Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 12.9s

Alternatives: 8

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 97.9%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.7%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.1%

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

   4. associate-/l/ 97.0%

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

   5. associate-*l* 97.6%

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

3. Simplified 97.6%

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

5. Final simplification 97.6%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.7%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.1%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 96.9%

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

   8. associate-*l* 97.2%

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

3. Simplified 97.2%

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

   1. associate-*r* 96.9%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. associate-/r* 97.1%

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

   5. *-commutative 97.1%

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

   6. associate-/r* 97.1%

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

   7. associate-/r* 96.8%

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

   8. *-un-lft-identity 96.8%

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

   9. times-frac 96.8%

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

   10. *-commutative 96.8%

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

   11. associate-*l* 96.7%

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

6. Applied egg-rr 96.7%

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

   1. associate-/l/ 96.8%

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

   2. frac-times 96.9%

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

   3. *-un-lft-identity 96.9%

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

   4. associate-*r* 97.1%

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

   5. *-commutative 97.1%

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

   6. associate-*r* 97.4%

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

8. Applied egg-rr 97.4%

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

9. Final simplification 97.4%

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

Alternative 4: 71.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.7%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.1%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 96.9%

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

   8. associate-*l* 97.2%

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

3. Simplified 97.2%

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

   1. associate-*r* 96.9%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. *-commutative 97.0%

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

   5. associate-*r* 97.6%

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

   6. *-commutative 97.6%

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

   7. associate-*r* 97.0%

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

   8. associate-/r* 97.1%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in x around 0 71.9%

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

   1. times-frac 72.1%

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

9. Applied egg-rr 72.1%

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

Alternative 5: 71.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.7%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.1%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 96.9%

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

   8. associate-*l* 97.2%

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

3. Simplified 97.2%

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

   1. associate-*r* 96.9%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. *-commutative 97.0%

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

   5. associate-*r* 97.6%

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

   6. *-commutative 97.6%

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

   7. associate-*r* 97.0%

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

   8. associate-/r* 97.1%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in x around 0 71.9%

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

8. Taylor expanded in tau around inf 72.1%

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

9. Final simplification 72.1%

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

Alternative 6: 71.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. add-log-exp 97.1%

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

   2. associate-*r* 96.8%

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

   3. associate-*r* 97.1%

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

   4. *-commutative 97.1%

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

   5. associate-*l* 96.8%

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

   6. *-commutative 96.8%

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

   7. associate-*l* 97.2%

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

4. Applied egg-rr 97.2%

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

   1. *-commutative 97.2%

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

   2. clear-num 97.1%

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

   3. rem-log-exp 97.8%

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

   4. frac-times 97.7%

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

   5. *-un-lft-identity 97.7%

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

   6. associate-*r* 97.0%

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

   7. *-commutative 97.0%

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

   8. associate-*r* 97.0%

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

   9. associate-*r* 97.1%

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

   10. *-commutative 97.1%

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

   11. associate-*r* 97.7%

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

6. Applied egg-rr 97.7%

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

7. Taylor expanded in x around 0 71.9%

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

8. Final simplification 71.9%

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

Alternative 7: 65.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.7%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.1%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 96.9%

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

   8. associate-*l* 97.2%

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

3. Simplified 97.2%

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

5. Taylor expanded in tau around 0 65.6%

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

Alternative 8: 64.2% accurate, 219.0× 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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.7%

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

   2. associate-/l* 97.8%

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

   3. associate-*l* 97.1%

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

   4. associate-/l/ 97.0%

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

   5. *-commutative 97.0%

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

   6. *-commutative 97.0%

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

   7. associate-*l* 96.9%

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

   8. associate-*l* 97.2%

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

3. Simplified 97.2%

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

5. Taylor expanded in x around 0 64.9%

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

Reproduce

herbie shell --seed 2024135 
(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))))