Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 9.8s

Alternatives: 10

Speedup: N/A×

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.2

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

4. Applied rewrites 97.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 97.9

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

6. Applied rewrites 97.9%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

Alternative 3: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

6. Applied rewrites 97.8%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

9. Applied rewrites 97.1%

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

Alternative 4: 70.6% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

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

   6. lift-/.f32 N/A

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

   7. clear-num N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 96.7%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 69.9

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

7. Applied rewrites 69.9%

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

9. Applied rewrites 70.2%

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

Alternative 5: 70.6% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

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

   6. lift-/.f32 N/A

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

   7. clear-num N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 96.7%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 69.9

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

7. Applied rewrites 69.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

9. Applied rewrites 70.2%

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

Alternative 6: 70.7% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

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

   6. lift-/.f32 N/A

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

   7. clear-num N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 96.7%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 69.9

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

7. Applied rewrites 69.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

9. Applied rewrites 70.2%

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

Alternative 7: 70.6% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.9%

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

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

   6. lift-/.f32 N/A

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

   7. clear-num N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 96.7%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 69.9

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

7. Applied rewrites 69.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f32 N/A

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

9. Applied rewrites 70.2%

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

Alternative 8: 70.6% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x))) (* (/ 1.0 tau) (/ (sin (* tau t_1)) t_1))))

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. Taylor expanded in x around 0

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

   1. lower-/.f32 70.2

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

7. Applied rewrites 70.2%

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

Alternative 9: 64.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \end{array} \]

FPCore

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

Derivation

1. Initial program 97.9%

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

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

6. Applied rewrites 96.9%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 63.9

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

9. Applied rewrites 63.9%

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

Alternative 10: 63.2% 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 97.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 63.2%

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

Reproduce

herbie shell --seed 2024326 
(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)))))