Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 3.8s

Alternatives: 12

Speedup: 1.0×

Specification

\[\left(\frac{1}{100000} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (x * single(pi)) * tau;
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (x * single(pi)) * tau;
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (tau * x) * ((float) M_PI);
	return (sinf(t_1) / t_1) * (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(tau * x) * Float32(pi))
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (tau * x) * single(pi);
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.3%

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.9%

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

5. Applied rewrites 97.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (tau * ((float) M_PI)) * x;
	float t_2 = -x * ((float) M_PI);
	return (sinf(t_1) * sinf(t_2)) / (t_1 * t_2);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(tau * Float32(pi)) * x)
	t_2 = Float32(Float32(-x) * Float32(pi))
	return Float32(Float32(sin(t_1) * sin(t_2)) / Float32(t_1 * t_2))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (tau * single(pi)) * x;
	t_2 = -x * single(pi);
	tmp = (sin(t_1) * sin(t_2)) / (t_1 * t_2);
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.2%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \pi\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\pi \cdot x\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.8%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

5. Applied rewrites 97.8%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(x \cdot \pi\right)}}{x \cdot \pi} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\pi \cdot x\right)}}{x \cdot \pi} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right)}{x \cdot \pi} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.2%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{x \cdot \pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\pi \cdot x}} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\mathsf{PI}\left(\right)} \cdot x} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)} \]

   16. lower-cbrt.f32 97.7%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)} \]

9. Applied rewrites 97.7%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)}} \]

11. Applied rewrites 97.7%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = tau * (((float) M_PI) * x);
	return (sinf((((float) M_PI) * x)) * sinf(t_1)) / ((((float) M_PI) * x) * t_1);
}

Julia

function code(x, tau)
	t_1 = Float32(tau * Float32(Float32(pi) * x))
	return Float32(Float32(sin(Float32(Float32(pi) * x)) * sin(t_1)) / Float32(Float32(Float32(pi) * x) * t_1))
end

MATLAB

function tmp = code(x, tau)
	t_1 = tau * (single(pi) * x);
	tmp = (sin((single(pi) * x)) * sin(t_1)) / ((single(pi) * x) * t_1);
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. lower-/.f32 N/A

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

3. Applied rewrites 97.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f32 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 97.8%

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 97.8%

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

5. Applied rewrites 97.8%

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

Alternative 4: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = x * (((float) M_PI) * tau);
	return (sinf((x * ((float) M_PI))) / ((t_1 * ((float) M_PI)) * x)) * sinf(t_1);
}

Julia

function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * tau))
	return Float32(Float32(sin(Float32(x * Float32(pi))) / Float32(Float32(t_1 * Float32(pi)) * x)) * sin(t_1))
end

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = (sin((x * single(pi))) / ((t_1 * single(pi)) * x)) * sin(t_1);
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.2%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \pi\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\pi \cdot x\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.8%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

5. Applied rewrites 97.8%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(x \cdot \pi\right)}}{x \cdot \pi} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\pi \cdot x\right)}}{x \cdot \pi} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right)}{x \cdot \pi} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.2%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{x \cdot \pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\pi \cdot x}} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\mathsf{PI}\left(\right)} \cdot x} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)} \]

   16. lower-cbrt.f32 97.7%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)} \]

9. Applied rewrites 97.7%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)}} \]

11. Applied rewrites 97.7%

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

13. Applied rewrites 97.3%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (tau * ((float) M_PI)) * x;
	return sinf(t_1) * (sinf((x * ((float) M_PI))) / ((t_1 * x) * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(tau * Float32(pi)) * x)
	return Float32(sin(t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(Float32(t_1 * x) * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (tau * single(pi)) * x;
	tmp = sin(t_1) * (sin((x * single(pi))) / ((t_1 * x) * single(pi)));
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.2%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \pi\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\pi \cdot x\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.8%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

5. Applied rewrites 97.8%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(x \cdot \pi\right)}}{x \cdot \pi} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\pi \cdot x\right)}}{x \cdot \pi} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right)}{x \cdot \pi} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.2%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{x \cdot \pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\pi \cdot x}} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\mathsf{PI}\left(\right)} \cdot x} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)} \]

   16. lower-cbrt.f32 97.7%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)} \]

9. Applied rewrites 97.7%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)}} \]

11. Applied rewrites 97.3%

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

Alternative 6: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return sinf((tau * (((float) M_PI) * x))) * (sinf((((float) M_PI) * x)) / (((tau * ((float) M_PI)) * ((float) M_PI)) * (x * x)));
}

Julia

function code(x, tau)
	return Float32(sin(Float32(tau * Float32(Float32(pi) * x))) * Float32(sin(Float32(Float32(pi) * x)) / Float32(Float32(Float32(tau * Float32(pi)) * Float32(pi)) * Float32(x * x))))
end

MATLAB

function tmp = code(x, tau)
	tmp = sin((tau * (single(pi) * x))) * (sin((single(pi) * x)) / (((tau * single(pi)) * single(pi)) * (x * x)));
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.2%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \pi\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\pi \cdot x\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.8%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

5. Applied rewrites 97.8%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(x \cdot \pi\right)}}{x \cdot \pi} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\pi \cdot x\right)}}{x \cdot \pi} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right)}{x \cdot \pi} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.2%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{x \cdot \pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\pi \cdot x}} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\mathsf{PI}\left(\right)} \cdot x} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)} \]

   16. lower-cbrt.f32 97.7%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)} \]

9. Applied rewrites 97.7%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)}} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    2. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\pi} \cdot x\right)}\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    3. associate-*r* N/A

       \[\leadsto \frac{\sin \left(\color{blue}{\left(\left({\pi}^{\frac{2}{3}} \cdot \sqrt[3]{\pi}\right) \cdot x\right)} \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    4. lift-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left(\left(\color{blue}{{\pi}^{\frac{2}{3}}} \cdot \sqrt[3]{\pi}\right) \cdot x\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    5. lift-cbrt.f32 N/A

       \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\sqrt[3]{\pi}}\right) \cdot x\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    6. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot x\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    7. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{2}{3} + \frac{1}{3}\right)}} \cdot x\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\color{blue}{1}} \cdot x\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    9. unpow1 N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{\pi} \cdot x\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    10. *-commutative N/A

        \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot \pi\right)} \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    11. lift-*.f32 97.2%

        \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot \pi\right)} \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

11. Applied rewrites 97.2%

    \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \pi\right) \cdot tau\right)}}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]
12. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    2. lift-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\pi} \cdot x\right)}\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    3. associate-*r* N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\color{blue}{\left(\left({\pi}^{\frac{2}{3}} \cdot \sqrt[3]{\pi}\right) \cdot x\right)} \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    4. lift-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(\left(\color{blue}{{\pi}^{\frac{2}{3}}} \cdot \sqrt[3]{\pi}\right) \cdot x\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    5. lift-cbrt.f32 N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\sqrt[3]{\pi}}\right) \cdot x\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    6. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot x\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    7. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{2}{3} + \frac{1}{3}\right)}} \cdot x\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left({\pi}^{\color{blue}{1}} \cdot x\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    9. unpow1 N/A

       \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(\color{blue}{\pi} \cdot x\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    10. *-commutative N/A

        \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \pi\right)} \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

    11. lift-*.f32 97.9%

        \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \pi\right)} \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

13. Applied rewrites 97.9%

    \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \pi\right) \cdot tau}} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)} \]

15. Applied rewrites 96.8%

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

Alternative 7: 79.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  (*
 (sin (* PI x))
 (+ (* -1/6 (* (pow tau 2) (* x PI))) (/ 1 (* x PI)))))

C

float code(float x, float tau) {
	return sinf((((float) M_PI) * x)) * ((-0.16666666666666666f * (powf(tau, 2.0f) * (x * ((float) M_PI)))) + (1.0f / (x * ((float) M_PI))));
}

Julia

function code(x, tau)
	return Float32(sin(Float32(Float32(pi) * x)) * Float32(Float32(Float32(-0.16666666666666666) * Float32((tau ^ Float32(2.0)) * Float32(x * Float32(pi)))) + Float32(Float32(1.0) / Float32(x * Float32(pi)))))
end

MATLAB

function tmp = code(x, tau)
	tmp = sin((single(pi) * x)) * ((single(-0.16666666666666666) * ((tau ^ single(2.0)) * (x * single(pi)))) + (single(1.0) / (x * single(pi))));
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. mult-flip N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

3. Applied rewrites 97.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \sin \left(\pi \cdot x\right) \cdot \color{blue}{\frac{1}{x \cdot \pi}} \]
5. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 64.4%

      \[\leadsto \sin \left(\pi \cdot x\right) \cdot \frac{1}{x \cdot \pi} \]

6. Applied rewrites 64.4%

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

7. Taylor expanded in tau around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 79.2%

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

9. Applied rewrites 79.2%

   \[\leadsto \sin \left(\pi \cdot x\right) \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot \left(x \cdot \pi\right)\right) + \frac{1}{x \cdot \pi}\right)} \]
10. Add Preprocessing

Alternative 8: 64.5% accurate, 1.9× speedup

Math

\[\frac{\frac{1}{\pi} \cdot \sin \left(\pi \cdot x\right)}{x} \]

FPCore

(FPCore (x tau)
  :precision binary32
  (/ (* (/ 1 PI) (sin (* PI x))) x))

C

float code(float x, float tau) {
	return ((1.0f / ((float) M_PI)) * sinf((((float) M_PI) * x))) / x;
}

Julia

function code(x, tau)
	return Float32(Float32(Float32(Float32(1.0) / Float32(pi)) * sin(Float32(Float32(pi) * x))) / x)
end

MATLAB

function tmp = code(x, tau)
	tmp = ((single(1.0) / single(pi)) * sin((single(pi) * x))) / x;
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f32 N/A

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

3. Applied rewrites 97.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{\pi}} \cdot \sin \left(\pi \cdot x\right)}{x} \]
5. Step-by-step derivation

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 64.4%

      \[\leadsto \frac{\frac{1}{\pi} \cdot \sin \left(\pi \cdot x\right)}{x} \]

6. Applied rewrites 64.4%

   \[\leadsto \frac{\color{blue}{\frac{1}{\pi}} \cdot \sin \left(\pi \cdot x\right)}{x} \]
7. Add Preprocessing

Alternative 9: 64.4% accurate, 2.0× speedup

Math

\[1 \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

FPCore

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

C

float code(float x, float tau) {
	return 1.0f * (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) * (sin((x * single(pi))) / (x * single(pi)));
end

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
3. Step-by-step derivation

4. Applied rewrites 64.5%

   \[\leadsto \color{blue}{1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
5. Add Preprocessing

Alternative 10: 63.7% accurate, 4.5× speedup

Math

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

FPCore

(FPCore (x tau)
  :precision binary32
  (let* ((t_1 (* (* (* x (* PI tau)) PI) x))) (* (/ 1 t_1) t_1)))

C

float code(float x, float tau) {
	float t_1 = ((x * (((float) M_PI) * tau)) * ((float) M_PI)) * x;
	return (1.0f / t_1) * t_1;
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(Float32(x * Float32(Float32(pi) * tau)) * Float32(pi)) * x)
	return Float32(Float32(Float32(1.0) / t_1) * t_1)
end

MATLAB

function tmp = code(x, tau)
	t_1 = ((x * (single(pi) * tau)) * single(pi)) * x;
	tmp = (single(1.0) / t_1) * t_1;
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. lower-/.f32 N/A

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

3. Applied rewrites 97.1%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-PI.f32 63.4%

      \[\leadsto \frac{tau \cdot \left({x}^{2} \cdot {\pi}^{2}\right)}{\left(\left(\pi \cdot x\right) \cdot x\right) \cdot \left(tau \cdot \pi\right)} \]

6. Applied rewrites 63.4%

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

8. Applied rewrites 63.6%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \pi\right) \cdot x} \cdot \left(\left(\left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \pi\right) \cdot x\right)} \]
9. Add Preprocessing

Alternative 11: 63.6% accurate, 258.0× speedup

Math

\[1 \]

FPCore

(FPCore (x tau)
  :precision binary32
  1)

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 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
2. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.2%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \left(\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \pi\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\pi \cdot x\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.8%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

5. Applied rewrites 97.8%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(x \cdot \pi\right)}}{x \cdot \pi} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\pi \cdot x\right)}}{x \cdot \pi} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right)}{x \cdot \pi} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x\right)}{x \cdot \pi} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}}{x \cdot \pi} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}}{x \cdot \pi} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)\right)}{x \cdot \pi} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}\right)}{x \cdot \pi} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)\right)}{x \cdot \pi} \]

   16. lower-cbrt.f32 97.2%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)\right)}{x \cdot \pi} \]

7. Applied rewrites 97.2%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}}{x \cdot \pi} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{x \cdot \pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\pi \cdot x}} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\mathsf{PI}\left(\right)} \cdot x} \]

   4. add-cube-cbrt N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot x} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   7. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   8. pow1/3 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   9. lift-PI.f32 N/A

      \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   10. pow1/3 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   11. pow-prod-up N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   12. lower-pow.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

   15. lift-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot x\right)} \]

   16. lower-cbrt.f32 97.7%

       \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{{\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot x\right)} \]

9. Applied rewrites 97.7%

   \[\leadsto \frac{\sin \left(\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau\right)}{\left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right) \cdot tau} \cdot \frac{\sin \left({\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}{\color{blue}{{\pi}^{\frac{2}{3}} \cdot \left(\sqrt[3]{\pi} \cdot x\right)}} \]

11. Applied rewrites 97.7%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 63.7%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x tau)
  :name "Lanczos kernel"
  :precision binary32
  :pre (and (and (<= 1/100000 x) (<= x 1)) (and (<= 1 tau) (<= tau 5)))
  (* (/ (sin (* (* x PI) tau)) (* (* x PI) tau)) (/ (sin (* x PI)) (* x PI))))