Lanczos kernel

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Percentage Accurate: 97.9% → 97.9%
Time: 17.2s
Precision: binary32
Cost: 19680

?

\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
↓
\[\begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \]
(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(Float32(x * 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 := \left(x \cdot \pi\right) \cdot tau\\
\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 22 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.

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 97.6%

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

  2. Final simplification97.6%

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

Alternatives

Alternative 1
Accuracy98.0%
Cost19680
\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \frac{\sin t_1}{t_1} \end{array} \]
Alternative 2
Accuracy97.2%
Cost19616
\[\sin \left(x \cdot \pi\right) \cdot \frac{\frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{tau}}{{\left(x \cdot \pi\right)}^{2}} \]
Alternative 3
Accuracy97.2%
Cost19616
\[\sin \left(\pi \cdot \left(x \cdot tau\right)\right) \cdot \frac{\sin \left(x \cdot \pi\right)}{tau \cdot {\left(x \cdot \pi\right)}^{2}} \]
Alternative 4
Accuracy97.2%
Cost19616
\[\sin \left(x \cdot \pi\right) \cdot \frac{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{tau}}{{\left(x \cdot \pi\right)}^{2}} \]
Alternative 5
Accuracy97.7%
Cost19616
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{tau \cdot \frac{{\left(x \cdot \pi\right)}^{2}}{\sin \left(x \cdot \pi\right)}} \]
Alternative 6
Accuracy79.6%
Cost16608
\[\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \cdot \left(1 + -0.16666666666666666 \cdot \left(\left(tau \cdot tau\right) \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right)\right)\right) \]
Alternative 7
Accuracy84.7%
Cost16608
\[\left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right)\right) \cdot \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \left(\pi \cdot tau\right)} \]
Alternative 8
Accuracy85.3%
Cost16608
\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right)\right) \end{array} \]
Alternative 9
Accuracy79.4%
Cost16480
\[\sin \left(x \cdot \pi\right) \cdot \left(-0.16666666666666666 \cdot \left(\left(x \cdot \pi\right) \cdot {tau}^{2}\right) + \frac{1}{x \cdot \pi}\right) \]
Alternative 10
Accuracy79.2%
Cost16448
\[\sin \left(x \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \pi \cdot \left(x \cdot \left(tau \cdot tau\right)\right), \frac{\frac{1}{\pi}}{x}\right) \]
Alternative 11
Accuracy79.4%
Cost16448
\[\sin \left(x \cdot \pi\right) \cdot \mathsf{fma}\left(-0.16666666666666666, \left(x \cdot \pi\right) \cdot \left(tau \cdot tau\right), \frac{1}{x \cdot \pi}\right) \]
Alternative 12
Accuracy78.9%
Cost10016
\[\mathsf{fma}\left(-0.16666666666666666 \cdot \left({\pi}^{2} \cdot \left(1 + tau \cdot tau\right)\right), x \cdot x, 1\right) \]
Alternative 13
Accuracy78.9%
Cost10016
\[\mathsf{fma}\left({\pi}^{2} \cdot \left(-0.16666666666666666 + -0.16666666666666666 \cdot \left(tau \cdot tau\right)\right), x \cdot x, 1\right) \]
Alternative 14
Accuracy70.8%
Cost9888
\[\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{\left(x \cdot \pi\right) \cdot tau} \]
Alternative 15
Accuracy70.8%
Cost9888
\[\frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{x \cdot \left(\pi \cdot tau\right)} \]
Alternative 16
Accuracy71.0%
Cost9888
\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \end{array} \]
Alternative 17
Accuracy71.0%
Cost9888
\[\begin{array}{l} t_1 := \left(x \cdot \pi\right) \cdot tau\\ \frac{\sin t_1}{t_1} \end{array} \]
Alternative 18
Accuracy64.6%
Cost9824
\[\mathsf{fma}\left(x \cdot -0.16666666666666666, x \cdot {\pi}^{2}, 1\right) \]
Alternative 19
Accuracy64.6%
Cost6688
\[1 + -0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{2}\right) \]
Alternative 20
Accuracy64.6%
Cost6624
\[1 + {\left(x \cdot \pi\right)}^{2} \cdot -0.16666666666666666 \]
Alternative 21
Accuracy63.7%
Cost32
\[1 \]

Reproduce?

herbie shell --seed 2023160 
(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))))