Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 11.0s

Alternatives: 9

Speedup: 1.0×

Specification

\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

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 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* 97.6%

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

   2. associate-*l* 98.1%

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

3. Simplified 98.1%

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

5. Final simplification 98.1%

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

Alternative 2: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(x \cdot \pi\right) \cdot \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau \cdot {\left(x \cdot \pi\right)}^{2}} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (* (sin (* x PI)) (/ (sin (* x (* PI tau))) (* tau (pow (* x PI) 2.0)))))

C

float code(float x, float tau) {
	return sinf((x * ((float) M_PI))) * (sinf((x * (((float) M_PI) * tau))) / (tau * powf((x * ((float) M_PI)), 2.0f)));
}

Julia

function code(x, tau)
	return Float32(sin(Float32(x * Float32(pi))) * Float32(sin(Float32(x * Float32(Float32(pi) * tau))) / Float32(tau * (Float32(x * Float32(pi)) ^ Float32(2.0)))))
end

MATLAB

function tmp = code(x, tau)
	tmp = sin((x * single(pi))) * (sin((x * (single(pi) * tau))) / (tau * ((x * single(pi)) ^ single(2.0))));
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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* 97.8%

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

   2. frac-times 97.9%

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

   3. *-commutative 97.9%

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

   4. associate-*l* 97.4%

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

   5. *-commutative 97.4%

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

   6. *-commutative 97.4%

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

   7. associate-*l* 98.0%

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in x around inf 96.8%

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

   1. associate-/l* 96.9%

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{\frac{tau \cdot \left({x}^{2} \cdot {\pi}^{2}\right)}{\sin \left(x \cdot \pi\right)}}} \]

   2. associate-*r* 97.2%

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \pi\right)}}{\frac{tau \cdot \left({x}^{2} \cdot {\pi}^{2}\right)}{\sin \left(x \cdot \pi\right)}} \]

   3. *-commutative 97.2%

      \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \pi\right)}{\frac{tau \cdot \left({x}^{2} \cdot {\pi}^{2}\right)}{\sin \left(x \cdot \pi\right)}} \]

   4. *-commutative 97.2%

      \[\leadsto \frac{\sin \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}}{\frac{tau \cdot \left({x}^{2} \cdot {\pi}^{2}\right)}{\sin \left(x \cdot \pi\right)}} \]

   5. *-commutative 97.2%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\frac{\color{blue}{\left({x}^{2} \cdot {\pi}^{2}\right) \cdot tau}}{\sin \left(x \cdot \pi\right)}} \]

   6. *-commutative 97.2%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\frac{\color{blue}{\left({\pi}^{2} \cdot {x}^{2}\right)} \cdot tau}{\sin \left(x \cdot \pi\right)}} \]

   7. unpow2 97.2%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\frac{\left(\color{blue}{\left(\pi \cdot \pi\right)} \cdot {x}^{2}\right) \cdot tau}{\sin \left(x \cdot \pi\right)}} \]

   8. unpow2 97.2%

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

   9. swap-sqr 97.3%

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

   10. unpow2 97.3%

       \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\frac{\color{blue}{{\left(\pi \cdot x\right)}^{2}} \cdot tau}{\sin \left(x \cdot \pi\right)}} \]

   11. *-commutative 97.3%

       \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\frac{{\left(\pi \cdot x\right)}^{2} \cdot tau}{\sin \color{blue}{\left(\pi \cdot x\right)}}} \]

   12. *-commutative 97.3%

       \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\frac{\color{blue}{tau \cdot {\left(\pi \cdot x\right)}^{2}}}{\sin \left(\pi \cdot x\right)}} \]

   13. associate-/r/ 97.1%

       \[\leadsto \color{blue}{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{tau \cdot {\left(\pi \cdot x\right)}^{2}} \cdot \sin \left(\pi \cdot x\right)} \]

7. Simplified 97.2%

   \[\leadsto \color{blue}{\sin \left(x \cdot \pi\right) \cdot \frac{\sin \left(x \cdot \left(tau \cdot \pi\right)\right)}{tau \cdot {\left(x \cdot \pi\right)}^{2}}} \]

8. Final simplification 97.2%

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

Alternative 3: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(x \cdot \pi\right) \cdot \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau \cdot {\left(x \cdot \pi\right)}^{2}} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (* (sin (* x PI)) (/ (sin (* tau (* x PI))) (* tau (pow (* x PI) 2.0)))))

C

float code(float x, float tau) {
	return sinf((x * ((float) M_PI))) * (sinf((tau * (x * ((float) M_PI)))) / (tau * powf((x * ((float) M_PI)), 2.0f)));
}

Julia

function code(x, tau)
	return Float32(sin(Float32(x * Float32(pi))) * Float32(sin(Float32(tau * Float32(x * Float32(pi)))) / Float32(tau * (Float32(x * Float32(pi)) ^ Float32(2.0)))))
end

MATLAB

function tmp = code(x, tau)
	tmp = sin((x * single(pi))) * (sin((tau * (x * single(pi)))) / (tau * ((x * single(pi)) ^ single(2.0))));
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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* 97.7%

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

   2. div-inv 97.7%

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

4. Applied egg-rr 97.7%

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

5. Taylor expanded in x around inf 96.8%

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

   1. *-commutative 96.8%

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \pi\right) \cdot \sin \left(tau \cdot \left(x \cdot \pi\right)\right)}}{tau \cdot \left({x}^{2} \cdot {\pi}^{2}\right)} \]

   2. *-commutative 96.8%

      \[\leadsto \frac{\sin \left(x \cdot \pi\right) \cdot \sin \color{blue}{\left(\left(x \cdot \pi\right) \cdot tau\right)}}{tau \cdot \left({x}^{2} \cdot {\pi}^{2}\right)} \]

   3. associate-*r* 97.1%

      \[\leadsto \frac{\sin \left(x \cdot \pi\right) \cdot \sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{tau \cdot \left({x}^{2} \cdot {\pi}^{2}\right)} \]

   4. unpow2 97.1%

      \[\leadsto \frac{\sin \left(x \cdot \pi\right) \cdot \sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {\pi}^{2}\right)} \]

   5. unpow2 97.1%

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

   6. swap-sqr 97.3%

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

   7. unpow2 97.3%

      \[\leadsto \frac{\sin \left(x \cdot \pi\right) \cdot \sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau \cdot \color{blue}{{\left(x \cdot \pi\right)}^{2}}} \]

   8. associate-*r/ 97.2%

      \[\leadsto \color{blue}{\sin \left(x \cdot \pi\right) \cdot \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau \cdot {\left(x \cdot \pi\right)}^{2}}} \]

   9. associate-*r* 97.4%

      \[\leadsto \sin \left(x \cdot \pi\right) \cdot \frac{\sin \color{blue}{\left(\left(x \cdot \pi\right) \cdot tau\right)}}{tau \cdot {\left(x \cdot \pi\right)}^{2}} \]

7. Simplified 97.4%

   \[\leadsto \color{blue}{\sin \left(x \cdot \pi\right) \cdot \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{tau \cdot {\left(x \cdot \pi\right)}^{2}}} \]

8. Final simplification 97.4%

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

Alternative 4: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{\frac{tau \cdot {\left(x \cdot \pi\right)}^{2}}{\sin \left(x \cdot \pi\right)}} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (/ (sin (* tau (* x PI))) (/ (* tau (pow (* x PI) 2.0)) (sin (* x PI)))))

C

float code(float x, float tau) {
	return sinf((tau * (x * ((float) M_PI)))) / ((tau * powf((x * ((float) M_PI)), 2.0f)) / sinf((x * ((float) M_PI))));
}

Julia

function code(x, tau)
	return Float32(sin(Float32(tau * Float32(x * Float32(pi)))) / Float32(Float32(tau * (Float32(x * Float32(pi)) ^ Float32(2.0))) / sin(Float32(x * Float32(pi)))))
end

MATLAB

function tmp = code(x, tau)
	tmp = sin((tau * (x * single(pi)))) / ((tau * ((x * single(pi)) ^ single(2.0))) / sin((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. frac-times 97.9%

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

   2. associate-/l* 97.8%

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

   3. *-commutative 97.8%

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

   4. associate-*l* 97.3%

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

   5. *-commutative 97.3%

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

   6. associate-*l* 97.3%

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

   7. pow2 97.3%

      \[\leadsto \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\frac{tau \cdot \color{blue}{{\left(x \cdot \pi\right)}^{2}}}{\sin \left(x \cdot \pi\right)}} \]

4. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\frac{tau \cdot {\left(x \cdot \pi\right)}^{2}}{\sin \left(x \cdot \pi\right)}}} \]

5. Taylor expanded in x around inf 97.5%

   \[\leadsto \frac{\color{blue}{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}}{\frac{tau \cdot {\left(x \cdot \pi\right)}^{2}}{\sin \left(x \cdot \pi\right)}} \]

6. Final simplification 97.5%

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

Alternative 5: 78.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma((x ^ Float32(2.0)), Float32(Float32(-0.16666666666666666) * Float32(Float32((tau ^ Float32(2.0)) + Float32(1.0)) * (Float32(pi) ^ Float32(2.0)))), Float32(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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 98.0%

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

   2. associate-*l/ 98.0%

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

   3. *-commutative 98.0%

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

   4. associate-*l* 97.4%

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

   5. *-commutative 97.4%

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

   6. associate-*l* 98.0%

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in x around 0 78.1%

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

   1. +-commutative 78.1%

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

   2. fma-def 78.1%

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

   3. distribute-lft-out 78.1%

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

   4. distribute-lft1-in 78.1%

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

7. Simplified 78.1%

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

8. Final simplification 78.1%

   \[\leadsto \mathsf{fma}\left({x}^{2}, -0.16666666666666666 \cdot \left(\left({tau}^{2} + 1\right) \cdot {\pi}^{2}\right), 1\right) \]
9. Add Preprocessing

Alternative 6: 70.7% 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 98.0%

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

   1. associate-*l* 97.6%

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

   2. associate-*l* 98.1%

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

3. Simplified 98.1%

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

5. Taylor expanded in x around 0 70.8%

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

6. Final simplification 70.8%

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

Alternative 7: 64.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(-0.16666666666666666), (Float32(x * Float32(pi)) ^ Float32(2.0)), Float32(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. Add Preprocessing

3. Taylor expanded in x around 0 85.4%

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

   1. +-commutative 85.4%

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

   2. fma-def 85.4%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {x}^{2} \cdot {\pi}^{2}, 1\right)} \]

   3. unpow2 85.4%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\left(x \cdot x\right)} \cdot {\pi}^{2}, 1\right) \]

   4. unpow2 85.4%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(-0.16666666666666666, \left(x \cdot x\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}, 1\right) \]

   5. swap-sqr 85.4%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)}, 1\right) \]

   6. unpow2 85.4%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{\left(x \cdot \pi\right)}^{2}}, 1\right) \]

5. Simplified 85.4%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(x \cdot \pi\right)}^{2}, 1\right)} \]

6. Taylor expanded in tau around 0 64.5%

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

   1. +-commutative 64.5%

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

   2. unpow2 64.5%

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

   3. unpow2 64.5%

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

   4. swap-sqr 64.5%

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

   5. unpow2 64.5%

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

   6. fma-udef 64.5%

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

8. Simplified 64.5%

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

9. Final simplification 64.5%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, {\left(x \cdot \pi\right)}^{2}, 1\right) \]
10. Add Preprocessing

Alternative 8: 64.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (((x * single(pi)) ^ single(2.0)) * single(-0.16666666666666666));
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. Taylor expanded in x around 0 85.4%

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

   1. +-commutative 85.4%

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

   2. fma-def 85.4%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {x}^{2} \cdot {\pi}^{2}, 1\right)} \]

   3. unpow2 85.4%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\left(x \cdot x\right)} \cdot {\pi}^{2}, 1\right) \]

   4. unpow2 85.4%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(-0.16666666666666666, \left(x \cdot x\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}, 1\right) \]

   5. swap-sqr 85.4%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\left(x \cdot \pi\right) \cdot \left(x \cdot \pi\right)}, 1\right) \]

   6. unpow2 85.4%

      \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{\left(x \cdot \pi\right)}^{2}}, 1\right) \]

5. Simplified 85.4%

   \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(x \cdot \pi\right)}^{2}, 1\right)} \]

6. Taylor expanded in tau around 0 64.5%

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

   1. +-commutative 64.5%

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

   2. unpow2 64.5%

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

   3. unpow2 64.5%

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

   4. swap-sqr 64.5%

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

   5. unpow2 64.5%

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

   6. fma-udef 64.5%

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

8. Simplified 64.5%

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

   1. fma-udef 64.5%

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

   2. *-commutative 64.5%

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

10. Applied egg-rr 64.5%

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

11. Final simplification 64.5%

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

Alternative 9: 63.2% accurate, 615.0× 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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 98.0%

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

   2. associate-*l/ 98.0%

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

   3. *-commutative 98.0%

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

   4. associate-*l* 97.4%

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

   5. *-commutative 97.4%

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

   6. associate-*l* 98.0%

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in x around 0 63.5%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 63.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))))