Editing Floating-point arithmetic (section)

== Notes ==
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<ref group="nb" name="NB_1">Computer hardware does not necessarily compute the exact value; it simply has to produce the equivalent rounded result as though it had computed the infinitely precise result.</ref>
<ref group="nb" name="NB_2">The enormous complexity of modern [[division algorithm]]s once led to a famous error. An early version of the [[Intel Pentium]] chip was shipped with a [[FDIV|division instruction]] that, on rare occasions, gave slightly incorrect results. Many computers had been shipped before the error was discovered. Until the defective computers were replaced, patched versions of compilers were developed that could avoid the failing cases. See ''[[Pentium FDIV bug]]''.</ref>
<ref group="nb" name="NB_3">But an attempted computation of cos(π) yields −1 exactly. Since the derivative is nearly zero near π, the effect of the inaccuracy in the argument is far smaller than the spacing of the floating-point numbers around −1, and the rounded result is exact.</ref>
<ref group="nb" name="NB_4">[[William Morton Kahan|William Kahan]] notes: "Except in extremely uncommon situations, extra-precise arithmetic generally attenuates risks due to roundoff at far less cost than the price of a competent error-analyst."</ref>
<ref group="nb" name="NB_5">The [[Taylor expansion]] of this function demonstrates that it is well-conditioned near 1: A(x) = 1 − (x−1)/2 + (x−1)^2/12 − (x−1)^4/720 + (x−1)^6/30240 − (x−1)^8/1209600 + ... for &#124;x−1&#124; < π.</ref>
<ref group="nb" name="NB_6">If [[long double]] is [[IEEE quad precision]] then full double precision is retained; if long double is [[IEEE double extended precision]] then additional, but not full precision is retained.</ref>
<ref group="nb" name="NB_7">The equivalence of the two forms can be verified algebraically by noting that the [[denominator]] of the fraction in the second form is the [[conjugate (algebra)|conjugate]] of the [[numerator]] of the first. By multiplying the top and bottom of the first expression by this conjugate, one obtains the second expression.</ref>
<ref group="nb" name="NB_8">Octal (base-8) floating-point arithmetic is used in the [[Ferranti Atlas]] (1962)<!-- Beebe -->, [[Burroughs B5500]] (1964), [[Burroughs B5700]] (1971)<!-- Beebe -->, [[Burroughs B6700]] (1971)<!-- Beebe --> and [[Burroughs B7700]] (1972)<!-- Beebe --> computers.</ref>
<ref group="nb" name="NB_9">[[Hexadecimal floating-point|Hexadecimal (base-16) floating-point]] arithmetic is used in the [[IBM System 360]] (1964) and [[IBM System 370|370]] (1970) as well as various newer IBM machines, in the [[RCA Spectra 70]] (1964), the Siemens 4004 (1965), 7.700 (1974), 7.800, 7.500 (1977) series mainframes and successors, the Unidata 7.000 series mainframes, the [[Manchester MU5]] (1972), the [[Heterogeneous Element Processor|HEP]] (1982) computers, and in 360/370-compatible mainframe families made by Fujitsu, Amdahl and Hitachi. It is also used<!-- according to Beebe --> in the [[Illinois ILLIAC III]] (1966), [[Data General Eclipse S/200]] (ca. 1974), [[Gould Powernode 9080]] (1980s), [[Interdata 8/32]] (1970s), the [[SEL System 85|SEL Systems 85]] and [[SEL System 86|86]] as well as the [[SDS Sigma 5]] (1967), [[SDS Sigma 7|7]] (1966) and [[Xerox Sigma 9]] (1970).</ref>
<ref group="nb" name="NB_10">Base-65536 floating-point arithmetic is used in the [[MANIAC II]] (1956)<!-- Los Alamos --> computer.</ref>
<ref group="nb" name="NB_11">Quaternary (base-4) floating-point arithmetic is used in the [[Illinois ILLIAC II]] (1962) computer.<!-- according to Beebe --> It is also used in the Digital Field System DFS IV and V high-resolution site survey systems.</ref>
<ref group="nb" name="NB_12">Base-256 floating-point arithmetic is used in the [[Rice Institute R1]] computer (since 1958).<!-- according to Beebe --></ref>
<ref group="nb" name="NB_Significand">{{anchor|NB-Significand}}The ''[[significand]]'' of a floating-point number is also called ''mantissa'' by some authors&mdash;not to be confused with the [[mantissa (logarithm)|mantissa]] of a [[logarithm]]. Somewhat vague, terms such as ''coefficient'' or ''argument'' are also used by some. The usage of the term ''fraction'' by some authors is potentially misleading as well. The term ''characteristic'' (as used e.g. by [[Control Data Corporation|CDC]]) is ambiguous, as it was historically also used to specify some form of [[#NB-Exponent|exponent]] of floating-point numbers.</ref>
<ref group="nb" name="NB_Exponent">{{anchor|NB-Exponent}}The ''[[exponent]]'' of a floating-point number is sometimes also referred to as ''scale''. The term ''characteristic'' (for ''[[biased exponent]]'', ''exponent bias'', or ''excess n representation'') is ambiguous, as it was historically also used to specify the [[#NB-Significand|significand]] of floating-point numbers.</ref>
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