{0.5 \cdot x + 1} \]

      7. pow-flip99.0%

         \[\leadsto \left(0 + \color{blue}{{x}^{\left(-0.5\right)}}\right) - \frac{1}{0.5 \cdot x + 1} \]

      8. metadata-eval99.0%

         \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) - \frac{1}{0.5 \cdot x + 1} \]

   4. Applied egg-rr99.0%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{0.5 \cdot x + 1} \]
   5. Step-by-step derivation

      1. +-lft-identity99.0%

         \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{0.5 \cdot x + 1} \]

   6. Simplified99.0%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{0.5 \cdot x + 1} \]

   if 1.44999999999999996 < x

   1. Initial program 40.4%

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

      1. frac-sub40.5%

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

      2. div-inv40.5%

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

      3. *-un-lft-identity40.5%

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

      4. +-commutative40.5%

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

      5. *-rgt-identity40.5%

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

      6. metadata-eval40.5%

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

      7. frac-times40.5%

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

      8. un-div-inv40.5%

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

      9. pow1/240.5%

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

      10. pow-flip40.5%

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

      11. metadata-eval40.5%

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

      12. +-commutative40.5%

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

   3. Applied egg-rr40.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   4. Step-by-step derivation

      1. associate-*r/40.5%

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

      2. *-rgt-identity40.5%

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

      3. times-frac40.5%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      4. div-sub40.5%

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

      5. *-inverses40.5%

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

      6. /-rgt-identity40.5%

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

   5. Simplified40.5%

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

   6. Taylor expanded in x around inf 98.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right)} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*r/98.0%

         \[\leadsto \left(\color{blue}{\frac{0.5 \cdot 1}{x}} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      2. metadata-eval98.0%

         \[\leadsto \left(\frac{\color{blue}{0.5}}{x} - 0.375 \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      3. associate-*r/98.0%

         \[\leadsto \left(\frac{0.5}{x} - \color{blue}{\frac{0.375 \cdot 1}{{x}^{2}}}\right) \cdot {x}^{-0.5} \]

      4. metadata-eval98.0%

         \[\leadsto \left(\frac{0.5}{x} - \frac{\color{blue}{0.375}}{{x}^{2}}\right) \cdot {x}^{-0.5} \]

      5. unpow298.0%

         \[\leadsto \left(\frac{0.5}{x} - \frac{0.375}{\color{blue}{x \cdot x}}\right) \cdot {x}^{-0.5} \]

   8. Simplified98.0%

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)} \cdot {x}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if x < 1.67999999999999994

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 98.5%

      \[\leadsto \frac{1}{\sqrt{x}} - \frac{1}{\color{blue}{0.5 \cdot x + 1}} \]
   3. Step-by-step derivation

      1. add-log-exp5.1%

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

      2. *-un-lft-identity5.1%

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

      3. log-prod5.1%

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

      4. metadata-eval5.1%

         \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right) - \frac{1}{0.5 \cdot x + 1} \]

      5. add-log-exp98.5%

         \[\leadsto \left(0 + \color{blue}{\frac{1}{\sqrt{x}}}\right) - \frac{1}{0.5 \cdot x + 1} \]

      6. pow1/298.5%

         \[\leadsto \left(0 + \frac{1}{\color{blue}{{x}^{0.5}}}\right) - \frac{1}{0.5 \cdot x + 1} \]

      7. pow-flip99.0%

         \[\leadsto \left(0 + \color{blue}{{x}^{\left(-0.5\right)}}\right) - \frac{1}{0.5 \cdot x + 1} \]

      8. metadata-eval99.0%

         \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) - \frac{1}{0.5 \cdot x + 1} \]

   4. Applied egg-rr99.0%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{0.5 \cdot x + 1} \]
   5. Step-by-step derivation

      1. +-lft-identity99.0%

         \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{0.5 \cdot x + 1} \]

   6. Simplified99.0%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{0.5 \cdot x + 1} \]

   if 1.67999999999999994 < x

   1. Initial program 40.4%

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

      1. frac-sub40.5%

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

      2. div-inv40.5%

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

      3. *-un-lft-identity40.5%

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

      4. +-commutative40.5%

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

      5. *-rgt-identity40.5%

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

      6. metadata-eval40.5%

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

      7. frac-times40.5%

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

      8. un-div-inv40.5%

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

      9. pow1/240.5%

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

      10. pow-flip40.5%

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

      11. metadata-eval40.5%

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

      12. +-commutative40.5%

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

   3. Applied egg-rr40.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   4. Step-by-step derivation

      1. associate-*r/40.5%

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

      2. *-rgt-identity40.5%

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

      3. times-frac40.5%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      4. div-sub40.5%

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

      5. *-inverses40.5%

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

      6. /-rgt-identity40.5%

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

   5. Simplified40.5%

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

   6. Taylor expanded in x around inf 96.5%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*l/96.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-0.5}}{x}} \]

   8. Applied egg-rr96.6%

      \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-0.5}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.68:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot {x}^{-0.5}}{x}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.5%

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

      1. inv-pow99.5%

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

      2. pow1/299.5%

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

      3. pow-pow100.0%

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

      4. add-exp-log92.6%

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

      5. pow-exp92.6%

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

      6. metadata-eval92.6%

         \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr92.6%

      \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

   4. Taylor expanded in x around 0 98.9%

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

   if 1 < x

   1. Initial program 40.4%

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

      1. frac-sub40.5%

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

      2. div-inv40.5%

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

      3. *-un-lft-identity40.5%

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

      4. +-commutative40.5%

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

      5. *-rgt-identity40.5%

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

      6. metadata-eval40.5%

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

      7. frac-times40.5%

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

      8. un-div-inv40.5%

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

      9. pow1/240.5%

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

      10. pow-flip40.5%

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

      11. metadata-eval40.5%

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

      12. +-commutative40.5%

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

   3. Applied egg-rr40.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   4. Step-by-step derivation

      1. associate-*r/40.5%

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

      2. *-rgt-identity40.5%

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

      3. times-frac40.5%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      4. div-sub40.5%

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

      5. *-inverses40.5%

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

      6. /-rgt-identity40.5%

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

   5. Simplified40.5%

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

   6. Taylor expanded in x around inf 96.5%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*l/96.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-0.5}}{x}} \]

   8. Applied egg-rr96.6%

      \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-0.5}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot {x}^{-0.5}}{x}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   1. Initial program 99.5%

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

      1. inv-pow99.5%

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

      2. pow1/299.5%

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

      3. pow-pow100.0%

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

      4. add-exp-log92.6%

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

      5. pow-exp92.6%

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

      6. metadata-eval92.6%

         \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr92.6%

      \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{{x}^{-0.5} - 1} \]

   if 0.660000000000000031 < x

   1. Initial program 40.4%

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

      1. frac-sub40.5%

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

      2. div-inv40.5%

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

      3. *-un-lft-identity40.5%

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

      4. +-commutative40.5%

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

      5. *-rgt-identity40.5%

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

      6. metadata-eval40.5%

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

      7. frac-times40.5%

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

      8. un-div-inv40.5%

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

      9. pow1/240.5%

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

      10. pow-flip40.5%

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

      11. metadata-eval40.5%

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

      12. +-commutative40.5%

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

   3. Applied egg-rr40.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   4. Step-by-step derivation

      1. associate-*r/40.5%

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

      2. *-rgt-identity40.5%

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

      3. times-frac40.5%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      4. div-sub40.5%

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

      5. *-inverses40.5%

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

      6. /-rgt-identity40.5%

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

   5. Simplified40.5%

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

   6. Taylor expanded in x around inf 96.5%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \cdot {x}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} \cdot {x}^{-0.5}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   1. Initial program 99.5%

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

      1. inv-pow99.5%

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

      2. pow1/299.5%

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

      3. pow-pow100.0%

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

      4. add-exp-log92.6%

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

      5. pow-exp92.6%

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

      6. metadata-eval92.6%

         \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr92.6%

      \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{{x}^{-0.5} - 1} \]

   if 0.660000000000000031 < x

   1. Initial program 40.4%

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

      1. frac-sub40.5%

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

      2. div-inv40.5%

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

      3. *-un-lft-identity40.5%

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

      4. +-commutative40.5%

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

      5. *-rgt-identity40.5%

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

      6. metadata-eval40.5%

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

      7. frac-times40.5%

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

      8. un-div-inv40.5%

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

      9. pow1/240.5%

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

      10. pow-flip40.5%

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

      11. metadata-eval40.5%

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

      12. +-commutative40.5%

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

   3. Applied egg-rr40.5%

      \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
   4. Step-by-step derivation

      1. associate-*r/40.5%

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

      2. *-rgt-identity40.5%

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

      3. times-frac40.5%

         \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

      4. div-sub40.5%

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

      5. *-inverses40.5%

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

      6. /-rgt-identity40.5%

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

   5. Simplified40.5%

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

   6. Taylor expanded in x around inf 96.5%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \cdot {x}^{-0.5} \]
   7. Step-by-step derivation

      1. associate-*l/96.6%

         \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-0.5}}{x}} \]

   8. Applied egg-rr96.6%

      \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{-0.5}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot {x}^{-0.5}}{x}\\ \end{array} \]

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 99.5%

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

      1. inv-pow99.5%

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

      2. pow1/299.5%

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

      3. pow-pow100.0%

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

      4. add-exp-log92.6%

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

      5. pow-exp92.6%

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

      6. metadata-eval92.6%

         \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr92.6%

      \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{{x}^{-0.5} - 1} \]

   if 0.80000000000000004 < x

   1. Initial program 40.4%

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

      1. inv-pow40.4%

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

      2. pow1/240.4%

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

      3. pow-pow29.9%

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

      4. add-exp-log8.2%

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

      5. pow-exp8.3%

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

      6. metadata-eval8.3%

         \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr8.3%

      \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

   4. Taylor expanded in x around inf 5.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \]
   5. Step-by-step derivation

      1. pow1/25.5%

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

      2. inv-pow5.5%

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

      3. pow-pow5.5%

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

      4. metadata-eval5.5%

         \[\leadsto {x}^{\color{blue}{-0.5}} \]

      5. sqr-pow5.5%

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

      6. pow-prod-down37.2%

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

      7. metadata-eval37.2%

         \[\leadsto {\left(x \cdot x\right)}^{\color{blue}{-0.25}} \]

   6. Applied egg-rr37.2%

      \[\leadsto \color{blue}{{\left(x \cdot x\right)}^{-0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

1. Initial program 69.9%

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

   1. inv-pow69.9%

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

   2. pow1/269.9%

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

   3. pow-pow65.0%

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

   4. add-exp-log50.4%

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

   5. pow-exp50.4%

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

   6. metadata-eval50.4%

      \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

3. Applied egg-rr50.4%

   \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

4. Taylor expanded in x around inf 51.0%

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

5. Final simplification51.0%

   \[\leadsto \sqrt{\frac{1}{x}} \]

1. Initial program 69.9%

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

   1. frac-sub70.0%

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

   2. div-inv70.0%

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

   3. *-un-lft-identity70.0%

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

   4. +-commutative70.0%

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

   5. *-rgt-identity70.0%

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

   6. metadata-eval70.0%

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

   7. frac-times70.0%

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

   8. un-div-inv70.0%

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

   9. pow1/270.0%

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

   10. pow-flip70.2%

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

   11. metadata-eval70.2%

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

   12. +-commutative70.2%

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

3. Applied egg-rr70.2%

   \[\leadsto \color{blue}{\left(\sqrt{1 + x} - \sqrt{x}\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}} \]
4. Step-by-step derivation

   1. associate-*r/70.2%

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

   2. *-rgt-identity70.2%

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

   3. times-frac70.2%

      \[\leadsto \color{blue}{\frac{\sqrt{1 + x} - \sqrt{x}}{\sqrt{1 + x}} \cdot \frac{{x}^{-0.5}}{1}} \]

   4. div-sub70.3%

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

   5. *-inverses70.3%

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

   6. /-rgt-identity70.3%

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

5. Simplified70.3%

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

6. Taylor expanded in x around 0 51.1%

   \[\leadsto \color{blue}{1} \cdot {x}^{-0.5} \]

7. Final simplification51.1%

   \[\leadsto {x}^{-0.5} \]

1. Initial program 69.9%

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

   1. inv-pow69.9%

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

   2. pow1/269.9%

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

   3. pow-pow65.0%

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

   4. add-exp-log50.4%

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

   5. pow-exp50.4%

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

   6. metadata-eval50.4%

      \[\leadsto e^{\log x \cdot \color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

3. Applied egg-rr50.4%

   \[\leadsto \color{blue}{e^{\log x \cdot -0.5}} - \frac{1}{\sqrt{x + 1}} \]

4. Taylor expanded in x around 0 51.1%

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

5. Taylor expanded in x around inf 3.8%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

6. Final simplification3.8%

   \[\leadsto x \cdot 0.5 \]

herbie shell --seed 2023166 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))