Average Error: 0.7 → 0.7
Time: 10.0s
Precision: binary32
\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
↓
\[\begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ t_2 := \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin t_2}{t_2} \end{array} \]
\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 := \left(x \cdot \pi\right) \cdot tau\\
t_2 := \mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\\
\frac{\sin t_1}{t_1} \cdot \frac{\sin t_2}{t_2}
\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 (* (* x PI) tau)) (t_2 (expm1 (log1p (* x PI)))))
   (* (/ (sin t_1) t_1) (/ (sin t_2) t_2))))
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 = (x * ((float) M_PI)) * tau;
	float t_2 = expm1f(log1pf(x * ((float) M_PI)));
	return (sinf(t_1) / t_1) * (sinf(t_2) / t_2);
}

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 expm1-log1p-u_binary320.7

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

  3. Applied expm1-log1p-u_binary320.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2022068 
(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))))