point vs. rate for mortgage

n: morgage term in number of years
m: number of years to closing
p: paid point

let t= n *annual_rate
s= m* annual_rate

p is equivalent to an increase of t to d+dt
Assuming no closing cost then

dt ~ c*p
c=t*(exp(t)-1)/(exp(t)-exp(t-s)-s)

So paying point p is equivalent to an increase of annual rate of p*(1/n)*c

example 1:
annual rate = 0.06
term = 30 years
closing until 30 years

t = 0.06*30 = 1.8
s = 0.06*30 = 1.8
c = 1.8 *(exp(1.8)-1)/(exp(1.8)-exp(1.8-1.8)-1.8) = 2.797
suppose paid point is 0.01 (1 point)
then the effective annual rate of this mortgage is
0.06 + 0.01 * (1/30) * 2.797 = 0.06093 = 6.093%

example 2:
Everything same as example 1 except that
closing afte 10 years
s = 0.06*10 = 0.6
c = 1.8 *(exp(1.8)-1)/(exp(1.8)-exp(1.8-0.6)-0.6) = 4.268
then the effective annual rate of this mortgage is
0.06 + 0.01 * (1/30) * 4.268 = 0.06142 = 6.142%

example 3:
Everything same as example 1 except that
closing afte 5 years
s = 0.06*5 = 0.3
c = 1.8 *(exp(1.8)-1)/(exp(1.8)-exp(1.8-0.3)-0.3) = 7.169
then the effective annual rate of this mortgage is
0.06 + 0.01 * (1/30) * 7.169 = 0.06239 = 6.239%