Lanczos kernel

?

Percentage Accurate: 97.9% → 97.9%
Time: 16.5s
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}

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 18 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 98.1%

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

  2. Simplified98.2%

    \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}} \]
    Step-by-step derivation

    [Start]98.1

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

    associate-*l* [=>]97.7

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

    associate-*l* [=>]98.2

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

  3. Final simplification98.2%

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

Alternatives

Alternative 1
Accuracy97.1%
Cost19616
\[\sin \left(x \cdot \left(\pi \cdot tau\right)\right) \cdot \frac{\frac{\sin \left(x \cdot \pi\right)}{tau}}{{\left(x \cdot \pi\right)}^{2}} \]
Alternative 2
Accuracy97.1%
Cost19616
\[\sin \left(x \cdot \pi\right) \cdot \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau \cdot {\left(x \cdot \pi\right)}^{2}} \]
Alternative 3
Accuracy97.3%
Cost19616
\[\frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{\frac{tau}{\frac{\sin \left(x \cdot \pi\right)}{{\left(x \cdot \pi\right)}^{2}}}} \]
Alternative 4
Accuracy97.4%
Cost19616
\[\frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{\frac{{\left(x \cdot \pi\right)}^{2}}{\frac{\sin \left(x \cdot \pi\right)}{tau}}} \]
Alternative 5
Accuracy97.4%
Cost19616
\[\frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right) \cdot \sin \left(x \cdot \pi\right)}{tau \cdot {\left(x \cdot \pi\right)}^{2}} \]
Alternative 6
Accuracy85.3%
Cost16608
\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + -0.16666666666666666 \cdot \left({\pi}^{2} \cdot \left(x \cdot x\right)\right)\right) \end{array} \]
Alternative 7
Accuracy85.3%
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 8
Accuracy79.6%
Cost13536
\[\begin{array}{l} t_1 := {\pi}^{2} \cdot \left(x \cdot x\right)\\ \left(1 + -0.16666666666666666 \cdot t_1\right) \cdot \left(1 + -0.16666666666666666 \cdot \left(t_1 \cdot \left(tau \cdot tau\right)\right)\right) \end{array} \]
Alternative 9
Accuracy78.8%
Cost10016
\[\mathsf{fma}\left(-0.16666666666666666 \cdot \left({\pi}^{2} \cdot \left(1 + tau \cdot tau\right)\right), x \cdot x, 1\right) \]
Alternative 10
Accuracy78.8%
Cost9920
\[1 + -0.16666666666666666 \cdot {\left(\left(x \cdot \pi\right) \cdot \mathsf{hypot}\left(1, tau\right)\right)}^{2} \]
Alternative 11
Accuracy69.9%
Cost9888
\[\mathsf{fma}\left(-0.16666666666666666, \left(x \cdot x\right) \cdot {\left(\pi \cdot tau\right)}^{2}, 1\right) \]
Alternative 12
Accuracy64.8%
Cost9856
\[1 + -0.16666666666666666 \cdot {\left({\left(x \cdot \pi\right)}^{6}\right)}^{0.3333333333333333} \]
Alternative 13
Accuracy64.8%
Cost9824
\[1 + -0.16666666666666666 \cdot \sqrt[3]{{\left(x \cdot \pi\right)}^{6}} \]
Alternative 14
Accuracy64.8%
Cost9824
\[\mathsf{fma}\left(x \cdot -0.16666666666666666, x \cdot {\pi}^{2}, 1\right) \]
Alternative 15
Accuracy64.8%
Cost9760
\[\mathsf{fma}\left(-0.16666666666666666, {\left(x \cdot \pi\right)}^{2}, 1\right) \]
Alternative 16
Accuracy64.8%
Cost6624
\[1 + {\left(x \cdot \pi\right)}^{2} \cdot -0.16666666666666666 \]
Alternative 17
Accuracy63.8%
Cost32
\[1 \]

Reproduce?

herbie shell --seed 2023161 
(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))))