computer science

IEEE 754

IEEE 754 is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand IEEE 754 rather than just read about it. In short: The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portably.

Key takeaways

IEEE 754 belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect IEEE 754 to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of IEEE 754 from memory before moving on to harder problems.

Reference excerpt

The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portably. Many hardware floating-point units use the IEEE 754 standard. The standard defines:

arithmetic formats: sets of binary and decimal floating-point data, which consist of finite numbers (including signed zeros and subnormal numbers), infinities, and special "not a number" values (NaNs) interchange formats: encodings (bit strings) that may be used to exchange floating-point data in an efficient and compact form rounding rules: properties to be satisfied when rounding numbers during arithmetic and conversions operations: arithmetic and other operations (such as trigonometric functions) on arithmetic formats exception handling: indications of exceptional conditions (such as division by zero, overflow, etc.) IEEE 754-2008, published in August 2008, includes nearly all of the original IEEE 754-1985 standard, plus the IEEE 854-1987 (Radix-Independent Floating-Point Arithmetic) standard. The current version, IEEE 754-2019, was published in July 2019. It is a minor revision of the previous version, incorporating mainly clarifications, defect fixes and new recommended operations.

History

The need for a floating-point standard arose from chaos in the business and scientific computing industry in the 1960s and 1970s. IBM used a hexadecimal floating-point format with seven bits always used for the exponent regardless of precision. CDC and Cray computers used ones' complement representation, which admits a value of +0 and −0. CDC 60-bit computers did not have full 60-bit adders, so integer arithmetic was limited to 48 bits of precision from the floating-point unit. Exception processing from divide-by-zero was different on different computers. Moving data between systems and even repeating the same calculations on different systems was often difficult. The first IEEE standard for floating-point arithmetic, IEEE 754-1985, was published in 1985. It covered only binary floating-point arithmetic. A new version, IEEE 754-2008, was published in August 2008, following a seven-year revision process, chaired by Dan Zuras and edited by Mike Cowlishaw. It replaced both IEEE 754-1985 (Binary Floating-Point Arithmetic) and IEEE 854-1987 (Radix-Independent Floating-Point Arithmetic) standards. The binary formats in the original standard are included in this new standard along with three new basic formats, one binary and two decimal. To conform to the current standard, an implementation must implement at least one of the basic formats as both an arithmetic format and an interchange format. The international standard ISO/IEC/IEEE 60559:2011 (with content identical to IEEE 754-2008) has been approved for adoption through ISO/IEC JTC 1/SC 25 under the ISO/IEEE PSDO Agreement and published. The current version, IEEE 754-2019 published in July 2019, is derived from and replaces IEEE 754-2008, following a revision process started in September 2015, chaired by David G. Hough and edited by Mike Cowlishaw. It incorporates mainly clarifications (e.g. totalOrder) and defect fixes (e.g. minNum), but also includes some new recommended operations (e.g. augmentedAddition). The international standard ISO/IEC 60559:2020 (with content identical to IEEE 754-2019) has been approved for adoption through ISO/IEC JTC 1/SC 25 and published. The next projected revision of the standard is in 2029.

Formats IEEE 754 defines a format as "a set of representations of numerical values and symbols, perhaps accompanied by an encoding". A floating-point format is specified by

a base (also called radix) b, which is either 2 (binary) or 10 (decimal) in IEEE 754; a precision p; an exponent range from emin to emax, with emin = 1 − emax, or equivalently emin = − (emax − 1), for all IEEE 754 formats. A format comprises

Finite numbers, which can be described by three integers: s = a sign (zero or one), c = a significand (also called a coefficient or mantissa) having no more than p digits when written in base b (i.e., an integer in the range through 0 to bp − 1), and q = an exponent such that emin ≤ q + p − 1 ≤ emax. The numerical value of such a finite number is (−1)s × c × bq. Moreover, there are two zero values, called signed zeros: the sign bit specifies whether a zero is +0 (positive zero) or −0 (negative zero). Two infinities: +∞ and −∞. Two kinds of NaN (not-a-number): a quiet NaN (qNaN) and a signaling NaN (sNaN). For example, if b = 10, p = 7, and emax = 96, then emin = −95, the significand satisfies 0 ≤ c ≤ 9999999, and the exponent satisfies −101 ≤ q ≤ 90. Consequently, the smallest non-zero positive number that can be represented is 1×10−101, and the largest is 9999999×1090 (9.999999×1096), so the full range of numbers is −9.999999×1096 through 9.999999×1096. The numbers −b1−emax and b1−emax (here, −1×10−95 and 1×10−95) are the smallest (in magnitude) normal numbers; non-zero numbers between these smallest numbers are called subnormal numbers.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with IEEE 754

Start with the simplest possible case. Write down what IEEE 754 claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to IEEE 754 before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about IEEE 754 ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of IEEE 754

In research: IEEE 754 appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses IEEE 754 in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: IEEE 754 is common in secondary-school and first-year university syllabi. It links to neighbouring topics Binary arithmetic, Computer arithmetic, Floating point types, so understanding it makes those chapters shorter.
In everyday life: Look for IEEE 754 outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study IEEE 754 in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what IEEE 754 means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain IEEE 754 out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is IEEE 754 in simple terms?

Why does IEEE 754 matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study IEEE 754?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on IEEE 754.