Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 14.2s

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 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
3. Add Preprocessing

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = (sin((x * single(pi))) / (x * single(pi))) * (sin(t_1) / 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. *-commutative 98.0%

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

   2. associate-*l* 97.3%

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

   3. *-commutative 97.3%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = single(pi) * (x * tau);
	tmp = (sin(t_1) / (x * single(pi))) * (sin((x * single(pi))) / 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. associate-*l/ 97.8%

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

   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* 97.3%

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

   4. associate-/l/ 97.3%

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

   5. *-commutative 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

   1. associate-*r* 97.2%

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

   2. associate-*r* 97.3%

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

   3. *-commutative 97.3%

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

   4. *-commutative 97.3%

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

   5. associate-*r* 97.8%

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

   6. associate-*r/ 97.9%

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

6. Applied egg-rr 97.4%

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

   1. associate-/l/ 97.3%

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

   2. *-commutative 97.3%

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

   3. associate-*r* 97.4%

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

   4. *-commutative 97.4%

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

   5. associate-*r* 97.8%

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

   6. *-commutative 97.8%

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

   7. associate-*l/ 98.0%

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

   8. frac-times 97.7%

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

   9. *-commutative 97.7%

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

   10. *-commutative 97.7%

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

8. Applied egg-rr 97.8%

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

Alternative 4: 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 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. associate-*l/ 97.8%

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

   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* 97.3%

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

   4. associate-/l/ 97.3%

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

   5. associate-*l* 97.8%

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

3. Simplified 97.8%

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

5. Final simplification 97.8%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = sin(t_1) * (sin((x * single(pi))) / (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. associate-*l/ 97.8%

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

   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* 97.3%

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

   4. associate-/l/ 97.3%

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

   5. *-commutative 97.3%

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

   6. *-commutative 97.3%

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

   7. associate-*l* 97.2%

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

   8. associate-*l* 97.4%

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

3. Simplified 97.4%

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

Alternative 6: 85.0% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (* x PI) tau)))
   (*
    (/ (sin t_1) t_1)
    (+ 1.0 (* -0.16666666666666666 (* (* 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) * (1.0f + (-0.16666666666666666f * ((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(Float32(1.0) + Float32(Float32(-0.16666666666666666) * Float32(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) * (single(1.0) + (single(-0.16666666666666666) * ((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
3. Step-by-step derivation

   1. expm1-log1p-u 97.7%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)\right)} \]

   2. expm1-undefine 97.7%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)} - 1\right)} \]

4. Applied egg-rr 97.7%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)} - 1\right)} \]

5. Taylor expanded in x around 0 85.3%

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

   1. unpow2 85.3%

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

   2. unpow2 85.3%

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

   3. swap-sqr 85.3%

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

   4. unpow2 85.3%

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

7. Simplified 85.3%

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

   1. unpow2 85.3%

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

9. Applied egg-rr 85.3%

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

Alternative 7: 85.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = (single(1.0) + (single(-0.16666666666666666) * ((x * single(pi)) * (x * single(pi))))) * (sin(t_1) / 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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 97.7%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)\right)} \]

   2. expm1-undefine 97.7%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)} - 1\right)} \]

4. Applied egg-rr 97.7%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)} - 1\right)} \]

5. Taylor expanded in x around 0 85.3%

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

   1. unpow2 85.3%

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

   2. unpow2 85.3%

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

   3. swap-sqr 85.3%

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

   4. unpow2 85.3%

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

7. Simplified 85.3%

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

   1. unpow2 85.3%

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

9. Applied egg-rr 85.3%

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

    1. *-un-lft-identity 85.3%

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

    2. associate-*r* 84.8%

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

    3. *-commutative 84.8%

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

    4. times-frac 84.5%

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

    5. *-commutative 84.5%

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

    6. associate-*r* 84.7%

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

11. Applied egg-rr 84.7%

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

    1. associate-*l/ 84.7%

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

    2. *-lft-identity 84.7%

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

    3. associate-/l/ 84.8%

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

    4. remove-double-neg 84.8%

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

    5. associate-*r* 84.8%

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

    6. *-commutative 84.8%

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

    7. distribute-lft-neg-out 84.8%

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

    8. distribute-lft-neg-in 84.8%

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

    9. associate-*r* 85.3%

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

    10. distribute-lft-neg-out 85.3%

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

    11. *-commutative 85.3%

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

    12. remove-double-neg 85.3%

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

    13. *-commutative 85.3%

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

13. Simplified 85.3%

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

14. Final simplification 85.3%

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

Alternative 8: 79.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (single(1.0) + (single(-0.16666666666666666) * ((x * single(pi)) * (x * single(pi))))) * (single(1.0) + (single(-0.16666666666666666) * ((x * (single(pi) * tau)) ^ single(2.0))));
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. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 97.7%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)\right)} \]

   2. expm1-undefine 97.7%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)} - 1\right)} \]

4. Applied egg-rr 97.7%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)} - 1\right)} \]

5. Taylor expanded in x around 0 85.3%

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

   1. unpow2 85.3%

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

   2. unpow2 85.3%

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

   3. swap-sqr 85.3%

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

   4. unpow2 85.3%

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

7. Simplified 85.3%

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

   1. unpow2 85.3%

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

9. Applied egg-rr 85.3%

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

10. Taylor expanded in x around 0 78.8%

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

    1. *-commutative 78.8%

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

    2. unpow2 78.8%

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

    3. unpow2 78.8%

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

    4. swap-sqr 78.8%

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

    5. unpow2 78.8%

       \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(\left(\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)\right) \cdot \color{blue}{\left(tau \cdot tau\right)}\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)\right)\right) \]

    6. swap-sqr 78.8%

       \[\leadsto \left(1 + -0.16666666666666666 \cdot \color{blue}{\left(\left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \left(\left(x \cdot \pi\right) \cdot tau\right)\right)}\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)\right)\right) \]

    7. *-commutative 78.8%

       \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(\left(\color{blue}{\left(\pi \cdot x\right)} \cdot tau\right) \cdot \left(\left(x \cdot \pi\right) \cdot tau\right)\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)\right)\right) \]

    8. associate-*r* 78.8%

       \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(\color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)} \cdot \left(\left(x \cdot \pi\right) \cdot tau\right)\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)\right)\right) \]

    9. *-commutative 78.8%

       \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(\left(\pi \cdot \left(x \cdot tau\right)\right) \cdot \left(\color{blue}{\left(\pi \cdot x\right)} \cdot tau\right)\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)\right)\right) \]

    10. associate-*r* 78.8%

        \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(\left(\pi \cdot \left(x \cdot tau\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)\right)\right) \]

    11. unpow2 78.8%

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

12. Simplified 78.8%

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

13. Final simplification 78.8%

    \[\leadsto \left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot {\left(x \cdot \left(\pi \cdot tau\right)\right)}^{2}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024179 
(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))))