Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 11.2s

Alternatives: 9

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

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

FPCore

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

Derivation

1. Initial program 97.7%

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

3. Final simplification 97.7%

   \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \frac{\sin \left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau\right)}{\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau} \]
4. 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 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 97.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. frac-times N/A

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

8. Applied rewrites 97.6%

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

9. Final simplification 97.6%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*l/ N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f32 N/A

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

6. Applied rewrites 97.6%

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

8. Applied rewrites 97.1%

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

9. Final simplification 97.1%

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

Alternative 4: 71.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{PI}\left(\right) \cdot x\\ t_2 := t\_1 \cdot tau\\ t_3 := \sqrt{\mathsf{PI}\left(\right)}\\ \frac{\frac{\frac{\frac{t\_1}{t\_2} \cdot \sin t\_2}{x}}{t\_3}}{t\_3} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)) (t_2 (* t_1 tau)) (t_3 (sqrt (PI))))
   (/ (/ (/ (* (/ t_1 t_2) (sin t_2)) x) t_3) t_3)))

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. add-sqr-sqrt N/A

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

   7. associate-*r* N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 97.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f32 N/A

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

   5. lower-*.f32 96.6

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

6. Applied rewrites 96.6%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 70.0

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

9. Applied rewrites 70.0%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. associate-/r* N/A

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

    5. lower-/.f32 N/A

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

11. Applied rewrites 70.3%

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

12. Final simplification 70.3%

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

Alternative 5: 71.3% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 97.6

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

4. Applied rewrites 97.6%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 70.2%

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

8. Final simplification 70.2%

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

Alternative 6: 71.3% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. 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. lower-/.f32 N/A

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

4. Applied rewrites 97.4%

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

5. Taylor expanded in x around 0

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

   1. lower-PI.f32 70.2

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

7. Applied rewrites 70.2%

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

8. Final simplification 70.2%

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

Alternative 7: 71.3% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 70.2%

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

6. Final simplification 70.2%

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

Alternative 8: 64.8% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 61.9%

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

6. Taylor expanded in tau around 0

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

8. Applied rewrites 61.9%

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

10. Applied rewrites 63.1%

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

11. Final simplification 63.1%

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

Alternative 9: 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 97.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 62.2%

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

Reproduce

herbie shell --seed 2024264 
(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)))))