How to store variable's "exact" value?

[Bobu]

Hey,

I hope I'm able to explain my programming issue clearly:

I have written a solver for differential equations.
Now I solved it, with different initial conditions, which are calculated within several loops; for instance:

for(double value_a = a1; a <= a2; a+=2.2E-20*a_steps){
    for(double value_b = b1; b <= b2; b+=2.2E-20*b_steps){
      function_to_solve_my_differential_equation(value_a,value_b,...);
    }
}

Then I store the data of the solved differential equation; with the remark which initial conditions I have used to solve it:

Solution with initial conditions:
a_value = 2.343234443243E-16 (e.g.)
 b_value = 2.876344242354E-13 (e.g.)
... huge amount of data ...
... huge amount of data ...
...........................

Suppose, I want to reproduce my data again: I solve my differential equation again with the initial conditions, which I had printed out in the data-file I produced before:

function_to_solve_my_differential_equation(2.343234443243E-16, 2.876344242354E-13,...)

But, surprise, then the "huge amount of data" I get is not _exactly_ the same as before!

I very, very strongly assume, that the problem is, that the printed initial conditions (a_value, b_value) are not "exactly" the "internal", "exact" ones, with which my differential equation is solved, although I printed them out with "precision(16)", cause double should be exact to - more or less - 16 decimals...
I think that's why after many loops, the numerical inacurracy of conversion between decimal to "internal" binary representation accumulates.

I hope I explained my problem clearly: I want to output the initial conditions produced by the two for-loops "exactly", so that i can call my differential-equation-solver again, to reproduce the data I got via going through the initial conditions within the loops.

I tried to store the initial values produced by the loops as binary data, but that is very, very unpracticable for my needs.

[codekiddy]

in visual studio long double is the same ad double.
also precision is not always the same.

not shore what what IDE are you using.

[Caligulaminus]

double x = 123.45;
//file opened in binary mode
file.write((const char*)&x, sizeof(double)); // people will flame me for the c-style cast. But just to show what I mean...
/*
...
*/
double y;
file.read((char*)&y, sizeof(double))// now y should be guaranteed to be exactly equal to x 

[Bobu]

I'm using Eclipse under Ubuntu 10.04.
But I think my problem is IDE and platform-independent.

I also tried to output my initial conditions (a_value, b_value...) with a much higher precision, but then nothing changed, as expected, cause the additional decimals just contain rubbish.

[JLBorges]

> How to store variable's "exact" value?

You can not store "exact" values in a double - floating point is not an "exact" representation of real numbers.
http://en.wikipedia.org/wiki/Floating_point#Representable_numbers.2C_conversion_and_rounding.

For example addition is not commutative.

#include <iostream>

int main()
{
    double a = 1.0e+20 ;
    double b = 1.0e-20 ;
    double c = - a ;
    std::cout << "a+b+c == " << a+b+c << '\n'
              << "a+c+b == " << a+c+b << '\n' ;

    double d = 1234567890.9876543210 ;
    double e = 1234567890.98765432219 ;
    std::cout << std::boolalpha << "d == e ? " << (d==e) << '\n' ;
}

Output on my implementation:

a+b+c == 0
a+c+b == 1e-020
d == e ? true

For precision that is limited only by available memory, use an arbitrary-precision arithmetic library; for example GMP http://en.wikipedia.org/wiki/GNU_Multi-Precision_Library

[Bobu]

@JLBorges: Thx for your reply!! I'm aware of that, that the very exact cannot be stored in doubles. That's why I wrote "exact" in quotes. I just wanted to know, how to print out the values, with which a function is called, when the values themselves were calculated via several steps. Hereby, the problem is, that the values printed out (by e.g. cout << ... , write(...),...) are not the ones with which my program operates, because of "inaccurate" conversion between "internal" binary data and printed decimal data.
Thx for the link to the multi-precision library. I will consider this in my next project!!

@Caligulaminus: Thx too!! I was afraid, that this the only possibility. I have implemented that, and it works fine for one variable. I have to extend it, so that in one data file I can write and read tab-separated binary data, which seems to be not so trivial as with non-binary data.