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Topic: Arithmetic underflow

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	IEEE Arithmetic
		Underflow occurs, roughly speaking, when the result of an arithmetic operation is so small that it cannot be stored in its intended destination format without suffering a rounding error that is larger than usual.
		The presence of subnormal numbers in the arithmetic means that untrapped underflow (which implies loss of accuracy) cannot occur on addition or subtraction.
		When subnormal numbers are added to the representable set and gradual underflow is implemented, the worst effect of one inexact or underflowed result is to introduce an error no greater than the distance to one of the representable neighbors of the computed result.
	docs.sun.com /source/806-3568/ncg_math.html (5560 words)

	Basic arithmetic coding by Arturo Campos
		Arithmetic coding, is entropy coder widely used, the only problem is it's speed, but compression tends to be better than Huffman can achieve.
		The idea behind arithmetic coding is to have a probability line, 0-1, and assign to every symbol a range in this line based on its probability, the higher the probability, the higher range which assigns to it.
		The algorithm of arithmetic coding makes that if ever the msb of both high and low match are equal, then they'll never change, this is how can output the higher bits of the output infinite number, and continue working with just 16 bits.
	www.arturocampos.com /ac_arithmetic.html (1638 words)

	Arithmetic Encyclopedia (Site not responding. Last check: )
		Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations.
		The prehistory of arithmetic is limited by a very small number of small artifacts indicating a clear conception of addition and subtraction, the best-known being the Ishango Bone from Africa, dating from 18,000 BC.
		Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one to its right.
	www.hallencyclopedia.com /topic/Arithmetic.html (1801 words)

	Arithmetic underflow - Wikipedia, the free encyclopedia
		Arithmetic underflow (or simply underflow) occurs in a digital computer, when a calculation produces a result that is less than a given register or storage location can store or represent.
		Underflow is analogous to overflow except that the magnitude is too large rather than too small.
		As specified in IEEE 754 the underflow condition is only signalled if there is also a loss of accuracy.
	en.wikipedia.org /wiki/Arithmetic_underflow (180 words)

	Numerical Computation Guide: 2 - IEEE Arithmetic (Site not responding. Last check: )
		A floating-point format is a data structure specifying the fields that comprise a floating-point numeral, the layout of those fields, and their arithmetic interpretation.
		Underflow occurs, roughly speaking, when the result of an arithmetic operation is so small that it cannot be stored in its intended destination format without suffering a rounding error that is larger than usual.
		The presence of subnormal numbers in the arithmetic means that untrapped underflow (which implies loss of accuracy) cannot occur on addition or subtraction.
	cch.loria.fr /documentation/IEEE754/numerical_comp_guide/ncg_math.doc.html (5218 words)

	Arithmetic Code Discussion and Implementation
		Arithmetic coding is similar to Huffman coding; they both achieve their compression by reducing the average number of bits required to represent a symbol.
		The standard form of arithmetic coding's encoding is based on fractional ranges on a probability line between 0 and 1.
		The standard form of arithmetic coding's decoding is also based on fractional ranges on a probability line between 0 and 1.
	michael.dipperstein.com /arithmetic/index.html (3334 words)

	Elementary Arithmetic
		The first and most important consideration is the choice of number representation for arithmetic calculations.
		Underflow is the same problem at the bottom end of the scale.
		There are two ways to do this: you can divide it by 100 and add the result to the original value; or you can multiply by 101 and then divide by 100.
	www.erasmatazz.com /library/JCGD_Volume_4/Elementary_Arithmetic.html (1422 words)

	Data Types and Scaling (Fixed-Point Blockset)
		Although the IEEE Standard 754 specifies practices and procedures to deal with exceptional arithmetic conditions in a consistent manner, microprocessor manufacturers may handle these conditions in ways that depart from the standard.
		When the exponent of the result is too small (i.e., a negative exponent with too large a magnitude), the result is denormalized by right-shifting the fraction and leaving the exponent at its minimum value.
		Gradual underflow fills that gap and reduces the impact of exponent underflow to a level comparable with round off among the normalized numbers.
	www.weizmann.ac.il /matlab/toolbox/fixpoint/c3_ty15a.html (352 words)

	Quantization and Quantized Filtering (Filter Design Toolbox) (Site not responding. Last check: )
		In addition to specifying a floating-point format, the IEEE 754 Standard for binary floating-point arithmetic specifies practices and procedures so that predictable results are produced independent of the hardware platform.
		When the exponent of the result is too small (such as a negative exponent whose magnitude is too large), the result is denormalized by right-shifting the fraction and leaving the exponent at its minimum value.
		Gradual underflow fills that gap and reduces the impact of exponent underflow to a level comparable with roundoff among the normalized numbers.
	www.weizmann.ac.il /matlab/toolbox/filterdesign/quant_23.html (180 words)

	Decimal Arithmetic - FAQ 4
		With a normalized arithmetic, alignment of the opreands is needed more often, and an extra step is required at the end of every calculation to determine whether normalization is required and then effect it if it is. These unnecessary steps requires extra code (in software) or extra circuitry (in hardware).
		It also gives a helpful indication that an underflow may have occurred (after an underflow which rounds to zero, the exponent of the zero will always be the smallest possible exponent, whereas zeros which are the results of normal calculations will have exponents appropriately related to the operands).
		In their arithmetic, all numbers are to be treated as exact, with the proviso that if the number of decimal places becomes unduly large, some of them may be eliminated by rounding off in the course of the calculation.
	www2.hursley.ibm.com /decimal/decifaq4.html (3464 words)

	HP-UX floating-point guide for HP Integrity servers - HP DSPP (Site not responding. Last check: )
		But most arithmetic operations on floating-numbers have mathematical results that fall into the gaps between the computerâs floating-point numbers, and therefore must suffer a roundoff error in order to be represented in the computer.
		Underflow signifies that the result may have less precision than is usual for the format or may be zero in lieu of a tiny mathematical value.
		By default, underflow is gradual, as specified by the floating-point standard.
	h21007.www2.hp.com /dspp/tech/tech_TechDocumentDetailPage_IDX/1,1701,8359,00.html (5563 words)

	IEEE 754R committee minutes from September 19, 2002
		Underflowed results of sums and differences are exact, so we need not worry about rounding them; we just need to put them in the proper format.
		The effects of flushed underflow are subtle, and bugs caused by misapplication of flush-to-zero are difficult to diagnose.
		The underflowed values contributed not at all to the sum, but underflow occurred with sufficient frequency that the entire benchmark was 6X slower than it should have been.
	grouper.ieee.org /groups/754/meeting-minutes/02-09-19.html (3779 words)

	Untitled
		The output from an arithmetic coding process is a single number less than 1 and greater than or equal to 0.
		We then have to set an underflow counter to remember that we threw away a digit, and we aren't quite sure whether it was going to end up as a 0 or a 9.
		The underflow digits will be all 9s or 0s, depending on whether High and Low converged to the higher or lower value.
	www.dogma.net /markn/articles/arith/part1.htm (5152 words)

	Computer Arithmetic Demonstration and Test Facility
		The adoption of the IEEE Standard for Binary Floating-Point Arithmetic in 1985 was the culmination of many years of development.
		It stipulates the definition of parameters for the arithmetic and the reaction of the system when the arithmetic fails due to overflow, underflow, or division by zero.
		The new arithmetics are less prone to failure than conventional arithmetic, a potentially important characteristic in critical applications with requirements for very high reliability.
	math.nist.gov /mcsd/Reports/95/yearly/node49.html (392 words)

	The Java Lesson 5: Arithmetic operations, conversions, and casts
		Five arithmetic operations are part of the Java language.
		Note: The arithmetic assignment operators always result in the first operand (the one on the left) being modified.
		Arithmetic expressions can be large and complex with several variables, constants, and operators.
	javafaq.nu /java-article396.html (1596 words)

	GLOSSARY OF COMPUTERIZED SYSTEM AND SOFTWARE DEVELOPMENT TERMINOLOGY
		The [high speed] circuits within the CPU which are responsible for performing the arithmetic and logical operations of a computer.
		(ISO) In an arithmetic operation, a result whose absolute value is too small to be represented within the range of the numeration system in use.
		As ADD, SUBTRACT, MULTIPLY, and DIVIDE are the primary operations of arithmetic, AND, OR, and NOT are the primary operations of Boolean Logic.
	www.fda.gov /ora/inspect_ref/igs/gloss.html (15293 words)

	What Every Computer Scientist Should Know About Floating-Point Arithmetic
		Sign/magnitude is the system used for the sign of the significand in the IEEE formats: one bit is used to hold the sign, the rest of the bits represent the magnitude of the number.
		Although it is true that the reciprocal of the largest number will underflow, underflow is usually less serious than overflow.
		When a program is moved between two machines and both support IEEE arithmetic, then if any intermediate result differs, it must be because of software bugs, not from differences in arithmetic.
	docs.sun.com /source/806-3568/ncg_goldberg.html (8200 words)

	IEEE 754: Frequently Asked Questions (Site not responding. Last check: )
		The short answer is that gradual underflow preserves more mathematical identities.
		A longer answer involves preserving relative error bounds in the face of underflow.
		The 854 standard encompasses decimal arithmetic, but there is little hardware support outside of desktop calculators.
	www.cs.berkeley.edu /~ejr/Projects/ieee754/faq.html (489 words)

	FixedPointArithmetic - GCC Wiki
		NOTE: special behaviors involving the fixed-point arithmetic are that when mixing integer data with fixed-point data or mixing two different fixed-point data types for binary operators, the compiler should not convert one data type to another data type (ex.
		Therefore, having the fixed-point arithmetic supports in the compiler, the saturation add example can be written by using the "_Sat _Fract" data type and the "+" operator as follows.
		Without the fixed-point arithmetic supports, programmers need to write explicit C code to perform fractional multiplication (which requires an extra left shift after integer multiplication) and detect the saturation.
	gcc.gnu.org /wiki/FixedPointArithmetic (1049 words)

	[No title] (Site not responding. Last check: )
		Exact arithmetic is vital in accounting applications where rounding errors may introduce monetary losses that cannot be reconciled.
		Arithmetic Instructions The 80287's arithmetic instruction set (table 2-2) provides a wealth of variations on the basic add, subtract, multiply, and divide operations, and a number of other useful functions.
		The 80287's basic arithmetic instructions (addition, subtraction, multiplication, and division) are designed to encourage the development of very efficient algorithms.
	www.ragestorm.net /downloads/287intel.txt (6716 words)

	11.2.3 The FPU Instruction Set
		They will set the underflow exception bit when storing an 80 bit value into a 32 or 64 bit memory variable, but the value is too small to fit into the destination operand.
		These instructions set the underflow exception bit if the result is too small (i.e., less than one but greater than zero or less than zero but greater than -1).
		These instructions convert their 16 or 32 bit integer operands to an 80 bit extended precision floating point value and then use this value as the source operand for the specified operation.
	webster.cs.ucr.edu /AoA/Windows/HTML/RealArithmetica2.html (4992 words)

	Computer Arithmetic - IEEE Floating Point Number
		Underflow and overflow during calulations were (and remain) of special concern.
		Underflow occurs when an operation results in a value too small to be represented.
		Considerations of overflow and underflow led the IEEE 754 members to decide that they needed special positive and negative infinity values to indicate overflows, and both positive and negative zero to indicate underflows.
	www.hal-pc.org /~clyndes/computer-arithmetic/floats.html (1660 words)

	Floating Point Arithmetic
		As can be seen, a floating point arithmetic unit needs to be able to add and subtract exponents, and to shift, add, and subtract mantissas.
		Floating Point on MIPS -- -------- ----- -- ---- A. MIPS floating point arithmetic is performed by a coprocessor - which is typically part of the main CPU chip, though having its own dedicated registers and arithmetic circuitry.
		Floating point arithmetic (single and double precision versions.) add.s add.d sub.s sub.d mul.s mul.d div.s div.d These all use R format, but interpret the register specifier fields as designating floating point registers.
	www.cs.gordon.edu /local/courses/cs311/lectures-2003/floating_point.html (956 words)

	Chapter Eleven Real Arithmetic
		By the conclusion of this chapter you should be able to translate arithmetic expressions and assignment statements involving floating point operands from high level languages like Pascal and C/C++ into 80x86 assembly language.
		Underflow occurs when the result is too small to fit in the destination operand.
		Of course, this is always better than underflowing the denormalized value to zero (which could make the computation even less accurate), but you must keep in mind that if you work with very small values you may lose some accuracy in your computations.
	webster.cs.ucr.edu /AoA/Linux/HTML/RealArithmetic.html (2647 words)

	Definition of Arithmetic underflow
		In a digital computer, the condition that occurs when a calculation produces a result that is less than a given register or storage location can store or represent.
		As specified in IEEE 754 the underflow condition is only signalled if there is also a loss of accuracy.
		However if the user is trapping on underflow, this happens regardless of consideration for loss of precision.
	www.wordiq.com /definition/Arithmetic_underflow (195 words)

	Chapter 5: Expressions -- Valvano
		An important issue when performing arithmetic calculations on integer values is the problem of underflow and overflow.
		It is important to remember that arithmetic operations (addition, subtraction, multiplication, division, and shifting) have constraints when performed with finite precision on a microcomputer.
		If the overflow bit is set after a signed operation the result is adjusted to the largest (ceiling) or smallest (floor) possible signed number depending on whether it was a -128 to 127 cross over (N=0) or 127 to -128 cross over (N=1).
	www.ece.utexas.edu /~valvano/embed/chap5/chap5.htm (3614 words)

	Decimal Arithmetic - Scope
		Exceptional conditions, such as overflow, underflow, undefined results, and other exceptional situations which may occur during operations.
		This specification does not require that values returned after overflow and underflow change if the exception trap-enabler is set, and the criteria for underflow similarly do not change if its trap-enabler is set (the Subnormal condition has been added to allow the alternative underflow condition to be detected).
		Dependence on the trap-enabler setting is difficult to make thread-safe, and also the IEEE 854 definition does not generalize to the power operator (the required replacement value may itself overflow or underflow).
	www.gobosoft.com /eiffel/gobo/math/decimal/dascope.html (330 words)

	[No title] (Site not responding. Last check: )
		The arithmetic operations defined on a computer are of necessity inexact.
		The model we use for floating-point arithmetic is the IEEE-754 double-precision standard; it is used on all of the computers the computation is run on.
		We ask the system to signal an error if underflow occurs; and the error bounds we use are only valid when there is no underflow.
	math.princeton.edu /~annals/issues/2003/verification/corona/roundoff.w (367 words)

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