Code_ The Hidden Language of Computer Hardware and Software - Charles Petzold [143]
This is actually easier than it might first seem. In decimal notation, digits to the right of the decimal point represent negative powers of ten. In binary notation, digits to the right of the binary point (which is simply a period and looks just like a decimal point) represent negative powers of two. For example, this binary number
101.1101
can be converted to decimal using this formula:
1 x 4 +
0 x 2 +
1 x 1 +
1 ÷ 2 +
1 ÷ 4 +
0 ÷ 8 +
1 ÷ 16
The division signs can be replaced with negative powers of two:
1 x 22 +
0 x 21 +
1 x 20 +
1 x 2-1 +
1 x 2-2 +
0 x 2-3 +
1 x 2-4
Or the negative powers of two can be calculated by starting at 1 and repeatedly dividing by 2:
1 x 4+
0 x 2+
1 x 1+
1 x 0.5+
1 x 0.25+
0 x 0.125+
1 x 0.0625
By this calculation, the decimal equivalent of 101.1101 is 5.8125.
In decimal scientific notation, the normalized significand should be greater than or equal to 1 but less than 10. Similarly, the normalized significand of numbers in binary scientific notation is always greater than or equal to 1 but less than binary 10, which is 2 in decimal. So in binary scientific notation, the number
101.1101
is expressed as
1.011101 x 22
One interesting implication of this rule is that a normalized binary floating-point number always has a 1 and nothing else at the left of the binary point.
Most contemporary computers and computer programs that deal with floating-point numbers use a standard established by the IEEE (the Institute of Electrical and Electronics Engineers) in 1985, a standard also recognized by ANSI (the American National Standards Institute). ANSI/IEEE Std 754-1985 is called the IEEE Standard for Binary Floating-Point Arithmetic. It's not very lengthy as standards go—just 18 pages—but gives the basics of encoding binary floating-point numbers in a convenient manner.
The IEEE floating-point standard defines two basic formats: single precision, which requires 4 bytes, and double precision, which requires 8 bytes.
Let's look at the single-precision format first. It has three parts: a 1-bit sign (0 for positive and 1 for negative), an 8-bit exponent, and a 23-bit significand fraction arranged like this, with the least-significant bits on the right:
s = 1-Bit Sign
e = 8-Bit Exponent
f = 23-Bit Significand Fraction
That's a total of 32 bits, or 4 bytes. Because the significand of a normalized binary floating-point number always has a 1 to the left of the binary point, that bit is not included in the storage of floating-point numbers in the IEEE format. The 23-bit fractional part of the significand is the only part stored. So even though only 23 bits are used to store the significand, the precision is said to be 24 bits. We'll get a feel for what 24-bit precision means in a moment.
The 8-bit exponent part can range from 0 through 255. This is called a biased exponent, which means that you must subtract a number—called the bias—from the exponent in order to determine the signed exponent that actually applies. For single-precision floating-point numbers, this bias is 127.
The exponents 0 and 255 are used for special purposes that I'll describe shortly. If the exponent ranges from 1 through 254, the number represented by particular values of s (the sign bit), e (the exponent), and f (the significand fraction) is
(–1)s x 1.f x 2e-127
That negative 1 to the s power is a mathematician's annoyingly clever way of saying, "If s is 0, the number is positive (because anything to the 0 power equals 1); and if s is 1, the number is negative (because –1 to the 1 power is –1)."
The next part of the expression is 1.f, which means a 1 followed by a binary point, followed by the 23 bits of the significand fraction. This is multiplied by 2 to a power. The exponent is the 8-bit biased exponent stored in memory minus 127.
Notice that I haven't mentioned any way to express a very common number that we seem to have forgotten about, namely 0. That's one of the special cases, which are these:
If e equals 0, and f