Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 12.9s

Alternatives: 11

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. Final simplification 98.0%

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

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float tau) {
	return (sinf((x * ((float) M_PI))) / (x * ((float) M_PI))) * (sinf(((x * ((float) M_PI)) * tau)) / (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(Float32(x * Float32(pi)) * tau)) / Float32(x * Float32(Float32(pi) * tau))))
end

MATLAB

function tmp = code(x, tau)
	tmp = (sin((x * single(pi))) / (x * single(pi))) * (sin(((x * single(pi)) * tau)) / (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* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around inf 97.5%

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

6. Final simplification 97.5%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

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

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(Float32(sin(Float32(x * Float32(pi))) / 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. Step-by-step derivation

   1. expm1-log1p-u 97.9%

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

4. Applied egg-rr 97.9%

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

   1. clear-num 97.8%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \pi}{\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}} \]

   2. expm1-log1p-u 97.9%

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

   3. inv-pow 97.9%

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

6. Applied egg-rr 97.9%

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

   1. unpow-1 97.9%

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

   2. clear-num 98.0%

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

   3. associate-/r* 97.8%

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

8. Applied egg-rr 97.8%

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

9. Taylor expanded in x around inf 97.8%

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

    1. *-commutative 97.8%

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

    2. associate-*r* 97.4%

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

    3. *-commutative 97.4%

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

    4. *-commutative 97.4%

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

    5. associate-*r* 97.9%

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

    6. *-commutative 97.9%

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

11. Simplified 97.9%

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

12. Final simplification 97.9%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (* PI tau))))
   (* (/ (sin (* 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 (sinf((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(sin(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 = (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. associate-*l* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 5: 70.6% accurate, 1.8× speedup

Math

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

FPCore

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

MATLAB

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

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

4. Applied egg-rr 97.9%

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

   1. clear-num 97.8%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \pi}{\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}} \]

   2. expm1-log1p-u 97.9%

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

   3. inv-pow 97.9%

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

6. Applied egg-rr 97.9%

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

   1. unpow-1 97.9%

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

   2. clear-num 98.0%

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

   3. associate-/r* 97.8%

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

8. Applied egg-rr 97.8%

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

9. Taylor expanded in x around 0 71.4%

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

10. Final simplification 71.4%

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

Alternative 6: 70.7% 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%

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

4. Applied egg-rr 97.9%

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

   1. clear-num 97.8%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\frac{1}{\frac{x \cdot \pi}{\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}} \]

   2. expm1-log1p-u 97.9%

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

   3. inv-pow 97.9%

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

6. Applied egg-rr 97.9%

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

7. Taylor expanded in x around 0 71.4%

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

8. Final simplification 71.4%

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

Alternative 7: 64.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (sin((x * single(pi))) / x) * (single(1.0) / 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* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around 0 64.5%

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

   1. associate-/r* 64.6%

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

   2. div-inv 64.6%

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

7. Applied egg-rr 64.6%

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

8. Final simplification 64.6%

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

Alternative 8: 64.3% 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* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around 0 64.5%

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

6. Final simplification 64.5%

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

Alternative 9: 64.2% accurate, 2.0× speedup

Math

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

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(Float32(sin(Float32(x * Float32(pi))) / 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* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around 0 64.5%

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

   1. *-un-lft-identity 64.5%

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

   2. times-frac 64.5%

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

7. Applied egg-rr 64.5%

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

   1. *-commutative 64.5%

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

   2. associate-*l/ 64.4%

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

   3. div-inv 64.6%

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

9. Applied egg-rr 64.6%

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

10. Final simplification 64.6%

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

Alternative 10: 63.5% accurate, 31.3× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \pi}{x \cdot \pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = (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* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around 0 64.5%

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

6. Taylor expanded in x around 0 63.8%

   \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \pi}}{x \cdot \pi} \]

7. Final simplification 63.8%

   \[\leadsto \frac{x \cdot \pi}{x \cdot \pi} \]
8. Add Preprocessing

Alternative 11: 63.4% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = x * (single(1.0) / x);
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* 97.4%

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

   2. associate-*l* 98.0%

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

3. Simplified 98.0%

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

5. Taylor expanded in x around 0 64.5%

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

   1. *-un-lft-identity 64.5%

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

   2. times-frac 64.5%

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

7. Applied egg-rr 64.5%

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

8. Taylor expanded in x around 0 63.8%

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

9. Final simplification 63.8%

   \[\leadsto x \cdot \frac{1}{x} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))