Lanczos kernel

Average Error: 0.7 → 0.7

Time: 21.7s

Precision: binary32

Cost: 19808

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Derivation

1. Initial program 0.7

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

2. Simplified 0.7

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

   [Start] 0.7	\[ \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   rational.json-simplify-2 [=>] 0.7	\[ \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\color{blue}{tau \cdot \left(x \cdot \pi\right)}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   rational.json-simplify-2 [=>] 0.7	\[ \frac{\sin \color{blue}{\left(tau \cdot \left(x \cdot \pi\right)\right)}}{tau \cdot \left(x \cdot \pi\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   rational.json-simplify-2 [=>] 0.7	\[ \frac{\sin \left(tau \cdot \color{blue}{\left(\pi \cdot x\right)}\right)}{tau \cdot \left(x \cdot \pi\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   rational.json-simplify-43 [=>] 0.9	\[ \frac{\sin \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}}{tau \cdot \left(x \cdot \pi\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   rational.json-simplify-2 [=>] 0.9	\[ \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{tau \cdot \color{blue}{\left(\pi \cdot x\right)}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   rational.json-simplify-43 [=>] 0.7	\[ \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{\color{blue}{\pi \cdot \left(x \cdot tau\right)}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Applied egg-rr 0.8

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

4. Applied egg-rr 0.8

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

5. Applied egg-rr 0.7

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

6. Final simplification 0.7

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

Alternatives

Alternative 1
Error	0.9
Cost	19680

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

Alternative 2
Error	0.7
Cost	19680

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

Alternative 3
Error	0.7
Cost	19680

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

Alternative 4
Error	4.8
Cost	16736

\[\begin{array}{l} t_1 := \pi \cdot \left(tau \cdot x\right)\\ \left(-1 \cdot \left(-1 + \left(1 - \frac{\sin t_1}{t_1}\right)\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) \end{array} \]

Alternative 5
Error	4.9
Cost	16544

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

Alternative 6
Error	4.8
Cost	16544

\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + -0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) \end{array} \]

Alternative 7
Error	4.8
Cost	16544

\[\begin{array}{l} t_1 := \pi \cdot \left(x \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + -0.16666666666666666 \cdot {\left(x \cdot \pi\right)}^{2}\right) \end{array} \]

Alternative 8
Error	6.6
Cost	13344

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

Alternative 9
Error	6.6
Cost	13344

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

Alternative 10
Error	9.3
Cost	9952

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

Alternative 11
Error	9.3
Cost	9952

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

Alternative 12
Error	9.7
Cost	6752

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

Alternative 13
Error	11.4
Cost	6688

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

Alternative 14
Error	11.7
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023073 
(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))))