Average Error: 0.7 → 0.7
Time: 12.7s
Precision: binary32
\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
↓
\[\begin{array}{l} t_1 := \pi \cdot \left(tau \cdot x\right)\\ \frac{\sin t_1}{t_1} \cdot \frac{1}{\frac{\pi \cdot x}{\sin \left(\pi \cdot x\right)}} \end{array} \]
\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{1}{\frac{\pi \cdot x}{\sin \left(\pi \cdot x\right)}}
\end{array}

(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) (/ 1.0 (/ (* PI x) (sin (* PI x)))))))
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) * (1.0f / ((((float) M_PI) * x) / sinf(((float) M_PI) * x)));
}

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 clear-num_binary320.7

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

  3. Taylor expanded in x around inf 0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2021280 
(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))))