TITLE: Easy functions?
QUESTION [21 upvotes]: Let $f$ be an analytic function, and suppose that we want to compute
$f(x)$. The input consists of the digits of $x$ and the output of
a rational number approximating $f(x)$. A function $f$ is called easy if
there is
an algorithm which computes $f(x)$ with accuracy $2^{-n}$
using $n^{1+o(1)}$
arithmetic operations.
It is known that elementary functions like $e^x,\log x$ are easy.

Is it known (proven) about any reasonable function
  that it is hard (not easy)?

For an algorithm, using the AGM, showing that $e^x$ is easy, a reference is
D. Newman, Rational approximation versus fast computer methods,
Lectures on approximation and value distribution, pp. 149.174,
Sém. Math. Sup., 79, Presses Univ. Montréal, Montreal, Que., 1982.
EDIT1: The same paper contains a proof that multiplication is easy (fast multiplication), 
and if $f$ is easy then
the inverse function is easy (Newton's method).
EDIT2: I understand that with our present knowledge we cannot compute Euler's constant
efficiently. But I don't know a proof that this is impossible.
Remark. I am mostly interested in analytic functions, even "special functions". Are they all easy?

REPLY [3 votes]: If we consider constant functions (trivially analytic...), then we could change the original question to: Are there "reasonable" real numbers that are not computable in quadratic time?
As there is a time hierachy theorem on the real numbers (Norbert Th. Müller: Subpolynomial Complexity Classes of Real Functions and Real Numbers. ICALP 1986: 284-293), there exist numbers in qubic time that are not in quadratic time. Whether there numbers are as "reasonable" as $\pi$ or $e$, however, is another question...