Lanczos kernel

Average Error: 2.12% → 2.11%

Time: 15.8s

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

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

↓

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

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 (* x (* PI tau))))
   (* (/ (sin t_1) t_1) (/ (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 = 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)
	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(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)
	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 = x * (single(pi) * tau);
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Derivation

1. Initial program 2.12

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

2. Simplified 2.11

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

   [Start] 2.12	\[ \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   associate-*l* [=>] 2.73	\[ \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   associate-*l* [=>] 2.11	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\color{blue}{x \cdot \left(\pi \cdot tau\right)}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

3. Final simplification 2.11

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

Alternatives

Alternative 1
Error	2.9%
Cost	19616

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

Alternative 2
Error	2.88%
Cost	19616

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

Alternative 3
Error	2.89%
Cost	19616

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

Alternative 4
Error	2.52%
Cost	19616

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

Alternative 5
Error	21.06%
Cost	16608

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

Alternative 6
Error	15.24%
Cost	16608

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

Alternative 7
Error	21.31%
Cost	16480

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

Alternative 8
Error	21.6%
Cost	16448

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

Alternative 9
Error	21.87%
Cost	10016

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

Alternative 10
Error	21.87%
Cost	6880

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

Alternative 11
Error	35.97%
Cost	6624

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

Alternative 12
Error	36.91%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023089 
(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))))