Lanczos kernel

Percentage Accurate: 98.0% → 97.9%

Time: 6.9s

Alternatives: 12

Speedup: N/A×

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.5%

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

6. Applied rewrites 97.5%

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

8. Applied rewrites 97.6%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.5%

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

6. Applied rewrites 97.5%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

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

   4. lower-/.f32 N/A

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

4. Applied rewrites 97.5%

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

6. Applied rewrites 97.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 97.2

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.2

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

8. Applied rewrites 97.2%

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

Alternative 4: 85.4% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 96.9

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 84.1

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

7. Applied rewrites 84.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-*.f32 84.2

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 84.2

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f32 N/A

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

   16. lower-*.f32 84.8

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f32 84.8

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

9. Applied rewrites 84.8%

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

Alternative 5: 84.8% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 97.1

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

4. Applied rewrites 97.1%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 84.2

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

7. Applied rewrites 84.2%

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

Alternative 6: 79.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot tau, \left(\left(-\mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (*
  (*
   (fma
    (fma (* 0.16666666666666666 tau) (* (* (- (PI)) (PI)) (* x x)) 0.0)
    tau
    1.0)
   (/ tau tau))
  (fma (* -0.16666666666666666 (* x x)) (* (PI) (PI)) 1.0)))

Derivation

1. Initial program 97.6%

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

   1. remove-double-neg N/A

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

   2. lift-sin.f32 N/A

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

   3. sin-neg-rev N/A

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

   4. sin-+PI-rev N/A

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

   5. lower-sin.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r* N/A

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

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

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

   11. lift-PI.f32 N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-neg.f32 81.2

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

4. Applied rewrites 81.2%

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

5. Taylor expanded in tau around 0

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

7. Applied rewrites 78.9%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot {x}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot {x}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   5. unpow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   7. unpow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot \left(x \cdot x\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot \left(x \cdot x\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \]

   9. lower-PI.f32 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot \left(x \cdot x\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \]

   10. lower-PI.f32 79.4

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \]

10. Applied rewrites 79.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot tau, \left(-\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)} \]

11. Final simplification 79.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot tau, \left(\left(-\mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot x\right), 0\right), tau, 1\right) \cdot \frac{tau}{tau}\right) \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \]
12. Add Preprocessing

Alternative 7: 79.6% accurate, 4.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 96.9

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 84.1

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

7. Applied rewrites 84.1%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 79.4

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

10. Applied rewrites 79.4%

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

11. Final simplification 79.4%

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

Alternative 8: 78.9% accurate, 7.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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 78.5%

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

7. Applied rewrites 78.5%

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

Alternative 9: 78.9% accurate, 7.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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 78.5%

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

7. Applied rewrites 78.5%

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

Alternative 10: 70.1% accurate, 8.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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 78.5%

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

6. Taylor expanded in tau around inf

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

8. Applied rewrites 70.3%

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

Alternative 11: 64.9% accurate, 11.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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 78.5%

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

6. Taylor expanded in tau around 0

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

8. Applied rewrites 65.0%

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

Alternative 12: 64.0% 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 97.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 64.4%

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

Reproduce

herbie shell --seed 2025010 
(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)))))