Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 9.4s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   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. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 97.9

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

4. Applied rewrites 97.9%

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

Alternative 3: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   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. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 97.9

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

4. Applied rewrites 97.9%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f32 N/A

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

7. Applied rewrites 96.7%

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

8. Final simplification 96.7%

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

Alternative 4: 79.2% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   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. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 97.9

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

4. Applied rewrites 97.9%

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

6. Applied rewrites 97.0%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower--.f32 N/A

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

9. Applied rewrites 79.6%

   \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(tau \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right) - \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{-x} \]
10. Add Preprocessing

Alternative 5: 71.2% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-/.f32 N/A

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

   7. frac-times N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

4. Applied rewrites 97.5%

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

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 71.1

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

7. Applied rewrites 71.1%

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

Alternative 6: 64.6% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   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. associate-*r/ N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. times-frac N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in x around 0

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

   1. lower-PI.f32 64.3

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

7. Applied rewrites 64.3%

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

Alternative 7: 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 98.0%

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

   1. lift-*.f32 N/A

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

   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. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 97.9

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

4. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. associate-/l/ N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 97.8%

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

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 64.3

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

9. Applied rewrites 64.3%

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

Alternative 8: 63.8% accurate, 258.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 63.5%

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

6. Final simplification 63.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(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)))))