Lanczos kernel

Percentage Accurate: 97.9% → 97.7%

Time: 10.1s

Alternatives: 10

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ \frac{\sin t\_2}{t\_2} \cdot \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)))
   (* (/ (sin t_2) t_2) (/ (sin t_1) t_1))))

Initial Program: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ \frac{\sin t\_2}{t\_2} \cdot \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)))
   (* (/ (sin t_2) t_2) (/ (sin t_1) t_1))))

Alternative 1: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ \frac{\sin t\_2}{\frac{t\_1}{\sin t\_1} \cdot t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)))
   (/ (sin t_2) (* (/ t_1 (sin t_1)) t_2))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   14. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   15. lift-/.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

6. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   6. frac-times N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

   7. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)} \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

8. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}} \]

9. Final simplification 98.0%

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)} \]
10. Add Preprocessing

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin t\_2}{t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)))
   (* (/ (sin t_1) t_1) (/ (sin t_2) t_2))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   14. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   15. lift-/.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

6. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

7. Final simplification 98.0%

   \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]
8. Add Preprocessing

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ \frac{\sin t\_2}{t\_1} \cdot \frac{\sin t\_1}{t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)))
   (* (/ (sin t_2) t_1) (/ (sin t_1) t_2))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   4. frac-times N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   7. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{1 \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   8. associate-*l/ N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   9. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x}} \]

5. Final simplification 97.9%

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{x \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]
6. Add Preprocessing

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ \frac{\sin t\_1 \cdot \sin t\_2}{t\_2 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)))
   (/ (* (sin t_1) (sin t_2)) (* t_2 t_1))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   14. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   15. lift-/.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

6. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   6. frac-times N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

   7. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)} \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

8. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}} \]

   2. *-rgt-identity N/A

      \[\leadsto \frac{\color{blue}{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot 1}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot 1}{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot 1}{\color{blue}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot 1}{\color{blue}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)} \cdot \frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot 1}{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau\right)} \cdot \frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot 1}{\left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot tau\right) \cdot \frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   8. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot 1}{\left(\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau\right) \cdot \frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot 1}{\left(\left(x \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot tau\right) \cdot \frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   10. frac-times N/A

       \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   11. lift-/.f32 N/A

       \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

10. Applied rewrites 97.8%

    \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \left(-\sin \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right)}} \]

11. Final simplification 97.8%

    \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)} \]
12. Add Preprocessing

Alternative 5: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ \frac{\sin t\_1 \cdot \sin \left(t\_1 \cdot tau\right)}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x\right) \cdot tau\right) \cdot x} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))))
   (/ (* (sin t_1) (sin (* t_1 tau))) (* (* (* (* (PI) (PI)) x) tau) x))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   14. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   15. lift-/.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

6. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   6. frac-times N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

   7. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)} \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

8. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}} \]

10. Applied rewrites 96.9%

    \[\leadsto \color{blue}{\frac{\left(-\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)\right) \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(-x\right) \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x\right) \cdot tau\right)}} \]

11. Final simplification 96.9%

    \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x\right) \cdot tau\right) \cdot x} \]
12. Add Preprocessing

Alternative 6: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ \frac{\sin t\_1 \cdot \sin \left(t\_1 \cdot tau\right)}{\left(x \cdot x\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))))
   (/ (* (sin t_1) (sin (* t_1 tau))) (* (* x x) (* (* (PI) tau) (PI))))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   14. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   15. lift-/.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

6. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

8. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\left(tau \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right)}} \]

9. Final simplification 96.8%

   \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot x\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right)} \]
10. Add Preprocessing

Alternative 7: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ \left(1 + {t\_1}^{2} \cdot -0.16666666666666666\right) \cdot \frac{\sin \left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right)}{t\_1 \cdot tau} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))))
   (*
    (+ 1.0 (* (pow t_1 2.0) -0.16666666666666666))
    (/ (sin (* (* x tau) (PI))) (* t_1 tau)))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lower-*.f32 97.5

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

4. Applied rewrites 97.5%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot {x}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   5. associate-*r* N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot x}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot x}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{6} \cdot x\right)} \cdot x, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\left(\frac{-1}{6} \cdot x\right) \cdot x, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\left(\frac{-1}{6} \cdot x\right) \cdot x, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \]

   10. lower-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\left(\frac{-1}{6} \cdot x\right) \cdot x, \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \]

   11. lower-PI.f32 32.9

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left(\left(-0.16666666666666666 \cdot x\right) \cdot x, \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

7. Applied rewrites 32.9%

   \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot x\right) \cdot x, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)} \]
8. Step-by-step derivation

9. Applied rewrites 86.1%

   \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \left(-0.16666666666666666 \cdot {\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2} + \color{blue}{1}\right) \]

10. Final simplification 86.1%

    \[\leadsto \left(1 + {\left(x \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot -0.16666666666666666\right) \cdot \frac{\sin \left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]
11. Add Preprocessing

Alternative 8: 70.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x (PI)) tau))) (/ (sin t_1) t_1)))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   3. clear-num N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\sin \left(tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   14. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   15. lift-/.f32 N/A

       \[\leadsto \frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

6. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   4. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   6. frac-times N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

   7. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)} \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

8. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

    3. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

    4. lower-PI.f32 72.2

       \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right)} \]

11. Applied rewrites 72.2%

    \[\leadsto \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\color{blue}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

12. Final simplification 72.2%

    \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]
13. Add Preprocessing

Alternative 9: 64.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(-0.16666666666666666 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right) + 1 \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+ (* (* (* -0.16666666666666666 (PI)) (PI)) (* x x)) 1.0))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}} + 1 \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}, {x}^{2}, 1\right)} \]

5. Applied rewrites 65.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666\right), x \cdot x, 1\right)} \]

6. Taylor expanded in tau around 0

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

8. Applied rewrites 64.9%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666, \color{blue}{x} \cdot x, 1\right) \]
9. Step-by-step derivation

10. Applied rewrites 66.8%

    \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-0.16666666666666666 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{1} \]

11. Final simplification 66.8%

    \[\leadsto \left(\left(-0.16666666666666666 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right) + 1 \]
12. Add Preprocessing

Alternative 10: 63.1% accurate, 258.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

float code(float x, float tau) {
	return 1.0f;
}

Fortran

real(4) function code(x, tau)
    real(4), intent (in) :: x
    real(4), intent (in) :: tau
    code = 1.0e0
end function

Julia

function code(x, tau)
	return Float32(1.0)
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0);
end

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 65.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (x tau)
  :name "Lanczos kernel"
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0)) (and (<= 1.0 tau) (<= tau 5.0)))
  (* (/ (sin (* (* x (PI)) tau)) (* (* x (PI)) tau)) (/ (sin (* x (PI))) (* x (PI)))))