Lanczos kernel

Average Accuracy: 97.9% → 97.9%

Time: 17.9s

Precision: binary32

Cost: 26080

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot \left(\pi \cdot tau\right)\\ \frac{\sin t_1}{t_1} \cdot \frac{\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}{x \cdot \pi} \end{array} \]

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) (/ (sin (expm1 (log1p (* x PI)))) (* x PI)))))

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) * (sinf(expm1f(log1pf((x * ((float) M_PI))))) / (x * ((float) M_PI)));
}

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(sin(expm1(log1p(Float32(x * Float32(pi))))) / Float32(x * Float32(pi))))
end

Derivation

1. Initial program 97.8%

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

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] 97.8	\[ \frac{\sin \left(\left(x \cdot \pi\right) \cdot tau\right)}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   associate-*l* [=>] 97.2	\[ \frac{\sin \color{blue}{\left(x \cdot \left(\pi \cdot tau\right)\right)}}{\left(x \cdot \pi\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   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 97.9%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}{x \cdot \pi} \]
   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 [=>] 97.9	\[ \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}}{x \cdot \pi} \]

4. Final simplification 97.9%

   \[\leadsto \frac{\sin \left(x \cdot \left(\pi \cdot tau\right)\right)}{x \cdot \left(\pi \cdot tau\right)} \cdot \frac{\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \pi\right)\right)\right)}{x \cdot \pi} \]

Alternatives

Alternative 1
Accuracy	97.2%
Cost	19680

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

Alternative 2
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 3
Accuracy	97.1%
Cost	19616

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

Alternative 4
Accuracy	85.5%
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 5
Accuracy	85.5%
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 6
Accuracy	85.5%
Cost	16544

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

Alternative 7
Accuracy	85.3%
Cost	16512

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

Alternative 8
Accuracy	85.0%
Cost	13312

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

Alternative 9
Accuracy	79.0%
Cost	13280

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

Alternative 10
Accuracy	79.0%
Cost	10016

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

Alternative 11
Accuracy	71.3%
Cost	9952

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

Alternative 12
Accuracy	71.3%
Cost	9888

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

Alternative 13
Accuracy	71.3%
Cost	9888

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

Alternative 14
Accuracy	71.3%
Cost	9888

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

Alternative 15
Accuracy	64.9%
Cost	6688

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

Alternative 16
Accuracy	64.9%
Cost	6688

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

Alternative 17
Accuracy	64.9%
Cost	6624

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

Alternative 18
Accuracy	63.9%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023157 
(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))))