machine precision and max and min value of a double-precision type

Question

(1) I have met several cases where epsilon is added to a non-negative variable to guarantee nonzero value. So I wonder why not add the minimum value that the data type can r

Answer 1

1. Epsilons are added to test equality between two values that should be equal, but aren't because of rounding errors. While you could use the smallest positive value for epsilon, it wouldn't be optimal, because it's simply too small. The rounding errors caused by floating point arithmetic almost always exceed that smallest value, so a larger epsilon is needed. How large depends on your desired accuracy.

2. I don't understand the question. Are the reciprocals useful for what? I can't think of any reason why they would be useful.

3. In general, dividing by very small values is a bad idea as it will cause very large rounding errors. I'm not sure what you mean by adding an upper bound. Just avoid dividing by small values wherever possible.

Answer 2

Epsilon

Epsilon is the smallest value that can be added to 1.0 and produce a result that's distinguishable from 1.0. As Poita_ implied, this is useful for dealing with rounding errors. The situation is pretty simple: a normal floating point number has precision that remains fixed, regardless of the magnitude of the number. To put that slightly differently, it always computes to the same number of significant digits. For example, a typical implementation of double will have around 15 significant digits (which translates to Epsilon = ~1e-15). If you're working with a number in the range 10e-200, the smallest change it can represent will be around 10e-215. If you're working with a number in the range 10e+200, the smallest change it can represent will be around 1e+185.

Meaningful use of Epsilon normally requires scaling it to the range of the numbers you're working with, and using that to define a range you're willing to accept as probably due to rounding errors, so if two numbers fall within that range, you assume they're probably really equal. For example, with Epsilon of 1e-15, you might decide to treat numbers that fall within 1e-14 of each other as equal (i.e. on significant digit has been lost to rounding).

The smallest number that can be represented will normally be dramatically smaller than that. With that same typical double, it's usually going to be around 1e-308. This would be equivalent to Epsilon if you were using fixed point numbers instead of floating point numbers. For example, at one time quite a few people used fixed-point for various graphics. A typical version was a 16-bit bit integer broken into a something like 10 bits before the decimal point and six bits after the decimal point. Such a number can represent numbers from roughly 0 to 1024, with about two (decimal) digits after the decimal point. Alternatively, you can treat it as signed, running from (roughly) -512 to +512, again with around two digits after the decimal point.

In this case, the scaling factor is fixed, so the smallest difference that can be represented between two numbers is also fixed -- i.e. the difference between 1024 and the next larger number is exactly the same as the difference between 0 and the next larger number.

Reciprocals

I'm not sure exactly why you're concerned with computing reciprocals of extremely large or extremely small numbers. IEEE floating point uses denormals, which means numbers close to the limits of the range lose precision. Basically, a number is divided into an exponent and a significand. The exponent contains the magnitude of the number, and the significand contains the significant digits. Each is represented with a specified number of bits. In the usual case, numbers are normalized, which means they're vaguely similar to the scientific notation we all learned in school. In scientific notation, you always adjust the significand and exponent so there's exactly one place before the decimal point, so (for example) 140 becomes 1.4e2, 20030 becomes 2.003e4, and so on.

Think of this as the "normalized" form of a floating point number. Assume, however, that you're limited t an exponent having 2 digits, so it can only run from -99 to +99. Also assume that you can have a maximum of 15 significant digits. Within those limitations, you could produce a number like 0.00001002e-99. This lets you represent a number smaller than 1e-99, at the expense of losing some precision -- instead of 15 digits of precision, you've used 5 digits of your significand to represent magnitude, so you're left with only 10 digits that are really significant.

Except that it's in binary instead of decimal, IEEE floating point works roughly that way. As you approach the end of the range, the numbers have less and less precision, until (at the very end of the range) you have only one bit of precision left.

If you take that number that has only one bit of precision, and take its reciprocal you get an extremely large number -- but since you only started with one bit of precision, the result can only have one bit of precision as well. Although slightly better than no result at all, it's still pretty close to meaningless. You've reached the limit of what the number of bits can represent; about the only way to cure the problem is to use more bits.

There's not really any one point at which a reciprocal (or other computation) "stops making sense". It's not really a hard line where one result makes sense, and another doesn't. Rather, it's a slope, where one result might have 15 digits of precision, another 10 and a third only 1. What "makes sense" or not is mostly how you interpret that result. To get meaningful results, you need a fair idea of how many digits in your final result are really meaningful.

Answer 3

You need to understand how floating point numbers are represented in the CPU. In the data type, 1 bit is reserved for the sign, i.e. whether it is a positive or negative number, (yes you can have positive and negative 0 in floating point numbers,) then a number of bits is reserved for the significand (or mantissa,) these are the significant digits in the floating point number and finally a number of bits is reserved for the exponent. The value of the floating point number now is:

-1^sign * significand * 2^exponent

1. This means the smallest number is a very small value, namely the smalles significand with the lowest exponent. The rounding error however is much larger and depends on the magnitude of the number, namely the smallest number with a given exponent. The epsilon is the difference between 1.0 and the next representable larger value. That's why epsilon is used in code that is robust for rounding errors, and really you should scale the epsilon with the magnitude of the numbers you work with if you do it right. The smallest representable value is not really of any significant use normally.

2. You're seeing the difference between the normalized and denormalized minimum. The problem is that due to the way the significand is used it is possible to make a larger negative exponent than a positive one, say the bit pattern of the significand is all zeros except the last bit, which is one, then the exponent is effectively lowered by the number of bits in the significand. For the maximum you cannot do this, even if you set the significand to all ones, the effective exponent will still only be the exponent that is given. i.e. think of the difference between 0.000001e-10 and 9.999999e+10, the first is much smaller than the second is big. The first is actually 1e-16 while the second is approx 1e+11.

3. It depends on the precision of the floating point number of course. In the case of double precision, the difference between the maximum and the next smaller value is already huge, (along the lines of 10^292,) so your rounding errors will be very big. If the value is too small you will simply get inf instead, as you already saw. Really, there is no strict answer, it depends entirely on the precision of numbers you need. Given that the rounding error is approx epsilon*magnitude, the reciprocal of (1/epsilon) already has a rounding error of around 1.0 if you need numbers to be accurate to 1e-3 then even epsilon would be too big to divide by.

See these wikipedia pages on IEEE754 and Machine epsilon for some background info.