Lanczos kernel

Percentage Accurate: 97.9% → 97.8%

Time: 8.4s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.0

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.0

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. div-inv N/A

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

   14. frac-times N/A

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

6. Applied rewrites 98.0%

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

8. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]
9. 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 98.0%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.0

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.0

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. div-inv N/A

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

   14. frac-times N/A

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

6. Applied rewrites 98.0%

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

8. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

10. Applied rewrites 97.8%

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

11. Final simplification 97.8%

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

Alternative 4: 71.1% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.0

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.0

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. div-inv N/A

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

   14. frac-times N/A

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

6. Applied rewrites 97.6%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 70.9

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

9. Applied rewrites 70.9%

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

    1. lift-/.f32 N/A

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

    2. lift-/.f32 N/A

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

    3. clear-num N/A

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

    4. lift-*.f32 N/A

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

    5. lift-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lift-*.f32 N/A

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

    8. associate-/r* N/A

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

    9. lower-/.f32 N/A

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

11. Applied rewrites 71.0%

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

Alternative 5: 71.1% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.0

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.0

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 70.9%

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

Alternative 6: 64.7% accurate, 10.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 98.0

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 98.0

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in x around 0

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

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

7. Applied rewrites 63.2%

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

8. Taylor expanded in tau around 0

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

10. Applied rewrites 63.2%

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

12. Applied rewrites 64.2%

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

Alternative 7: 63.7% accurate, 258.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 63.2%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 63.2%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(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)))))