Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 8.5s

Alternatives: 23

Speedup: 0.5×

Specification

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

Math

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

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

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

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(tau \cdot \pi\right) \cdot x\\ t_2 := \sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)\\ \frac{\sin t\_1}{t\_1} \cdot \frac{\sin t\_2}{t\_2} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

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

   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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-cbrt.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-cbrt.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-cbrt.f32 96.1

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

7. Applied rewrites 96.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-cbrt.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-cbrt.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-cbrt.f32 97.8

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

9. Applied rewrites 97.8%

   \[\leadsto \frac{\sin \left(\left(tau \cdot \pi\right) \cdot x\right)}{\left(tau \cdot \pi\right) \cdot x} \cdot \frac{\sin \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)\right)}{\color{blue}{\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot x\right)\right)}} \]
10. Add Preprocessing

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

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 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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%

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

7. Applied rewrites 97.6%

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

Alternative 4: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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%

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

7. Applied rewrites 97.3%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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%

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

7. Applied rewrites 97.3%

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

Alternative 6: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(x \cdot \pi\right) \cdot \frac{\sin \left(\left(\pi \cdot tau\right) \cdot x\right)}{\left(\pi \cdot \left(\left(x \cdot \pi\right) \cdot x\right)\right) \cdot tau} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-cbrt.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-cbrt.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-cbrt.f32 96.1

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

7. Applied rewrites 96.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-cbrt.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-cbrt.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-cbrt.f32 97.8

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

9. Applied rewrites 97.8%

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

11. Applied rewrites 97.1%

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

Alternative 7: 83.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   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%

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

7. Applied rewrites 97.3%

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

8. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 83.9

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

10. Applied rewrites 83.9%

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

Alternative 8: 70.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 70.4

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

8. Applied rewrites 70.4%

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

10. Applied rewrites 70.4%

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

Alternative 9: 70.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 70.4

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

8. Applied rewrites 70.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f32 N/A

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

10. Applied rewrites 70.3%

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

Alternative 10: 70.4% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* tau x) PI))) (* (/ (sin t_1) t_1) (/ (* 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) * ((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(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) * ((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. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 70.4

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

8. Applied rewrites 70.4%

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

Alternative 11: 70.4% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* tau (* x PI)))) (* (/ (sin t_1) t_1) (/ (* 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) * ((x * ((float) M_PI)) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(tau * Float32(x * Float32(pi)))
	return Float32(Float32(sin(t_1) / t_1) * Float32(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) * ((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. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 70.4

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

8. Applied rewrites 70.4%

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

9. Taylor expanded in x around inf

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

    1. lower-/.f32 N/A

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

    2. lower-sin.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 N/A

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

    6. lower-*.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. lower-PI.f32 70.4

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

11. Applied rewrites 70.4%

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

Alternative 12: 70.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.3

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

3. Applied rewrites 97.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 97.9

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

5. Applied rewrites 97.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-cbrt.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-cbrt.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-cbrt.f32 96.1

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

7. Applied rewrites 96.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-cbrt.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-cbrt.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-cbrt.f32 97.8

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

9. Applied rewrites 97.8%

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

11. Applied rewrites 97.8%

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

12. Taylor expanded in x around 0

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

    1. lower-*.f32 N/A

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

    2. lower-PI.f32 70.4

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

14. Applied rewrites 70.4%

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

Alternative 13: 70.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* tau x) PI))) (* (/ (sin t_1) (* t_1 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 * 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) / Float32(t_1 * x)) * Float32(Float32(Float32(pi) * x) / Float32(pi)))
end

MATLAB

function tmp = code(x, tau)
	t_1 = (tau * x) * single(pi);
	tmp = (sin(t_1) / (t_1 * 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. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 70.4

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

8. Applied rewrites 70.4%

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

10. Applied rewrites 70.2%

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

Alternative 14: 70.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 70.4

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

8. Applied rewrites 70.4%

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

10. Applied rewrites 70.2%

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

Alternative 15: 70.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   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. Taylor expanded in x around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 70.4

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

8. Applied rewrites 70.4%

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

10. Applied rewrites 70.2%

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

Alternative 16: 70.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{tau \cdot x} \cdot \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{\pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*r* N/A

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

   9. times-frac N/A

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

   10. lower-*.f32 N/A

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

3. Applied rewrites 97.2%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 70.2

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

6. Applied rewrites 70.2%

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

Alternative 17: 70.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sin \left(\pi \cdot \left(x \cdot tau\right)\right) \cdot \frac{1}{tau \cdot \left(x \cdot \pi\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

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

7. Applied rewrites 97.3%

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

8. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 70.2

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

10. Applied rewrites 70.2%

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

Alternative 18: 63.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}} \end{array} \]

FPCore

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

C

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

Julia

function code(x, tau)
	return Float32(Float32(1.0) / 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) / (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. 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. frac-2neg N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. div-flip N/A

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

   9. lower-special-/.f32 N/A

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

   10. lower-special-/.f32 N/A

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

3. Applied rewrites 96.9%

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

4. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 63.7

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

6. Applied rewrites 63.7%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r/ N/A

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

   9. lift-sin.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. div-flip N/A

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

   13. lower-special-/.f32 N/A

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

   14. lower-special-/.f32 N/A

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

   15. lower-/.f32 N/A

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

8. Applied rewrites 63.7%

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

Alternative 19: 63.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{x \cdot \pi}{\sin \left(x \cdot \pi\right)}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) / ((x * single(pi)) / sin((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. frac-2neg N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. div-flip N/A

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

   9. lower-special-/.f32 N/A

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

   10. lower-special-/.f32 N/A

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

3. Applied rewrites 96.9%

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

4. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 63.7

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

6. Applied rewrites 63.7%

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

Alternative 20: 63.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + 0.16666666666666666 \cdot e^{\log \left(\pi \cdot x\right) \cdot 2}} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (/ 1.0 (+ 1.0 (* 0.16666666666666666 (exp (* (log (* PI x)) 2.0))))))

C

float code(float x, float tau) {
	return 1.0f / (1.0f + (0.16666666666666666f * expf((logf((((float) M_PI) * x)) * 2.0f))));
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) * exp(Float32(log(Float32(Float32(pi) * x)) * Float32(2.0))))))
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) / (single(1.0) + (single(0.16666666666666666) * exp((log((single(pi) * x)) * single(2.0)))));
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

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

   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. frac-2neg N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. div-flip N/A

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

   9. lower-special-/.f32 N/A

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

   10. lower-special-/.f32 N/A

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

3. Applied rewrites 96.9%

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

4. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 63.7

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

6. Applied rewrites 63.7%

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

7. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 63.7

      \[\leadsto \frac{1}{1 + 0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)} \]

9. Applied rewrites 63.7%

   \[\leadsto \frac{1}{1 + \color{blue}{0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)}} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

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

    2. lift-pow.f32 N/A

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

    3. lift-pow.f32 N/A

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

    4. pow-prod-down N/A

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

    5. lift-*.f32 N/A

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

    6. pow-to-exp N/A

       \[\leadsto \frac{1}{1 + \frac{1}{6} \cdot e^{\log \left(x \cdot \pi\right) \cdot 2}} \]

    7. lower-special-exp.f32 N/A

       \[\leadsto \frac{1}{1 + \frac{1}{6} \cdot e^{\log \left(x \cdot \pi\right) \cdot 2}} \]

    8. lower-special-*.f32 N/A

       \[\leadsto \frac{1}{1 + \frac{1}{6} \cdot e^{\log \left(x \cdot \pi\right) \cdot 2}} \]

    9. lower-special-log.f32 63.7

       \[\leadsto \frac{1}{1 + 0.16666666666666666 \cdot e^{\log \left(x \cdot \pi\right) \cdot 2}} \]

    10. lift-*.f32 N/A

        \[\leadsto \frac{1}{1 + \frac{1}{6} \cdot e^{\log \left(x \cdot \pi\right) \cdot 2}} \]

    11. *-commutative N/A

        \[\leadsto \frac{1}{1 + \frac{1}{6} \cdot e^{\log \left(\pi \cdot x\right) \cdot 2}} \]

    12. lift-*.f32 63.7

        \[\leadsto \frac{1}{1 + 0.16666666666666666 \cdot e^{\log \left(\pi \cdot x\right) \cdot 2}} \]

11. Applied rewrites 63.7%

    \[\leadsto \frac{1}{1 + 0.16666666666666666 \cdot e^{\log \left(\pi \cdot x\right) \cdot 2}} \]
12. Add Preprocessing

Alternative 21: 63.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 - -0.16666666666666666 \cdot \left(\left(\left(x \cdot x\right) \cdot \pi\right) \cdot \pi\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) / (single(1.0) - (single(-0.16666666666666666) * (((x * x) * single(pi)) * 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. frac-2neg N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. div-flip N/A

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

   9. lower-special-/.f32 N/A

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

   10. lower-special-/.f32 N/A

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

3. Applied rewrites 96.9%

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

4. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 63.7

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

6. Applied rewrites 63.7%

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

7. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 63.7

      \[\leadsto \frac{1}{1 + 0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)} \]

9. Applied rewrites 63.7%

   \[\leadsto \frac{1}{1 + \color{blue}{0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)}} \]
10. Step-by-step derivation

    1. lift-+.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. fp-cancel-sign-sub-inv N/A

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

    4. lower--.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. metadata-eval 63.7

       \[\leadsto \frac{1}{1 - -0.16666666666666666 \cdot \left({x}^{2} \cdot {\color{blue}{\pi}}^{2}\right)} \]

    7. lift-*.f32 N/A

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

    8. lift-pow.f32 N/A

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

    9. pow2 N/A

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

    10. associate-*r* N/A

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

    11. lower-*.f32 N/A

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

    12. lower-*.f32 63.7

        \[\leadsto \frac{1}{1 - -0.16666666666666666 \cdot \left(\left({x}^{2} \cdot \pi\right) \cdot \pi\right)} \]

    13. lift-pow.f32 N/A

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

    14. unpow2 N/A

        \[\leadsto \frac{1}{1 - \frac{-1}{6} \cdot \left(\left(\left(x \cdot x\right) \cdot \pi\right) \cdot \pi\right)} \]

    15. lower-*.f32 63.7

        \[\leadsto \frac{1}{1 - -0.16666666666666666 \cdot \left(\left(\left(x \cdot x\right) \cdot \pi\right) \cdot \pi\right)} \]

11. Applied rewrites 63.7%

    \[\leadsto \frac{1}{1 - -0.16666666666666666 \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \pi\right) \cdot \pi\right)}} \]
12. Add Preprocessing

Alternative 22: 63.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return 1.0f / fmaf(((x * x) * 0.16666666666666666f), (((float) M_PI) * ((float) M_PI)), 1.0f);
}

Julia

function code(x, tau)
	return Float32(Float32(1.0) / fma(Float32(Float32(x * x) * Float32(0.16666666666666666)), Float32(Float32(pi) * Float32(pi)), Float32(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

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

   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. frac-2neg N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. div-flip N/A

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

   9. lower-special-/.f32 N/A

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

   10. lower-special-/.f32 N/A

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

3. Applied rewrites 96.9%

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

4. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 63.7

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

6. Applied rewrites 63.7%

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

7. Taylor expanded in x around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-pow.f32 N/A

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

   6. lower-PI.f32 63.7

      \[\leadsto \frac{1}{1 + 0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)} \]

9. Applied rewrites 63.7%

   \[\leadsto \frac{1}{1 + \color{blue}{0.16666666666666666 \cdot \left({x}^{2} \cdot {\pi}^{2}\right)}} \]
10. Step-by-step derivation

    1. lift-+.f32 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. lift-pow.f32 N/A

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

    6. pow2 N/A

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

    7. lift-*.f32 N/A

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

    8. associate-*r* N/A

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

    9. lower-fma.f32 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f32 63.7

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

    12. lift-pow.f32 N/A

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

    13. unpow2 N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{6}, \pi \cdot \pi, 1\right)} \]

    14. lower-*.f32 63.7

        \[\leadsto \frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.16666666666666666, \pi \cdot \pi, 1\right)} \]

11. Applied rewrites 63.7%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.16666666666666666, \pi \cdot \color{blue}{\pi}, 1\right)} \]
12. Add Preprocessing

Alternative 23: 62.9% accurate, 94.3× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 62.9%

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

Reproduce

herbie shell --seed 2025151 
(FPCore (x tau)
  :name "Lanczos kernel"
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0)) (and (<= 1.0 tau) (<= tau 5.0)))
  (* (/ (sin (* (* x PI) tau)) (* (* x PI) tau)) (/ (sin (* x PI)) (* x PI))))