Is hardcode float precise if it can be represented by binary format in IEEE 754?

Question

for example, 0 , 0.5, 0.15625 , 1 , 2 , 3... are values converted from IEEE 754. Are their hardcode version precise?

for example:

is

float a=0;
if(a==0){
    return true;
}

always return true? other example:

float a=0.5;
float b=0.25;
float c=0.125;

is a * b always equal to 0.125 and a * b==c always true? And one more example:

int a=123;
float b=0.5;

is a * b always be 61.5? or in general, is integer multiply by IEEE 754 binary float precise?

Or a more general question: if the value is hardcode and both the value and result can be represented by binary format in IEEE 754 (e.g.:0.5 - 0.125), is the value precise?

Answer 1

There is no inherent fuzzyness in floating-point numbers. It's just that some, but not all, real numbers can't be exactly represented.

Compare with a fixed-width decimal representation, let's say with three digits. The integer 1 can be represented, using 1.00, and 1/10 can be represented, using 0.10, but 1/3 can only be approximated, using 0.33.

If we instead use binary digits, the integer 1 would be represented as 1.00 (binary digits), 1/2 as 0.10, 1/4 as 0.01, but 1/3 can (again) only be approximated.

There are some things to remember, though:

• It's not the same numbers as with decimal digits. 1/10 can be written exactly as 0.1 using decimal digits, but not using binary digits, no matter how many you use (short of infinity).
• In practice, it is difficult to keep track of which numbers can be
  represented and which can't. 0.5 can, but 0.4 can't. So when you need exact numbers, such as (often) when working with money, you shouldn't use floating-point numbers.
• According to some sources, some processors do strange things internally when performing floating-point calculations on numbers that can't be exactly represented, causing results to vary in a way that is, in practice, unpredictable.

(My opinion is that it's actually a reasonable first approximation to say that yes, floating-point numbers are inherently fuzzy, so unless you are sure your particular application can handle that, stay away from them.)

For more details than you probably need or want, read the famous What Every Computer Scientist Should Know About Floating-Point Arithmetic. Also, this somewhat more accessible website: The Floating-Point Guide.

Answer 2

No, but as Thomas Padron-McCarthy says, some numbers can be exactly represented using binary but not all of them can.

This is the way I explain it to non-developers who I work with (like Mahmut Ali I also work on an very old financial package): Imagine having a very large cake that is cut into 256 slices. Now you can give 1 person the whole cake, 2 people half of the slices but soon as you decide to split it between 3 you can't - it's either 85 or 86 - you can't split the cake any further. The same is with floating point. You can only get exact numbers on some representations - some numbers can only be closely approximated.