# The smallest floating point x such that x+2=x

Hi! I want to calculate the smallest floating point x such that x+2=x. I've written a programme in C that does not actually determines the smallest floating point x such that x+2=x, but rather determines a power of two within an interval that contains the smallest floating point x. Does anyone have any ideas how can I improve this solution or find out a new one? Is there any built-in function in MATLAB and/or C to get accurate approximations of that x?

Code:
#include <stdio.h>

int main( int argc, char **argv )
{
float x = 1.0f;

printf( "current x \t 2 + current x\n" );
do {
printf( "%G\t\t %.20f\n", x, (2.0f + x) );
x *= 2.0f;
}
while ((float)(2.0 + x) != x);

printf( "\nCalculated x: %G\n", x );
return 0;
}

Code:
Calculated x: 3.68935E+019

Might I suggest you approach this from the bit level? You may however not be a bit-head and won't like this idea. But if you're interested, for example, on a 32-bit machine, say floating point numbers are double-wide so 64 bits, what is the smallest and next largest number centered at 2 in 64-bit floating-point format that can be coded? What are the bit-values for the exponent and mantissa for these numbers? There has to be a definite bit-pattern for this.

Mark44
Mentor
Hi! I want to calculate the smallest floating point x such that x+2=x. I've written a programme in C that does not actually determines the smallest floating point x such that x+2=x, but rather determines a power of two within an interval that contains the smallest floating point x. Does anyone have any ideas how can I improve this solution or find out a new one?
Your "solution" is not a solution at all, so there is lots of room for improvement. The value you calculated, 3.68935E+019, is NOT a small number.

In your code, the while loop repeatedly multiplies your starting value by 2. You should be dividing it by 2 in each loop iteration.
Is there any built-in function in MATLAB and/or C to get accurate approximations of that x?

Code:
#include <stdio.h>

int main( int argc, char **argv )
{
float x = 1.0f;

printf( "current x \t 2 + current x\n" );
do {
printf( "%G\t\t %.20f\n", x, (2.0f + x) );
x *= 2.0f;
}
while ((float)(2.0 + x) != x);

printf( "\nCalculated x: %G\n", x );
return 0;
}

Code:
Calculated x: 3.68935E+019

Might I suggest you approach this from the bit level? You may however not be a bit-head and won't like this idea. But if you're interested, for example, on a 32-bit machine, say floating point numbers are double-wide so 64 bits, what is the smallest and next largest number centered at 2 in 64-bit floating-point format that can be coded? What are the bit-values for the exponent and mantissa for these numbers? There has to be a definite bit-pattern for this.

Thank you for your answer. I like your approach but I'm a bit confused. Could you be more specific? In 64-bit floating-point format, sign bit = 1 bit, exponent width = 11 bits and mantissa = 52 bit.
So, I suppose that: $$(2)_{10} =(0.1 \underbrace{000...0}_\text{51})_2 \times 2^2$$
What's next?

Your "solution" is not a solution at all, so there is lots of room for improvement. The value you calculated, 3.68935E+019, is NOT a small number.

In your code, the while loop repeatedly multiplies your starting value by 2. You should be dividing it by 2 in each loop iteration.

I'm searching for the smallest floating point x so that x+2=x. Obviously, x won't be a small number. If I divide it by 2 in each iteration, the programme loops endlessly.

Last edited:
rcgldr
Homework Helper
Single precision floating point numbers have a sign bit, an 8 bit exponent, and a 24 bit signifcand where the upper bit is assumed to be 1 so only the lower 23 bits are stored in the number.

http://en.wikipedia.org/wiki/Single_precision_floating-point_format

For single precicion, (x+2) (truncated) = x when x = 2^25 = 33554432, which in single precision format will be encoded as ((25+127) << 23) = hex 4c000000. Trying to add 2 to this number will result in the 2 shifted off to the right of the signifcand when adding, assuming the compiler doesn't include some rounding up function internally.

For double precision, the signficand has 53 bits (52 stored), so you'd want
2^54 = 18014398509481984 which will be stored as ((54+1023)<< 52) = hex 4350000000000000

http://en.wikipedia.org/wiki/Double_precision_floating-point_format

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SteamKing
Staff Emeritus
Homework Helper
Each time you enter the test loop, instead of multiplying the old x value by 2, why not divide by 2?

D H
Staff Emeritus
Your "solution" is not a solution at all, so there is lots of room for improvement. The value you calculated, 3.68935E+019, is NOT a small number.

In your code, the while loop repeatedly multiplies your starting value by 2. You should be dividing it by 2 in each loop iteration.
He wants a largish number. He is looking for the smallest x such that x+2 == x. You are thinking of the largest x such that x+2 == 2.

phil.st:
That number is a bit big; it looks like a double rather than a float. I get 33554432 rather than 3.68935E+019.

@rcgldr: MATLAB gives me the following results:
Code:
>> format long e
>> x = 2^54

x =

1.801439850948198e+016

>> tf = isequal(x,x+2)

tf =

1

>>

You're right! However, I don't understand why the smallest x is equal to 1.801439850948198e+016.

@D H: To be honest, I'm totally confused at this point.

D H
Staff Emeritus
What machine, what compiler are you using?

What machine, what compiler are you using?

1) Intel i3 CPU M 350 @ 2.27GHz, 3.00 GB RAM, Windows 7 x64
2) Dev-C++ 5.0 beta 9.2 (4.9.9.2) with Mingw/GCC, MATLAB

D H
Staff Emeritus
@rcgldr: MATLAB gives me the following results:
...
You're right! However, I don't understand why the smallest x is equal to 1.801439850948198e+016.
Because that is exactly what you should get for a double.
You haven't told MATLAB to use single precision, so it is using the default double precision.

You are probably compiling optimized, and your compiler is (erroneously) eliminating the cast to float, instead using the internal 80 bit doubles for the comparison. Try compiling unoptimized, and also tell the compiler to stop using floating point registers so much:
Code:
#include <stdio.h>

int main( int argc, char **argv )
{
float x = 1.0f;
volatile float xp2 = x + 2.0f;

printf( "current x \t 2 + current x\n" );
do {
printf( "%.1f\t\t %.1f\n", x, xp2 );
x *= 2.0f;
xp2 = x + 2.0f;
}
while (xp2 != x);

printf( "\nCalculated x: %.1f\n", x );
return 0;
}

Thank you for your answer. I like your approach but I'm a bit confused. Could you be more specific? In 64-bit floating-point format, sign bit = 1 bit, exponent width = 11 bits and mantissa = 52 bit.
So, I suppose that: $$(2)_{10} =(0.1 \underbrace{000...0}_\text{51})_2 \times 2^2$$
What's next?
.

Ok, sorry I mis-read your post. I thought you were looking for x+2=2 but I think the bit-analysis I suggested would still work for x+2=x which I think the others in here are doing.

Mark44
Mentor
Ok, sorry I mis-read your post. I thought you were looking for x+2=2 ...
As did I.

Because that is exactly what you should get for a double.
You haven't told MATLAB to use single precision, so it is using the default double precision.

You are probably compiling optimized, and your compiler is (erroneously) eliminating the cast to float, instead using the internal 80 bit doubles for the comparison. Try compiling unoptimized, and also tell the compiler to stop using floating point registers so much:
Code:
#include <stdio.h>

int main( int argc, char **argv )
{
float x = 1.0f;
volatile float xp2 = x + 2.0f;

printf( "current x \t 2 + current x\n" );
do {
printf( "%.1f\t\t %.1f\n", x, xp2 );
x *= 2.0f;
xp2 = x + 2.0f;
}
while (xp2 != x);

printf( "\nCalculated x: %.1f\n", x );
return 0;
}

Thanks man! I appreciate your help. Everything is ok now!

rcgldr
Homework Helper
You are probably compiling optimized, and your compiler is (erroneously) eliminating the cast to float, instead using the internal 80 bit doubles

update - DH is correct - I forgot the orignal value for the C program (not the Matlab program) corresponds to 2^65, which works for x+2 == 2 in Intel 80 bit extended precision, which has a 64 bit significand:

http://en.wikipedia.org/wiki/Extended_precision

Last edited:
D H
Staff Emeritus
You are probably compiling optimized, and your compiler is (erroneously) eliminating the cast to float, instead using the internal 80 bit doubles for the comparison.
That's unlikely, since 2^54 works for x+2 == 2, which works for a 64 bit double, but not for an internal 80 bit value. Note that IEEE floats and doubles can usually be compared directly as if they were integers (except for issues like NAN's).
I stand by what I said. Look at the stated value in the original post:
Code:
Calculated x: 3.68935E+019

2^54 is 1.8014398509481984×1016, considerably smaller than the stated value.

2^65 is 3.6893488147419103232×1019, which is the stated value. The Intel 80 bit extended precision format has a 64 bit significand and does not use normalized numbers as does the IEEE floating point standard.

rcgldr
Homework Helper
Look at the stated value in the original post:
2^65 is 3.6893488147419103232×1019, which is the stated value. The Intel 80 bit extended precision format has a 64 bit significand and does not use normalized numbers as does the IEEE floating point standard.
You're correct, I forgot about the value in the original post. I updated my previous post. Wiki article for Intel 80 bit extended precision format:

http://en.wikipedia.org/wiki/Extended_precision

Note that IEEE floats and doubles can usually be compared directly as if they were integers (except for issues like NAN's).
Already quoted but removed from previous post. This is only true for positive IEEE floating point numbers with less than maximum exponent values (used to represent infinity or NAN). For negative IEEE floating piont numbers, only the sign bit is set (as opposed to using the equivalent of a two's complement format), so you'd have to reverse the sense of a compare. A compiler is going to use a floating point compare to compare floating point numbers, so sorry for this bit of off topic trivia. (Must remember to not post when I'm tired).

You are probably compiling optimized, and your compiler is (erroneously) eliminating the cast to float, instead using the internal 80 bit doubles for the comparison.
I tested the program with various versions of Microsoft compiliers, and it didn't matter if optimizer was enabled (/Ox) or disabled (/Od) : 16 bit Visual C++ 2.2 used Intel 80 bit extended precision and stopped at 2^65, 32 bit Visual C++ 4.0 and Visual Studio 2005 used 64 bit double precision and stopped at 2^54. Forcing the compiler to do 32 bit integer (long) compares via pointers and casting solved the problem even with optimizer enabled:

Code:
#include <stdio.h>

int main(int argc, char **argv )
{
float x = 1.0f;
float y;

do {
x *= 2.0f;
y = x + 2.0f;
}
while (*(long *)(&y) != *(long *)(&x));

printf( "\nCalculated x: %21.0f\n", x );
return 0;
}

This while statement would work with optimizer disabled, but failed with optimizer enabled:

Code:
    while (y != x);

Last edited: