Lanczos kernel

Average Error: 0.6 → 0.6

Time: 6.8s

Precision: binary32

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

Math

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

↓

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

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

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

Error

Bits error versus x

Bits error versus tau

Derivation

1. Initial program 0.6

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

2. Taylor expanded in x around inf 0.6

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

3. Final simplification 0.6

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

Reproduce

herbie shell --seed 2021310 
(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))))