x - floor((x+3.14)/3.14)*3.14

if x < -1.999999999999994e-310

1. Initial program 7.1%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   2. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   3. lift-+.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   4. flip-+N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}}{x - \frac{7070651414971679}{2251799813685248}}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   5. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{x - \frac{7070651414971679}{2251799813685248}}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   6. associate-*l*N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   7. lower-*.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   8. sub-negN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)\right)} \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   9. lower-fma.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   10. metadata-evalN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{49994111432041006387552372079041}{5070602400912917605986812821504}}\right)\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   11. metadata-evalN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   12. lower-*.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \color{blue}{\left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   13. inv-powN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left(\color{blue}{{\left(x - \frac{7070651414971679}{2251799813685248}\right)}^{-1}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   14. lower-pow.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left(\color{blue}{{\left(x - \frac{7070651414971679}{2251799813685248}\right)}^{-1}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   15. lower--.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left({\color{blue}{\left(x - \frac{7070651414971679}{2251799813685248}\right)}}^{-1} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   16. metadata-eval56.3

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, -9.8596\right) \cdot \left({\left(x - 3.14\right)}^{-1} \cdot \color{blue}{0.3184713375796178}\right)\right\rfloor \cdot 3.14 \]

4. Applied rewrites56.3%

   \[\leadsto x - \left\lfloor \color{blue}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot \left({\left(x - 3.14\right)}^{-1} \cdot 0.3184713375796178\right)}\right\rfloor \cdot 3.14 \]

if -1.999999999999994e-310 < x

1. Initial program 47.4%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   2. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   3. lift-+.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   4. flip-+N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}}{x - \frac{7070651414971679}{2251799813685248}}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   5. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{x - \frac{7070651414971679}{2251799813685248}}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   6. associate-*l*N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   7. lower-*.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   8. sub-negN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)\right)} \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   9. lower-fma.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   10. metadata-evalN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{49994111432041006387552372079041}{5070602400912917605986812821504}}\right)\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   11. metadata-evalN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   12. lower-*.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \color{blue}{\left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   13. inv-powN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left(\color{blue}{{\left(x - \frac{7070651414971679}{2251799813685248}\right)}^{-1}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   14. lower-pow.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left(\color{blue}{{\left(x - \frac{7070651414971679}{2251799813685248}\right)}^{-1}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   15. lower--.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left({\color{blue}{\left(x - \frac{7070651414971679}{2251799813685248}\right)}}^{-1} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   16. metadata-eval7.4

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, -9.8596\right) \cdot \left({\left(x - 3.14\right)}^{-1} \cdot \color{blue}{0.3184713375796178}\right)\right\rfloor \cdot 3.14 \]

4. Applied rewrites7.4%

   \[\leadsto x - \left\lfloor \color{blue}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot \left({\left(x - 3.14\right)}^{-1} \cdot 0.3184713375796178\right)}\right\rfloor \cdot 3.14 \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x - \frac{7070651414971679}{2251799813685248} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor } \]
6. Step-by-step derivation

   1. cancel-sign-sub-invN/A

      \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right)\right) \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor } \]

   2. metadata-evalN/A

      \[\leadsto x + \color{blue}{\frac{-7070651414971679}{2251799813685248}} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor \]

   3. +-commutativeN/A

      \[\leadsto \color{blue}{\frac{-7070651414971679}{2251799813685248} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor + x} \]

   4. *-commutativeN/A

      \[\leadsto \color{blue}{\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{-7070651414971679}{2251799813685248}} + x \]

   5. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right)} \]

   6. lower-floor.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor }, \frac{-7070651414971679}{2251799813685248}, x\right) \]

   7. associate-*r/N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{\frac{2251799813685248}{7070651414971679} \cdot \left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   8. lower-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{\frac{2251799813685248}{7070651414971679} \cdot \left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   9. *-commutativeN/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \frac{2251799813685248}{7070651414971679}}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   10. lower-*.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \frac{2251799813685248}{7070651414971679}}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   11. sub-negN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} + \left(\mathsf{neg}\left(\frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)\right)\right)} \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   12. unpow2N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(\frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)\right)\right) \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   13. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\left(x \cdot x + \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   14. lower-fma.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right)} \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   15. lower--.f6448.3

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}{\color{blue}{x - 3.14}}\right\rfloor , -3.14, x\right) \]

7. Applied rewrites48.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}{x - 3.14}\right\rfloor , -3.14, x\right)} \]

if x < -2.7e-77

1. Initial program 9.7%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64N/A

      \[\leadsto \color{blue}{x - \left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}} \]

   2. sub-negN/A

      \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \]

   3. +-commutativeN/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}\right)\right) + x} \]

   4. lift-*.f64N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}}\right)\right) + x \]

   5. distribute-rgt-neg-inN/A

      \[\leadsto \color{blue}{\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right)\right)} + x \]

   6. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right)} \]

   7. lift-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   8. clear-numN/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{1}{\frac{\frac{7070651414971679}{2251799813685248}}{x + \frac{7070651414971679}{2251799813685248}}}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   9. associate-/r/N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot \left(x + \frac{7070651414971679}{2251799813685248}\right)}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   10. lift-+.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   11. distribute-lft-inN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot x + \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot \frac{7070651414971679}{2251799813685248}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   12. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot x + \color{blue}{\frac{2251799813685248}{7070651414971679}} \cdot \frac{7070651414971679}{2251799813685248}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   13. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot x + \color{blue}{1}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   14. lower-fma.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\mathsf{fma}\left(\frac{1}{\frac{7070651414971679}{2251799813685248}}, x, 1\right)}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   15. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \mathsf{fma}\left(\color{blue}{\frac{2251799813685248}{7070651414971679}}, x, 1\right)\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   16. metadata-eval10.9

       \[\leadsto \mathsf{fma}\left(\left\lfloor \mathsf{fma}\left(0.3184713375796178, x, 1\right)\right\rfloor , \color{blue}{-3.14}, x\right) \]

4. Applied rewrites10.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \mathsf{fma}\left(0.3184713375796178, x, 1\right)\right\rfloor , -3.14, x\right)} \]

if -2.7e-77 < x < -1.999999999999994e-310

1. Initial program 4.3%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   2. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   3. lift-+.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   4. flip-+N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}}{x - \frac{7070651414971679}{2251799813685248}}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   5. associate-*l/N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   6. clear-numN/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   7. lower-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   8. lower-/.f64N/A

      \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   9. lower--.f64N/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\color{blue}{x - \frac{7070651414971679}{2251799813685248}}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   10. lower-*.f64N/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   11. sub-negN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\color{blue}{\left(x \cdot x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   12. lower-fma.f64N/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   13. metadata-evalN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{49994111432041006387552372079041}{5070602400912917605986812821504}}\right)\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   14. metadata-evalN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\mathsf{fma}\left(x, x, \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   15. metadata-eval4.3

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - 3.14}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot \color{blue}{0.3184713375796178}}}\right\rfloor \cdot 3.14 \]

4. Applied rewrites4.3%

   \[\leadsto x - \left\lfloor \color{blue}{\frac{1}{\frac{x - 3.14}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}}}\right\rfloor \cdot 3.14 \]

5. Taylor expanded in x around -inf

   \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{2499411177879333246441435816434822626729211414583438076751479681}{25711008708143844408671393477458601640355247900524685364822016} \cdot \frac{1}{x} - \frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992}}{x} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x} - \frac{7070651414971679}{2251799813685248}}{x}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

7. Applied rewrites100.0%

   \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{\frac{\frac{-9.8596 - \frac{\frac{97.21171216000002}{x} - 30.959144000000002}{x}}{x} + 3.14}{x}}}\right\rfloor \cdot 3.14 \]

if -1.999999999999994e-310 < x

1. Initial program 47.4%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   2. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   3. lift-+.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   4. flip-+N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}}{x - \frac{7070651414971679}{2251799813685248}}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   5. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{x - \frac{7070651414971679}{2251799813685248}}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   6. associate-*l*N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   7. lower-*.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   8. sub-negN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)\right)} \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   9. lower-fma.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   10. metadata-evalN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{49994111432041006387552372079041}{5070602400912917605986812821504}}\right)\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   11. metadata-evalN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   12. lower-*.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \color{blue}{\left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   13. inv-powN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left(\color{blue}{{\left(x - \frac{7070651414971679}{2251799813685248}\right)}^{-1}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   14. lower-pow.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left(\color{blue}{{\left(x - \frac{7070651414971679}{2251799813685248}\right)}^{-1}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   15. lower--.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left({\color{blue}{\left(x - \frac{7070651414971679}{2251799813685248}\right)}}^{-1} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   16. metadata-eval7.4

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, -9.8596\right) \cdot \left({\left(x - 3.14\right)}^{-1} \cdot \color{blue}{0.3184713375796178}\right)\right\rfloor \cdot 3.14 \]

4. Applied rewrites7.4%

   \[\leadsto x - \left\lfloor \color{blue}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot \left({\left(x - 3.14\right)}^{-1} \cdot 0.3184713375796178\right)}\right\rfloor \cdot 3.14 \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x - \frac{7070651414971679}{2251799813685248} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor } \]
6. Step-by-step derivation

   1. cancel-sign-sub-invN/A

      \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right)\right) \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor } \]

   2. metadata-evalN/A

      \[\leadsto x + \color{blue}{\frac{-7070651414971679}{2251799813685248}} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor \]

   3. +-commutativeN/A

      \[\leadsto \color{blue}{\frac{-7070651414971679}{2251799813685248} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor + x} \]

   4. *-commutativeN/A

      \[\leadsto \color{blue}{\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{-7070651414971679}{2251799813685248}} + x \]

   5. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right)} \]

   6. lower-floor.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor }, \frac{-7070651414971679}{2251799813685248}, x\right) \]

   7. associate-*r/N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{\frac{2251799813685248}{7070651414971679} \cdot \left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   8. lower-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{\frac{2251799813685248}{7070651414971679} \cdot \left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   9. *-commutativeN/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \frac{2251799813685248}{7070651414971679}}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   10. lower-*.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \frac{2251799813685248}{7070651414971679}}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   11. sub-negN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} + \left(\mathsf{neg}\left(\frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)\right)\right)} \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   12. unpow2N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(\frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)\right)\right) \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   13. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\left(x \cdot x + \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   14. lower-fma.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right)} \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   15. lower--.f6448.3

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}{\color{blue}{x - 3.14}}\right\rfloor , -3.14, x\right) \]

7. Applied rewrites48.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}{x - 3.14}\right\rfloor , -3.14, x\right)} \]

Final simplification50.6%

if x < -5.50000000000000032e-103

1. Initial program 9.4%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64N/A

      \[\leadsto \color{blue}{x - \left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}} \]

   2. sub-negN/A

      \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \]

   3. +-commutativeN/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}\right)\right) + x} \]

   4. lift-*.f64N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}}\right)\right) + x \]

   5. distribute-rgt-neg-inN/A

      \[\leadsto \color{blue}{\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right)\right)} + x \]

   6. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right)} \]

   7. lift-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   8. clear-numN/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{1}{\frac{\frac{7070651414971679}{2251799813685248}}{x + \frac{7070651414971679}{2251799813685248}}}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   9. associate-/r/N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot \left(x + \frac{7070651414971679}{2251799813685248}\right)}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   10. lift-+.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   11. distribute-lft-inN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot x + \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot \frac{7070651414971679}{2251799813685248}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   12. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot x + \color{blue}{\frac{2251799813685248}{7070651414971679}} \cdot \frac{7070651414971679}{2251799813685248}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   13. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot x + \color{blue}{1}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   14. lower-fma.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\mathsf{fma}\left(\frac{1}{\frac{7070651414971679}{2251799813685248}}, x, 1\right)}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   15. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \mathsf{fma}\left(\color{blue}{\frac{2251799813685248}{7070651414971679}}, x, 1\right)\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   16. metadata-eval10.5

       \[\leadsto \mathsf{fma}\left(\left\lfloor \mathsf{fma}\left(0.3184713375796178, x, 1\right)\right\rfloor , \color{blue}{-3.14}, x\right) \]

4. Applied rewrites10.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \mathsf{fma}\left(0.3184713375796178, x, 1\right)\right\rfloor , -3.14, x\right)} \]

if -5.50000000000000032e-103 < x < -1.999999999999994e-310

1. Initial program 4.1%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   2. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   3. lift-+.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   4. flip-+N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}}{x - \frac{7070651414971679}{2251799813685248}}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   5. associate-*l/N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   6. clear-numN/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   7. lower-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   8. lower-/.f64N/A

      \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   9. lower--.f64N/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\color{blue}{x - \frac{7070651414971679}{2251799813685248}}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   10. lower-*.f64N/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   11. sub-negN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\color{blue}{\left(x \cdot x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   12. lower-fma.f64N/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   13. metadata-evalN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{49994111432041006387552372079041}{5070602400912917605986812821504}}\right)\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   14. metadata-evalN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\mathsf{fma}\left(x, x, \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   15. metadata-eval4.1

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - 3.14}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot \color{blue}{0.3184713375796178}}}\right\rfloor \cdot 3.14 \]

4. Applied rewrites4.1%

   \[\leadsto x - \left\lfloor \color{blue}{\frac{1}{\frac{x - 3.14}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}}}\right\rfloor \cdot 3.14 \]

5. Taylor expanded in x around inf

   \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{\frac{\left(\frac{7070651414971679}{2251799813685248} + \frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992}}{{x}^{2}}\right) - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}}{x}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]
6. Step-by-step derivation

   1. lower-/.f64N/A

      \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{\frac{\left(\frac{7070651414971679}{2251799813685248} + \frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992}}{{x}^{2}}\right) - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}}{x}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   2. associate--l+N/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\color{blue}{\frac{7070651414971679}{2251799813685248} + \left(\frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992}}{{x}^{2}} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}\right)}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   3. +-commutativeN/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\color{blue}{\left(\frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992}}{{x}^{2}} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}\right) + \frac{7070651414971679}{2251799813685248}}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   4. unpow2N/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\left(\frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992}}{\color{blue}{x \cdot x}} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}\right) + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   5. associate-/r*N/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\left(\color{blue}{\frac{\frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992}}{x}}{x}} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}\right) + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   6. metadata-evalN/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\left(\frac{\frac{\color{blue}{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992} \cdot 1}}{x}}{x} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}\right) + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   7. associate-*r/N/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\left(\frac{\color{blue}{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992} \cdot \frac{1}{x}}}{x} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}\right) + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   8. associate-*r/N/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\left(\frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992} \cdot \frac{1}{x}}{x} - \color{blue}{\frac{\frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot 1}{x}}\right) + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   9. metadata-evalN/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\left(\frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992} \cdot \frac{1}{x}}{x} - \frac{\color{blue}{\frac{49994111432041006387552372079041}{5070602400912917605986812821504}}}{x}\right) + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   10. div-subN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{\color{blue}{\frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992} \cdot \frac{1}{x} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x}} + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   11. lower-+.f64N/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{\color{blue}{\frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992} \cdot \frac{1}{x} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x} + \frac{7070651414971679}{2251799813685248}}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   12. lower-/.f64N/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{\color{blue}{\frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992} \cdot \frac{1}{x} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x}} + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   13. lower--.f64N/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{\frac{\color{blue}{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992} \cdot \frac{1}{x} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}}{x} + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   14. associate-*r/N/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{\frac{\color{blue}{\frac{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992} \cdot 1}{x}} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x} + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   15. metadata-evalN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{\frac{\frac{\color{blue}{\frac{353490934737212534922304384685375867763364479839}{11417981541647679048466287755595961091061972992}}}{x} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x} + \frac{7070651414971679}{2251799813685248}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   16. lower-/.f64100.0

       \[\leadsto x - \left\lfloor \frac{1}{\frac{\frac{\color{blue}{\frac{30.959144000000002}{x}} - 9.8596}{x} + 3.14}{x}}\right\rfloor \cdot 3.14 \]

7. Applied rewrites100.0%

   \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{\frac{\frac{\frac{30.959144000000002}{x} - 9.8596}{x} + 3.14}{x}}}\right\rfloor \cdot 3.14 \]

if -1.999999999999994e-310 < x

1. Initial program 47.4%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   2. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   3. lift-+.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   4. flip-+N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}}{x - \frac{7070651414971679}{2251799813685248}}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   5. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{x - \frac{7070651414971679}{2251799813685248}}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   6. associate-*l*N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   7. lower-*.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   8. sub-negN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)\right)} \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   9. lower-fma.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   10. metadata-evalN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{49994111432041006387552372079041}{5070602400912917605986812821504}}\right)\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   11. metadata-evalN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   12. lower-*.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \color{blue}{\left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   13. inv-powN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left(\color{blue}{{\left(x - \frac{7070651414971679}{2251799813685248}\right)}^{-1}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   14. lower-pow.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left(\color{blue}{{\left(x - \frac{7070651414971679}{2251799813685248}\right)}^{-1}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   15. lower--.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left({\color{blue}{\left(x - \frac{7070651414971679}{2251799813685248}\right)}}^{-1} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   16. metadata-eval7.4

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, -9.8596\right) \cdot \left({\left(x - 3.14\right)}^{-1} \cdot \color{blue}{0.3184713375796178}\right)\right\rfloor \cdot 3.14 \]

4. Applied rewrites7.4%

   \[\leadsto x - \left\lfloor \color{blue}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot \left({\left(x - 3.14\right)}^{-1} \cdot 0.3184713375796178\right)}\right\rfloor \cdot 3.14 \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x - \frac{7070651414971679}{2251799813685248} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor } \]
6. Step-by-step derivation

   1. cancel-sign-sub-invN/A

      \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right)\right) \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor } \]

   2. metadata-evalN/A

      \[\leadsto x + \color{blue}{\frac{-7070651414971679}{2251799813685248}} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor \]

   3. +-commutativeN/A

      \[\leadsto \color{blue}{\frac{-7070651414971679}{2251799813685248} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor + x} \]

   4. *-commutativeN/A

      \[\leadsto \color{blue}{\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{-7070651414971679}{2251799813685248}} + x \]

   5. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right)} \]

   6. lower-floor.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor }, \frac{-7070651414971679}{2251799813685248}, x\right) \]

   7. associate-*r/N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{\frac{2251799813685248}{7070651414971679} \cdot \left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   8. lower-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{\frac{2251799813685248}{7070651414971679} \cdot \left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   9. *-commutativeN/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \frac{2251799813685248}{7070651414971679}}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   10. lower-*.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \frac{2251799813685248}{7070651414971679}}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   11. sub-negN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} + \left(\mathsf{neg}\left(\frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)\right)\right)} \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   12. unpow2N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(\frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)\right)\right) \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   13. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\left(x \cdot x + \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   14. lower-fma.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right)} \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   15. lower--.f6448.3

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}{\color{blue}{x - 3.14}}\right\rfloor , -3.14, x\right) \]

7. Applied rewrites48.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}{x - 3.14}\right\rfloor , -3.14, x\right)} \]

Final simplification48.4%

if x < -2.39999999999999987e-154

1. Initial program 8.8%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64N/A

      \[\leadsto \color{blue}{x - \left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}} \]

   2. sub-negN/A

      \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \]

   3. +-commutativeN/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}\right)\right) + x} \]

   4. lift-*.f64N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248}}\right)\right) + x \]

   5. distribute-rgt-neg-inN/A

      \[\leadsto \color{blue}{\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right)\right)} + x \]

   6. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right)} \]

   7. lift-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   8. clear-numN/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{1}{\frac{\frac{7070651414971679}{2251799813685248}}{x + \frac{7070651414971679}{2251799813685248}}}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   9. associate-/r/N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot \left(x + \frac{7070651414971679}{2251799813685248}\right)}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   10. lift-+.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   11. distribute-lft-inN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot x + \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot \frac{7070651414971679}{2251799813685248}}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   12. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot x + \color{blue}{\frac{2251799813685248}{7070651414971679}} \cdot \frac{7070651414971679}{2251799813685248}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   13. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{1}{\frac{7070651414971679}{2251799813685248}} \cdot x + \color{blue}{1}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   14. lower-fma.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\mathsf{fma}\left(\frac{1}{\frac{7070651414971679}{2251799813685248}}, x, 1\right)}\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   15. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \mathsf{fma}\left(\color{blue}{\frac{2251799813685248}{7070651414971679}}, x, 1\right)\right\rfloor , \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right), x\right) \]

   16. metadata-eval9.8

       \[\leadsto \mathsf{fma}\left(\left\lfloor \mathsf{fma}\left(0.3184713375796178, x, 1\right)\right\rfloor , \color{blue}{-3.14}, x\right) \]

4. Applied rewrites9.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \mathsf{fma}\left(0.3184713375796178, x, 1\right)\right\rfloor , -3.14, x\right)} \]

if -2.39999999999999987e-154 < x < -1.999999999999994e-310

1. Initial program 3.9%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   2. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   3. lift-+.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   4. flip-+N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}}{x - \frac{7070651414971679}{2251799813685248}}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   5. associate-*l/N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   6. clear-numN/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   7. lower-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   8. lower-/.f64N/A

      \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   9. lower--.f64N/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\color{blue}{x - \frac{7070651414971679}{2251799813685248}}}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   10. lower-*.f64N/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   11. sub-negN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\color{blue}{\left(x \cdot x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   12. lower-fma.f64N/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   13. metadata-evalN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{49994111432041006387552372079041}{5070602400912917605986812821504}}\right)\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   14. metadata-evalN/A

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - \frac{7070651414971679}{2251799813685248}}{\mathsf{fma}\left(x, x, \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   15. metadata-eval3.9

       \[\leadsto x - \left\lfloor \frac{1}{\frac{x - 3.14}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot \color{blue}{0.3184713375796178}}}\right\rfloor \cdot 3.14 \]

4. Applied rewrites3.9%

   \[\leadsto x - \left\lfloor \color{blue}{\frac{1}{\frac{x - 3.14}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}}}\right\rfloor \cdot 3.14 \]

5. Taylor expanded in x around inf

   \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{\frac{\frac{7070651414971679}{2251799813685248} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}}{x}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]
6. Step-by-step derivation

   1. lower-/.f64N/A

      \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{\frac{\frac{7070651414971679}{2251799813685248} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}}{x}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   2. lower--.f64N/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\color{blue}{\frac{7070651414971679}{2251799813685248} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot \frac{1}{x}}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   3. associate-*r/N/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\frac{7070651414971679}{2251799813685248} - \color{blue}{\frac{\frac{49994111432041006387552372079041}{5070602400912917605986812821504} \cdot 1}{x}}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   4. metadata-evalN/A

      \[\leadsto x - \left\lfloor \frac{1}{\frac{\frac{7070651414971679}{2251799813685248} - \frac{\color{blue}{\frac{49994111432041006387552372079041}{5070602400912917605986812821504}}}{x}}{x}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   5. lower-/.f64100.0

      \[\leadsto x - \left\lfloor \frac{1}{\frac{3.14 - \color{blue}{\frac{9.8596}{x}}}{x}}\right\rfloor \cdot 3.14 \]

7. Applied rewrites100.0%

   \[\leadsto x - \left\lfloor \frac{1}{\color{blue}{\frac{3.14 - \frac{9.8596}{x}}{x}}}\right\rfloor \cdot 3.14 \]

if -1.999999999999994e-310 < x

1. Initial program 47.4%

   \[x - \left\lfloor \frac{x + 3.14}{3.14}\right\rfloor \cdot 3.14 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x + \frac{7070651414971679}{2251799813685248}}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   2. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   3. lift-+.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x + \frac{7070651414971679}{2251799813685248}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   4. flip-+N/A

      \[\leadsto x - \left\lfloor \color{blue}{\frac{x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}}{x - \frac{7070651414971679}{2251799813685248}}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   5. div-invN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \frac{1}{x - \frac{7070651414971679}{2251799813685248}}\right)} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   6. associate-*l*N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   7. lower-*.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x - \frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   8. sub-negN/A

      \[\leadsto x - \left\lfloor \color{blue}{\left(x \cdot x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)\right)} \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   9. lower-fma.f64N/A

      \[\leadsto x - \left\lfloor \color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(\frac{7070651414971679}{2251799813685248} \cdot \frac{7070651414971679}{2251799813685248}\right)\right)} \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   10. metadata-evalN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \mathsf{neg}\left(\color{blue}{\frac{49994111432041006387552372079041}{5070602400912917605986812821504}}\right)\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   11. metadata-evalN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   12. lower-*.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \color{blue}{\left(\frac{1}{x - \frac{7070651414971679}{2251799813685248}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)}\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   13. inv-powN/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left(\color{blue}{{\left(x - \frac{7070651414971679}{2251799813685248}\right)}^{-1}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   14. lower-pow.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left(\color{blue}{{\left(x - \frac{7070651414971679}{2251799813685248}\right)}^{-1}} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   15. lower--.f64N/A

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \left({\color{blue}{\left(x - \frac{7070651414971679}{2251799813685248}\right)}}^{-1} \cdot \frac{1}{\frac{7070651414971679}{2251799813685248}}\right)\right\rfloor \cdot \frac{7070651414971679}{2251799813685248} \]

   16. metadata-eval7.4

       \[\leadsto x - \left\lfloor \mathsf{fma}\left(x, x, -9.8596\right) \cdot \left({\left(x - 3.14\right)}^{-1} \cdot \color{blue}{0.3184713375796178}\right)\right\rfloor \cdot 3.14 \]

4. Applied rewrites7.4%

   \[\leadsto x - \left\lfloor \color{blue}{\mathsf{fma}\left(x, x, -9.8596\right) \cdot \left({\left(x - 3.14\right)}^{-1} \cdot 0.3184713375796178\right)}\right\rfloor \cdot 3.14 \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x - \frac{7070651414971679}{2251799813685248} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor } \]
6. Step-by-step derivation

   1. cancel-sign-sub-invN/A

      \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{7070651414971679}{2251799813685248}\right)\right) \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor } \]

   2. metadata-evalN/A

      \[\leadsto x + \color{blue}{\frac{-7070651414971679}{2251799813685248}} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor \]

   3. +-commutativeN/A

      \[\leadsto \color{blue}{\frac{-7070651414971679}{2251799813685248} \cdot \left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor + x} \]

   4. *-commutativeN/A

      \[\leadsto \color{blue}{\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor \cdot \frac{-7070651414971679}{2251799813685248}} + x \]

   5. lower-fma.f64N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right)} \]

   6. lower-floor.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left\lfloor \frac{2251799813685248}{7070651414971679} \cdot \frac{{x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor }, \frac{-7070651414971679}{2251799813685248}, x\right) \]

   7. associate-*r/N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{\frac{2251799813685248}{7070651414971679} \cdot \left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   8. lower-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \color{blue}{\frac{\frac{2251799813685248}{7070651414971679} \cdot \left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)}{x - \frac{7070651414971679}{2251799813685248}}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   9. *-commutativeN/A

      \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \frac{2251799813685248}{7070651414971679}}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   10. lower-*.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} - \frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right) \cdot \frac{2251799813685248}{7070651414971679}}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   11. sub-negN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\left({x}^{2} + \left(\mathsf{neg}\left(\frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)\right)\right)} \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   12. unpow2N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(\frac{49994111432041006387552372079041}{5070602400912917605986812821504}\right)\right)\right) \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   13. metadata-evalN/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\left(x \cdot x + \color{blue}{\frac{-49994111432041006387552372079041}{5070602400912917605986812821504}}\right) \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   14. lower-fma.f64N/A

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\color{blue}{\mathsf{fma}\left(x, x, \frac{-49994111432041006387552372079041}{5070602400912917605986812821504}\right)} \cdot \frac{2251799813685248}{7070651414971679}}{x - \frac{7070651414971679}{2251799813685248}}\right\rfloor , \frac{-7070651414971679}{2251799813685248}, x\right) \]

   15. lower--.f6448.3

       \[\leadsto \mathsf{fma}\left(\left\lfloor \frac{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}{\color{blue}{x - 3.14}}\right\rfloor , -3.14, x\right) \]

7. Applied rewrites48.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left\lfloor \frac{\mathsf{fma}\left(x, x, -9.8596\right) \cdot 0.3184713375796178}{x - 3.14}\right\rfloor , -3.14, x\right)} \]

Final simplification44.7%

Initial program 27.4%

Applied rewrites28.1%

Initial program 27.4%

Taylor expanded in x around 0

Applied rewrites25.0%

Applied rewrites25.0%