Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 4.5s

Alternatives: 11

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 98.1%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

4. Applied rewrites 98.1%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-sin.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lift-*.f32 N/A

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

4. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

6. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-sin.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f32 N/A

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

8. Applied rewrites 97.6%

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

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\\ \frac{\sin t\_1}{t\_1} \cdot \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}, -0.16666666666666666, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x (PI)) tau)))
   (* (/ (sin t_1) t_1) (fma (pow (* (PI) x) 2.0) -0.16666666666666666 1.0))))

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 85.5

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}, -0.16666666666666666, 1\right) \]

5. Applied rewrites 85.5%

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}, -0.16666666666666666, 1\right)} \]
6. Add Preprocessing

Alternative 5: 79.4% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}, -0.16666666666666666, 1\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 80.1

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

8. Applied rewrites 80.1%

   \[\leadsto \mathsf{fma}\left({\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}, -0.16666666666666666, 1\right) \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}, -0.16666666666666666, 1\right)} \]
9. Add Preprocessing

Alternative 6: 79.4% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}, -0.16666666666666666, 1\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 79.8

       \[\leadsto \mathsf{fma}\left({\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}^{2}, -0.16666666666666666, 1\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

7. Applied rewrites 79.8%

   \[\leadsto \mathsf{fma}\left({\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}^{2}, -0.16666666666666666, 1\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 80.1

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

10. Applied rewrites 80.1%

    \[\leadsto \mathsf{fma}\left({\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}^{2}, -0.16666666666666666, 1\right) \cdot \color{blue}{\mathsf{fma}\left({\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}, -0.16666666666666666, 1\right)} \]
11. Add Preprocessing

Alternative 7: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\\ t_2 := x \cdot \mathsf{PI}\left(\right)\\ \mathsf{fma}\left(t\_1 \cdot t\_1, -0.16666666666666666, 1\right) \cdot \frac{\sin t\_2}{t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x tau) (PI))) (t_2 (* x (PI))))
   (* (fma (* t_1 t_1) -0.16666666666666666 1.0) (/ (sin t_2) t_2))))

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}, -0.16666666666666666, 1\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
6. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{6}, 1\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. lower-*.f32 79.8

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

   4. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{6}, 1\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{6}, 1\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lower-*.f32 79.8

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

   7. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{6}, 1\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-1}{6}, 1\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   9. lower-*.f32 79.8

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

7. Applied rewrites 79.8%

   \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right), -0.16666666666666666, 1\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
8. Add Preprocessing

Alternative 8: 78.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 78.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left({\left(\mathsf{PI}\left(\right) \cdot tau\right)}^{2} + \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), x \cdot x, 1\right)} \]
6. Add Preprocessing

Alternative 9: 69.5% accurate, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}^{2}, -0.16666666666666666, 1\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 79.8

       \[\leadsto \mathsf{fma}\left({\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}^{2}, -0.16666666666666666, 1\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

7. Applied rewrites 79.8%

   \[\leadsto \mathsf{fma}\left({\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}^{2}, -0.16666666666666666, 1\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 70.2%

    \[\leadsto \mathsf{fma}\left({\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}^{2}, -0.16666666666666666, 1\right) \cdot \color{blue}{1} \]
11. Add Preprocessing

Alternative 10: 63.9% 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 98.1%

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

3. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 64.9

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

5. Applied rewrites 64.9%

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

Alternative 11: 63.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(x, tau)
use fmin_fmax_functions
    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 98.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 64.1%

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

Reproduce

herbie shell --seed 2025051 
(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)))))