Lanczos kernel

Average Error: 0.7 → 0.8

Time: 16.9s

Precision: binary32

Cost: 19680

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

Math

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

↓

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

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) (* PI x)) (/ (sin (* PI x)) t_1))))

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) / (((float) M_PI) * x)) * (sinf((((float) M_PI) * x)) / t_1);
}

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) / Float32(Float32(pi) * x)) * Float32(sin(Float32(Float32(pi) * x)) / t_1))
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) / (single(pi) * x)) * (sin((single(pi) * x)) / t_1);
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 0.8

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

   [Start] 0.7	\[ \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   associate-*l/ [=>] 0.7	\[ \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{\left(x \cdot \pi\right) \cdot tau}} \]
   times-frac [=>] 0.8	\[ \color{blue}{\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau}} \]
   *-commutative [=>] 0.8	\[ \frac{\sin \left(\color{blue}{\left(\pi \cdot x\right)} \cdot tau\right)}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau} \]
   associate-*l* [=>] 0.9	\[ \frac{\sin \color{blue}{\left(\pi \cdot \left(x \cdot tau\right)\right)}}{x \cdot \pi} \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}}{tau} \]
   associate-/r* [<=] 0.9	\[ \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x \cdot \pi} \cdot \color{blue}{\frac{\sin \left(x \cdot \pi\right)}{\left(x \cdot \pi\right) \cdot tau}} \]
   *-commutative [=>] 0.9	\[ \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\left(\pi \cdot x\right)} \cdot tau} \]
   associate-*l* [=>] 0.8	\[ \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{x \cdot \pi} \cdot \frac{\sin \left(x \cdot \pi\right)}{\color{blue}{\pi \cdot \left(x \cdot tau\right)}} \]

3. Final simplification 0.8

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

Alternatives

Alternative 1
Error	0.9
Cost	19616

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

Alternative 2
Error	0.9
Cost	19616

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

Alternative 3
Error	0.9
Cost	19616

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

Alternative 4
Error	0.9
Cost	19616

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

Alternative 5
Error	0.9
Cost	19616

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

Alternative 6
Error	4.8
Cost	16608

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

Alternative 7
Error	4.8
Cost	13376

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

Alternative 8
Error	6.3
Cost	10016

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

Alternative 9
Error	9.4
Cost	9952

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

Alternative 10
Error	9.4
Cost	9888

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

Alternative 11
Error	9.4
Cost	9888

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

Alternative 12
Error	9.8
Cost	6816

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

Alternative 13
Error	11.4
Cost	6688

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

Alternative 14
Error	11.4
Cost	6688

\[1 + -0.16666666666666666 \cdot \left(\pi \cdot \left(x \cdot \left(\pi \cdot x\right)\right)\right) \]

Alternative 15
Error	11.4
Cost	6688

\[1 + -0.16666666666666666 \cdot \left(x \cdot \left(\pi \cdot \left(\pi \cdot x\right)\right)\right) \]

Alternative 16
Error	11.4
Cost	6624

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

Alternative 17
Error	11.7
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023067 
(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))))