A Variable Precision Hardware Acceleration for Scientific Computing Andrea Bocco
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Lab 7: Floating-Point Addition 0.0
Lab 7: Floating-Point Addition 0.0 Introduction In this lab, you will write a MIPS assembly language function that performs floating-point addition. You will then run your program using PCSpim (just as you did in Lab 6). For testing, you are provided a program that calls your function to compute the value of the mathematical constant e. For those with no assembly language experience, this will be a long lab, so plan your time accordingly. Background You should be familiar with the IEEE 754 Floating-Point Standard, which is described in Section 3.6 of your book. (Hopefully you have read that section carefully!) Here we will be dealing only with single precision floating- point values, which are formatted as follows (this is also described in the “Floating-Point Representation” subsec- tion of Section 3.6 in your book): Sign Exponent (8 bits) Significand (23 bits) 31 30 29 ... 24 23 22 21 ... 0 Remember that the exponent is biased by 127, which means that an exponent of zero is represented by 127 (01111111). The exponent is not encoded using 2s-complement. The significand is always positive, and the sign bit is kept separately. Note that the actual significand is 24 bits long; the first bit is always a 1 and thus does not need to be stored explicitly. This will be important to remember when you write your function! There are several details of IEEE 754 that you will not have to worry about in this lab. For example, the expo- nents 00000000 and 11111111 are reserved for special purposes that are described in your book (representing zero, denormalized numbers and NaNs). -
Arithmetic Algorithms for Extended Precision Using Floating-Point Expansions Mioara Joldes, Olivier Marty, Jean-Michel Muller, Valentina Popescu
Arithmetic algorithms for extended precision using floating-point expansions Mioara Joldes, Olivier Marty, Jean-Michel Muller, Valentina Popescu To cite this version: Mioara Joldes, Olivier Marty, Jean-Michel Muller, Valentina Popescu. Arithmetic algorithms for extended precision using floating-point expansions. IEEE Transactions on Computers, Institute of Electrical and Electronics Engineers, 2016, 65 (4), pp.1197 - 1210. 10.1109/TC.2015.2441714. hal- 01111551v2 HAL Id: hal-01111551 https://hal.archives-ouvertes.fr/hal-01111551v2 Submitted on 2 Jun 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. IEEE TRANSACTIONS ON COMPUTERS, VOL. , 201X 1 Arithmetic algorithms for extended precision using floating-point expansions Mioara Joldes¸, Olivier Marty, Jean-Michel Muller and Valentina Popescu Abstract—Many numerical problems require a higher computing precision than the one offered by standard floating-point (FP) formats. One common way of extending the precision is to represent numbers in a multiple component format. By using the so- called floating-point expansions, real numbers are represented as the unevaluated sum of standard machine precision FP numbers. This representation offers the simplicity of using directly available, hardware implemented and highly optimized, FP operations. -
Floating Points
Jin-Soo Kim ([email protected]) Systems Software & Architecture Lab. Seoul National University Floating Points Fall 2018 ▪ How to represent fractional values with finite number of bits? • 0.1 • 0.612 • 3.14159265358979323846264338327950288... ▪ Wide ranges of numbers • 1 Light-Year = 9,460,730,472,580.8 km • The radius of a hydrogen atom: 0.000000000025 m 4190.308: Computer Architecture | Fall 2018 | Jin-Soo Kim ([email protected]) 2 ▪ Representation • Bits to right of “binary point” represent fractional powers of 2 • Represents rational number: 2i i i–1 k 2 bk 2 k=− j 4 • • • 2 1 bi bi–1 • • • b2 b1 b0 . b–1 b–2 b–3 • • • b–j 1/2 1/4 • • • 1/8 2–j 4190.308: Computer Architecture | Fall 2018 | Jin-Soo Kim ([email protected]) 3 ▪ Examples: Value Representation 5-3/4 101.112 2-7/8 10.1112 63/64 0.1111112 ▪ Observations • Divide by 2 by shifting right • Multiply by 2 by shifting left • Numbers of form 0.111111..2 just below 1.0 – 1/2 + 1/4 + 1/8 + … + 1/2i + … → 1.0 – Use notation 1.0 – 4190.308: Computer Architecture | Fall 2018 | Jin-Soo Kim ([email protected]) 4 ▪ Representable numbers • Can only exactly represent numbers of the form x / 2k • Other numbers have repeating bit representations Value Representation 1/3 0.0101010101[01]…2 1/5 0.001100110011[0011]…2 1/10 0.0001100110011[0011]…2 4190.308: Computer Architecture | Fall 2018 | Jin-Soo Kim ([email protected]) 5 Fixed Points ▪ p.q Fixed-point representation • Use the rightmost q bits of an integer as representing a fraction • Example: 17.14 fixed-point representation -
Hacking in C 2020 the C Programming Language Thom Wiggers
Hacking in C 2020 The C programming language Thom Wiggers 1 Table of Contents Introduction Undefined behaviour Abstracting away from bytes in memory Integer representations 2 Table of Contents Introduction Undefined behaviour Abstracting away from bytes in memory Integer representations 3 – Another predecessor is B. • Not one of the first programming languages: ALGOL for example is older. • Closely tied to the development of the Unix operating system • Unix and Linux are mostly written in C • Compilers are widely available for many, many, many platforms • Still in development: latest release of standard is C18. Popular versions are C99 and C11. • Many compilers implement extensions, leading to versions such as gnu18, gnu11. • Default version in GCC gnu11 The C programming language • Invented by Dennis Ritchie in 1972–1973 4 – Another predecessor is B. • Closely tied to the development of the Unix operating system • Unix and Linux are mostly written in C • Compilers are widely available for many, many, many platforms • Still in development: latest release of standard is C18. Popular versions are C99 and C11. • Many compilers implement extensions, leading to versions such as gnu18, gnu11. • Default version in GCC gnu11 The C programming language • Invented by Dennis Ritchie in 1972–1973 • Not one of the first programming languages: ALGOL for example is older. 4 • Closely tied to the development of the Unix operating system • Unix and Linux are mostly written in C • Compilers are widely available for many, many, many platforms • Still in development: latest release of standard is C18. Popular versions are C99 and C11. • Many compilers implement extensions, leading to versions such as gnu18, gnu11. -
Floating Point Formats
Telemetry Standard RCC Document 106-07, Appendix O, September 2007 APPENDIX O New FLOATING POINT FORMATS Paragraph Title Page 1.0 Introduction..................................................................................................... O-1 2.0 IEEE 32 Bit Single Precision Floating Point.................................................. O-1 3.0 IEEE 64 Bit Double Precision Floating Point ................................................ O-2 4.0 MIL STD 1750A 32 Bit Single Precision Floating Point............................... O-2 5.0 MIL STD 1750A 48 Bit Double Precision Floating Point ............................. O-2 6.0 DEC 32 Bit Single Precision Floating Point................................................... O-3 7.0 DEC 64 Bit Double Precision Floating Point ................................................. O-3 8.0 IBM 32 Bit Single Precision Floating Point ................................................... O-3 9.0 IBM 64 Bit Double Precision Floating Point.................................................. O-4 10.0 TI (Texas Instruments) 32 Bit Single Precision Floating Point...................... O-4 11.0 TI (Texas Instruments) 40 Bit Extended Precision Floating Point................. O-4 LIST OF TABLES Table O-1. Floating Point Formats.................................................................................... O-1 Telemetry Standard RCC Document 106-07, Appendix O, September 2007 This page intentionally left blank. ii Telemetry Standard RCC Document 106-07, Appendix O, September 2007 APPENDIX O FLOATING POINT -
The Hexadecimal Number System and Memory Addressing
C5537_App C_1107_03/16/2005 APPENDIX C The Hexadecimal Number System and Memory Addressing nderstanding the number system and the coding system that computers use to U store data and communicate with each other is fundamental to understanding how computers work. Early attempts to invent an electronic computing device met with disappointing results as long as inventors tried to use the decimal number sys- tem, with the digits 0–9. Then John Atanasoff proposed using a coding system that expressed everything in terms of different sequences of only two numerals: one repre- sented by the presence of a charge and one represented by the absence of a charge. The numbering system that can be supported by the expression of only two numerals is called base 2, or binary; it was invented by Ada Lovelace many years before, using the numerals 0 and 1. Under Atanasoff’s design, all numbers and other characters would be converted to this binary number system, and all storage, comparisons, and arithmetic would be done using it. Even today, this is one of the basic principles of computers. Every character or number entered into a computer is first converted into a series of 0s and 1s. Many coding schemes and techniques have been invented to manipulate these 0s and 1s, called bits for binary digits. The most widespread binary coding scheme for microcomputers, which is recog- nized as the microcomputer standard, is called ASCII (American Standard Code for Information Interchange). (Appendix B lists the binary code for the basic 127- character set.) In ASCII, each character is assigned an 8-bit code called a byte. -
Extended Precision Floating Point Arithmetic
Extended Precision Floating Point Numbers for Ill-Conditioned Problems Daniel Davis, Advisor: Dr. Scott Sarra Department of Mathematics, Marshall University Floating Point Number Systems Precision Extended Precision Patriot Missile Failure Floating point representation is based on scientific An increasing number of problems exist for which Patriot missile defense modules have been used by • The precision, p, of a floating point number notation, where a nonzero real decimal number, x, IEEE double is insufficient. These include modeling the U.S. Army since the mid-1960s. On February 21, system is the number of bits in the significand. is expressed as x = ±S × 10E, where 1 ≤ S < 10. of dynamical systems such as our solar system or the 1991, a Patriot protecting an Army barracks in Dha- This means that any normalized floating point The values of S and E are known as the significand • climate, and numerical cryptography. Several arbi- ran, Afghanistan failed to intercept a SCUD missile, number with precision p can be written as: and exponent, respectively. When discussing float- trary precision libraries have been developed, but leading to the death of 28 Americans. This failure E ing points, we are interested in the computer repre- x = ±(1.b1b2...bp−2bp−1)2 × 2 are too slow to be practical for many complex ap- was caused by floating point rounding error. The sentation of numbers, so we must consider base 2, or • The smallest x such that x > 1 is then: plications. David Bailey’s QD library may be used system measured time in tenths of seconds, using binary, rather than base 10. -
POINTER (IN C/C++) What Is a Pointer?
POINTER (IN C/C++) What is a pointer? Variable in a program is something with a name, the value of which can vary. The way the compiler and linker handles this is that it assigns a specific block of memory within the computer to hold the value of that variable. • The left side is the value in memory. • The right side is the address of that memory Dereferencing: • int bar = *foo_ptr; • *foo_ptr = 42; // set foo to 42 which is also effect bar = 42 • To dereference ted, go to memory address of 1776, the value contain in that is 25 which is what we need. Differences between & and * & is the reference operator and can be read as "address of“ * is the dereference operator and can be read as "value pointed by" A variable referenced with & can be dereferenced with *. • Andy = 25; • Ted = &andy; All expressions below are true: • andy == 25 // true • &andy == 1776 // true • ted == 1776 // true • *ted == 25 // true How to declare pointer? • Type + “*” + name of variable. • Example: int * number; • char * c; • • number or c is a variable is called a pointer variable How to use pointer? • int foo; • int *foo_ptr = &foo; • foo_ptr is declared as a pointer to int. We have initialized it to point to foo. • foo occupies some memory. Its location in memory is called its address. &foo is the address of foo Assignment and pointer: • int *foo_pr = 5; // wrong • int foo = 5; • int *foo_pr = &foo; // correct way Change the pointer to the next memory block: • int foo = 5; • int *foo_pr = &foo; • foo_pr ++; Pointer arithmetics • char *mychar; // sizeof 1 byte • short *myshort; // sizeof 2 bytes • long *mylong; // sizeof 4 byts • mychar++; // increase by 1 byte • myshort++; // increase by 2 bytes • mylong++; // increase by 4 bytes Increase pointer is different from increase the dereference • *P++; // unary operation: go to the address of the pointer then increase its address and return a value • (*P)++; // get the value from the address of p then increase the value by 1 Arrays: • int array[] = {45,46,47}; • we can call the first element in the array by saying: *array or array[0]. -
Subtyping Recursive Types
ACM Transactions on Programming Languages and Systems, 15(4), pp. 575-631, 1993. Subtyping Recursive Types Roberto M. Amadio1 Luca Cardelli CNRS-CRIN, Nancy DEC, Systems Research Center Abstract We investigate the interactions of subtyping and recursive types, in a simply typed λ-calculus. The two fundamental questions here are whether two (recursive) types are in the subtype relation, and whether a term has a type. To address the first question, we relate various definitions of type equivalence and subtyping that are induced by a model, an ordering on infinite trees, an algorithm, and a set of type rules. We show soundness and completeness between the rules, the algorithm, and the tree semantics. We also prove soundness and a restricted form of completeness for the model. To address the second question, we show that to every pair of types in the subtype relation we can associate a term whose denotation is the uniquely determined coercion map between the two types. Moreover, we derive an algorithm that, when given a term with implicit coercions, can infer its least type whenever possible. 1This author's work has been supported in part by Digital Equipment Corporation and in part by the Stanford-CNR Collaboration Project. Page 1 Contents 1. Introduction 1.1 Types 1.2 Subtypes 1.3 Equality of Recursive Types 1.4 Subtyping of Recursive Types 1.5 Algorithm outline 1.6 Formal development 2. A Simply Typed λ-calculus with Recursive Types 2.1 Types 2.2 Terms 2.3 Equations 3. Tree Ordering 3.1 Subtyping Non-recursive Types 3.2 Folding and Unfolding 3.3 Tree Expansion 3.4 Finite Approximations 4. -
Effectiveness of Floating-Point Precision on the Numerical Approximation by Spectral Methods
Mathematical and Computational Applications Article Effectiveness of Floating-Point Precision on the Numerical Approximation by Spectral Methods José A. O. Matos 1,2,† and Paulo B. Vasconcelos 1,2,∗,† 1 Center of Mathematics, University of Porto, R. Dr. Roberto Frias, 4200-464 Porto, Portugal; [email protected] 2 Faculty of Economics, University of Porto, R. Dr. Roberto Frias, 4200-464 Porto, Portugal * Correspondence: [email protected] † These authors contributed equally to this work. Abstract: With the fast advances in computational sciences, there is a need for more accurate compu- tations, especially in large-scale solutions of differential problems and long-term simulations. Amid the many numerical approaches to solving differential problems, including both local and global methods, spectral methods can offer greater accuracy. The downside is that spectral methods often require high-order polynomial approximations, which brings numerical instability issues to the prob- lem resolution. In particular, large condition numbers associated with the large operational matrices, prevent stable algorithms from working within machine precision. Software-based solutions that implement arbitrary precision arithmetic are available and should be explored to obtain higher accu- racy when needed, even with the higher computing time cost associated. In this work, experimental results on the computation of approximate solutions of differential problems via spectral methods are detailed with recourse to quadruple precision arithmetic. Variable precision arithmetic was used in Tau Toolbox, a mathematical software package to solve integro-differential problems via the spectral Tau method. Citation: Matos, J.A.O.; Vasconcelos, Keywords: floating-point arithmetic; variable precision arithmetic; IEEE 754-2008 standard; quadru- P.B. -
MIPS Floating Point
CS352H: Computer Systems Architecture Lecture 6: MIPS Floating Point September 17, 2009 University of Texas at Austin CS352H - Computer Systems Architecture Fall 2009 Don Fussell Floating Point Representation for dynamically rescalable numbers Including very small and very large numbers, non-integers Like scientific notation –2.34 × 1056 +0.002 × 10–4 normalized +987.02 × 109 not normalized In binary yyyy ±1.xxxxxxx2 × 2 Types float and double in C University of Texas at Austin CS352H - Computer Systems Architecture Fall 2009 Don Fussell 2 Floating Point Standard Defined by IEEE Std 754-1985 Developed in response to divergence of representations Portability issues for scientific code Now almost universally adopted Two representations Single precision (32-bit) Double precision (64-bit) University of Texas at Austin CS352H - Computer Systems Architecture Fall 2009 Don Fussell 3 IEEE Floating-Point Format single: 8 bits single: 23 bits double: 11 bits double: 52 bits S Exponent Fraction x = (!1)S "(1+Fraction)"2(Exponent!Bias) S: sign bit (0 ⇒ non-negative, 1 ⇒ negative) Normalize significand: 1.0 ≤ |significand| < 2.0 Always has a leading pre-binary-point 1 bit, so no need to represent it explicitly (hidden bit) Significand is Fraction with the “1.” restored Exponent: excess representation: actual exponent + Bias Ensures exponent is unsigned Single: Bias = 127; Double: Bias = 1203 University of Texas at Austin CS352H - Computer Systems Architecture Fall 2009 Don Fussell 4 Single-Precision Range Exponents 00000000 and 11111111 reserved Smallest -
FPGA Based Quadruple Precision Floating Point Arithmetic for Scientific Computations
International Journal of Advanced Computer Research (ISSN (print): 2249-7277 ISSN (online): 2277-7970) Volume-2 Number-3 Issue-5 September-2012 FPGA Based Quadruple Precision Floating Point Arithmetic for Scientific Computations 1Mamidi Nagaraju, 2Geedimatla Shekar 1Department of ECE, VLSI Lab, National Institute of Technology (NIT), Calicut, Kerala, India 2Asst.Professor, Department of ECE, Amrita Vishwa Vidyapeetham University Amritapuri, Kerala, India Abstract amounts, and fractions are essential to many computations. Floating-point arithmetic lies at the In this project we explore the capability and heart of computer graphics cards, physics engines, flexibility of FPGA solutions in a sense to simulations and many models of the natural world. accelerate scientific computing applications which Floating-point computations suffer from errors due to require very high precision arithmetic, based on rounding and quantization. Fast computers let IEEE 754 standard 128-bit floating-point number programmers write numerically intensive programs, representations. Field Programmable Gate Arrays but computed results can be far from the true results (FPGA) is increasingly being used to design high due to the accumulation of errors in arithmetic end computationally intense microprocessors operations. Implementing floating-point arithmetic in capable of handling floating point mathematical hardware can solve two separate problems. First, it operations. Quadruple Precision Floating-Point greatly speeds up floating-point arithmetic and Arithmetic is important in computational fluid calculations. Implementing a floating-point dynamics and physical modelling, which require instruction will require at a generous estimate at least accurate numerical computations. However, twenty integer instructions, many of them conditional modern computers perform binary arithmetic, operations, and even if the instructions are executed which has flaws in representing and rounding the on an architecture which goes to great lengths to numbers.