Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 10.2s

Alternatives: 8

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.2

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

4. Applied rewrites 97.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 97.8

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 97.8

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

6. Applied rewrites 97.8%

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

7. Final simplification 97.8%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

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))))

Derivation

1. Initial program 97.8%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

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 97.8%

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

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.2

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

4. Applied rewrites 97.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 97.8

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 97.8

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

6. Applied rewrites 97.8%

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

8. Applied rewrites 97.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. frac-times N/A

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

10. Applied rewrites 97.7%

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

11. Final simplification 97.7%

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

Alternative 4: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.2

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

4. Applied rewrites 97.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 97.8

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 97.8

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

6. Applied rewrites 97.8%

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

8. Applied rewrites 97.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

10. Applied rewrites 97.0%

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

11. Final simplification 97.0%

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

Alternative 5: 70.4% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 72.0%

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

6. Final simplification 72.0%

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

Alternative 6: 70.3% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. lower-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 97.0

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 97.0

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

4. Applied rewrites 97.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 71.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

9. Applied rewrites 71.9%

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

10. Final simplification 71.9%

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

Alternative 7: 63.9% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 64.3%

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

6. Taylor expanded in tau around 0

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

8. Applied rewrites 64.6%

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

10. Applied rewrites 65.6%

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

11. Final simplification 65.6%

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

Alternative 8: 62.9% 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 97.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 64.6%

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

Reproduce

herbie shell --seed 2024268 
(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)))))