Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 9.9s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 98.0% 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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)) (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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.7%

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

8. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. div-inv N/A

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

   6. lift-/.f32 N/A

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

   7. clear-num N/A

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

   8. lower-*.f32 N/A

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

10. Applied rewrites 97.8%

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

11. Final simplification 97.8%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)) (t_2 (* t_1 tau)))
   (/ (* (sin t_2) (sin t_1)) (* t_2 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. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.7%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

8. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. distribute-lft-neg-out N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lift-*.f32 N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lift-*.f32 N/A

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

   13. lift-neg.f32 N/A

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

   14. distribute-lft-neg-out N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. remove-double-neg N/A

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

   18. lower-*.f32 97.6

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

10. Applied rewrites 97.6%

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

11. Final simplification 97.6%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.7%

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

8. Applied rewrites 97.3%

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

9. Final simplification 97.3%

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

Alternative 4: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.7%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

8. Applied rewrites 97.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. distribute-lft-neg-out N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-neg.f32 N/A

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

   12. distribute-lft-neg-out N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. sqr-neg N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

10. Applied rewrites 97.2%

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

11. Final simplification 97.2%

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

Alternative 5: 71.2% accurate, 1.6× speedup

Math

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

FPCore

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

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 72.2

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

7. Applied rewrites 72.2%

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

8. Final simplification 72.2%

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

Alternative 6: 71.2% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)) (t_2 (* t_1 tau))) (/ t_1 (* (/ t_2 (sin t_2)) 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. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.7%

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

8. Applied rewrites 97.6%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 72.2

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

11. Applied rewrites 72.2%

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

Alternative 7: 71.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* (PI) x) tau))) (* (/ 1.0 (* t_1 (PI))) (* (PI) (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. 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. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. frac-times N/A

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

   7. div-inv N/A

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

   8. lower-*.f32 N/A

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

4. Applied rewrites 97.4%

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

5. Taylor expanded in x around 0

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

   1. lower-PI.f32 72.2

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

7. Applied rewrites 72.2%

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

8. Final simplification 72.2%

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

Alternative 8: 64.5% accurate, 2.0× speedup

Math

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

FPCore

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

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.2%

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

7. Taylor expanded in x around 0

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

   1. lower-PI.f32 65.7

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

9. Applied rewrites 65.7%

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

10. Final simplification 65.7%

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

Alternative 9: 64.6% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x))) (/ (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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f32 N/A

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 97.7%

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

7. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 65.7

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

9. Applied rewrites 65.7%

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

Alternative 10: 63.9% 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 65.0%

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

Reproduce

herbie shell --seed 2024296 
(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)))))