Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 13.2s

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: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. expm1-log1p-u 97.9%

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

   2. expm1-undefine 82.4%

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

4. Applied egg-rr 82.4%

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

   1. expm1-define 97.9%

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

6. Simplified 97.9%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}{x \cdot \pi} \]
7. Step-by-step derivation

   1. *-commutative 97.9%

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

   2. expm1-log1p-u 98.0%

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

   3. associate-/r* 97.8%

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

   4. *-commutative 97.8%

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

   5. associate-*r* 97.4%

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

   6. frac-times 97.3%

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

8. Applied egg-rr 97.7%

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

   1. associate-/l* 97.7%

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

   2. associate-*r* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*r* 97.3%

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

   5. associate-*r* 97.4%

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

   6. *-commutative 97.4%

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

   7. associate-*r* 97.7%

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

10. Applied egg-rr 97.7%

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

11. Final simplification 97.7%

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

Alternative 3: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = sin(((x * single(pi)) * tau)) * (sin((x * single(pi))) / ((x * single(pi)) * (x * (single(pi) * tau))));
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* 97.7%

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

   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* 97.6%

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

3. Simplified 97.6%

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

5. 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%

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

Alternative 4: 70.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. expm1-log1p-u 97.9%

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

   2. expm1-undefine 82.4%

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

4. Applied egg-rr 82.4%

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

   1. expm1-define 97.9%

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

6. Simplified 97.9%

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

7. Taylor expanded in x around 0 71.0%

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

   1. frac-times 71.0%

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

   2. *-commutative 71.0%

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

   3. associate-*r* 70.7%

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

   4. *-commutative 70.7%

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

   5. associate-*r* 71.0%

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

9. Applied egg-rr 71.0%

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

10. Final simplification 71.0%

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

Alternative 5: 70.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   2. expm1-undefine 82.4%

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

4. Applied egg-rr 82.4%

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

   1. expm1-define 97.9%

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

6. Simplified 97.9%

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

7. Taylor expanded in x around 0 71.0%

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

   1. *-inverses 71.0%

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

   2. *-commutative 71.0%

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

   3. *-un-lft-identity 71.0%

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

   4. clear-num 71.0%

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

   5. *-commutative 71.0%

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

   6. associate-*r* 70.6%

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

   7. *-commutative 70.6%

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

   8. associate-*r* 71.0%

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

9. Applied egg-rr 71.0%

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

Alternative 6: 70.6% accurate, 2.0× speedup

Math

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

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(Float32(x * 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 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 97.9%

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

   2. expm1-undefine 82.4%

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

4. Applied egg-rr 82.4%

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

   1. expm1-define 97.9%

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

6. Simplified 97.9%

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

7. Taylor expanded in x around 0 71.0%

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

8. Taylor expanded in x around inf 71.0%

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

9. Final simplification 71.0%

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

Alternative 7: 63.8% 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 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* 97.7%

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

   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* 97.3%

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

   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 64.7%

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

Alternative 8: 63.0% 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 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* 97.7%

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

   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* 97.3%

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

   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 63.9%

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

Reproduce

herbie shell --seed 2024145 
(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))))