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76-421: BLISS is a typeless block - structured programming language based on expressions rather than statements, and includes constructs for exception handling , coroutines , and macros . It does not include a goto statement. The name is variously said to be short for Basic Language for Implementation of System Software or System Software Implementation Language, Backwards . However, in his 2015 oral history for

152-434: A data type (or simply type ) is a collection or grouping of data values , usually specified by a set of possible values, a set of allowed operations on these values, and/or a representation of these values as machine types. A data type specification in a program constrains the possible values that an expression , such as a variable or a function call, might take. On literal data, it tells the compiler or interpreter how

228-402: A structured programming model would tend to not include code, and are called plain old data structures . Data types may be categorized according to several factors: The terminology varies - in the literature, primitive, built-in, basic, atomic, and fundamental may be used interchangeably. All data in computers based on digital electronics is represented as bits (alternatives 0 and 1) on

304-421: A to forall r. (forall a. f a -> r) -> r or a similar type. A refinement type is a type endowed with a predicate which is assumed to hold for any element of the refined type. For instance, the type of natural numbers greater than 5 may be written as { n ∈ N | n > 5 } {\displaystyle \{n\in \mathbb {N} \,|\,n>5\}} A dependent type

380-461: A computer system specially designed to carry out operations on floating-point numbers. A number representation specifies some way of encoding a number, usually as a string of digits. There are several mechanisms by which strings of digits can represent numbers. In standard mathematical notation, the digit string can be of any length, and the location of the radix point is indicated by placing an explicit "point" character (dot or comma) there. If

456-597: A corresponding native type does not exist on the target platform, the compiler will break them down into code using types that do exist. For instance, if a 32-bit integer is requested on a 16 bit platform, the compiler will tacitly treat it as an array of two 16 bit integers. Floating point data types represent certain fractional values ( rational numbers , mathematically). Although they have predefined limits on both their maximum values and their precision, they are sometimes misleadingly called reals (evocative of mathematical real numbers ). They are typically stored internally in

532-565: A family of function types differentiated by argument and return types, such as the type Int -> Bool denoting functions taking an integer and returning a Boolean. In C, a function is not a first-class data type but function pointers can be manipulated by the program. Java and C++ originally did not have function values but have added them in C++11 and Java 8. A type constructor builds new types from old ones, and can be thought of as an operator taking zero or more types as arguments and producing

608-497: A few details. Five of these formats are called basic formats , and others are termed extended precision formats and extendable precision format . Three formats are especially widely used in computer hardware and languages: Increasing the precision of the floating-point representation generally reduces the amount of accumulated round-off error caused by intermediate calculations. Other IEEE formats include: Any integer with absolute value less than 2 can be exactly represented in

684-475: A language-defined machine representation. The C programming language , for instance, supplies types such as Booleans, integers, floating-point numbers, etc., but the precise bit representations of these types are implementation-defined. The only C type with a precise machine representation is the char type that represents a byte. The Boolean type represents the values true and false . Although only two values are possible, they are more often represented as

760-408: A number, the base (10) need not be stored, since it will be the same for the entire range of supported numbers, and can thus be inferred. Symbolically, this final value is: s b p − 1 × b e , {\displaystyle {\frac {s}{b^{\,p-1}}}\times b^{e},} where s is the significand (ignoring any implied decimal point), p

836-419: A numeric string, such as "1234" . These numeric strings are usually considered distinct from numeric values such as 1234 , although some languages automatically convert between them. A union type definition will specify which of a number of permitted subtypes may be stored in its instances, e.g. "float or long integer". In contrast with a record , which could be defined to contain a float and an integer,

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912-617: A pointer whose value was never a valid memory address would cause a program to crash. To ameliorate this potential problem, a pointer type is typically considered distinct from the corresponding integer type, even if the underlying representation is the same. Functional programming languages treat functions as a distinct datatype and allow values of this type to be stored in variables and passed to functions. Some multi-paradigm languages such as JavaScript also have mechanisms for treating functions as data. Most contemporary type systems go beyond JavaScript's simple type "function object" and have

988-484: A precision limited only by the available memory and computational resources on the system. Bignum implementations of arithmetic operations on machine-sized values are significantly slower than the corresponding machine operations. The enumerated type has distinct values, which can be compared and assigned, but which do not necessarily have any particular concrete representation in the computer's memory; compilers and interpreters can represent them arbitrarily. For example,

1064-409: A sequence of characters used to store words or plain text , most often textual markup languages representing formatted text . Characters may be a letter of some alphabet , a digit, a blank space, a punctuation mark, etc. Characters are drawn from a character set such as ASCII . Character and string types can have different subtypes according to the character encoding. The original 7-bit wide ASCII

1140-1118: A syntax to be used that could be entered through a typewriter , as was the case of his Electromechanical Arithmometer in 1920. In 1938, Konrad Zuse of Berlin completed the Z1 , the first binary, programmable mechanical computer ; it uses a 24-bit binary floating-point number representation with a 7-bit signed exponent, a 17-bit significand (including one implicit bit), and a sign bit. The more reliable relay -based Z3 , completed in 1941, has representations for both positive and negative infinities; in particular, it implements defined operations with infinity, such as 1 / ∞ = 0 {\displaystyle ^{1}/_{\infty }=0} , and it stops on undefined operations, such as 0 × ∞ {\displaystyle 0\times \infty } . Zuse also proposed, but did not complete, carefully rounded floating-point arithmetic that includes ± ∞ {\displaystyle \pm \infty } and NaN representations, anticipating features of

1216-427: A system-software project usually develops its own input/output or builds on basic monitor I/O or screen management services ... it permits access to machine-specific features, because system software often requires this. BLISS has characteristics that are unusual among high-level languages. A name ... is uniformly interpreted as the address of that segment rather than the value of the segment ... Also, BLISS

1292-421: A type. Product types, function types, power types and list types can be made into type constructors. Universally-quantified and existentially-quantified types are based on predicate logic . Universal quantification is written as ∀ x . f ( x ) {\displaystyle \forall x.f(x)} or forall x. f x and is the intersection over all types x of the body f x , i.e.

1368-413: A union may only contain one subtype at a time. A tagged union (also called a variant , variant record, discriminated union, or disjoint union) contains an additional field indicating its current type for enhanced type safety. An algebraic data type (ADT) is a possibly recursive sum type of product types . A value of an ADT consists of a constructor tag together with zero or more field values, with

1444-536: A value. The value of an expression can be either used or discarded in BLISS ;... Finally, BLISS includes a macro facility that provides a level of capability usually found only in macro-assemblers. The BLISS language has the following characteristics: The following example is taken verbatim from the Bliss Language Manual : Data type In computer science and computer programming ,

1520-415: A word rather as a single bit as it requires more machine instructions to store and retrieve an individual bit. Many programming languages do not have an explicit Boolean type, instead using an integer type and interpreting (for instance) 0 as false and other values as true. Boolean data refers to the logical structure of how the language is interpreted to the machine language. In this case a Boolean 0 refers to

1596-500: Is a rational number , because it can be represented as one integer divided by another; for example 1.45 × 10 is (145/100)×1000 or 145,000 /100. The base determines the fractions that can be represented; for instance, 1/5 cannot be represented exactly as a floating-point number using a binary base, but 1/5 can be represented exactly using a decimal base ( 0.2 , or 2 × 10 ). However, 1/3 cannot be represented exactly by either binary (0.010101...) or decimal (0.333...), but in base 3 , it

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1672-544: Is a data type that does not specify the concrete representation of the data. Instead, a formal specification based on the data type's operations is used to describe it. Any implementation of a specification must fulfill the rules given. For example, a stack has push/pop operations that follow a Last-In-First-Out rule, and can be concretely implemented using either a list or an array. Abstract data types are used in formal semantics and program verification and, less strictly, in design . The main non-composite, derived type

1748-524: Is a smallest positive normal floating-point number, which has a 1 as the leading digit and 0 for the remaining digits of the significand, and the smallest possible value for the exponent. There is a largest floating-point number, which has B − 1 as the value for each digit of the significand and the largest possible value for the exponent. In addition, there are representable values strictly between −UFL and UFL. Namely, positive and negative zeros , as well as subnormal numbers . The IEEE standardized

1824-492: Is a type whose definition depends on a value. Two common examples of dependent types are dependent functions and dependent pairs. The return type of a dependent function may depend on the value (not just type) of one of its arguments. A dependent pair may have a second value of which the type depends on the first value. An intersection type is a type containing those values that are members of two specified types. For example, in Java

1900-837: Is also possible to define a dependent intersection type, denoted ( x : σ ) ∩ τ {\displaystyle (x:\sigma )\cap \tau } , where the type τ {\displaystyle \tau } may depend on the term variable x {\displaystyle x} . Some programming languages represent the type information as data, enabling type introspection and reflection . In contrast, higher order type systems , while allowing types to be constructed from other types and passed to functions as values, typically avoid basing computational decisions on them. For convenience, high-level languages and databases may supply ready-made "real world" data types, for instance times, dates, and monetary values (currency). These may be built-in to

1976-482: Is an "expression language" rather than a "statement language". This means that every construct of the language that is not a declaration is an expression. Expressions produce a value as well as possibly causing an action such as modification of storage, transfer of control, or execution of a program loop. For example, the counterpart of an assignment "statement" in BLISS is, strictly speaking, an expression that itself has

2052-440: Is denoted i32 and panics on overflow in debug mode. Most programming languages also allow the programmer to define additional data types, usually by combining multiple elements of other types and defining the valid operations of the new data type. For example, a programmer might create a new data type named " complex number " that would include real and imaginary parts, or a color data type represented by three bytes denoting

2128-432: Is from 2  ≈ 2 × 10 to approximately 2  ≈ 2 × 10 . The number of normal floating-point numbers in a system ( B , P , L , U ) where is 2 ( B − 1 ) ( B P − 1 ) ( U − L + 1 ) {\displaystyle 2\left(B-1\right)\left(B^{P-1}\right)\left(U-L+1\right)} . There

2204-447: Is not a floating-point number in base ten with five digits—it needs six digits. The nearest floating-point number with only five digits is 12.346. And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits. In practice, most floating-point systems use base two , though base ten ( decimal floating point ) is also common. Floating-point arithmetic operations, such as addition and division, approximate

2280-496: Is often used to allow very small and very large real numbers that require fast processing times. The result of this dynamic range is that the numbers that can be represented are not uniformly spaced; the difference between two consecutive representable numbers varies with their exponent. Over the years, a variety of floating-point representations have been used in computers. In 1985, the IEEE 754 Standard for Floating-Point Arithmetic

2356-399: Is similar in concept to scientific notation. Logically, a floating-point number consists of: To derive the value of the floating-point number, the significand is multiplied by the base raised to the power of the exponent , equivalent to shifting the radix point from its implied position by a number of places equal to the value of the exponent—to the right if the exponent is positive or to

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2432-1358: Is stored in memory using the IEEE 754 encoding, this becomes the significand s . The significand is assumed to have a binary point to the right of the leftmost bit. So, the binary representation of π is calculated from left-to-right as follows: ( ∑ n = 0 p − 1 bit n × 2 − n ) × 2 e = ( 1 × 2 − 0 + 1 × 2 − 1 + 0 × 2 − 2 + 0 × 2 − 3 + 1 × 2 − 4 + ⋯ + 1 × 2 − 23 ) × 2 1 ≈ 1.57079637 × 2 ≈ 3.1415927 {\displaystyle {\begin{aligned}&\left(\sum _{n=0}^{p-1}{\text{bit}}_{n}\times 2^{-n}\right)\times 2^{e}\\={}&\left(1\times 2^{-0}+1\times 2^{-1}+0\times 2^{-2}+0\times 2^{-3}+1\times 2^{-4}+\cdots +1\times 2^{-23}\right)\times 2^{1}\\\approx {}&1.57079637\times 2\\\approx {}&3.1415927\end{aligned}}} where p

2508-411: Is the pointer , a data type whose value refers directly to (or "points to") another value stored elsewhere in the computer memory using its address . It is a primitive kind of reference . (In everyday terms, a page number in a book could be considered a piece of data that refers to another one). Pointers are often stored in a format similar to an integer; however, attempting to dereference or "look up"

2584-458: Is the precision ( 24 in this example), n is the position of the bit of the significand from the left (starting at 0 and finishing at 23 here) and e is the exponent ( 1 in this example). It can be required that the most significant digit of the significand of a non-zero number be non-zero (except when the corresponding exponent would be smaller than the minimum one). This process is called normalization . For binary formats (which uses only

2660-621: Is the precision (the number of digits in the significand), b is the base (in our example, this is the number ten ), and e is the exponent. Historically, several number bases have been used for representing floating-point numbers, with base two ( binary ) being the most common, followed by base ten ( decimal floating point ), and other less common varieties, such as base sixteen ( hexadecimal floating point ), base eight (octal floating point ), base four (quaternary floating point ), base three ( balanced ternary floating point ) and even base 256 and base 65,536 . A floating-point number

2736-469: Is trivial (0.1 or 1×3 ) . The occasions on which infinite expansions occur depend on the base and its prime factors . The way in which the significand (including its sign) and exponent are stored in a computer is implementation-dependent. The common IEEE formats are described in detail later and elsewhere, but as an example, in the binary single-precision (32-bit) floating-point representation, p = 24 {\displaystyle p=24} , and so

2812-688: The English Electric DEUCE . The arithmetic is actually implemented in software, but with a one megahertz clock rate, the speed of floating-point and fixed-point operations in this machine were initially faster than those of many competing computers. The mass-produced IBM 704 followed in 1954; it introduced the use of a biased exponent . For many decades after that, floating-point hardware was typically an optional feature, and computers that had it were said to be "scientific computers", or to have " scientific computation " (SC) capability (see also Extensions for Scientific Computation (XSC)). It

2888-729: The PDP-10 , PDP-11 , VAX , DEC PRISM , MIPS , DEC Alpha , and Intel IA-32 , The language did not become popular among customers and few had the compiler, but DEC used it heavily in-house into the 1980s; most of the utility programs for the OpenVMS operating system were written in BLISS-32. The DEC BLISS compiler has been ported to the IA-64 and x86-64 architectures as part of the ports of OpenVMS to these platforms. The x86-64 BLISS compiler uses LLVM as its backend code generator , replacing

2964-480: The Python programming language , int represents an arbitrary-precision integer which has the traditional numeric operations such as addition, subtraction, and multiplication. However, in the Java programming language , the type int represents the set of 32-bit integers ranging in value from −2,147,483,648 to 2,147,483,647, with arithmetic operations that wrap on overflow . In Rust this 32-bit integer type

3040-590: The ADT corresponds to a product type similar to a tuple or record. A constructor with no fields corresponds to the empty product (unit type). If all constructors have no fields then the ADT corresponds to an enumerated type . One common ADT is the option type , defined in Haskell as data Maybe a = Nothing | Just a . Some types are very useful for storing and retrieving data and are called data structures . Common data structures include: An abstract data type

3116-505: The Babbage Institute's Computer Security History Project, Wulf claimed that the acronym was originally based on the name "Bill's Language for Implementing System Software." The original Carnegie Mellon compiler was notable for its extensive use of optimizations , and formed the basis of the classic book The Design of an Optimizing Compiler . Digital Equipment Corporation (DEC) developed and maintained BLISS compilers for

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3192-703: The IEEE Standard by four decades. In contrast, von Neumann recommended against floating-point numbers for the 1951 IAS machine , arguing that fixed-point arithmetic is preferable. The first commercial computer with floating-point hardware was Zuse's Z4 computer, designed in 1942–1945. In 1946, Bell Laboratories introduced the Model ;V , which implemented decimal floating-point numbers . The Pilot ACE has binary floating-point arithmetic, and it became operational in 1950 at National Physical Laboratory, UK . Thirty-three were later sold commercially as

3268-608: The Spanish engineer Leonardo Torres Quevedo published Essays on Automatics , where he designed a special-purpose electromechanical calculator based on Charles Babbage 's analytical engine and described a way to store floating-point numbers in a consistent manner. He stated that numbers will be stored in exponential format as n x 10 m {\displaystyle ^{m}} , and offered three rules by which consistent manipulation of floating-point numbers by machines could be implemented. For Torres, " n will always be

3344-513: The amounts each of red, green, and blue, and a string representing the color's name. Data types are used within type systems , which offer various ways of defining, implementing, and using them. In a type system, a data type represents a constraint placed upon the interpretation of data, describing representation, interpretation and structure of values or objects stored in computer memory. The type system uses data type information to check correctness of computer programs that access or manipulate

3420-627: The class Boolean implements both the Serializable and the Comparable interfaces. Therefore, an object of type Boolean is a member of the type Serializable & Comparable . Considering types as sets of values, the intersection type σ ∩ τ {\displaystyle \sigma \cap \tau } is the set-theoretic intersection of σ {\displaystyle \sigma } and τ {\displaystyle \tau } . It

3496-568: The computer representation for binary floating-point numbers in IEEE 754 (a.k.a. IEC 60559) in 1985. This first standard is followed by almost all modern machines. It was revised in 2008 . IBM mainframes support IBM's own hexadecimal floating point format and IEEE 754-2008 decimal floating point in addition to the IEEE 754 binary format. The Cray T90 series had an IEEE version, but the SV1 still uses Cray floating-point format. The standard provides for many closely related formats, differing in only

3572-420: The corresponding real number arithmetic operations by rounding any result that is not a floating-point number itself to a nearby floating-point number. For example, in a floating-point arithmetic with five base-ten digits, the sum 12.345 + 1.0001 = 13.3451 might be rounded to 13.345. The term floating point refers to the fact that the number's radix point can "float" anywhere to the left, right, or between

3648-483: The creation of the IEEE 754 standard once the 32-bit (or 64-bit) word had become commonplace. This standard was significantly based on a proposal from Intel, which was designing the i8087 numerical coprocessor; Motorola, which was designing the 68000 around the same time, gave significant input as well. In 1989, mathematician and computer scientist William Kahan was honored with the Turing Award for being

3724-568: The data. A compiler may use the static type of a value to optimize the storage it needs and the choice of algorithms for operations on the value. In many C compilers the float data type, for example, is represented in 32 bits , in accord with the IEEE specification for single-precision floating point numbers . They will thus use floating-point-specific microprocessor operations on those values (floating-point addition, multiplication, etc.). Most data types in statistics have comparable types in computer programming, and vice versa, as shown in

3800-503: The digits 0 and 1 ), this non-zero digit is necessarily 1 . Therefore, it does not need to be represented in memory, allowing the format to have one more bit of precision. This rule is variously called the leading bit convention , the implicit bit convention , the hidden bit convention , or the assumed bit convention . The floating-point representation is by far the most common way of representing in computers an approximation to real numbers. However, there are alternatives: In 1914,

3876-474: The following table: Parnas, Shore & Weiss (1976) identified five definitions of a "type" that were used—sometimes implicitly—in the literature: The definition in terms of a representation was often done in imperative languages such as ALGOL and Pascal , while the definition in terms of a value space and behaviour was used in higher-level languages such as Simula and CLU . Types including behavior align more closely with object-oriented models, whereas

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3952-410: The form a × 2 (where a and b are integers), but displayed in familiar decimal form. Fixed point data types are convenient for representing monetary values. They are often implemented internally as integers, leading to predefined limits. For independence from architecture details, a Bignum or arbitrary precision numeric type might be supplied. This represents an integer or rational to

4028-422: The four suits in a deck of playing cards may be four enumerators named CLUB , DIAMOND , HEART , SPADE , belonging to an enumerated type named suit . If a variable V is declared having suit as its data type, one can assign any of those four values to it. Some implementations allow programmers to assign integer values to the enumeration values, or even treat them as type-equivalent to integers. Strings are

4104-492: The given number is scaled by a power of 10 , so that it lies within a specific range—typically between 1 and 10, with the radix point appearing immediately after the first digit. As a power of ten, the scaling factor is then indicated separately at the end of the number. For example, the orbital period of Jupiter 's moon Io is 152,853.5047 seconds, a value that would be represented in standard-form scientific notation as 1.528535047 × 10 seconds. Floating-point representation

4180-405: The language or implemented as composite types in a library. Floating point In computing , floating-point arithmetic ( FP ) is arithmetic on subsets of real numbers formed by a signed string of a fixed number of digits in some base , called a significand , scaled by an integer exponent of that base. Numbers of this form are called floating-point numbers . For example,

4256-453: The left if the exponent is negative. Using base-10 (the familiar decimal notation) as an example, the number 152,853.5047 , which has ten decimal digits of precision, is represented as the significand 1,528,535,047 together with 5 as the exponent. To determine the actual value, a decimal point is placed after the first digit of the significand and the result is multiplied by 10 to give 1.528535047 × 10 , or 152,853.5047 . In storing such

4332-475: The logic False. True is always a non zero, especially a one which is known as Boolean 1. Almost all programming languages supply one or more integer data types. They may either supply a small number of predefined subtypes restricted to certain ranges (such as short and long and their corresponding unsigned variants in C/C++); or allow users to freely define subranges such as 1..12 (e.g. Pascal / Ada ). If

4408-592: The lowest level. The smallest addressable unit of data is usually a group of bits called a byte (usually an octet , which is 8 bits). The unit processed by machine code instructions is called a word (as of 2011 , typically 32 or 64 bits). Machine data types expose or make available fine-grained control over hardware, but this can also expose implementation details that make code less portable. Hence machine types are mainly used in systems programming or low-level programming languages . In higher-level languages most data types are abstracted in that they do not have

4484-411: The mainframe level was an ongoing problem by the early 1970s for those writing and maintaining higher-level source code; these manufacturer floating-point standards differed in the word sizes, the representations, and the rounding behavior and general accuracy of operations. Floating-point compatibility across multiple computing systems was in desperate need of standardization by the early 1980s, leading to

4560-441: The notion of data type, though the possible data types are often restricted by considerations of simplicity, computability, or regularity. An explicit data type declaration typically allows the compiler to choose an efficient machine representation, but the conceptual organization offered by data types should not be discounted. Different languages may use different data types or similar types with different semantics. For example, in

4636-567: The number 2469/200 is a floating-point number in base ten with five digits: 2469 / 200 = 12.345 = 12345 ⏟ significand × 10 ⏟ base − 3 ⏞ exponent {\displaystyle 2469/200=12.345=\!\underbrace {12345} _{\text{significand}}\!\times \!\underbrace {10} _{\text{base}}\!\!\!\!\!\!\!\overbrace {{}^{-3}} ^{\text{exponent}}} However, unlike 2469/200 = 12.345, 7716/625 = 12.3456

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4712-400: The number and type of the field values fixed by the constructor. The set of all possible values of an ADT is the set-theoretic disjoint union (sum), of the sets of all possible values of its variants (product of fields). Values of algebraic types are analyzed with pattern matching, which identifies a value's constructor and extracts the fields it contains. If there is only one constructor, then

4788-420: The primary architect behind this proposal; he was aided by his student Jerome Coonen and a visiting professor, Harold Stone . Among the x86 innovations are these: A floating-point number consists of two fixed-point components, whose range depends exclusively on the number of bits or digits in their representation. Whereas components linearly depend on their range, the floating-point range linearly depends on

4864-476: The programmer intends to use the data. Most programming languages support basic data types of integer numbers (of varying sizes), floating-point numbers (which approximate real numbers ), characters and Booleans . A data type may be specified for many reasons: similarity, convenience, or to focus the attention. It is frequently a matter of good organization that aids the understanding of complex definitions. Almost all programming languages explicitly include

4940-491: The proprietary GEM backend used for Alpha and IA-64. BLISS has many of the features of other modern high-level languages. It has block structure, an automatic stack, and mechanisms for defining and calling recursive routines ... provides a variety of predefined data structures and ... facilities for testing and iteration ... On the other hand, BLISS omits certain features of other high-level languages. It does not have built-in facilities for input/output, because

5016-457: The radix point is not specified, then the string implicitly represents an integer and the unstated radix point would be off the right-hand end of the string, next to the least significant digit. In fixed-point systems, a position in the string is specified for the radix point. So a fixed-point scheme might use a string of 8 decimal digits with the decimal point in the middle, whereby "00012345" would represent 0001.2345. In scientific notation ,

5092-437: The same number of digits (e.g. six), the first digit of n will be of order of tenths, the second of hundredths, etc, and one will write each quantity in the form: n ; m ." The format he proposed shows the need for a fixed-sized significand as is presently used for floating-point data, fixing the location of the decimal point in the significand so that each representation was unique, and how to format such numbers by specifying

5168-496: The significand is a string of 24 bits . For instance, the number π 's first 33 bits are: 11001001   00001111   1101101 0 _   10100010   0. {\displaystyle 11001001\ 00001111\ 1101101{\underline {0}}\ 10100010\ 0.} In this binary expansion, let us denote the positions from 0 (leftmost bit, or most significant bit) to 32 (rightmost bit). The 24-bit significand will stop at position 23, shown as

5244-417: The significand range and exponentially on the range of exponent component, which attaches outstandingly wider range to the number. On a typical computer system, a double-precision (64-bit) binary floating-point number has a coefficient of 53 bits (including 1 implied bit), an exponent of 11 bits, and 1 sign bit. Since 2 = 1024, the complete range of the positive normal floating-point numbers in this format

5320-408: The significant digits of the number. This position is indicated by the exponent, so floating point can be considered a form of scientific notation . A floating-point system can be used to represent, with a fixed number of digits, numbers of very different orders of magnitude — such as the number of meters between galaxies or between protons in an atom . For this reason, floating-point arithmetic

5396-500: The single-precision format, and any integer with absolute value less than 2 can be exactly represented in the double-precision format. Furthermore, a wide range of powers of 2 times such a number can be represented. These properties are sometimes used for purely integer data, to get 53-bit integers on platforms that have double-precision floats but only 32-bit integers. The standard specifies some special values, and their representation: positive infinity ( +∞ ), negative infinity ( −∞ ),

5472-547: The underlined bit 0 above. The next bit, at position 24, is called the round bit or rounding bit . It is used to round the 33-bit approximation to the nearest 24-bit number (there are specific rules for halfway values , which is not the case here). This bit, which is 1 in this example, is added to the integer formed by the leftmost 24 bits, yielding: 11001001   00001111   1101101 1 _ . {\displaystyle 11001001\ 00001111\ 1101101{\underline {1}}.} When this

5548-436: The value is of type f x for every x . Existential quantification written as ∃ x . f ( x ) {\displaystyle \exists x.f(x)} or exists x. f x and is the union over all types x of the body f x , i.e. the value is of type f x for some x . In Haskell, universal quantification is commonly used, but existential types must be encoded by transforming exists a. f

5624-408: Was established, and since the 1990s, the most commonly encountered representations are those defined by the IEEE. The speed of floating-point operations, commonly measured in terms of FLOPS , is an important characteristic of a computer system , especially for applications that involve intensive mathematical calculations. A floating-point unit (FPU, colloquially a math coprocessor ) is a part of

5700-400: Was found to be limited, and superseded by 8, 16 and 32-bit sets, which can encode a wide variety of non-Latin alphabets (such as Hebrew and Chinese ) and other symbols. Strings may be of either variable length or fixed length, and some programming languages have both types. They may also be subtyped by their maximum size. Since most character sets include the digits , it is possible to have

5776-1010: Was not until the launch of the Intel i486 in 1989 that general-purpose personal computers had floating-point capability in hardware as a standard feature. The UNIVAC 1100/2200 series , introduced in 1962, supported two floating-point representations: The IBM 7094 , also introduced in 1962, supported single-precision and double-precision representations, but with no relation to the UNIVAC's representations. Indeed, in 1964, IBM introduced hexadecimal floating-point representations in its System/360 mainframes; these same representations are still available for use in modern z/Architecture systems. In 1998, IBM implemented IEEE-compatible binary floating-point arithmetic in its mainframes; in 2005, IBM also added IEEE-compatible decimal floating-point arithmetic. Initially, computers used many different representations for floating-point numbers. The lack of standardization at

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