Lanczos kernel

Average Error: 0.7 → 0.7

Time: 18.0s

Precision: binary32

Cost: 19680

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

Math

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

↓

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

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 (* tau PI))))
   (* (/ (sin t_1) t_1) (/ (sin (* 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 * (tau * ((float) M_PI));
	return (sinf(t_1) / t_1) * (sinf((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(tau * Float32(pi)))
	return Float32(Float32(sin(t_1) / t_1) * Float32(sin(Float32(x * Float32(pi))) / Float32(x * Float32(pi))))
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 * (tau * single(pi));
	tmp = (sin(t_1) / t_1) * (sin((x * single(pi))) / (x * single(pi)));
end

Derivation

1. Initial program 0.7

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

2. Taylor expanded in x around inf 0.7

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

3. Simplified 0.7

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \left(tau \cdot \pi\right)\right)}{x \cdot \left(tau \cdot \pi\right)}} \cdot \frac{\sin \left(x \cdot \pi\right)}{x \cdot \pi} \]
   Proof
   (/.f32 (sin.f32 (*.f32 x (*.f32 tau (PI.f32)))) (*.f32 x (*.f32 tau (PI.f32)))): 0 points increase in error, 0 points decrease in error
   (/.f32 (sin.f32 (Rewrite<= *-commutative_binary32 (*.f32 (*.f32 tau (PI.f32)) x))) (*.f32 x (*.f32 tau (PI.f32)))): 0 points increase in error, 0 points decrease in error
   (/.f32 (sin.f32 (Rewrite<= associate-*r*_binary32 (*.f32 tau (*.f32 (PI.f32) x)))) (*.f32 x (*.f32 tau (PI.f32)))): 43 points increase in error, 33 points decrease in error
   (/.f32 (sin.f32 (*.f32 tau (Rewrite=> *-commutative_binary32 (*.f32 x (PI.f32))))) (*.f32 x (*.f32 tau (PI.f32)))): 0 points increase in error, 0 points decrease in error
   (/.f32 (sin.f32 (*.f32 tau (*.f32 x (PI.f32)))) (Rewrite<= *-commutative_binary32 (*.f32 (*.f32 tau (PI.f32)) x))): 0 points increase in error, 0 points decrease in error
   (/.f32 (sin.f32 (*.f32 tau (*.f32 x (PI.f32)))) (Rewrite<= associate-*r*_binary32 (*.f32 tau (*.f32 (PI.f32) x)))): 34 points increase in error, 47 points decrease in error

4. Final simplification 0.7

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

Alternatives

Alternative 1
Error	0.8
Cost	19680

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

Alternative 2
Error	0.8
Cost	19680

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

Alternative 3
Error	0.8
Cost	19680

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

Alternative 4
Error	0.7
Cost	19680

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

Alternative 5
Error	0.7
Cost	19680

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

Alternative 6
Error	0.9
Cost	19616

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

Alternative 7
Error	0.9
Cost	19616

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

Alternative 8
Error	0.9
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 9
Error	0.9
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 10
Error	0.9
Cost	19616

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

Alternative 11
Error	4.8
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 12
Error	6.8
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 13
Error	6.8
Cost	10016

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

Alternative 14
Error	9.3
Cost	9952

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

Alternative 15
Error	9.2
Cost	9888

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

Alternative 16
Error	9.2
Cost	9888

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

Alternative 17
Error	11.3
Cost	9824

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

Alternative 18
Error	11.3
Cost	6624

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

Alternative 19
Error	11.6
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022318 
(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))))