floating point
Let an agent explain floating-point representation and its formats, relay the IEEE 754 standard, rounding and error behaviour from computing references, describe reduced-precision formats used in machine learning, and distinguish floating point from fixed point, integers, decimal arithmetic and arbitrary precision.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent explain floating-point representation and its formats, relay the IEEE 754 standard, rounding and error behaviour from computing references, describe reduced-precision formats used in machine learning, and distinguish floating point from fixed point, integers, decimal arithmetic and arbitrary precision.
A method of representing real numbers in computers with a significand, a base and an exponent so that the radix point can float, standardised by IEEE 754 in single-precision 32-bit and double-precision 64-bit binary formats and in decimal formats including binary integer decimal encoding, with smaller formats such as minifloats and FP6 used in machine learning and block floating-point sharing one exponent across several values; floating point trades exact representation for range and is the basis of most scientific and graphics computing.
What it is for: Representing real numbers in computing.
It can be explain representation and formats; relay the standard and errors; describe reduced-precision formats; distinguish related representations.
Distinguishing features
Floating radix point
Standardised formats
Rounding error
Special values
What it looks like
Not a visible object; a binary encoding.
Physical character
IEEE 754 first published: 1985 year - revised 2008 and 2019
single precision: 32 bits - 24-bit significand
double precision: 64 bits - 53-bit significand
decimal digits, double: about 15-17 count
How it is recognised
Sign, exponent and significand encoding
Single precision, double precision, minifloat, FP6, block floating-point, binary integer decimal
Fixed point has a fixed radix; integers have no fraction; arbitrary precision grows as needed
Related models
is a kind of - in registry terms
is a kind of - in registry terms
is standardised by - the floating-point standard
is contrasted with - with a fixed radix
In practice
Families and kinds
binary formats: half, single, double and quadruple precision
decimal formats with binary integer decimal and densely packed decimal encodings
reduced-precision formats such as bfloat16, FP8, FP6 and minifloats
block floating-point
extended and non-IEEE formats
Standards and regulation
IEEE 754-2019
ISO/IEC 60559
Language standards specifying floating-point behaviour
Failure modes and hazards
Rounding and cancellation errors
Comparing floats for exact equality
Overflow, underflow and NaN propagation
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Understand What floating point is.
Science.
Definition
Definition and encoding.
Definition
Definition.
- What is floating-point representation, and how do sign, exponent and significand encode a number? definition
- Is the question about the concept, a format, numerical error or a programming language? boundary
Formats
Formats.
Formats
Formats.
- What are single and double precision, decimal formats, minifloats, FP6 and block floating-point? definition
- Which entry fits the specific format? action
Standard The standard.
Regulation.
IEEE 754
IEEE 754.
IEEE 754
IEEE 754.
- What does IEEE 754 specify about formats, rounding modes, exceptions and special values? provenance
- Which references are standard? provenance
History
History.
History
History.
- How did floating point develop from early machines to the standard, with Kahan s role attributed? provenance
- Which sources are cited? provenance
Use Numerical practice.
Practice.
Errors
Errors and pitfalls.
Errors
Errors.
- How do rounding, cancellation and accumulation errors arise, and how are they managed? action
- Which entry fits numerical analysis? action
Machine learning
Reduced precision.
Machine learning
Machine learning.
- Why do machine learning systems use bfloat16, FP8, FP6 and block floating-point? provenance
- Which entry fits mixed-precision training? action
Context Hardware and alternatives.
Context.
Hardware
Hardware.
Hardware
Hardware.
- How do processors and GPUs implement floating-point units? provenance
- Which entry fits floating-point unit? action
Alternatives
Alternatives.
Alternatives
Alternatives.
- When are fixed point, decimal, rational or arbitrary precision arithmetic preferred? provenance
- Which entry fits arbitrary-precision arithmetic? action
What the second pass must settle
- Should IEEE 754, double precision and bfloat16 be separate primary entries?
- How should computing references be linked?
- The registry entry has merged aliases naming specific formats; should they be split off?