Average Error: 0.7 → 0.7
Time: 7.7s
Precision: binary32
\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
↓
\[\begin{array}{l} t_1 := \pi \cdot \left(tau \cdot x\right)\\ t_2 := \sin t_1\\ \sqrt[3]{\frac{{t_2}^{2} \cdot t_2}{{t_1}^{3}}} \cdot \sqrt[3]{\frac{{\sin \left(x \cdot \pi\right)}^{3}}{{\left(x \cdot \pi\right)}^{3}}} \end{array} \]
(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))) (t_2 (sin t_1)))
   (*
    (cbrt (/ (* (pow t_2 2.0) t_2) (pow t_1 3.0)))
    (cbrt (/ (pow (sin (* x PI)) 3.0) (pow (* x PI) 3.0))))))
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);
	float t_2 = sinf(t_1);
	return cbrtf(((powf(t_2, 2.0f) * t_2) / powf(t_1, 3.0f))) * cbrtf((powf(sinf((x * ((float) M_PI))), 3.0f) / powf((x * ((float) M_PI)), 3.0f)));
}

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(Float32(pi) * Float32(tau * x))
	t_2 = sin(t_1)
	return Float32(cbrt(Float32(Float32((t_2 ^ Float32(2.0)) * t_2) / (t_1 ^ Float32(3.0)))) * cbrt(Float32((sin(Float32(x * Float32(pi))) ^ Float32(3.0)) / (Float32(x * Float32(pi)) ^ Float32(3.0)))))
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 := \pi \cdot \left(tau \cdot x\right)\\
t_2 := \sin t_1\\
\sqrt[3]{\frac{{t_2}^{2} \cdot t_2}{{t_1}^{3}}} \cdot \sqrt[3]{\frac{{\sin \left(x \cdot \pi\right)}^{3}}{{\left(x \cdot \pi\right)}^{3}}}
\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. Taylor expanded in x around inf 0.7

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

  3. Applied egg-rr0.7

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

  4. Applied egg-rr0.7

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

  5. Applied egg-rr0.7

    \[\leadsto \sqrt[3]{\frac{\color{blue}{{\sin \left(\pi \cdot \left(tau \cdot x\right)\right)}^{2} \cdot \sin \left(\pi \cdot \left(tau \cdot x\right)\right)}}{{\left(\pi \cdot \left(tau \cdot x\right)\right)}^{3}}} \cdot \sqrt[3]{\frac{{\sin \left(x \cdot \pi\right)}^{3}}{{\left(x \cdot \pi\right)}^{3}}} \]

  6. Final simplification0.7

    \[\leadsto \sqrt[3]{\frac{{\sin \left(\pi \cdot \left(tau \cdot x\right)\right)}^{2} \cdot \sin \left(\pi \cdot \left(tau \cdot x\right)\right)}{{\left(\pi \cdot \left(tau \cdot x\right)\right)}^{3}}} \cdot \sqrt[3]{\frac{{\sin \left(x \cdot \pi\right)}^{3}}{{\left(x \cdot \pi\right)}^{3}}} \]

Reproduce

herbie shell --seed 2022152 
(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))))