Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 12.8s

Alternatives: 9

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 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/ 97.9%

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

   2. associate-/l* 97.7%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. associate-*l* 97.7%

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

3. Simplified 97.7%

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

5. Final simplification 97.7%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

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

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 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/ 97.9%

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

   2. associate-/l* 97.7%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.4%

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

   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 4: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((x * single(pi))) / ((x * single(pi)) * (x * (single(pi) * tau)))) * sin(((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. Step-by-step derivation

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.7%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. associate-*l* 97.7%

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

3. Simplified 97.7%

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

5. Taylor expanded in x around inf 97.2%

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

6. Final simplification 97.2%

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

Alternative 5: 70.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = sin(t_1) * (single(1.0) / 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/ 97.9%

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

   2. associate-/l* 97.7%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. associate-*l* 97.7%

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

3. Simplified 97.7%

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

   1. expm1-log1p-u 96.8%

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

6. Applied egg-rr 96.8%

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

7. Taylor expanded in x around 0 70.8%

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

   1. associate-*r* 70.8%

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

   2. *-commutative 70.8%

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

   3. associate-*r* 71.0%

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

9. Simplified 71.0%

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

10. Final simplification 71.0%

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

Alternative 6: 70.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin((x * (single(pi) * tau))) * (single(1.0) / ((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. Step-by-step derivation

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.7%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. associate-*l* 97.7%

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

3. Simplified 97.7%

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

   1. expm1-log1p-u 96.8%

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

6. Applied egg-rr 96.8%

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

7. Taylor expanded in x around 0 70.8%

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

8. Final simplification 70.8%

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

Alternative 7: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. associate-*l/ 97.9%

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

   2. associate-/l* 97.7%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.4%

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

   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. Taylor expanded in tau around 0 64.4%

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

   1. *-un-lft-identity 64.4%

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

   2. times-frac 64.3%

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

7. Applied egg-rr 64.3%

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

8. Taylor expanded in x around 0 64.8%

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

   1. unpow2 64.8%

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

   2. unpow2 64.8%

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

   3. swap-sqr 64.8%

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

   4. unpow2 64.8%

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

10. Simplified 64.8%

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

Alternative 8: 64.2% 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/ 97.9%

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

   2. associate-/l* 97.7%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.4%

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

   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. Taylor expanded in tau around 0 64.4%

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

Alternative 9: 63.4% 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/ 97.9%

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

   2. associate-/l* 97.7%

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

   3. associate-*l* 97.2%

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

   4. associate-/l/ 97.2%

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

   5. *-commutative 97.2%

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

   6. *-commutative 97.2%

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

   7. associate-*l* 97.4%

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

   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. Taylor expanded in x around 0 63.6%

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

Reproduce

herbie shell --seed 2024172 
(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))))