Average Error: 0.7 → 0.7
Time: 18.2s
Precision: binary32
\[10^{-05} \leq x \land x \leq 1 \land 1 \leq tau \land tau \leq 5\]
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]
↓
\[\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]
\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 (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))))
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

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. Using strategy rm

  3. Applied associate-/r*_binary320.7

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

  4. Simplified0.7

    \[\leadsto \frac{\color{blue}{\frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{\pi \cdot x}}}{tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity_binary320.7

    \[\leadsto \frac{\frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{\pi \cdot x}}{\color{blue}{1 \cdot tau}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]

  7. Applied *-un-lft-identity_binary320.7

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{\pi \cdot x}}}{1 \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]

  8. Applied times-frac_binary320.7

    \[\leadsto \color{blue}{\left(\frac{1}{1} \cdot \frac{\frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{\pi \cdot x}}{tau}\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]

  9. Simplified0.7

    \[\leadsto \left(\color{blue}{1} \cdot \frac{\frac{\sin \left(tau \cdot \left(\pi \cdot x\right)\right)}{\pi \cdot x}}{tau}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]

  10. Simplified0.7

    \[\leadsto \left(1 \cdot \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)}}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]

  11. Final simplification0.7

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

Reproduce

herbie shell --seed 2021174 
(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))))