Floating-Point (FP) Numbers
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Bibliography [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Applied Math. Series 55 (National Bureau of Standards, Washington, D.C., 1964) [2] R.C. Agarwal, J.C. Cooley, F.G. Gustavson, J.B. Shearer, G. Slishman, B. Tuckerman, New scalar and vector elementary functions for the IBM system/370. IBM J. Res. Dev. 30(2), 126–144 (1986) [3] H.M. Ahmed, Efficient elementary function generation with multipliers, in Proceedings of the 9th IEEE Symposium on Computer Arithmetic (1989), pp. 52–59 [4] H.M. Ahmed, J.M. Delosme, M. Morf, Highly concurrent computing structures for matrix arithmetic and signal processing. Computer 15(1), 65–82 (1982) [5] H. Alt, Comparison of arithmetic functions with respect to Boolean circuits, in Proceedings of the 16th ACM STOC (1984), pp. 466–470 [6] American National Standards Institute and Institute of Electrical and Electronic Engineers. IEEE Standard for Binary Floating-Point Arithmetic. ANSI/IEEE Standard 754–1985 (1985) [7] C. Ancourt, F. Irigoin, Scanning polyhedra with do loops, in Proceedings of the 3rd ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming (PPoPP’91), Apr 1991 (ACM Press, New York, NY, 1991), pp. 39–50 [8] I.J. Anderson, A distillation algorithm for floating-point summation. SIAM J. Sci. Comput. 20(5), 1797–1806 (1999) [9] M. Andrews, T. Mraz, Unified elementary function generator. Microprocess. Microsyst. 2(5), 270–274 (1978) [10] E. Antelo, J.D. Bruguera, J. Villalba, E. Zapata, Redundant CORDIC rotator based on parallel prediction, in Proceedings of the 12th IEEE Symposium on Computer Arithmetic, July 1995, pp. -
Design of 32 Bit Floating Point Addition and Subtraction Units Based on IEEE 754 Standard Ajay Rathor, Lalit Bandil Department of Electronics & Communication Eng
International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 2 Issue 6, June - 2013 Design Of 32 Bit Floating Point Addition And Subtraction Units Based On IEEE 754 Standard Ajay Rathor, Lalit Bandil Department of Electronics & Communication Eng. Acropolis Institute of Technology and Research Indore, India Abstract Precision format .Single precision format, Floating point arithmetic implementation the most basic format of the ANSI/IEEE described various arithmetic operations like 754-1985 binary floating-point arithmetic addition, subtraction, multiplication, standard. Double precision and quad division. In science computation this circuit precision described more bit operation so at is useful. A general purpose arithmetic unit same time we perform 32 and 64 bit of require for all operations. In this paper operation of arithmetic unit. Floating point single precision floating point arithmetic includes single precision, double precision addition and subtraction is described. This and quad precision floating point format paper includes single precision floating representation and provide implementation point format representation and provides technique of various arithmetic calculation. implementation technique of addition and Normalization and alignment are useful for subtraction arithmetic calculations. Low operation and floating point number should precision custom format is useful for reduce be normalized before any calculation [3]. the associated circuit costs and increasing their speed. I consider the implementation ofIJERT IJERT2. Floating Point format Representation 32 bit addition and subtraction unit for floating point arithmetic. Floating point number has mainly three formats which are single precision, double Keywords: Addition, Subtraction, single precision and quad precision. precision, doubles precision, VERILOG Single Precision Format: Single-precision HDL floating-point format is a computer number 1. -
Instruction Set Reference, A-M
Intel® 64 and IA-32 Architectures Software Developer’s Manual Volume 2A: Instruction Set Reference, A-M NOTE: The Intel 64 and IA-32 Architectures Software Developer's Manual consists of five volumes: Basic Architecture, Order Number 253665; Instruction Set Reference A-M, Order Number 253666; Instruction Set Reference N-Z, Order Number 253667; System Programming Guide, Part 1, Order Number 253668; System Programming Guide, Part 2, Order Number 253669. Refer to all five volumes when evaluating your design needs. Order Number: 253666-023US May 2007 INFORMATION IN THIS DOCUMENT IS PROVIDED IN CONNECTION WITH INTEL PRODUCTS. NO LICENSE, EXPRESS OR IMPLIED, BY ESTOPPEL OR OTHERWISE, TO ANY INTELLECTUAL PROPERTY RIGHTS IS GRANT- ED BY THIS DOCUMENT. EXCEPT AS PROVIDED IN INTEL’S TERMS AND CONDITIONS OF SALE FOR SUCH PRODUCTS, INTEL ASSUMES NO LIABILITY WHATSOEVER, AND INTEL DISCLAIMS ANY EXPRESS OR IMPLIED WARRANTY, RELATING TO SALE AND/OR USE OF INTEL PRODUCTS INCLUDING LIABILITY OR WARRANTIES RELATING TO FITNESS FOR A PARTICULAR PURPOSE, MERCHANTABILITY, OR INFRINGEMENT OF ANY PATENT, COPYRIGHT OR OTHER INTELLECTUAL PROPERTY RIGHT. INTEL PRODUCTS ARE NOT INTENDED FOR USE IN MEDICAL, LIFE SAVING, OR LIFE SUSTAINING APPLICATIONS. Intel may make changes to specifications and product descriptions at any time, without notice. Developers must not rely on the absence or characteristics of any features or instructions marked “reserved” or “undefined.” Improper use of reserved or undefined features or instructions may cause unpredictable be- havior or failure in developer's software code when running on an Intel processor. Intel reserves these fea- tures or instructions for future definition and shall have no responsibility whatsoever for conflicts or incompatibilities arising from their unauthorized use. -
Floating Point Representation (Unsigned) Fixed-Point Representation
Floating point representation (Unsigned) Fixed-point representation The numbers are stored with a fixed number of bits for the integer part and a fixed number of bits for the fractional part. Suppose we have 8 bits to store a real number, where 5 bits store the integer part and 3 bits store the fractional part: 1 0 1 1 1.0 1 1 $ 2& 2% 2$ 2# 2" 2'# 2'$ 2'% Smallest number: Largest number: (Unsigned) Fixed-point representation Suppose we have 64 bits to store a real number, where 32 bits store the integer part and 32 bits store the fractional part: "# "% + /+ !"# … !%!#!&. (#(%(" … ("% % = * !+ 2 + * (+ 2 +,& +,# "# "& & /# % /"% = !"#× 2 +!"&× 2 + ⋯ + !&× 2 +(#× 2 +(%× 2 + ⋯ + ("%× 2 0 ∞ (Unsigned) Fixed-point representation Range: difference between the largest and smallest numbers possible. More bits for the integer part ⟶ increase range Precision: smallest possible difference between any two numbers More bits for the fractional part ⟶ increase precision "#"$"%. '$'#'( # OR "$"%. '$'#'(') # Wherever we put the binary point, there is a trade-off between the amount of range and precision. It can be hard to decide how much you need of each! Scientific Notation In scientific notation, a number can be expressed in the form ! = ± $ × 10( where $ is a coefficient in the range 1 ≤ $ < 10 and + is the exponent. 1165.7 = 0.0004728 = Floating-point numbers A floating-point number can represent numbers of different order of magnitude (very large and very small) with the same number of fixed bits. In general, in the binary system, a floating number can be expressed as ! = ± $ × 2' $ is the significand, normally a fractional value in the range [1.0,2.0) . -
Floating Point Numbers and Arithmetic
Overview Floating Point Numbers & • Floating Point Numbers Arithmetic • Motivation: Decimal Scientific Notation – Binary Scientific Notation • Floating Point Representation inside computer (binary) – Greater range, precision • Decimal to Floating Point conversion, and vice versa • Big Idea: Type is not associated with data • MIPS floating point instructions, registers CS 160 Ward 1 CS 160 Ward 2 Review of Numbers Other Numbers •What about other numbers? • Computers are made to deal with –Very large numbers? (seconds/century) numbers 9 3,155,760,00010 (3.1557610 x 10 ) • What can we represent in N bits? –Very small numbers? (atomic diameter) -8 – Unsigned integers: 0.0000000110 (1.010 x 10 ) 0to2N -1 –Rationals (repeating pattern) 2/3 (0.666666666. .) – Signed Integers (Two’s Complement) (N-1) (N-1) –Irrationals -2 to 2 -1 21/2 (1.414213562373. .) –Transcendentals e (2.718...), π (3.141...) •All represented in scientific notation CS 160 Ward 3 CS 160 Ward 4 Scientific Notation Review Scientific Notation for Binary Numbers mantissa exponent Mantissa exponent 23 -1 6.02 x 10 1.0two x 2 decimal point radix (base) “binary point” radix (base) •Computer arithmetic that supports it called •Normalized form: no leadings 0s floating point, because it represents (exactly one digit to left of decimal point) numbers where binary point is not fixed, as it is for integers •Alternatives to representing 1/1,000,000,000 –Declare such variable in C as float –Normalized: 1.0 x 10-9 –Not normalized: 0.1 x 10-8, 10.0 x 10-10 CS 160 Ward 5 CS 160 Ward 6 Floating -
Floating Point Numbers Dr
Numbers Representaion Floating point numbers Dr. Tassadaq Hussain Riphah International University Real Numbers • Two’s complement representation deal with signed integer values only. • Without modification, these formats are not useful in scientific or business applications that deal with real number values. • Floating-point representation solves this problem. Floating-Point Representation • If we are clever programmers, we can perform floating-point calculations using any integer format. • This is called floating-point emulation, because floating point values aren’t stored as such; we just create programs that make it seem as if floating- point values are being used. • Most of today’s computers are equipped with specialized hardware that performs floating-point arithmetic with no special programming required. —Not embedded processors! Floating-Point Representation • Floating-point numbers allow an arbitrary number of decimal places to the right of the decimal point. For example: 0.5 0.25 = 0.125 • They are often expressed in scientific notation. For example: 0.125 = 1.25 10-1 5,000,000 = 5.0 106 Floating-Point Representation • Computers use a form of scientific notation for floating-point representation • Numbers written in scientific notation have three components: Floating-Point Representation • Computer representation of a floating-point number consists of three fixed-size fields: • This is the standard arrangement of these fields. Note: Although “significand” and “mantissa” do not technically mean the same thing, many people use these terms interchangeably. We use the term “significand” to refer to the fractional part of a floating point number. Floating-Point Representation • The one-bit sign field is the sign of the stored value. -
A Decimal Floating-Point Speciftcation
A Decimal Floating-point Specification Michael F. Cowlishaw, Eric M. Schwarz, Ronald M. Smith, Charles F. Webb IBM UK IBM Server Division P.O. Box 31, Birmingham Rd. 2455 South Rd., MS:P310 Warwick CV34 5JL. UK Poughkeepsie, NY 12601 USA [email protected] [email protected] Abstract ing is required. In the fixed point formats, rounding must be explicitly applied in software rather than be- Even though decimal arithmetic is pervasive in fi- ing provided by the hardware. To address these and nancial and commercial transactions, computers are other limitations, we propose implementing a decimal stdl implementing almost all arithmetic calculations floating-point format. But what should this format be? using binary arithmetic. As chip real estate becomes This paper discusses the issues of defining a decimal cheaper it is becoming likely that more computer man- floating-point format. ufacturers will provide processors with decimal arith- First, we consider the goals of the specification. It metic engines. Programming languages and databases must be compliant with standards already in place. are expanding the decimal data types available whale One standard we consider is the ANSI X3.274-1996 there has been little change in the base hardware. As (Programming Language REXX) [l]. This standard a result, each language and application is defining a contains a definition of an integrated floating-point and different arithmetic and few have considered the efi- integer decimal arithmetic which avoids the need for ciency of hardware implementations when setting re- two distinct data types and representations. The other quirements. relevant standard is the ANSI/IEEE 854-1987 (Radix- In this paper, we propose a decimal format which Independent Floating-point Arithmetic) [a]. -
Numerical Computing
Numerical computing How computers store real numbers and the problems that result Overview • Integers • Reals, floats, doubles etc • Arithmetical operations and rounding errors • We write: x = sqrt(2.0) – but how is this stored? L02 Numerical Computing 2 Mathematics vs Computers • Mathematics is an ideal world – integers can be as large as you want – real numbers can be as large or as small as you want – can represent every number exactly: 1, -3, 1/3, 1036237, 10-232322, √2, π, .... • Numbers range from - ∞ to +∞ – there is also infinite numbers in any interval • This not true on a computer – numbers have a limited range (integers and real numbers) – limited precision (real numbers) L02 Numerical Computing 3 Integers • We like to use base 10 – we only write the 10 characters 0,1,2,3,4,5,6,7,8,9 – use position to represent each power of 10 2 1 0 125 = 1 * 10 + 2 * 10 + 5 * 10 = 1*100 + 2*10 + 5*1 = 125 – represent positive or negative using a leading “+” or “-” • Computers are binary machines – can only store ones and zeros – minimum storage unit is 8 bits = 1 byte • Use base 2 1111101=1*26 +1*25 +1*24 +1*23 +1*22 +0*21 +1*20 =1*64 +1*32 +1*16 +1*8 +1*4 +0*2 +1*1 =125 L02 Numerical Computing 4 Storage and Range • Assume we reserve 1 byte (8 bits) for integers – minimum value 0 – maximum value 28 – 1 = 255 – if result is out of range we will overflow and get wrong answer! • Standard storage is 4 bytes = 32 bits – minimum value 0 – maximum value 232 – 1 = 4294967291 = 4 billion = 4G • Is this a problem? – question: what is a 32-bit operating -
Chap02: Data Representation in Computer Systems
CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 57 2.2 Positional Numbering Systems 58 2.3 Converting Between Bases 58 2.3.1 Converting Unsigned Whole Numbers 59 2.3.2 Converting Fractions 61 2.3.3 Converting between Power-of-Two Radices 64 2.4 Signed Integer Representation 64 2.4.1 Signed Magnitude 64 2.4.2 Complement Systems 70 2.4.3 Excess-M Representation for Signed Numbers 76 2.4.4 Unsigned Versus Signed Numbers 77 2.4.5 Computers, Arithmetic, and Booth’s Algorithm 78 2.4.6 Carry Versus Overflow 81 2.4.7 Binary Multiplication and Division Using Shifting 82 2.5 Floating-Point Representation 84 2.5.1 A Simple Model 85 2.5.2 Floating-Point Arithmetic 88 2.5.3 Floating-Point Errors 89 2.5.4 The IEEE-754 Floating-Point Standard 90 2.5.5 Range, Precision, and Accuracy 92 2.5.6 Additional Problems with Floating-Point Numbers 93 2.6 Character Codes 96 2.6.1 Binary-Coded Decimal 96 2.6.2 EBCDIC 98 2.6.3 ASCII 99 2.6.4 Unicode 99 2.7 Error Detection and Correction 103 2.7.1 Cyclic Redundancy Check 103 2.7.2 Hamming Codes 106 2.7.3 Reed-Soloman 113 Chapter Summary 114 CMPS375 Class Notes (Chap02) Page 1 / 25 Dr. Kuo-pao Yang 2.1 Introduction 57 This chapter describes the various ways in which computers can store and manipulate numbers and characters. Bit: The most basic unit of information in a digital computer is called a bit, which is a contraction of binary digit. -
Names for Standardized Floating-Point Formats
Names WORK IN PROGRESS April 19, 2002 11:05 am Names for Standardized Floating-Point Formats Abstract I lack the courage to impose names upon floating-point formats of diverse widths, precisions, ranges and radices, fearing the damage that can be done to one’s progeny by saddling them with ill-chosen names. Still, we need at least a list of the things we must name; otherwise how can we discuss them without talking at cross-purposes? These things include ... Wordsize, Width, Precision, Range, Radix, … all of which must be both declarable by a computer programmer, and ascertainable by a program through Environmental Inquiries. And precision must be subject to truncation by Coercions. Conventional Floating-Point Formats Among the names I have seen in use for floating-point formats are … float, double, long double, real, REAL*…, SINGLE PRECISION, DOUBLE PRECISION, double extended, doubled double, TEMPREAL, quad, … . Of these floating-point formats the conventional ones include finite numbers x all of the form x = (–1)s·m·ßn+1–P in which s, m, ß, n and P are integers with the following significance: s is the Sign bit, 0 or 1 according as x ≥ 0 or x ≤ 0 . ß is the Radix or Base, two for Binary and ten for Decimal. (I exclude all others). P is the Precision measured in Significant Digits of the given radix. m is the Significand or Coefficient or (wrongly) Mantissa of | x | running in the range 0 ≤ m ≤ ßP–1 . ≤ ≤ n is the Exponent, perhaps Biased, confined to some interval Nmin n Nmax . Other terms are in use. -
Largest Denormalized Negative Infinity ---Number with Hex
CSE2421 HOMEWORK #2 DUE DATE: MONDAY 11/5 11:59pm PROBLEM 2.84 Given a floating-point format with a k-bit exponent and an n-bit fraction, write formulas for the exponent E, significand M, the fraction f, and the value V for the quantities that follow. In addition, describe the bit representation. A. the number 7.0 B. the largest odd integer that can be represented exactly C. the reciprocal of the smallest positive normalized value PROBLEM 2.86 Consider a 16-bit floating-point representation based on the IEEE floating-point format, with one sign bit, seven exponent bits (k=7), and eight fraction bits (n=8). The exponent bias is 27-1 - 1 = 63. Fill in the table that follows for each of the numbers given, with the following instructions for each column: Hex: the four hexadecimal digits describing the encoded form. M: the value of the significand. This should be a number of the form x or x/y, where x is an integer and y is an integral power of 2. Examples include: 0, 67/64, and 1/256. E: the integer value of the exponent. V: the numeric value represented. Use the notation x or x * 2z, where x and z are integers. As an example, to represent the number 7/8, we would have s=0, M = 7/4 and E = -1. Our number would therefore have an exponent field of 0x3E (decimal value 63-1 = 62) and a significand field 0xC0 (binary 110000002) giving a hex representation of 3EC0. You need not fill in entries marked “---“. -
Chap02: Data Representation in Computer Systems
CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 57 2.2 Positional Numbering Systems 58 2.3 Converting Between Bases 58 2.3.1 Converting Unsigned Whole Numbers 59 2.3.2 Converting Fractions 61 2.3.3 Converting between Power-of-Two Radices 64 2.4 Signed Integer Representation 64 2.4.1 Signed Magnitude 64 2.4.2 Complement Systems 70 2.4.3 Excess-M Representation for Signed Numbers 76 2.4.4 Unsigned Versus Signed Numbers 77 2.4.5 Computers, Arithmetic, and Booth’s Algorithm 78 2.4.6 Carry Versus Overflow 81 2.4.7 Binary Multiplication and Division Using Shifting 82 2.5 Floating-Point Representation 84 2.5.1 A Simple Model 85 2.5.2 Floating-Point Arithmetic 88 2.5.3 Floating-Point Errors 89 2.5.4 The IEEE-754 Floating-Point Standard 90 2.5.5 Range, Precision, and Accuracy 92 2.5.6 Additional Problems with Floating-Point Numbers 93 2.6 Character Codes 96 2.6.1 Binary-Coded Decimal 96 2.6.2 EBCDIC 98 2.6.3 ASCII 99 2.6.4 Unicode 99 2.7 Error Detection and Correction 103 2.7.1 Cyclic Redundancy Check 103 2.7.2 Hamming Codes 106 2.7.3 Reed-Soloman 113 Chapter Summary 114 CMPS375 Class Notes (Chap02) Page 1 / 25 by Kuo-pao Yang 2.1 Introduction 57 This chapter describes the various ways in which computers can store and manipulate numbers and characters. Bit: The most basic unit of information in a digital computer is called a bit, which is a contraction of binary digit.