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Floating-Point Algebra


Simply stated, floating-point algebra is algebra performed on floating-point representations by any number of automated devices.

Traditionally, this definition is phrased so as to apply only to algebra performed on floating-point representations of real numbers (i.e., to finite elements of the collection of floating-point numbers) though several additional types of floating-point data including signed infinities and NaNs are also commonly allowed as inputs for such functions. In many widely-adopted standards, e.g., IEEE 754-2008, floating-point algebra falls under the larger heading of floating-point arithmetic.

operationfunctiondomainpossible exceptions
sinsin(x)(-infty,+infty)Invalid Operation (if |x|=infty); Underflow
coscos(x)(-infty,+infty)Invalid Operation (if |x|=infty); Underflow
tantan(x)(-infty,+infty)Invalid Operation (if |x|=infty); Underflow
sinPisin(pix)(-infty,+infty)Invalid Operation (if |x|=infty); Underflow; Several cases
cosPicos(pix)(-infty,+infty)Invalid Operation (if |x|=infty); Several cases
asinarcsin(x)[-1,+1]Invalid Operation (if |x|>1); Underflow
acosarccos(x)[-1,+1]Invalid Operation (if |x|>1)
atanarctan(x)[-infty,+infty]Underflow
atanPiarctan(x)/pi[-infty,+infty]Underflow
atan2(y,x)see below[-infty,+infty]^2Underflow; Several cases
atan2Pi(y,x)see below[-infty,+infty]^2Underflow
sinhsinh(x)[-infty,+infty]Overflow; Underflow
coshcosh(x)[-infty,+infty]Overflow
tanhtanh(x)[-infty,+infty]Underflow
asinharcsinh(x)[-infty,+infty]Underflow
acosharccosh(x)[+1,+infty]Invalid Operation (if x<1)
atanharctanh(x)[-1,+1]Underflow; Divide By Zero (if |x|=1); Invalid Operation (if |x|>1)

The above table summarizes the algebraic functions included in IEEE 754-2008 under the heading "recommended arithmetic operations." Note that trigonometric functions are included as well.

Note that the exact definition of the function atan2(y,x) is omitted from the table but is the angle subtended at the origin by the point (x,y) and the positive x-axis having range [-pi,+pi]; similarly, atan2Pi(y,x) is a normalized version of the same function having scaled range [-1,+1]. Other details and caveats of the functions mentioned throughout can be found in the documentation (IEEE Computer Society 2008, §5 and §9); the exceptions labeled "Several cases" are also addressed in detail (IEEE Computer Society 2008, pp 43-45).


See also

Algebra, Biased Exponent, Floating-Point Arithmetic, Floating-Point Exponent, Floating-Point Normal Number, Floating-Point Number, Floating-Point Preferred Exponent, Floating-Point Quantum, Floating-Point Representation, IEEE 754-2008, Interval Arithmetic, NaN, Quiet NaN, Signaling NaN, Significand, Subnormal Number

This entry contributed by Christopher Stover

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References

Goldberg, D. "What Every Computer Scientist Should Know About Floating-Point Arithmetic." ACM Comput. Surv. 23, 5-48, March 1991. http://docs.sun.com/source/806-3568/ncg_goldberg.html.Hauser, J. R. "Handling Floating-Point Exceptions in Numeric Programs." ACM Trans. Program. Lang. Sys. 18, 139-174, 1996. http://www.jhauser.us/publications/HandlingFloatingPointExceptions.html.IEEE Computer Society. "IEEE Standard for Floating-Point Arithmetic: IEEE Std 754-2008 (Revision of IEEE Std 754-1985)." 2008. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4610935.Severance, C. (Ed.). "IEEE 754: An Interview with William Kahan." Computer, 114-115, Mar. 1998.Stevenson, D. "A Proposed Standard for Binary Floating-Point Arithmetic: Draft 8.0 of IEEE Task P754." IEEE Comput. 14, 51-62, 1981.

Cite this as:

Stover, Christopher. "Floating-Point Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Floating-PointAlgebra.html

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