Question

Excerpt from a book:

A float value consists of a 24-bit
signed mantissa and an 8-bit signed
exponent. The precision is approximately seven decimal digits.
Values
range from
-3.402823 × 10^38 to 3.402823 × 10^38

How to calculate this range? Can someone explain the binary arithmetic?

Answer 1

I would definitely read the article to which Richard points. But if you need a simpler explanation, I hope this helps:

Basically, as you said, there is 1 sign bit, 8 bits for exponent, and 23 for fraction.
Then, using this equation (from Wikipedia)

N = (1 - 2s) * 2^(x-127) * (1 + m*2^-23)

where s is the sign bit, x is the exponent (minus the 127 bias), and m is the fractional part treated as a whole number (the equation above transforms the whole number into the appropriate fraction value).

Note, that the exponent value of 0xFF is reserved to represent infinity. So the largest exponent of a real value is 0xFE.

you see that the maximum value is

N = (1 - 2*0) * 2^(254-127) * (1 + (2^23 - 1) * 2^-23)

N = 1 * 2^127 * 1.999999

N = 3.4 x 10^34

The minimum value would be the same but with the sign bit set, which would simply negate the value to give you -3.4 X 10^34.

Q.E.D.