Average Error: 0.7 → 0.7
Time: 7.9s
Precision: binary32
\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
↓
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{tau \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \pi\right)\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
\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(\left(x \cdot \pi\right) \cdot tau\right)}{tau \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \pi\right)\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}

(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)) (* tau (log1p (expm1 (* x PI)))))
  (/ (sin (* x PI)) (* x 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)));
}
↓
float code(float x, float tau) {
	return (sinf((x * ((float) M_PI)) * tau) / (tau * log1pf(expm1f(x * ((float) M_PI))))) * (sinf(x * ((float) M_PI)) / (x * ((float) M_PI)));
}

Error

Bits error versus x

Bits error versus tau

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Applied log1p-expm1-u_binary320.7

    \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \pi\right)\right)} \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

  3. Simplified0.7

    \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\pi \cdot x\right)}\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

  4. Final simplification0.7

    \[\leadsto \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{tau \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \pi\right)\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]

Reproduce

herbie shell --seed 2022082 
(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))))