Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 14.1s

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. associate-*l* 97.7%

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

   3. *-commutative 97.7%

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

   4. associate-*l* 98.4%

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

3. Simplified 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around inf 97.6%

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

6. Final simplification 97.6%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 98.3%

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

3. Simplified 98.3%

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

5. Final simplification 98.3%

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

Alternative 4: 70.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around 0 72.7%

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

6. Taylor expanded in x around inf 72.3%

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

7. Final simplification 72.3%

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

Alternative 5: 71.0% 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 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around 0 72.7%

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

6. Final simplification 72.7%

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

Alternative 6: 64.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 98.3%

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

3. Simplified 98.3%

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

   1. clear-num 98.1%

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

   2. associate-/r/ 98.0%

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

   3. associate-*r* 97.5%

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

   4. *-commutative 97.5%

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

   5. associate-*r* 97.5%

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

   6. associate-*r* 97.8%

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

   7. *-commutative 97.8%

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

   8. associate-*r* 98.1%

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

6. Applied egg-rr 98.1%

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

   1. associate-*r* 97.8%

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

   2. *-commutative 97.8%

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

   3. associate-*r* 98.2%

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

   4. *-commutative 98.2%

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

   5. associate-*l/ 98.3%

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

   6. associate-*r* 97.6%

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

   7. *-un-lft-identity 97.6%

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

   8. associate-/r* 97.6%

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

8. Applied egg-rr 97.9%

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

9. Taylor expanded in x around 0 67.0%

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

10. Final simplification 67.0%

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

Alternative 7: 63.6% accurate, 16.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around 0 72.7%

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

6. Taylor expanded in x around 0 65.9%

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

   1. *-commutative 65.9%

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

   2. *-commutative 65.9%

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

   3. associate-*r* 65.9%

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

8. Simplified 65.9%

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

   1. associate-*r* 65.9%

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

   2. /-rgt-identity 65.9%

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

   3. *-commutative 65.9%

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

   4. associate-*r* 66.1%

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

   5. *-commutative 66.1%

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

   6. associate-/l* 66.1%

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

10. Applied egg-rr 66.1%

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

11. Final simplification 66.1%

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

Alternative 8: 63.6% 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 98.3%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around 0 72.7%

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

   1. associate-*r* 72.4%

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

   2. /-rgt-identity 72.4%

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

   3. *-commutative 72.4%

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

   4. associate-*r* 72.4%

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

   5. associate-/l* 72.5%

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

7. Applied egg-rr 72.5%

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

   1. associate-/r/ 72.4%

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

   2. /-rgt-identity 72.4%

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

   3. associate-*r* 72.4%

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

   4. *-commutative 72.4%

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

   5. associate-/l/ 72.4%

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

   6. *-un-lft-identity 72.4%

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

   7. times-frac 72.4%

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

9. Applied egg-rr 72.4%

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

10. Taylor expanded in x around 0 66.1%

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

11. Final simplification 66.1%

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

Reproduce

herbie shell --seed 2024020 
(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))))