Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 3.9s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

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: 98.0% 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: 98.0% 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

Derivation

1. Initial program 98.0%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l/ N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

5. Applied rewrites 97.8%

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

Alternative 3: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. frac-times N/A

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

3. Applied rewrites 97.8%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l/ N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

5. Applied rewrites 97.8%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/l/ N/A

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

   7. *-commutative N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-sin.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lift-PI.f32 N/A

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

7. Applied rewrites 97.7%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * tau) * Float32(pi))
	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 = (x * tau) * single(pi);
	tmp = sin((single(pi) * x)) * (sin(t_1) / ((t_1 * single(pi)) * x));
end

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l/ N/A

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

3. Applied rewrites 97.9%

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

5. Applied rewrites 97.3%

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

Alternative 6: 85.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * ((float) M_PI)) * tau;
	return (sinf(t_1) / t_1) * fmaf(powf((((float) M_PI) * x), 2.0f), -0.16666666666666666f, 1.0f);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * Float32(pi)) * tau)
	return Float32(Float32(sin(t_1) / t_1) * fma((Float32(Float32(pi) * x) ^ Float32(2.0)), Float32(-0.16666666666666666), Float32(1.0)))
end

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 85.3

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

4. Applied rewrites 85.3%

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

Alternative 7: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (x * tau) * ((float) M_PI);
	return sinf(t_1) * (fmaf(powf((((float) M_PI) * x), 2.0f), -0.16666666666666666f, 1.0f) / t_1);
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(x * tau) * Float32(pi))
	return Float32(sin(t_1) * Float32(fma((Float32(Float32(pi) * x) ^ Float32(2.0)), Float32(-0.16666666666666666), Float32(1.0)) / t_1))
end

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l/ N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 85.1

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

8. Applied rewrites 85.1%

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

Alternative 8: 84.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l/ N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 84.6

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

8. Applied rewrites 84.6%

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

   1. lift-/.f32 N/A

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

10. Applied rewrites 84.8%

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

Alternative 9: 84.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l/ N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 84.6

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

8. Applied rewrites 84.6%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 84.8

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

10. Applied rewrites 84.8%

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

Alternative 10: 84.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l/ N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 84.6

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

8. Applied rewrites 84.6%

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

9. Taylor expanded in tau around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower--.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. pow2 N/A

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

    7. lift-*.f32 N/A

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

    8. lift-*.f32 N/A

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

    9. lift-PI.f32 N/A

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

    10. lower-/.f32 N/A

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

    11. lift-PI.f32 N/A

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

    12. lower-*.f32 84.8

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

11. Applied rewrites 84.8%

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

Alternative 11: 84.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l/ N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 84.6

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

8. Applied rewrites 84.6%

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

   1. lift-/.f32 N/A

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

10. Applied rewrites 84.8%

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

Alternative 12: 79.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	float t_1 = (tau * x) * ((float) M_PI);
	return fmaf((t_1 * t_1), -0.16666666666666666f, 1.0f) * (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)));
}

Julia

function code(x, tau)
	t_1 = Float32(Float32(tau * x) * Float32(pi))
	return Float32(fma(Float32(t_1 * t_1), Float32(-0.16666666666666666), Float32(1.0)) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

Derivation

1. Initial program 98.0%

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

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 97.3

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

3. Applied rewrites 97.3%

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

4. Taylor expanded in x around 0

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

   1. frac-2neg N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. sin-+PI-rev N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

6. Applied rewrites 79.5%

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

   1. lift-pow.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. unpow-prod-down N/A

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

   6. *-commutative N/A

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

   7. unpow-prod-down N/A

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

   8. associate-*r* N/A

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

   9. pow2 N/A

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

   10. lower-*.f32 N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. associate-*r* N/A

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

   16. lift-*.f32 N/A

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

   17. lift-*.f32 N/A

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

   18. lift-PI.f32 79.5

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

8. Applied rewrites 79.5%

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

Alternative 13: 78.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

4. Applied rewrites 78.7%

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

Alternative 14: 64.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in tau around 0

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

   1. lower-/.f32 N/A

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

   2. lower-sin.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 64.4

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

4. Applied rewrites 64.4%

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

Alternative 15: 64.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*l/ N/A

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

3. Applied rewrites 97.9%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sin.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-sin.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 N/A

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

5. Applied rewrites 97.8%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 84.6

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

8. Applied rewrites 84.6%

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

9. Taylor expanded in x around 0

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

    1. associate-*r* N/A

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

    2. lower-*.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. lift-PI.f32 64.4

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

11. Applied rewrites 64.4%

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

Alternative 16: 63.6% accurate, 258.0× speedup

Math

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

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 63.6%

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

Reproduce

herbie shell --seed 2025093 
(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))))