Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 11.7s

Alternatives: 10

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 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around inf 97.2%

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

5. Final simplification 97.2%

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

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 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 97.9%

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

3. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 4: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. associate-/r* 97.8%

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

   2. div-inv 97.8%

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

5. Applied egg-rr 97.8%

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

   1. un-div-inv 97.8%

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

7. Applied egg-rr 97.8%

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

8. Taylor expanded in x around 0 79.3%

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

   1. +-commutative 79.3%

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

   2. fma-def 79.3%

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

   3. *-commutative 79.3%

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

   4. unpow2 79.3%

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

   5. unpow2 79.3%

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

   6. swap-sqr 79.3%

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

   7. unpow2 79.3%

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

   8. swap-sqr 79.3%

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

   9. associate-*r* 79.3%

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

   10. associate-*r* 79.3%

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

   11. unpow2 79.3%

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

   12. associate-*r* 79.3%

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

10. Simplified 79.3%

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

11. Final simplification 79.3%

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

Alternative 5: 70.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*l* 97.2%

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

   3. *-commutative 97.2%

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

   4. associate-*l* 98.0%

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

3. Simplified 98.0%

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

   1. associate-/r* 97.8%

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

   2. div-inv 97.8%

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

5. Applied egg-rr 97.8%

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

6. Taylor expanded in x around 0 71.2%

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

7. Final simplification 71.2%

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

Alternative 6: 64.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, tau)
	return fma(Float32(-0.16666666666666666), Float32(Float32(pi) * Float32(x * Float32(Float32(pi) * x))), Float32(1.0))
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 64.9%

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

5. Taylor expanded in x around 0 65.0%

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

   1. +-commutative 65.0%

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

   2. fma-def 65.0%

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

   3. unpow2 65.0%

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

   4. unpow2 65.0%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

7. Simplified 65.0%

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

   1. unpow2 65.0%

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

   2. associate-*r* 65.0%

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

9. Applied egg-rr 65.0%

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

10. Final simplification 65.0%

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

Alternative 7: 64.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 64.9%

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

5. Taylor expanded in x around 0 65.0%

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

   1. +-commutative 65.0%

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

   2. fma-def 65.0%

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

   3. unpow2 65.0%

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

   4. unpow2 65.0%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

7. Simplified 65.0%

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

8. Final simplification 65.0%

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

Alternative 8: 64.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0) + (single(-0.16666666666666666) * ((single(pi) * x) ^ single(2.0)));
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 64.9%

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

5. Taylor expanded in x around 0 65.0%

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

   1. +-commutative 65.0%

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

   2. fma-def 65.0%

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

   3. unpow2 65.0%

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

   4. unpow2 65.0%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

7. Simplified 65.0%

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

   1. fma-udef 65.0%

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

   2. *-commutative 65.0%

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

9. Applied egg-rr 65.0%

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

10. Final simplification 65.0%

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

Alternative 9: 4.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, tau)
	tmp = single(-0.16666666666666666) * ((single(pi) * x) ^ single(2.0));
end

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 64.9%

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

5. Taylor expanded in x around 0 65.0%

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

   1. +-commutative 65.0%

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

   2. fma-def 65.0%

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

   3. unpow2 65.0%

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

   4. unpow2 65.0%

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

   5. swap-sqr 65.0%

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

   6. unpow2 65.0%

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

7. Simplified 65.0%

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

8. Taylor expanded in x around inf 4.0%

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

   1. unpow2 4.0%

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

   2. unpow2 4.0%

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

   3. swap-sqr 4.0%

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

   4. unpow2 4.0%

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

10. Simplified 4.0%

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

11. Final simplification 4.0%

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

Alternative 10: 63.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 97.9%

   \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
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* 97.9%

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

3. Simplified 97.9%

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

4. Taylor expanded in x around 0 64.9%

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

5. Taylor expanded in x around 0 64.1%

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

6. Final simplification 64.1%

   \[\leadsto \frac{\pi \cdot x}{\pi \cdot x} \]

Reproduce

herbie shell --seed 2023333 
(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))))