MATLAB: Do I receive an inaccurate value of e (Euler’s Number) when I do exp(1) in MATLAB

Question

If I execute the following code to get e, Euler's Number to 100 decimal places,

digits(100)
one = vpa(1)

b = vpa(exp(one)) 

I get:

 b =
 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427

However, if I execute:

a = exp(1)

I get:

 a = 2.718281828459046

Note that this number is not a rounded or truncated version of Euler's constant e, and is not the best double precision value for e.

Answer 1

This is due to double precision roundoff error introduced due by the double 1 in exp(1). Note that if I execute the following,

eps(exp(1))

I get,

 ans =
     4.440892098500626e-016

which is enough error at the value exp(1) to allow for the difference in the theoretical and double precision values. Thus, the value provided by exp(1) is within eps of the theoretical value.