A few days ago I wrote about how to estimate 10^{x}. This is an analogous post for

exp(*x*) = *e*^{x}.

We will assume -0.5 ≤ *x* ≤ 0.5. You can bootstrap your way from there to other values of *x*. For example,

exp(1.3) = exp(1 + 0.3) = *e* exp(0.3)

and

exp(0.8) = exp(1 – 0.2) = *e* / exp(0.2).

I showed here that

log_{e}(*x*)≈ (2*x* − 2)/(*x* + 1)

for *x* between exp(-0.5) and exp(0.5).

Inverting both sides of the approximation shows

exp(*x*) ≈ (2 + *x*)/(2 − *x*)

for *x* between −0.5 and 0.5.

The maximum relative error in this approximation is less than 1.1% and occurs at *x* = 0.5. For *x* closer to the middle of the interval [−0.5, 0.5] the relative error is much smaller.

Here’s a plot of the relative error.

This was produced with the following Mathematica code.

re[x_] := (Exp[x] - (2 + x)/(2 - x))/Exp[x] Plot[re[x], {x, -0.5, 0.5}]

**Update**: The approximations from this post and several similar posts are all consolidated here.

Typo in log_e formula: 2nd – must be a +

(and (x) argument is missing).

Thanks!