Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 13.3s

Alternatives: 12

Speedup: N/A×

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

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

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(x * Float32(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 97.8%

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

   1. associate-*l* 97.1%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Final simplification 97.9%

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

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.1%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in x around inf 97.2%

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

6. Final simplification 97.2%

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

Alternative 3: 71.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.1%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in x around 0 72.2%

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

   1. associate-*r* 72.0%

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

   2. *-un-lft-identity 72.0%

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

   3. *-commutative 72.0%

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

   4. associate-*r* 71.9%

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

   5. times-frac 72.0%

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

   6. associate-*r* 72.0%

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

   7. *-commutative 72.0%

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

   8. associate-*r* 72.2%

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

7. Applied egg-rr 72.2%

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

8. Final simplification 72.2%

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

Alternative 4: 71.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.1%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in x around 0 72.2%

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

6. Taylor expanded in x around inf 71.9%

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

7. Final simplification 71.9%

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

Alternative 5: 71.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.1%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in x around 0 72.2%

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

6. Final simplification 72.2%

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

Alternative 6: 71.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in x around 0 72.2%

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

6. Final simplification 72.2%

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

Alternative 7: 64.7% 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 97.8%

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

   1. *-commutative 97.8%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 97.9%

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

3. Simplified 97.9%

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

   1. expm1-log1p-u 97.8%

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

6. Applied egg-rr 97.8%

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

7. Taylor expanded in x around 0 66.2%

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

8. Final simplification 66.2%

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

Alternative 8: 63.8% accurate, 16.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.1%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in x around 0 72.2%

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

6. Taylor expanded in x around 0 65.2%

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

   1. associate-*r* 65.4%

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

   2. /-rgt-identity 65.4%

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

   3. *-commutative 65.4%

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

   4. associate-*r* 65.3%

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

   5. *-commutative 65.3%

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

   6. associate-/l* 65.3%

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

8. Applied egg-rr 65.3%

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

9. Final simplification 65.3%

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

Alternative 9: 63.8% accurate, 16.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.1%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in x around 0 72.2%

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

6. Taylor expanded in x around 0 65.2%

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

   1. associate-*r* 65.4%

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

   2. /-rgt-identity 65.4%

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

   3. associate-/l* 65.4%

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

   4. *-commutative 65.4%

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

8. Applied egg-rr 65.4%

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

9. Final simplification 65.4%

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

Alternative 10: 63.8% accurate, 19.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.1%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in x around 0 72.2%

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

6. Taylor expanded in x around 0 65.2%

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

7. Final simplification 65.2%

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

Alternative 11: 63.8% accurate, 19.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

   1. associate-*l* 97.1%

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

   2. associate-*l* 97.9%

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

3. Simplified 97.9%

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

5. Taylor expanded in x around 0 72.2%

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

6. Taylor expanded in x around 0 65.2%

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

   1. *-commutative 65.2%

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

   2. *-commutative 65.2%

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

   3. associate-*r* 65.2%

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

8. Simplified 65.2%

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

9. Final simplification 65.2%

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

Alternative 12: 63.8% accurate, 19.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0 72.2%

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

4. Taylor expanded in x around 0 65.4%

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

   1. *-commutative 65.4%

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

   2. *-commutative 65.4%

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

   3. associate-*r* 65.3%

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

6. Simplified 65.3%

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

7. Final simplification 65.3%

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

Reproduce

herbie shell --seed 2024018 
(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))))