14 Ρ(1, P/G) IS THE VALUE OF Ρ SATISFYING P/G + Ρ(1 - LN Ρ) = 0. SUP...

6.14 ρ(1, p/g) is the value of ρ satisfying p/g + ρ(1 - ln ρ) = 0. Suppose we have a path

with n

stages, a path effort F, and a path parasitic delay P. If we add N-n

bufferes

of parasitic delay p and logical effort g, the best path delay is

(

¹

)

⁽

⁾

D N Fg =

+ + P N n p −

Differentiate this path delay with respect to N to find the number of stages that

minimizes delay. The best stage effort at this number of stages is ρ(g, p) = (Fg

)

. Substituting this into the derivative and simplify to find

D n g F

(

¹

)

¹

^{ln ln 0}

∂ =

  + −   + =

Fg p

∂  

N N N

ρ ρ

 −  + =

g p g p p

( , )

( , ) 1 ln 0

 

g

 

Assume the equality we are trying to prove is true and substitute it into the equation

above to obtain

ρ   − ρ   + =

(1, ) 1 ln (1, )

0

This is just the definition of ρ that we began with, so the substitution must have

been valid and the equality is proven.