Floating point versus fixed point: what are the pros/cons?

Question

Floating point type represents a number by storing its significant digits and its exponent separately on separate binary words so it fits in 16, 32, 64 or 128 bits.

Fixed point type stores numbers with 2 words, one representing the integer part, another representing the part past the radix, in negative exponents, 2^-1, 2^-2, 2^-3, etc.

Float are better because they have wider range in an exponent sense, but not if one wants to store number with more precision for a certain range, for example only using integer from -16 to 16, thus using more bits to hold digits past the radix.

In terms of performances, which one has the best performance, or are there cases where some is faster than the other ?

In video game programming, does everybody use floating point because the FPU makes it faster, or because the performance drop is just negligible, or do they make their own fixed type ?

Why isn't there any fixed type in C/C++ ?

Answer 1

That definition covers a very limited subset of fixed point implementations.

It would be more correct to say that in fixed point only the mantissa is stored and the exponent is a constant determined a-priori. There is no requirement for the binary point to fall inside the mantissa, and definitely no requirement that it fall on a word boundary. For example, all of the following are "fixed point":

• 64 bit mantissa, scaled by 2^-32 (this fits the definition listed in the question)
• 64 bit mantissa, scaled by 2^-33 (now the integer and fractional parts cannot be separated by an octet boundary)
• 32 bit mantissa, scaled by 2⁴ (now there is no fractional part)
• 32 bit mantissa, scaled by 2^-40 (now there is no integer part)

GPUs tend to use fixed point with no integer part (typically 32-bit mantissa scaled by 2^-32). Therefore APIs such as OpenGL and Direct3D often use floating-point types which are capable of holding these values. However, manipulating the integer mantissa is often more efficient so these APIs allow specifying coordinates (in texture space, color space, etc) this way as well.

As for your claim that C++ doesn't have a fixed point type, I disagree. All integer types in C++ are fixed point types. The exponent is often assumed to be zero, but this isn't required and I have quite a bit of fixed-point DSP code implemented in C++ this way.

Answer 2

At the code level, fixed-point arithmetic is simply integer arithmetic with an implied denominator.

For many simple arithmetic operations, fixed-point and integer operations are essentially the same. However, there are some operations which the intermediate values must be represented with a higher number of bits and then rounded off. For example, to multiply two 16-bit fixed-point numbers, the result must be temporarily stored in 32-bit before renormalizing (or saturating) back to 16-bit fixed-point.

When the software does not take advantage of vectorization (such as CPU-based SIMD or GPGPU), integer and fixed-point arithmeric is faster than FPU. When vectorization is used, the efficiency of vectorization matters a lot more, such that the performance differences between fixed-point and floating-point is moot.

Some architectures provide hardware implementations for certain math functions, such as sin, cos, atan, sqrt, for floating-point types only. Some architectures do not provide any hardware implementation at all. In both cases, specialized math software libraries may provide those functions by using only integer or fixed-point arithmetic. Often, such libraries will provide multiple level of precisions, for example, answers which are only accurate up to N-bits of precision, which is less than the full precision of the representation. The limited-precision versions may be faster than the highest-precision version.