Can a double represent all values a float can represent?

Question

There are certain int values that a float can not represent.

However, can a double represent all values a float can represent?

My intuition says yes, since double has more fractional bits & more exponent bits, but there might be some silly gotchas that I'm missing.

Answer 1

Yes.

It would probably help to know how floats and doubles work.

Without going too much into details...

Take the number 152853.5047 ( the revolution period of Jupiter's moon Io in seconds )

In scientific notation, this number is 0.1528535047 × 10^6

Since computers only understand 1 and 0, there is way to define .

The mantissa (1528535047) and the exponent (6) are stored within 32-bits... if I remember correctly, only 24-bits are for the mantissa, so floating point is usually more about precision than size. The larger the number, the less precise it can be.

1528535047 = 1011011000110111001100000000111 so you can only store the first 24-bits... the last three 1's are lopped off.

Since Integers are 32-bits, you're right, a floating point can't accurately contain it. less significant digits get lopped off the end.

Any integer with an absolute value of less than 2^24 ( 24-bits )can be stored without losing precision. (16,777,216)

This is how the bits are stored in a floating point number:

How floats are stores diagram http://phimuemue.wordpress.com/files/2009/06/576px-ieee-754-single-svg1.png

source One bit for the sign, 8-bits for the exponent and 23-bits for the mantissa. Therefore, to answer your question, since only 23-bits are reserved for the mantissa, a 32-bit integer can't be showed with precision. It will quickly start lopping off numbers ( from the right ) as there are more digits needed to display.

For a double, you're merely increasing the number of bits that it can store... in fact, it's called double precision so any number that can be shown as a float is capable of being shown as a double. Extra 0's are merely added to the mantissa.

For this reason, since a double takes up 64-bits, most people will use a double when converting from a 32-bit int to a double. A float would be good for converting a 16-bit short.

Answer 2

6.2.5/10 in n1256:

There are three real floating types, designated as float, double, and long double. The set of values of the type float is a subset of the set of values of the type double; the set of values of the type double is a subset of the set of values of the type long double.

(emphasis mine).

Whether the implementation uses IEEE754 or not is irrelevant, the C99 standard guarantees what you want.