CODEWITHSUNDEEP
mathfloating-point
SKLAK
SKLAK

Reputation: 3965

Floating point arithmetic rounding to nearest

I am a little confused about rounding to nearest in floating point arithmetic. Let a, b, and c be normalized double precision floating point numbers. Is it true that a+b=b+a where + is correctly rounded to nearest floating point addition? My initial guess is yes this is always true, but I don't completely understand rounding to nearest. Could someone give an example of when a+b != b+a using floating point addition with rounding to nearest?

Upvotes: 2

Views: 1327

Answers (2)

jerry
jerry

Reputation: 2611

As noted above, addition is commutative but not associative. The difference in rounding modes can be seen by running the following (MS Visual Studio) C++ code:

#include <iostream>
#include <float.h>

#pragma fenv_access(on)

using namespace std;

int main(int argc, char* argv[])
{
    float a = 1.75f, b = 1e-6f;

    cout.setf(ios::fixed,ios::floatfield);
    cout.precision(7);

    cout << "a = " << a << ", b = " << b << endl;

    _controlfp_s(NULL, _RC_DOWN,_MCW_RC);

    cout << "Result of a + b rounded down: " << a+b << endl;
    cout << "Result of a - b rounded down: " << a-b << endl;

    _controlfp_s(NULL, _RC_UP,_MCW_RC);
    cout << "Result of a + b rounded up: " << a+b << endl;
    cout << "Result of a - b rounded up: " << a-b << endl;

    _controlfp_s(NULL, _RC_NEAR,_MCW_RC);
    cout << "Result of a + b rounded to nearest: " << a+b << endl;
    cout << "Result of a - b rounded to nearest: " << a-b << endl;

    return 0;
}

Output:

a = 1.7500000, b = 0.0000010
Result of a + b rounded down: 1.7500010
Result of a - b rounded down: 1.7499989
Result of a + b rounded up: 1.7500011
Result of a - b rounded up: 1.7499990
Result of a + b rounded to nearest: 1.7500010
Result of a - b rounded to nearest: 1.7499990

Upvotes: 0

Eric Postpischil
Eric Postpischil

Reputation: 224596

Properly implemented IEEE-754 floating-point addition is commutative (a+b equals b+a) regardless of rounding mode.

The rounding mode affects how the exact mathematical result is rounded to fit into the destination format. Since the exact mathematical results of a+b and b+a are identical, they are rounded identically.

Upvotes: 3

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