computer science

Tapered floating point

Tapered floating point 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 Tapered floating point rather than just read about it. In short: In computing, tapered floating point (TFP) is a format similar to floating point, but with variable-sized entries for the significand and exponent instead of the fixed-length entries found in normal floating-point formats. In addition to this, tapered floating-point formats provide a fixed-size pointer entry indicating the number of digits in the exponent entry.

Key takeaways

Tapered floating point 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 Tapered floating point to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Tapered floating point from memory before moving on to harder problems.

Reference excerpt

In computing, tapered floating point (TFP) is a format similar to floating point, but with variable-sized entries for the significand and exponent instead of the fixed-length entries found in normal floating-point formats. In addition to this, tapered floating-point formats provide a fixed-size pointer entry indicating the number of digits in the exponent entry. The number of digits of the significand entry (including the sign) results from the difference of the fixed total length minus the length of the exponent and pointer entries. Thus numbers with a small exponent, i.e. whose order of magnitude is close to the one of 1, have a higher relative precision than those with a large exponent.

History The tapered floating-point scheme was first proposed by Robert Morris of Bell Laboratories in 1971, and refined with leveling by Masao Iri and Shouichi Matsui of University of Tokyo in 1981, and by Hozumi Hamada of Hitachi, Ltd. Alan Feldstein of Arizona State University and Peter Turner of Clarkson University described a tapered scheme resembling a conventional floating-point system except for the overflow or underflow conditions. In 2013, John Gustafson proposed the Unum number system, a variant of tapered floating-point arithmetic with an exact bit added to the representation and some interval interpretation to the non-exact values.

See also Logarithmic number system (LNS) Symmetric level-index arithmetic (SLI)

References

Further reading Luk, Clement (1974-10-02) [1974-09-30]. "Microprogrammed significance arithmetic with tapered floating point representation". Conference record of the 7th annual workshop on Microprogramming - MICRO 7. Palo Alto, California, USA. pp. 248–252. doi:10.1145/800118.803869. ISBN 9781450374217.{{cite book}}: CS1 maint: location missing publisher (link) Azmi, Aquil M.; Lombardi, Fabrizio (1989-09-06). "On a tapered floating point system" (PDF). Proceedings of 9th Symposium on Computer Arithmetic. Santa Monica, California, USA: IEEE. pp. 2–9. doi:10.1109/ARITH.1989.72803. ISBN 0-8186-8963-3. S2CID 38180269. Archived (PDF) from the original on 2018-07-13. Retrieved 2018-07-13. Yokoo, Hidetoshi (August 1992). "Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field". IEEE Transactions on Computers. 41 (8). Washington, DC, USA: IEEE Computer Society: 1033–1039. doi:10.1109/12.156546. ISSN 0018-9340.. Previously published in: Yokoo, Hidetoshi (June 1991). Komerup, Peter; Matula, David W. (eds.). "Overflow/Underflow-Free Floating-Point Number Representations with Self-Delimiting Variable-Length Exponent Field". Proceedings of the 10th IEEE Symposium on Computer Arithmetic (ARITH 10). Washington, DC, USA: IEEE Computer Society: 110–117. Anuta, Michael A.; Lozier, Daniel W.; Turner, Peter R. (March–April 1996) [1995-11-15]. "The MasPar MP-1 As a Computer Arithmetic Laboratory". Journal of Research of the National Institute of Standards and Technology. 101 (2): 165–174. doi:10.6028/jres.101.018. PMC 4907584. PMID 27805123. Ray, Gary (2010-02-04). "Between Fixed and Floating Point". Electronic Systems Design Engineering incorporating Chip Design. Archived from the original on 2018-07-10. Retrieved 2018-07-09. Beebe, Nelson H. F. (2017-08-22). "Chapter H.8 - Unusual floating-point systems". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, Utah, USA: Springer International Publishing AG. p. 966. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721. […] representation with a moveable boundary between exponent and significand, sacrificing precision only when a larger range is needed (sometimes called tapered arithmetic) […]

Worked examples

Example 1 — a first encounter with Tapered floating point

Start with the simplest possible case. Write down what Tapered floating point 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 Tapered floating point 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 Tapered floating point 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 Tapered floating point

In research: Tapered floating point 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 Tapered floating point 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: Tapered floating point is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computer arithmetic, so understanding it makes those chapters shorter.
In everyday life: Look for Tapered floating point 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 Tapered floating point in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Tapered floating point 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 Tapered floating point out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Tapered floating point in simple terms?

Why does Tapered floating point 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 Tapered floating point?

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 Tapered floating point.