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	Floating point - Wikipedia, the free encyclopedia
		A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer.
		Rounding errors: unlike the fixed-point counterpart, the application of dither in a floating point environment is nearly impossible.
		Floating point representation is more likely to be appropriate when proportional accuracy over a range of scales is needed.
	en.wikipedia.org /wiki/Floating_point (1266 words)

	Floating point -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		In computers, floating-point numbers are sized by the number of (The cutting part of a drill; usually pointed and threaded and is replaceable in a brace or bitstock or drill press) bits used to store them.
		Floating point arithmetic is not (Click link for more info and facts about associative) associative.
		Floating point arithmetic is also not (Click link for more info and facts about distributive) distributive.
	www.absoluteastronomy.com /encyclopedia/f/fl/floating_point.htm (1479 words)

	Floating Point Numbers
		Floating point numbers are a numeric data type provided on most machines (hardware) and programming languages.
		Floating point provides a type to represent very large numbers and very small numbers at the expense of exactness, precision, or significant digits.
		That is, moving the radix point to the left in the mantissa increases the exponent to counter the decreasing value of the mantissa.
	faculty.juniata.edu /rhodes/org/ch14.htm (1098 words)

	Floating-point numbers - General view
		A possible implementation of round() for decimal floating-point numbers (represented in radix 10) is: e = INT(LOG10(X) + 1.0) (number of decimal digits in X) INT(X * (10(p-e)) + 0.5) round(X) = ---------------------- 10(p-e) The parameter p is the number of decimal digits in the representation.
		Floating Point numbers in practise ---------------------------------- In our finite machines, we can keep only a finite number of the binary digits of 'f' and 'e', let's say 'm' and 'n' digits respectively.
		The following table compares some floats used in practice, the REALn notation is a common extension to FORTRAN, 'n' is the number* of bytes used in the representation.
	www.ibiblio.org /pub/languages/fortran/ch4-1.html (1200 words)

	IEEE Standard 754 Floating-Point (Site not responding. Last check: 2007-11-07)
		Fixed point places a radix point somewhere in the middle of the digits, and is equivalent to using integers that represent portions of some unit.
		The range of positive floating point numbers can be split into normalized numbers (which preserve the full precision of the mantissa), and denormalized numbers (discussed later) which use only a portion of the fractions's precision.
		Since the sign of floating point numbers is given by a special leading bit, the range for negative numbers is given by the negation of the above values.
	stevehollasch.com /cgindex/coding/ieeefloat.html (1366 words)

	Math Forum - Ask Dr. Math
		In one method, the decimal point is assumed to be in a particular place, such as at the left-most or high-order end of the computer word, and the exponent of a power of 10 is adjoined to give you the value in scientific notation.
		The first part is the "mantissa" of the number, and the second is the "characteristic." This is what is meant by floating point numbers.
		Actually the terms "fixed point" and "floating point" are misnomers: with floating point numbers, the decimal location is fixed (in the examples, at the far left), and with fixed point numbers, the decimal location is floating (and in fact unspecified).
	mathforum.org /library/drmath/view/55946.html (352 words)

	Binary Numbers - Floating Point
		In this case, suppose that the numbers are denormalised to 3e2 and 0.00025e2, the sum is then easily calculated as 3.00025e2, which normalises to 0.300025e3.
		Floating point numbers arrived at in calculations are always approximations.
		A denormalised number is a way of allowing very small values (which don't have a 1 immediately after the binary point) and is used in specialised operations.
	www.cs.camosun.bc.ca /~deid/comp112/floatingpoint.html (1419 words)

	Comparing floating point numbers
		Because floating point math is imperfect you may not get an answer of exactly 10,000 - you may be off by one or two in the least significant bits of your result.
		The positive number closest to zero and the negative number closest to zero are extremely close to each other, yet this function will correctly calculate that they have a huge relative error of 2.0.
		For a normal float number a maxUlps of 1 is equivalent to a maxRelativeError of between 1/8,000,000 and 1/16,000,000.
	www.cygnus-software.com /papers/comparingfloats/comparingfloats.htm (3164 words)

	Inner Float Java Applet - an Interactive Illustration of Floating Point Numbers
		At that point, the value displayed in radix 10 is 8.3361e+06, which is the same as the radix 2 number.
		When a floating point operation underflows to negative zero, as when it underflows to positive zero, all bits of precision are lost.
		When a floating point operation overflows to negative infinity, as when it overflows to positive infinity, all bits of precision are lost.
	www.artima.com /insidejvm/applets/InnerFloat.html (456 words)

	Floating Point numbers considered harmful
		Arrrggghhh, there are many days when I wish that computers did not have floating point number representations, and that computer tools did not make it so easy for novice users to use floating point.
		Or use double precision floating point (64-bit) and you will find that it is represented as 0.0999999999999, which is again pretty close.
		Floating point number precision is not a measurement problem, 0.10 is always represented by the same number, one that is always a little off from the "true" value.
	www.pfarrell.com /misc/floating.html (226 words)

	An Overview of Floating Point Numbers. (Site not responding. Last check: 2007-11-07)
		In base ten, the number 123 is represented as one 100, two 10's and three 1's.
		Floating Point Round Off Error ------------------------------ Round off error is especially noticable in the smallest floating point data type available: the float.
		The float data type is four bytes in length, and uses these bytes to hold the mantissa, exponent, and sign of the number.
	community.borland.com /article/0,1410,15855,00.html (1387 words)

	Description of why floating point numbers may lose precision in Visual C++ (Site not responding. Last check: 2007-11-07)
		Floating point decimal values generally do not have an exact binary representation.
		The binary representation of the decimal number may not be exact.
		Because this is a very small number it is advisable that you employ user-defined tolerance for calculations involving very large numbers.
	support.microsoft.com /support/kb/articles/Q145/8/89.asp (453 words)

	Floating-Point Number Tutorial
		The floating-point numbers between.1 and.9 are separated by intervals of.1; the floating-point numbers between 1 and 9 are separated by intervals of 1; and the floating-point numbers between 10 and 90 are separated by intervals of 10.
		In addition to the 27 numbers that we had before, we now have nine numbers between.01 and.09 and nine numbers between 100 and 900.
		In this case, the closest floating-point number to the sum is 10.
	www.cs.utah.edu /~zachary/isp/applets/FP/FP.html (1560 words)

	Multiplying Floating Point Numbers (Site not responding. Last check: 2007-11-07)
		We'll do addition using the one byte floating point representation discussed in the other class notes.
		Unlike floating point addition, negative values are simple to take care of in floating point multiplication.
		Multiplying floating point requires you to add the exponents of two values, then multiply the mantissas, then renormalize that result, adjusting the exponent if necessary.
	www.cs.umd.edu /class/spring2003/cmsc311/Notes/BinMath/multFloat.html (485 words)

	PHP: Floating point numbers - Manual (Site not responding. Last check: 2007-11-07)
		At every operation that wii involve floats, ask yourself "what will happen in the real world if I get a fraction of a cent here" and if the answer is that this operation will generate a transaction in integer cents, do not try to carry fictional fractional accuracy that will only screw things up later.
		The author is not suggesting that some property of decimal numbers causes the behaviour, but that the property of finite binary representations of real numbers which does cause the problem is shared by finite decimal representations.
		I should point out that I originally thought this was an issue with the floats being stored as strings, so I forced them to be floats and they still didn't get evaluated properly (probably 2 different problems there).
	www.php.net /manual/en/language.types.float.php (1401 words)

	2.1.3. Floating-Point Numbers (Site not responding. Last check: 2007-11-07)
		However, the rough intent is that short floating-point numbers be precise to at least four decimal places (but also have a space-efficient representation); single floating-point numbers, to at least seven decimal places; and double floating-point numbers, to at least fourteen decimal places.
		Floating-point numbers are written in either decimal fraction or computerized scientific notation: an optional sign, then a non-empty sequence of digits with an embedded decimal point, then an optional decimal exponent specification.
		Ideally, short-format floating-point numbers should have an ``immediate'' representation that does not require heap allocation; single-format floating-point numbers should approximate IEEE proposed standard single-format floating-point numbers; and double-format floating-point numbers should approximate IEEE proposed standard double-format floating-point numbers [23,17,16].
	www.cs.berkeley.edu /~jaety/cs188/clm/node19.html (991 words)

	Computing with Floating Point Numbers
		Therefore, any number that has infinite number of digits such as 1/3, the square root of 2 and PI cannot be represented completely.
		If you know the way of converting a real number (in base 10) to a real number in other bases, say base 2 or 16, you should be able to verify that 0.1 in base 10 is converted to 0.199999999.....
		In this case, the denominator is equivalent to adding two positive numbers together and as a result the subtraction is implicitly removed.
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/overview/reals.html (1340 words)

	Fast Square Root Calc (Site not responding. Last check: 2007-11-07)
		The calculation of the square root of a floating-point number is a frequently encountered task.
		For example, the number 5.0 can be expressed in binary as 101.0, which means 101.0 x 2^0, which in turn is equal to 1.010 x 2^2, obtained by dividing the mantissa by 4 and multiplying 2^0 by 4.
		Given this representation, a first approximation to the square root of a number is obtained by dividing the exponent by 2.
	www.mactech.com /articles/mactech/Vol.14/14.01/FastSquareRootCalc (1277 words)

	Adding Floating Point Numbers
		Shift the radix point of the mantissa (signficand) Y left by x - y to compensate for the change in exponent.
		It basically consists of adjusting the number with the smaller exponent (call this Y) to that of the larger (call it X), and shifting the radix point of the mantissa of the Y left to compensate.
		Real floating point hardware uses more sophisticated means to round the summed result.
	www.cs.umd.edu /class/spring2003/cmsc311/Notes/BinMath/addFloat.html (901 words)

	What Every Computer Scientist Should Know About Floating-Point Arithmetic
		When adding two floating-point numbers, if their exponents are different, one of the significands will have to be shifted to make the radix points line up, slowing down the operation.
		Since floating-point numbers are always normalized, the most significant bit of the significand is always 1, and there is no reason to waste a bit of storage representing it.
		The section Guard Digits pointed out that computing the exact difference or sum of two floating-point numbers can be very expensive when their exponents are substantially different.
	docs.sun.com /source/806-3568/ncg_goldberg.html (8200 words)

	Floating point
		When using binary (b = 2), one bit can be saved if all numbers are required to be normalized.
		The leading digit of the significand of a normalized binary floating-point number is always non-zero; in particular it is always 1.
		This means that it does not need to be stored explicitly, for a normalized number it can be understood to be 1.
	www.brainyencyclopedia.com /encyclopedia/f/fl/floating_point_1.html (1282 words)

	IEEE floating-point representations of real numbers
		Such numbers are expressed in a slightly different form of scientific notation: The exponent is held fixed at -126, and the mantissa is a number greater than or equal to zero and less than one.
		Unnormalized numbers are stored less accurately than normalized ones (since there are fewer significant digits in the mantissa), but without this special convention for the all-zero exponent it would not be possible to represent them at all, and the designers of the IEEE standard felt that a degraded approximation is better than none.
		that precedes the binary point is once again a ``hidden bit.'' As in single-precision representations, the all-zero exponent is used for unnormalized numbers and (with an all-zero mantissa) for 0, and the all-one exponent is used for the pseudo-numbers positive infinity, negative infinity, and NaN.
	www.math.grin.edu /~stone/courses/fundamentals/IEEE-reals.html (2234 words)

	IEEE-754 References
		The S/390 G5 floating point unit by E. Schwarz and C. Krygowski, which appeared in the IBM Journal of Research and Development, vol.
		Since IEEE-754 floating-point numbers are stored in a signed magnitude form, the ranges and binary patterns of the positive and negative numbers are symmetric about the midpoint of the entire range of values (between the positive and negative zeros).
		As a result, essentially any statement made in regard to the positive numbers is also true of the negative numbers and vice versa.
	babbage.cs.qc.edu /courses/cs341/IEEE-754references.html (2017 words)

	Floating point numbers
		The size of a float is platform-dependent, although a maximum of ~1.8e308 with a precision of roughly 14 decimal digits is a common value (that's 64 bit IEEE format).
		So never trust floating number results to the last digit and never compare floating point numbers for equality.
		For values of other types, the conversion is the same as if the value would have been converted to integer and then to float.
	smithy.kiev.ua /doc/php/language.types.float.html (200 words)

	Floating point numbers (Site not responding. Last check: 2007-11-07)
		float value[3]; // an array of 3 floats int i; // a loop counter for (i = 0; i < 3; ++i) // loop 3 times
		All you have to do then is to take away the number left in the string with the one you created - Hence leaving you with the decimal parts..
		If you cant find it try searching for a hexadecimal number tutorial as hex is a formatting flag which tells cout to display numbers in hexadecimal....
	www.daniweb.com /techtalkforums/thread16191.html (1060 words)

	The IEEE standard for floating point arithmetic
		This standard specifies how single precision (32 bit) and double precision (64 bit) floating point numbers are to be represented, as well as how arithmetic should be carried out on them.
		The IEEE single precision floating point standard representation requires a 32 bit word, which may be represented as numbered from 0 to 31, left to right.
		The IEEE double precision floating point standard representation requires a 64 bit word, which may be represented as numbered from 0 to 63, left to right.
	www.psc.edu /general/software/packages/ieee/ieee.html (489 words)

	Floating Point Numbers
		All numbers in DScript are stored and manipulated in 8-byte IEEE floating-point number format.
		Although you can display numbers using 000 digit separators (e.g., 1,500.00), you should never enter a comma in the number literal because commas are used to separate arguments in a function.
		Depending on the country in which you live, you may not use a period as the decimal point and a comma as the list separator.
	www.vanguardsw.com /dphelp4/dph00187.htm (278 words)

	CCS - FAQ: What is the format of floating point numbers?
		See EX_FLOAT.C for a good example of using floats or float types variables.
		LSB in RAM MSB in RAM 0 00 00 00 00 1 7F 00 00 00 -1 7F 80 00 00 10 82 20 00 00 100 85 48 00 00 123.45 85 76 E6 66 123.45e20 C8 27 4E 53 123.45e-20 43 36 2E 17
		To view a floating point type in MPLAB, insert the first byte of the variable into the watch window and in properties select: 32-bit, MChip Float, High:Low.
	www.ccsinfo.com /faq/?21 (139 words)

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