Lanczos kernel

Percentage Accurate: 97.9% → 97.8%

Time: 11.0s

Alternatives: 11

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: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.1

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

6. Applied rewrites 97.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 97.6

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-/.f32 N/A

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

   11. clear-num N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*r/ N/A

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

8. Applied rewrites 97.7%

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

9. Final simplification 97.7%

   \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{\sin \left(x \cdot \left(tau \cdot \mathsf{PI}\left(\right)\right)\right)}{x \cdot \left(tau \cdot \mathsf{PI}\left(\right)\right)}}{x \cdot \mathsf{PI}\left(\right)} \]
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 97.7%

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

3. Final simplification 97.7%

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

Alternative 3: 97.6% 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 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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 97.5%

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

5. Final simplification 97.5%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-rgt-identity N/A

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

   6. clear-num N/A

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

   7. div-inv N/A

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

   8. *-rgt-identity N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 97.5%

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

8. Applied rewrites 97.4%

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

9. Final simplification 97.4%

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

Alternative 5: 70.6% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. *-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. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-PI.f32 N/A

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

   7. add-sqr-sqrt N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. frac-times N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 96.7%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 68.8

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

7. Applied rewrites 68.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

9. Applied rewrites 69.0%

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

10. Final simplification 69.0%

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

Alternative 6: 70.6% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. *-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. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-PI.f32 N/A

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

   7. add-sqr-sqrt N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. frac-times N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 96.7%

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

5. Taylor expanded in x around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-PI.f32 69.0

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

7. Applied rewrites 69.0%

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

8. Final simplification 69.0%

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

Alternative 7: 70.6% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. *-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. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-PI.f32 N/A

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

   7. add-sqr-sqrt N/A

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

   8. associate-/r* N/A

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

   9. lift-/.f32 N/A

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

   10. frac-times N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 96.7%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 68.8

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

7. Applied rewrites 68.8%

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

9. Applied rewrites 68.7%

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

    1. lift-/.f32 N/A

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

    2. lift-neg.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. distribute-neg-frac N/A

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

    5. associate-/l/ N/A

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

    6. lower-/.f32 N/A

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

    7. lower-neg.f32 N/A

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

    8. lower-*.f32 68.7

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

11. Applied rewrites 69.0%

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

12. Final simplification 69.0%

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

Alternative 8: 70.6% 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 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-rgt-identity N/A

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

   6. clear-num N/A

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

   7. div-inv N/A

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

   8. *-rgt-identity N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 97.5%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 68.9

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

9. Applied rewrites 68.9%

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

10. Final simplification 68.9%

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

Alternative 9: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(tau, tau, 1\right) \cdot \left({\left(x \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot -0.16666666666666666\right) + 1 \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+ (* (fma tau tau 1.0) (* (pow (* x (PI)) 2.0) -0.16666666666666666)) 1.0))

Derivation

1. Initial program 97.7%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 97.6

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.6

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 97.6

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 97.6

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

6. Applied rewrites 97.6%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]
8. 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

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

9. Applied rewrites 60.7%

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

11. Applied rewrites 37.4%

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

12. Final simplification 39.8%

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

Alternative 10: 63.9% accurate, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 61.8

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

5. Applied rewrites 61.8%

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

6. Final simplification 61.8%

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

Alternative 11: 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 97.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 61.1%

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

Reproduce

herbie shell --seed 2024283 
(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)))))