Lanczos kernel

Average Error: 2.11% → 2.11%

Time: 16.1s

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

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

↓

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

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 (* tau (* x PI))))
   (* (/ (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 = tau * (x * ((float) M_PI));
	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(tau * Float32(x * Float32(pi)))
	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 = tau * (x * single(pi));
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Derivation

1. Initial program 2.11

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

2. Final simplification 2.11

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

Alternatives

Alternative 1
Error	2.37%
Cost	19680

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

Alternative 2
Error	2.9%
Cost	19616

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

Alternative 3
Error	2.68%
Cost	19616

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

Alternative 4
Error	20.84%
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 5
Error	15.8%
Cost	16608

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

Alternative 6
Error	15.19%
Cost	16608

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

Alternative 7
Error	21.73%
Cost	13280

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

Alternative 8
Error	21.73%
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 9
Error	21.73%
Cost	10016

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

Alternative 10
Error	21.73%
Cost	6880

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

Alternative 11
Error	30.66%
Cost	6816

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

Alternative 12
Error	30.66%
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	35.79%
Cost	6688

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

Alternative 14
Error	35.79%
Cost	6624

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

Alternative 15
Error	36.79%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023090 
(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))))