Floating-point arithmetic in the context of IEEE 754

The IBM System/360 (S/360) is a family of computer systems announced by IBM on April 7, 1964, and delivered between 1965 and 1978. System/360 was the first family of computers designed to cover both commercial and scientific applications and a complete range of sizes from small, entry-level machines to large mainframes. The design distinguished between architecture and implementation, allowing IBM to release a suite of compatible designs at different prices. All but the only partially compatible Model 44 and the most expensive systems use microcode to implement the instruction set, which used 8-bit byte addressing with fixed-point binary, fixed-point decimal and hexadecimal floating-point calculations. The System/360 family introduced IBM's Solid Logic Technology (SLT), which packed more transistors onto a circuit card, allowing more powerful but smaller computers, but did not include integrated circuits, which IBM considered too immature.

System/360's chief architect was Gene Amdahl and the project was managed by Fred Brooks, responsible to Chairman Thomas J. Watson Jr. The commercial release was piloted by another of Watson's lieutenants, John R. Opel, who managed the launch of IBM's System/360 mainframe family in 1964. The slowest System/360 model announced in 1964, the Model 30, could perform up to 34,500 instructions per second, with memory from 8 to 64 KB. High-performance models came later. The 1967 IBM System/360 Model 91 could execute up to 16.6 million instructions per second. The larger 360 models could have up to 8 MB of main memory, though that much memory was unusual; a large installation might have as little as 256 KB of main storage, but 512 KB, 768 KB or 1024 KB was more common. Up to 8 megabytes of slower (8 microsecond) Large Capacity Storage (LCS) was also available for some models.

In high-level programming, a variable is an abstract storage or indirection location paired with an associated symbolic name, which contains some known or unknown quantity of data or object referred to as a value; or in simpler terms, a variable is a named container for a particular set of bits or type of data (like integer, float, string, etc...) or undefined. A variable can eventually be associated with or identified by a memory address. The variable name is the usual way to reference the stored value, in addition to referring to the variable itself, depending on the context. This separation of name and content allows the name to be used independently of the exact information it represents. The identifier in computer source code can be bound to a value during run time, and the value of the variable may thus change during the course of program execution.

Variables in programming may not directly correspond to the concept of variables in mathematics. The latter is abstract, having no reference to a physical object such as storage location. The value of a computing variable is not necessarily part of an equation or formula as in mathematics. Furthermore, the variables can also be constants if the value is defined statically. Variables in computer programming are frequently given long names to make them relatively descriptive of their use, whereas variables in mathematics often have terse, one- or two-character names for brevity in transcription and manipulation.

A coprocessor is a computer processor used to supplement the functions of the primary processor (the CPU). Operations performed by the coprocessor may be floating-point arithmetic, graphics, signal processing, string processing, cryptography or I/O interfacing with peripheral devices. By offloading processor-intensive tasks from the main processor, coprocessors can accelerate system performance. Coprocessors allow a line of computers to be customized, so that customers who do not need the extra performance do not need to pay for it.

Hexadecimal floating point (now called HFP by IBM) is a format for encoding floating-point numbers first introduced on the IBM System/360 computers, and supported on subsequent machines based on that architecture, as well as machines which were intended to be application-compatible with System/360.

In comparison to IEEE 754 floating point, the HFP format has a longer significand, and a shorter exponent. All HFP formats have 7 bits of exponent with a bias of 64. The normalized range of representable numbers is from 16 to 16 (approx. 5.39761 × 10 to 7.237005 × 10).

Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible.

Numerical linear algebra aims to solve problems of continuous mathematics using finite precision computers, so its applications to the natural and social sciences are as vast as the applications of continuous mathematics. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, telecommunication, computational finance, materials science simulations, structural biology, data mining, bioinformatics, and fluid dynamics. Matrix methods are particularly used in finite difference methods, finite element methods, and the modeling of differential equations. Noting the broad applications of numerical linear algebra, Lloyd N. Trefethen and David Bau, III argue that it is "as fundamental to the mathematical sciences as calculus and differential equations", even though it is a comparatively small field. Because many properties of matrices and vectors also apply to functions and operators, numerical linear algebra can also be viewed as a type of functional analysis which has a particular emphasis on practical algorithms.

C99 (C9X during its development, formally ISO/IEC 9899:1999) is a past version of the C programming language open standard. It extends the previous version (C90) with new features for the language and the standard library, and helps implementations make better use of available computer hardware, such as IEEE 754-1985 floating-point arithmetic, and compiler technology. The C11 version of the C programming language standard, published in 2011, updates C99.

A floating-point unit (FPU), numeric processing unit (NPU), colloquially math coprocessor, is a part of a computer system specially designed to carry out operations on floating-point numbers. Typical operations are addition, subtraction, multiplication, division, and square root. Modern designs generally include a fused multiply-add instruction, which was found to be very common in real-world code. Some FPUs can also perform various transcendental functions such as exponential or trigonometric calculations, but the accuracy can be low, so some systems prefer to compute these functions in software.

Floating-point operations were originally handled in software in early computers. Over time, manufacturers began to provide standardized floating-point libraries as part of their software collections. Some machines, those dedicated to scientific processing, would include specialized hardware to perform some of these tasks with much greater speed. The introduction of microcode in the 1960s allowed these instructions to be included in the system's instruction set architecture (ISA). Normally these would be decoded by the microcode into a series of instructions that were similar to the libraries, but on those machines with an FPU, they would instead be routed to that unit, which would perform them much faster. This allowed floating-point instructions to become universal while the floating-point hardware remained optional; for instance, on the PDP-11 one could add the floating-point processor unit at any time using plug-in expansion cards.