Floating Point Arithmetic Chapter 14
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Division by Zero Calculus and Singular Integrals
Proceedings of International Conference on Mechanical, Electrical and Medical Intelligent System 2017 Division by Zero Calculus and Singular Integrals Tsutomu Matsuura1, a and Saburou Saitoh2,b 1Faculty of Science and Technology, Gunma University, 1-5-1 Tenjin-cho, Kiryu 376-8515, Japan 2Institute of Reproducing Kernels, Kawauchi-cho, 5-1648-16, Kiryu 376-0041, Japan [email protected], [email protected] Keywords: division by zero, point at infinity, singularity, Laurent expansion, Hadamard finite part Abstract. In this paper, we will introduce the formulas log 0= log ∞= 0 (not as limiting values) in the meaning of the one point compactification of Aleksandrov and their fundamental applications, and we will also examine the relationship between Laurent expansion and the division by zero. Based on those examinations we give the interpretation for the Hadamard finite part of singular integrals by means of the division by zero calculus. In particular, we will know that the division by zero is our elementary and fundamental mathematics. 1. Introduction By a natural extension of the fractions b (1) a for any complex numbers a and b , we found the simple and beautiful result, for any complex number b b = 0, (2) 0 incidentally in [1] by the Tikhonov regularization for the Hadamard product inversions for matrices and we discussed their properties and gave several physical interpretations on the general fractions in [2] for the case of real numbers. The result is very special case for general fractional functions in [3]. The division by zero has a long and mysterious story over the world (see, for example, H. -
Fujitsu SPARC64™ X+/X Software on Chip Overview for Developers]
White paper [Fujitsu SPARC64™ X+/X Software on Chip Overview for Developers] White paper Fujitsu SPARC64™ X+/X Software on Chip Overview for Developers Page 1 of 13 www.fujitsu.com/sparc White paper [Fujitsu SPARC64™ X+/X Software on Chip Overview for Developers] Table of Contents Table of Contents 1 Software on Chip Innovative Technology 3 2 SPARC64™ X+/X SIMD Vector Processing 4 2.1 How Oracle Databases take advantage of SPARC64™ X+/X SIMD vector processing 4 2.2 How to use of SPARC64™ X+/X SIMD instructions in user applications 5 2.3 How to check if the system and operating system is capable of SIMD execution? 6 2.4 How to check if SPARC64™ X+/X SIMD instructions indeed have been generated upon compilation of a user source code? 6 2.5 Is SPARC64™ X+/X SIMD implementation compatible with Oracle’s SPARC SIMD? 6 3 SPARC64™ X+/X Decimal Floating Point Processing 8 3.1 How Oracle Databases take advantage of SPARC64™ X+/X Decimal Floating-Point 8 3.2 Decimal Floating-Point processing in user applications 8 4 SPARC64™ X+/X Extended Floating-Point Registers 9 4.1 How Oracle Databases take advantage of SPARC64™ X+/X Decimal Floating-Point 9 5 SPARC64™ X+/X On-Chip Cryptographic Processing Capabilities 10 5.1 How to use the On-Chip Cryptographic Processing Capabilities 10 5.2 How to use the On-Chip Cryptographic Processing Capabilities in user applications 10 6 Conclusions 12 Page 2 of 13 www.fujitsu.com/sparc White paper [Fujitsu SPARC64™ X+/X Software on Chip Overview for Developers] Software on Chip Innovative Technology 1 Software on Chip Innovative Technology Fujitsu brings together innovations in supercomputing, business computing, and mainframe computing in the Fujitsu M10 enterprise server family to help organizations meet their business challenges. -
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. -
Chapter 2 DIRECT METHODS—PART II
Chapter 2 DIRECT METHODS—PART II If we use finite precision arithmetic, the results obtained by the use of direct methods are contaminated with roundoff error, which is not always negligible. 2.1 Finite Precision Computation 2.2 Residual vs. Error 2.3 Pivoting 2.4 Scaling 2.5 Iterative Improvement 2.1 Finite Precision Computation 2.1.1 Floating-point numbers decimal floating-point numbers The essence of floating-point numbers is best illustrated by an example, such as that of a 3-digit floating-point calculator which accepts numbers like the following: 123. 50.4 −0.62 −0.02 7.00 Any such number can be expressed as e ±d1.d2d3 × 10 where e ∈{0, 1, 2}. (2.1) Here 40 t := precision = 3, [L : U] := exponent range = [0 : 2]. The exponent range is rather limited. If the calculator display accommodates scientific notation, e g., 3.46 3 −1.56 −3 then we might use [L : U]=[−9 : 9]. Some numbers have multiple representations in form (2.1), e.g., 2.00 × 101 =0.20 × 102. Hence, there is a normalization: • choose smallest possible exponent, • choose + sign for zero, e.g., 0.52 × 102 → 5.20 × 101, 0.08 × 10−8 → 0.80 × 10−9, −0.00 × 100 → 0.00 × 10−9. −9 Nonzero numbers of form ±0.d2d3 × 10 are denormalized. But for large-scale scientific computation base 2 is preferred. binary floating-point numbers This is an important matter because numbers like 0.2 do not have finite representations in base 2: 0.2=(0.001100110011 ···)2. -
Bit Nove Signal Interface Processor
www.audison.eu bit Nove Signal Interface Processor POWER SUPPLY CONNECTION Voltage 10.8 ÷ 15 VDC From / To Personal Computer 1 x Micro USB Operating power supply voltage 7.5 ÷ 14.4 VDC To Audison DRC AB / DRC MP 1 x AC Link Idling current 0.53 A Optical 2 sel Optical In 2 wire control +12 V enable Switched off without DRC 1 mA Mem D sel Memory D wire control GND enable Switched off with DRC 4.5 mA CROSSOVER Remote IN voltage 4 ÷ 15 VDC (1mA) Filter type Full / Hi pass / Low Pass / Band Pass Remote OUT voltage 10 ÷ 15 VDC (130 mA) Linkwitz @ 12/24 dB - Butterworth @ Filter mode and slope ART (Automatic Remote Turn ON) 2 ÷ 7 VDC 6/12/18/24 dB Crossover Frequency 68 steps @ 20 ÷ 20k Hz SIGNAL STAGE Phase control 0° / 180° Distortion - THD @ 1 kHz, 1 VRMS Output 0.005% EQUALIZER (20 ÷ 20K Hz) Bandwidth @ -3 dB 10 Hz ÷ 22 kHz S/N ratio @ A weighted Analog Input Equalizer Automatic De-Equalization N.9 Parametrics Equalizers: ±12 dB;10 pole; Master Input 102 dBA Output Equalizer 20 ÷ 20k Hz AUX Input 101.5 dBA OPTICAL IN1 / IN2 Inputs 110 dBA TIME ALIGNMENT Channel Separation @ 1 kHz 85 dBA Distance 0 ÷ 510 cm / 0 ÷ 200.8 inches Input sensitivity Pre Master 1.2 ÷ 8 VRMS Delay 0 ÷ 15 ms Input sensitivity Speaker Master 3 ÷ 20 VRMS Step 0,08 ms; 2,8 cm / 1.1 inch Input sensitivity AUX Master 0.3 ÷ 5 VRMS Fine SET 0,02 ms; 0,7 cm / 0.27 inch Input impedance Pre In / Speaker In / AUX 15 kΩ / 12 Ω / 15 kΩ GENERAL REQUIREMENTS Max Output Level (RMS) @ 0.1% THD 4 V PC connections USB 1.1 / 2.0 / 3.0 Compatible Microsoft Windows (32/64 bit): Vista, INPUT STAGE Software/PC requirements Windows 7, Windows 8, Windows 10 Low level (Pre) Ch1 ÷ Ch6; AUX L/R Video Resolution with screen resize min. -
Decimal Floating Point for Future Processors Hossam A
1 Decimal Floating Point for future processors Hossam A. H. Fahmy Electronics and Communications Department, Cairo University, Egypt Email: [email protected] Tarek ElDeeb, Mahmoud Yousef Hassan, Yasmin Farouk, Ramy Raafat Eissa SilMinds, LLC Email: [email protected] F Abstract—Many new designs for Decimal Floating Point (DFP) hard- Simple decimal fractions such as 1=10 which might ware units have been proposed in the last few years. To date, only represent a tax amount or a sales discount yield an the IBM POWER6 and POWER7 processors include internal units for infinitely recurring number if converted to a binary decimal floating point processing. We have designed and tested several representation. Hence, a binary number system with a DFP units including an adder, multiplier, divider, square root, and fused- multiply-add compliant with the IEEE 754-2008 standard. This paper finite number of bits cannot accurately represent such presents the results of using our units as part of a vector co-processor fractions. When an approximated representation is used and the anticipated gains once the units are moved on chip with the in a series of computations, the final result may devi- main processor. ate from the correct result expected by a human and required by the law [3], [4]. One study [5] shows that in a large billing application such an error may be up to 1 WHY DECIMAL HARDWARE? $5 million per year. Ten is the natural number base or radix for humans Banking, billing, and other financial applications use resulting in a decimal number system while a binary decimal extensively. -
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. -
Midterm-2020-Solution.Pdf
HONOR CODE Questions Sheet. A Lets C. [6 Points] 1. What type of address (heap,stack,static,code) does each value evaluate to Book1, Book1->name, Book1->author, &Book2? [4] 2. Will all of the print statements execute as expected? If NO, write print statement which will not execute as expected?[2] B. Mystery [8 Points] 3. When the above code executes, which line is modified? How many times? [2] 4. What is the value of register a6 at the end ? [2] 5. What is the value of register a4 at the end ? [2] 6. In one sentence what is this program calculating ? [2] C. C-to-RISC V Tree Search; Fill in the blanks below [12 points] D. RISCV - The MOD operation [8 points] 19. The data segment starts at address 0x10000000. What are the memory locations modified by this program and what are their values ? E Floating Point [8 points.] 20. What is the smallest nonzero positive value that can be represented? Write your answer as a numerical expression in the answer packet? [2] 21. Consider some positive normalized floating point number where p is represented as: What is the distance (i.e. the difference) between p and the next-largest number after p that can be represented? [2] 22. Now instead let p be a positive denormalized number described asp = 2y x 0.significand. What is the distance between p and the next largest number after p that can be represented? [2] 23. Sort the following minifloat numbers. [2] F. Numbers. [5] 24. What is the smallest number that this system can represent 6 digits (assume unsigned) ? [1] 25. -
2018-19 MAP 160 Byte File Layout Specifications
2018-19 MAP 160 Byte File Layout Specifications OVERVIEW: A) ISAC will provide an Eligibility Status File (ESF) record for each student to all schools listed as a college choice on the student’s Student Aid Report (SAR). The ESF records will be available daily as Record Type = 7. ESF records may be retrieved via the File Extraction option in MAP. B) Schools will transmit Payment Requests to ISAC via File Transfer Protocol (FTP) using the MAP 160byte layout and identify these with Record Type = 4. C) When payment requests are processed, ISAC will provide payment results to schools through the MAP system. The payment results records can be retrieved in the 160 byte format using the MAP Payment Results File Extraction Option. MAP results records have a Record Type = 5. The MAP Payment Results file contains some eligibility status data elements. Also, the same student record may appear on both the Payment Results and the Eligibility Status extract files. Schools may also use the Reports option in MAP to obtain payment results. D) To cancel Payment Requests, the school with the current Payment Results record on ISAC's Payment Database must transmit a matching record with MAP Payment Request Code = C, with the Requested Award Amount field equal to zero and the Enrollment Hours field equal to 0 along with other required data elements. These records must be transmitted to ISAC as Record Type = 4. E) Summary of Data Element Changes, revision (highlighted in grey) made to the 2018-19 layout. NONE 1 2018-19 MAP 160 Byte File Layout Specifications – 9/17 2018-19 MAP 160 Byte File Layout Specifications F) The following 160 byte record layout will be used for transmitting data between schools and ISAC.