?

Average Accuracy: 97.9% → 97.9%
Time: 18.8s
Precision: binary32
Cost: 19680

?

\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
↓
\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \]
(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))))
   (* (/ (sin t_1) t_1) (/ (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) {
	float t_1 = x * (((float) M_PI) * tau);
	return (sinf(t_1) / t_1) * (sinf((x * ((float) M_PI))) / (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(Float32(sin(t_1) / t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
end

function tmp = code(x, tau)
	tmp = (sin(((x * single(pi)) * tau)) / ((x * single(pi)) * tau)) * (sin((x * single(pi))) / (x * single(pi)));
end
↓
function tmp = code(x, tau)
	t_1 = x * (single(pi) * tau);
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(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)\\
\frac{\sin t_1}{t_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}
\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 97.9%

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

  2. Simplified97.9%

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

    [Start]97.9

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

    associate-*l* [=>]97.3

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

    associate-*l* [=>]97.9

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

  3. Final simplification97.9%

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

Alternatives

Alternative 1
Accuracy97.1%
Cost19616
\[\sin \left(x \cdot \pi\right) \cdot \left({\left(x \cdot \pi\right)}^{-2} \cdot \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau}\right) \]
Alternative 2
Accuracy97.3%
Cost19616
\[\sin \left(x \cdot \pi\right) \cdot \frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau \cdot {\left(x \cdot \pi\right)}^{2}} \]
Alternative 3
Accuracy97.2%
Cost19616
\[\frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{tau} \cdot \sin \left(x \cdot \pi\right)}{{\left(x \cdot \pi\right)}^{2}} \]
Alternative 4
Accuracy84.7%
Cost16608
\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + {\pi}^{2} \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right) \end{array} \]
Alternative 5
Accuracy84.7%
Cost16544
\[\begin{array}{l} t_1 := tau \cdot \left(x \cdot \pi\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + {\left(x \cdot \pi\right)}^{2} \cdot -0.16666666666666666\right) \end{array} \]
Alternative 6
Accuracy84.0%
Cost13312
\[\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\frac{tau}{\frac{1}{x \cdot \pi} + \left(x \cdot \pi\right) \cdot -0.16666666666666666}} \]
Alternative 7
Accuracy79.8%
Cost10016
\[e^{\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left({\pi}^{2} \cdot \left(1 + tau \cdot tau\right)\right)} \]
Alternative 8
Accuracy70.2%
Cost9888
\[\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau \cdot \left(x \cdot \pi\right)} \]
Alternative 9
Accuracy70.4%
Cost9888
\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \end{array} \]
Alternative 10
Accuracy70.4%
Cost9888
\[\begin{array}{l} t_1 := \pi \cdot \left(x \cdot tau\right)\\ \frac{\sin t_1}{t_1} \end{array} \]
Alternative 11
Accuracy70.4%
Cost9888
\[\begin{array}{l} t_1 := tau \cdot \left(x \cdot \pi\right)\\ \frac{\sin t_1}{t_1} \end{array} \]
Alternative 12
Accuracy69.1%
Cost6816
\[1 + \left(-0.16666666666666666 \cdot \left(tau \cdot tau\right)\right) \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right) \]
Alternative 13
Accuracy63.9%
Cost6752
\[1 + -0.16666666666666666 \cdot {\left(\left(x \cdot \pi + 1\right) + -1\right)}^{2} \]
Alternative 14
Accuracy63.9%
Cost6624
\[1 + {\left(x \cdot \pi\right)}^{2} \cdot -0.16666666666666666 \]
Alternative 15
Accuracy62.9%
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023151 
(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))))