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Non-integral mathematics 283

Listing 8.9 Inefficient representation of a binimal.

1 typedef struct{
2 int sign;
3 int exponent;
4 int mantissa;
5 } poorfloat;

4. If battery life is a consideration, then a fixed point implementation can have an enormous
advantage.

Note also from the table that the assembly language version of the fixed-point sine function
beats the identical C version by a significant margin. It does take considerable effort to write
assembly code that provides higher performance than a good optimizing compiler, but in
some situations, the extra effort is warranted. Section 9.8 will demonstrate that a good as-
sembly language programmer who is familiar with the floating point hardware can beat the
compiler by an even wider performance margin.

8.7 Floating point numbers

Sometimes we need more range than we can easily get from fixed precision. One approach to
solving this problem is to create an aggregate data type that can represent a fractional num-
ber by having fields for an exponent, a sign bit, and an integer mantissa. For example, in C,
we could represent a fractional number using the data structure shown in Listing 8.9.That
data structure, along with some subroutines for addition, subtraction, multiplication and divi-
sion, would provide the capability to perform arithmetic without explicitly tracking the radix
point. The subroutines for the basic arithmetic operations could do that, thereby freeing the
programmer to work at a higher level.

The structure shown in Listing 8.9 is a rather inefficient way to represent a fractional number,
and may create different data structures on different machines. It would be better to find some
way to pack all three fields into 32 bits. The C language includes the notion of bit fields. This
allows the programmer to specify exactly how many bits are to be used for each field within
a struct, Listing 8.10 shows a C data structure that consumes 32 bits on all machines and
architectures. It provides the same fields as the structure in Listing 8.9, but specifies exactly
how many bits each field consumes. The compiler will compress this data structure into 32
bits, regardless of the natural word size of the machine.
The method of representing fractional numbers as a sign, exponent, and mantissa is very pow-
erful, and IEEE has set standards for various floating point formats. These formats can be