IEEE Floating Point Operations

The IEEE floating-point standard (like IEEE 754) is designed to make sure that basic math operations like addition, subtraction, multiplication, and division give consistent and reliable results on all computers that follow the standard.

Exactly rounded results

When you add, subtract, multiply, or divide two floating-point numbers, the IEEE standard requires the result to be exactly rounded. This means the operation is done as if it were exact (with no rounding yet), and then the result is rounded to the nearest representable number (using a rule called "round-to-even", which helps avoid bias).

Guard digits and accuracy

Calculating exact results can be tricky and expensive, especially when the two numbers have very different sizes (like 1 and 1,000,000). To handle this, computers use extra digits—called guard digits—during calculations to avoid losing precision.

• Using one guard digit improves accuracy but might still give slightly wrong results in some cases.
• Adding a second guard digit and a "sticky bit" (which remembers if any non-zero bits were lost) makes the result as accurate as if the math was done exactly and then rounded - but still fast to compute.

This makes it possible to efficiently follow the IEEE standard, without needing perfect precision at every step.

Why precision matters

One big reason for having strict rules for precision is portability. If a program runs on two different computers, both using IEEE-compliant arithmetic, the results should always be the same. This is crucial for allowing programmers to safely assume that a bug in the program came from the program, not in how the calculations were performed.

Clearly defined behavior also makes it easier to reason about float-point arithmetic and write proofs to show that a certain floating point algorithm stays within an acceptable error bound.

Covered operations

The IEEE standard covers more than just the four basic operations (add, subtract, multiply, divide). It also covers mathematical operations such as square roots, remainders, conversions between integer and floating point, and most conversions between decimal and binary.

All of these operations are required to produce correctly rounded results, with the exception of rare cases in decimal-to-binary conversion where it is difficult to do so efficiently. This also applies to transcendental functions like exp(), log(), or sin().

Why not always round exactly?

Functions like transcendental functions (e.g. exp and sin) would create too much complexity and performance overhead.

Suppose you are trying to round exp(1.626) to 4 digits, which gives . Should it round to or ? The more digits you calculate, the longer it takes to be sure which one is correct. This is known as the table maker's dilemma, where you can't always tell where to round, even with lots of precision.

Also, there is no single algorithm that works best across all computer systems for these complex functions Different hardware may use lookup tables, CORDIC algorithms, and polynomial approximations to compute similarly-bounded results.

Because of this variety, the IEEE standard does not require exact rounding for these functions.

Improvements

Some researchers suggest adding operations like inner products (the sum of products) to the list of precisely specified operations, as these operations are extensively used in machine learning. Without precise rounding, the results of these computations can be significantly off. For example:

This should equal , but in IEEE arithmetic, it might come out as — a big mistake when working with tiny numbers. Fortunately, newer methods can compute inner products more accurately without needing hardware improvements to do so.