Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine? I failed to get an answer at https://math.stackexchange.com/questions/364061/can-all-programs-reducible-to-ones-with-only-arithmetic-operations-on-inputs-be, so I am asking here.
In https://math.stackexchange.com/questions/364018/can-all-programs-be-modeled-as-operations-of-elementary-arithmetic-operations-onmathematics and computabiltiy theory, I
asked:

we treat all inputs and intermediate results and
  final outputs as natural number. While
  algorithms/programs themselves are considered
  natural numbers, here we treat these programs/functions/algorithms as just computable
  functions. The question is, when the function
  operates on an input to produce an output, can we
  consider the operation of function as using only a
  number of arithmetic operations (addition,
  subtraction, multiplication and division) on an
  input? Or does the use of if/else make the
  aforementioned not true? If this is true, is the
  number of arithmetic operations polynomially
  proportional to the lowest time complexity bound
  possible for solving a problem? (That is, if the
  lowest time complexity is $\text{O(whatever)}$,
  then the number of arithmetic operations is
  $\text{O(whatever}^k)$ where $k$ is some
  rational number.)

I learned an answer to this, and now I would like
to present variation: If we limit our scope to
programs that can be modeled as operations of
arithmetic operations on inputs, can these
program be simulated by a machine that can only
do basic arithmetic processes on inputs
(multiplciation, division, subtraction, addition)
with polynomial overhead (That is, if the lowest
time complexity is $\text{O(whatever)}$, then the
number of arithmetic operations is
$\text{O(whatever}^k)$ where $k$ is some
rational number.) to the lowest possible time
complexity for solving a problem?
So suppose that there is a problem that have some computational complexity known for deterministic Turing machine. Then we construct a machine which can only use basic arithmetic operations - addition, subtraction, multiplication and division. Examplary problems that can be computed with this machine would be Fourier transform, or computing function $x^5+14x^4+13x^3+9x+1$ with $x$ given as input. Then we use this machine to solve the problem that we have knowledge of its computational complexity in deterministic Turing machine. What would complexity of solving the problem in this special machine be like? Would the problem be solved with only polynomial overhead over deterministic Turing machine? That's my question. Of course, I assume that the problem can be solved by the special machine.
Also, I know that we can only have unbounded recursions for some problems - there is no need for explaining that.
 A: If you look at the several definitions of computable functions, one that is close to what you are looking for is for instance recursive functions, defined via primitive recursive ones: http://en.wikipedia.org/wiki/Primitive_recursive
But the crucial point is that in this model, as in any other, you need an unbounded recursion (or "while") operator, which is not sure to terminate, and whose number of computation steps is not known a priori.
This recursion capacity is what makes computability and complexity interesting and hard, because it allows comptuable functions to have infinite runtime on some inputs. It is the reason why we cannot consider that computable functions are just compositions of basic arithmetic functions.
As for the restricted question, I'm not sure I completely understand it. If I remove the parentheses I get  "[...] can these program be simulated by a machine that can only do basic arithmetic processes on inputs with polynomial overhead to the lowest possible time complexity for solving a problem?", which is not clear to me.
