Average Error: 0.6 → 0.7
Time: 3.7s
Precision: binary32
\[10^{-05} \leq x \land x \leq 1 \land 1 \leq tau \land tau \leq 5\]
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]
↓
\[\left(\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{1}{tau}\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]
\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}
↓
\left(\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \pi} \cdot \frac{1}{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)) (/ 1.0 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))) * (1.0f / 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 0.6

    \[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]
  2. Using strategy rm

  3. Applied div-inv_binary32_4110.7

    \[\leadsto \color{blue}{\left(\sin \left(\left(x \cdot \pi\right) \cdot tau\right) \cdot \frac{1}{\left(x \cdot \pi\right) \cdot tau}\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity_binary32_4140.7

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

  6. Applied times-frac_binary32_4200.8

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

  7. Applied associate-*r*_binary32_3560.8

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

  8. Simplified0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2020299 
(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))))