Lanczos kernel

Average Error: 0.6 → 0.6

Time: 8.3s

Precision: binary32

\[10^{-05} \leq x \land x \leq 1 \land 1 \leq tau \land tau \leq 5\]

Math

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

↓

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

FPCore

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

↓

(FPCore (x tau)
 :precision binary32
 (* (/ (sin (* x (* PI tau))) (* x (* PI tau))) (/ (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) {
	return (sinf(x * (((float) M_PI) * tau)) / (x * (((float) M_PI) * tau))) * (sinf(x * ((float) M_PI)) / (x * ((float) M_PI)));
}

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 around inf 0.6

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

3. Final simplification 0.6

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

Reproduce

herbie shell --seed 2021173 
(FPCore (x tau)
  :name "Lanczos kernel"
  :precision binary32
  :pre (and (<= 1e-05 x 1.0) (<= 1.0 tau 5.0))
  (* (/ (sin (* (* x PI) tau)) (* (* x PI) tau)) (/ (sin (* x PI)) (* x PI))))