Quadruple-precision floating-point format: Difference between revisions

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== Double-double arithmetic ==

A common software technique to implement nearly quadruple precision using ''pairs'' of [[double-precision]] values is sometimes called '''double-double arithmetic'''.<ref name=Hida>Yozo Hida, X. Li, and D. H. Bailey, [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.4.5769 Quad-Double Arithmetic: Algorithms, Implementation, and Application], Lawrence Berkeley National Laboratory Technical Report LBNL-46996 (2000). Also Y. Hida et al., [httphttps://web.mit.edu/tabbott/Public/quaddouble-debian/qd-2.3.4-old/docs/qd.pdf Library for double-double and quad-double arithmetic] (2007).</ref><ref name="Shewchuk">J. R. Shewchuk, [https://www.cs.cmu.edu/~quake/robust.html Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates], [[Discrete & Computational Geometry]] 18: 305–363, 1997.</ref><ref name="Knuth-4.2.3-pr9">{{cite book |last=Knuth |first=D. E. |title=The Art of Computer Programming |edition=2nd |at=chapter 4.2.3. problem 9. }}</ref> Using pairs of IEEE double-precision values with 53-bit significands, double-double arithmetic provides operations on numbers with significands of at least<ref name=Hida/> {{nowrap|1=2 × 53 = 106 bits}} (actually 107 bits<ref>Robert Munafo. [httphttps://mrob.com/pub/math/f161.html F107 and F161 High-Precision Floating-Point Data Types] (2011).</ref> except for some of the largest values, due to the limited exponent range), only slightly less precise than the 113-bit significand of IEEE binary128 quadruple precision. The range of a double-double remains essentially the same as the double-precision format because the exponent has still 11 bits,<ref name=Hida /> significantly lower than the 15-bit exponent of IEEE quadruple precision (a range of {{nowrap|1.8 × 10³⁰⁸}} for double-double versus {{nowrap|1.2 × 10⁴⁹³²}} for binary128).

In particular, a double-double/quadruple-precision value ''q'' in the double-double technique is represented implicitly as a sum {{nowrap|1=''q'' = ''x'' + ''y''}} of two double-precision values ''x'' and ''y'', each of which supplies half of ''q''<nowiki/>'s significand.<ref name=Shewchuk/> That is, the pair {{nowrap|(''x'', ''y'')}} is stored in place of ''q'', and operations on ''q'' values {{nowrap|(+, −, ×, ...)}} are transformed into equivalent (but more complicated) operations on the ''x'' and ''y'' values. Thus, arithmetic in this technique reduces to a sequence of double-precision operations; since double-precision arithmetic is commonly implemented in hardware, double-double arithmetic is typically substantially faster than more general [[arbitrary-precision arithmetic]] techniques.<ref name=Hida/><ref name=Shewchuk/>

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In addition to the double-double arithmetic, it is also possible to generate triple-double or quad-double arithmetic if higher precision is required without any higher precision floating-point library. They are represented as a sum of three (or four) double-precision values respectively. They can represent operations with at least 159/161 and 212/215 bits respectively. A natural extension to an arbitrary number of terms (though limited by the exponent range) is called ''floating-point expansions''.

A similar technique can be used to produce a '''double-quad arithmetic''', which is represented as a sum of two quadruple-precision values. They can represent operations with at least 226 (or 227) bits.<ref>sourceware.org [httphttps://sourceware.org/legacy-ml/libc-alpha/2012-03/msg01024.html Re: The state of glibc libm]</ref>

== Implementations ==