Lanczos kernel

Average Accuracy: 97.9% → 97.9%

Time: 20.9s

Precision: binary32

Cost: 19808

\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \left(\left(-1 + \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) + 1\right) \end{array} \]

FPCore

(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) (+ (+ -1.0 (/ (sin (* x PI)) (* x PI))) 1.0))))

C

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) * ((-1.0f + (sinf((x * ((float) M_PI))) / (x * ((float) M_PI)))) + 1.0f);
}

Julia

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(Float32(Float32(-1.0) + Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi)))) + Float32(1.0)))
end

MATLAB

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) * ((single(-1.0) + (sin((x * single(pi))) / (x * single(pi)))) + single(1.0));
end

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. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}} \]
   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. Applied egg-rr 97.5%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)} - 1\right)} \]
   Proof

   [Start] 97.9	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   expm1-log1p-u [=>] 97.5	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)\right)} \]
   expm1-udef [=>] 97.5	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)} - 1\right)} \]

4. Applied egg-rr 97.9%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{\left(\left(-1 + \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) + 1\right)} \]
   Proof

   [Start] 97.5	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \left(e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)} - 1\right) \]
   sub-neg [=>] 97.5	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)} + \left(-1\right)\right)} \]
   +-commutative [=>] 97.5	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{\left(\left(-1\right) + e^{\mathsf{log1p}\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}\right)} \]
   log1p-udef [=>] 97.5	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \left(\left(-1\right) + e^{\color{blue}{\log \left(1 + \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}}\right) \]
   add-exp-log [<=] 97.5	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \left(\left(-1\right) + \color{blue}{\left(1 + \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right)}\right) \]
   +-commutative [=>] 97.5	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \left(\left(-1\right) + \color{blue}{\left(\frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} + 1\right)}\right) \]
   associate-+r+ [=>] 97.9	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \color{blue}{\left(\left(\left(-1\right) + \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) + 1\right)} \]
   metadata-eval [=>] 97.9	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \left(\left(\color{blue}{-1} + \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) + 1\right) \]

5. Final simplification 97.9%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \left(\left(-1 + \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi}\right) + 1\right) \]

Alternatives

Alternative 1
Accuracy	97.9%
Cost	19680

\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \end{array} \]

Alternative 2
Accuracy	97.4%
Cost	19616

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

Alternative 3
Accuracy	97.1%
Cost	19616

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

Alternative 4
Accuracy	97.1%
Cost	19616

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

Alternative 5
Accuracy	84.9%
Cost	16608

\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + -0.16666666666666666 \cdot \left({\pi}^{2} \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

Alternative 6
Accuracy	84.9%
Cost	16608

\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 \cdot {\pi}^{2}\right)\right) \end{array} \]

Alternative 7
Accuracy	84.9%
Cost	16608

\[\begin{array}{l} t_1 := tau \cdot \left(x \cdot \pi\right)\\ \frac{\sin t_1}{t_1} \cdot \left(1 + -0.16666666666666666 \cdot \left({\pi}^{2} \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

Alternative 8
Accuracy	84.3%
Cost	13312

\[\frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right)}{tau} \cdot \left(\left(x \cdot \pi\right) \cdot -0.16666666666666666 + \frac{\frac{1}{x}}{\pi}\right) \]

Alternative 9
Accuracy	84.3%
Cost	13312

\[\left(\frac{1}{x \cdot \pi} + \left(x \cdot \pi\right) \cdot -0.16666666666666666\right) \cdot \frac{\sin \left(\pi \cdot \left(x \cdot tau\right)\right)}{tau} \]

Alternative 10
Accuracy	84.5%
Cost	13312

\[\frac{\sin \left(tau \cdot \left(x \cdot \pi\right)\right) \cdot \left(\frac{1}{x \cdot \pi} + \left(x \cdot \pi\right) \cdot -0.16666666666666666\right)}{tau} \]

Alternative 11
Accuracy	80.0%
Cost	10016

\[e^{-0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot \left({\pi}^{2} \cdot \left(1 + tau \cdot tau\right)\right)\right)} \]

Alternative 12
Accuracy	78.4%
Cost	6880

\[1 + x \cdot \left(x \cdot \left({\pi}^{2} \cdot \left(-0.16666666666666666 + -0.16666666666666666 \cdot \left(tau \cdot tau\right)\right)\right)\right) \]

Alternative 13
Accuracy	78.4%
Cost	6880

\[1 + x \cdot \left(x \cdot \left({\pi}^{2} \cdot \left(-0.16666666666666666 + tau \cdot \left(tau \cdot -0.16666666666666666\right)\right)\right)\right) \]

Alternative 14
Accuracy	69.4%
Cost	6816

\[1 + x \cdot \left(x \cdot \left({\pi}^{2} \cdot \left(tau \cdot \left(tau \cdot -0.16666666666666666\right)\right)\right)\right) \]

Alternative 15
Accuracy	64.1%
Cost	6688

\[1 + -0.16666666666666666 \cdot \left({\pi}^{2} \cdot \left(x \cdot x\right)\right) \]

Alternative 16
Accuracy	64.1%
Cost	6688

\[1 + x \cdot \left({\pi}^{2} \cdot \left(x \cdot -0.16666666666666666\right)\right) \]

Alternative 17
Accuracy	64.1%
Cost	6624

\[1 + {\left(x \cdot \pi\right)}^{2} \cdot -0.16666666666666666 \]

Alternative 18
Accuracy	63.1%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023126 
(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))))