Average Error: 0.7 → 0.7
Time: 5.6s
Precision: binary32
\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
↓
\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \sqrt[3]{\frac{{\sin t_1}^{3}}{{t_1}^{3}}} \cdot \frac{\sin \left(x \cdot \pi\right)}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \pi\right)\right)} \end{array} \]
(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 (* x (* PI tau))))
   (*
    (cbrt (/ (pow (sin t_1) 3.0) (pow t_1 3.0)))
    (/ (sin (* x PI)) (log1p (expm1 (* 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) {
	float t_1 = x * (((float) M_PI) * tau);
	return cbrtf((powf(sinf(t_1), 3.0f) / powf(t_1, 3.0f))) * (sinf((x * ((float) M_PI))) / log1pf(expm1f((x * ((float) M_PI)))));
}

function code(x, tau)
	return Float32(Float32(sin(Float32(Float32(x * Float32(pi)) * tau)) / Float32(Float32(x * Float32(pi)) * tau)) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end
↓
function code(x, tau)
	t_1 = Float32(x * Float32(Float32(pi) * tau))
	return Float32(cbrt(Float32((sin(t_1) ^ Float32(3.0)) / (t_1 ^ Float32(3.0)))) * Float32(sin(Float32(x * Float32(pi))) / log1p(expm1(Float32(x * Float32(pi))))))
end

\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 := x \cdot \left(\pi \cdot tau\right)\\
\sqrt[3]{\frac{{\sin t_1}^{3}}{{t_1}^{3}}} \cdot \frac{\sin \left(x \cdot \pi\right)}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \pi\right)\right)}
\end{array}

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 egg-rr0.7

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

  3. Applied egg-rr0.7

    \[\leadsto \color{blue}{\sqrt[3]{\frac{{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}^{3}}{{\left(x \cdot \left(\pi \cdot tau\right)\right)}^{3}}}} \cdot \frac{\sin \left(x \cdot \pi\right)}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \pi\right)\right)} \]

  4. Final simplification0.7

    \[\leadsto \sqrt[3]{\frac{{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}^{3}}{{\left(x \cdot \left(\pi \cdot tau\right)\right)}^{3}}} \cdot \frac{\sin \left(x \cdot \pi\right)}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \pi\right)\right)} \]

Reproduce

herbie shell --seed 2022166 
(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))))