query: what is the default type for a floating point constant

pos: Double-precision floating-point format Double-precision floating-point format Double-precision floating-point format is a computer number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. Floating point is used to represent fractional values, or when a wider range is needed than is provided by fixed point (of the same bit width), even if at the cost of precision. Double precision may be chosen when the range or precision of single precision would be insufficient. In the IEEE 754-2008 standard, the 64-bit base-2 format is officially referred to as binary64; it was called

neg: C syntax suffixed by f or F to indicate a constant of type float, by l (letter l) or L to indicate type long double, or left unsuffixed for a double constant. The standard header file float.h defines the minimum and maximum values of the implementation's floating-point types float, double, and long double. It also defines other limits that are relevant to the processing of floating-point numbers. Every object has a storage class. This specifies most basically the storage "duration," which may be static (default for global), automatic (default for local), or dynamic (allocated), together with other features (linkage and register hint).

neg: Constant maturity credit default swap Constant maturity credit default swap A constant maturity credit default swap (CMCDS) is a type of credit derivative product, similar to a standard credit default swap (CDS). Addressing CMCDS typically requires prior understanding of credit default swaps. In a CMCDS the protection buyer does not makes periodic payments to the protection seller (these payments constitute the premium leg), and in return receives a payoff (protection or default leg) if an underlying financial instrument defaults. Differently from a standard CDS, the premium leg of a CMCDS does not pay a fixed and pre-agreed amount but a floating spread, using a traded

neg: Long double individual program (e.g. by the codice_23 function in Visual C++). The Intel C++ Compiler for x86, on the other hand, enables extended-precision mode by default. On OS X, long double is 80-bit extended precision In CORBA (from specification of 3.0, which uses "ANSI/IEEE Standard 754-1985" as its reference), "the long double data type represents an IEEE double-extended floating-point number, which has an exponent of at least 15 bits in length and a signed fraction of at least 64 bits", with GIOP/IIOP CDR, whose floating-point types "exactly follow the IEEE standard formats for floating point numbers", marshalling this as what seems

neg: Generic programming the standard library, which was inspired by C++'s standard template library. A "generic unit" is a package or a subprogram that takes one or more "generic formal parameters". A "generic formal parameter" is a value, a variable, a constant, a type, a subprogram, or even an instance of another, designated, generic unit. For generic formal types, the syntax distinguishes between discrete, floating-point, fixed-point, access (pointer) types, etc. Some formal parameters can have default values. To "instantiate" a generic unit, the programmer passes "actual" parameters for each formal. The generic instance then behaves just like any other unit. It is possible

neg: Commodore BASIC which it is based, Commodore BASIC is slower than native machine code. Test results have shown that copying 16 kilobytes from ROM to RAM takes less than a second in machine code, compared to over a minute in BASIC. To execute faster than the interpreter, programmers started using various techniques to speed up execution. One was to store often-used floating point values in variables rather than using literal values, as interpreting a variable name was faster than interpreting a literal number. Since floating point is default type for all commands, it's faster to use floating point numbers as arguments, rather

neg: NaN xxxx where "s" is the sign (most often ignored in applications) and the "x" sequence represents a non-zero number (the value zero encodes infinities). The first bit from "x" is used to determine the type of NaN: "quiet NaN" or "signaling NaN". The remaining bits encode a "payload" (most often ignored in applications). Floating-point operations other than ordered comparisons normally propagate a quiet NaN ("qNaN"). Most floating-point operations on a signaling NaN ("sNaN") signal the invalid operation exception, the default exception action is then the same as for qNaN operands and they produce a qNaN if producing a floating-point result.

neg: MAD (programming language) the constant, where 0 indicates floating point, 1 integer, 2 boolean, 3 function name, and 4 statement label. For GOM six additional modes are added: CHARACTER, SHORT INTEGER, BYTE INTEGER, LONG INTEGER, POINTER, and DYNAMIC RECORD. Alphabetic or character constants are stored as integers and written using the dollar sign as a delimiter ($ABCDEF$) with double dollar-signs used to enter a true dollar sign ($$$.56$ is 56 cents). Strings longer than six characters are represented using arrays. Variables may be implicitly or explicitly declared. By default all implicitly declared variables are assumed to be floating point. The NORMAL MODE IS

neg: GW-BASIC statement (e.g., codice_6). The data type of variables can be specified with a character at the end of the variable name: codice_7 is a string of characters, codice_8 is an integer, etc. Groups of variables can also be set to default types based on the initial letter of their name by use of the codice_9, etc., statements. The default type for undeclared variables not identified by such typing statements, is single-precision floating point (32-bit MBF). GW-BASIC allowed use of joystick and light pen input devices. GW-BASIC can read from and write to files and COM ports; it can also do

neg: Default constructor body. All fields are left at their initial value of 0 (integer types), 0.0 (floating-point types), codice_11 (codice_12 type), or codice_13 (reference types). A programmer-defined constructor that takes no parameters is also called a default constructor in C#, but not in Java. Default constructor In computer programming languages, the term default constructor can refer to a constructor that is automatically generated by the compiler in the absence of any programmer-defined constructors (e.g. in Java), and is usually a nullary constructor. In other languages (e.g. in C++) it is a constructor that can be called without having to provide any arguments,

neg: Floating-point arithmetic default "quiet" NaNs and, optionally, "signaling" NaNs. A signaling NaN in any arithmetic operation (including numerical comparisons) will cause an "invalid" exception to be signaled. The representation of NaNs specified by the standard has some unspecified bits that could be used to encode the type or source of error; but there is no standard for that encoding. In theory, signaling NaNs could be used by a runtime system to flag uninitialized variables, or extend the floating-point numbers with other special values without slowing down the computations with ordinary values, although such extensions are not common. It is a common misconception

neg: C mathematical functions types for fine-grained control of floating point environment. These functions can be used to control a variety of settings that affect floating-point computations, for example, the rounding mode, on what conditions exceptions occur, when numbers are flushed to zero, etc. The floating point environment functions and types are defined in codice_15 header (codice_16 in C++). C99 adds a new codice_17 keyword (and codice_18 convenience macro) that provides support for complex numbers. Any floating point type can be modified with codice_18, and is then defined as a pair of floating point numbers. Note that C99 and C++ do not implement complex

neg: Floating-point arithmetic until after a full floating-point expression or subroutine: without them exceptional conditions that could not be otherwise ignored would require explicit testing immediately after every floating-point operation. By default, an operation always returns a result according to specification without interrupting computation. For instance, 1/0 returns +∞, while also setting the divide-by-zero flag bit (this default of ∞ is designed so as to often return a finite result when used in subsequent operations and so be safely ignored). The original IEEE 754 standard, however, failed to recommend operations to handle such sets of arithmetic exception flag bits. So while these were

neg: Real data type is 65,536 (2), the hexadecimal number 0x12345678 means 0x12345678/65536 or 305419896/65536 or 4660 + 22136/65536 or about 4660.33777. See fixed-point arithmetic. A "floating point" type is a compromise between the flexibility of a general rational type and the speed of fixed-point arithmetic. It uses some of the bits in the data type to specify a power of two for the denominator. See floating point and IEEE Floating Point Standard. Similar to fixed-point or floating-point numbers, except with a denominator that is a power of 10 instead of a power of 2. Real data type A real data type is a

neg: Double-precision floating-point format languages in different ways such as the following. On processors with only dynamic precision, such as x86 without SSE2 (or when SSE2 is not used, for compatibility purpose) and with extended precision used by default, software may have difficulties to fulfill some requirements. C and C++ offer a wide variety of arithmetic types. Double precision is not required by the standards (except by the optional annex F of C99, covering IEEE 754 arithmetic), but on most systems, the codice_7 type corresponds to double precision. However, on 32-bit x86 with extended precision by default, some compilers may not conform to the

neg: Fortran 95 language features corresponding form of "literal constant". The numeric types codice_3 and codice_4 can only be signed (there is no concept of sign for type codice_5). Integer literal constants of the default kind take the form 1 0 -999 32767 +10 Kind can be defined as a named constant. If the desired range is ±10, the portable syntax for defining the appropriate kind, codice_11 is INTEGER, PARAMETER :: two_bytes = SELECTED_INT_KIND(4) that allows subsequent definition of constants of the form -1234_two_bytes +1_two_bytes Here, codice_11 is the kind type parameter; it can also be an explicit default integer literal constant, like -1234_2 but

neg: Floating-point arithmetic Java, in which an "exception" is an alternative flow of control, closer to what is termed a "trap" in IEEE 754 terminology). Here, the required default method of handling exceptions according to IEEE 754 is discussed (the IEEE 754 optional trapping and other "alternate exception handling" modes are not discussed). Arithmetic exceptions are (by default) required to be recorded in "sticky" status flag bits. That they are "sticky" means that they are not reset by the next (arithmetic) operation, but stay set until explicitly reset. The use of "sticky" flags thus allows for testing of exceptional conditions to be delayed

neg: C Sharp (programming language) types are all primitive types, such as (a signed 32-bit integer), (a 32-bit IEEE floating-point number), (a 16-bit Unicode code unit), and (identifies a specific point in time with nanosecond precision). Other examples are (enumerations) and (user defined structures). In contrast, reference types have the notion of referential identity - each instance of a reference type is inherently distinct from every other instance, even if the data within both instances is the same. This is reflected in default equality and inequality comparisons for reference types, which test for referential rather than structural equality, unless the corresponding operators are overloaded (such

neg: Half-precision floating-point format default, 1/3 rounds down like for double precision, because of the odd number of bits in the significand. So the bits beyond the rounding point are codice_1 which is less than 1/2 of a unit in the last place. ARM processors support (via a floating point control register bit) an "alternative half-precision" format, which does away with the special case for an exponent value of 31 (11111). It is almost identical to the IEEE format, but there is no encoding for infinity or NaNs; instead, an exponent of 31 encodes normalized numbers in the range 65536 to 131008. Half-precision floating-point

neg: Fast inverse square root and calculating the reciprocal via floating-point division. The algorithm was designed with the IEEE 754-1985 32-bit floating-point specification in mind, but investigation from Chris Lomont showed that it could be implemented in other floating-point specifications. The advantages in speed offered by the fast inverse square root kludge came from treating the longword containing the floating point number as an integer then subtracting it from a specific constant, 0x5F3759DF. The purpose of the constant is not immediately clear to someone viewing the code, so, like other such constants found in code, it is often called a magic number. This integer subtraction

neg: IEEE 754 between different representations of NaNs and between the same decimal floating point number encoded in different ways. The standard defines five exceptions, each of which returns a default value and has a corresponding status flag that (except in certain cases of underflow) is raised when the exception occurs. No other exception handling is required, but additional non-default alternatives are recommended (see below). The five possible exceptions are: These are the same five exceptions as were defined in IEEE 754-1985, but the "division by zero" exception has been extended to operations other than the division. For decimal floating point, there are

neg: Type inference neutral algorithmically, as illustrated by the prior floating-point precision issue. An algorithm of intermediate generality implicitly declares "result2" as a floating-point variable, and the addition implicitly converts codice_6 to a floating point. This can be correct if the calling contexts never supply a floating point argument. Such a situation shows the difference between "type inference", which does not involve type conversion, and implicit type conversion, which forces data to a different data type, often without restrictions. Finally, a significant downside of complex type-inference algorithm is that the resulting type inference resolution is not going to be obvious to humans (notably

neg: Data type compilers the 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.). The depth of type constraints and the manner of their evaluation affect the "typing" of the language. A programming language may further associate an operation with varying concrete algorithms on each type in the case of type polymorphism. Type theory is the study of type systems, although the concrete type systems of programming languages originate from practical issues of computer architecture, compiler implementation, and

neg: Intel C++ Compiler loop interchange, loop fusion, loop unrolling, loop distribution, data prefetch, and more. Interprocedural optimization applies typical compiler optimizations (such as constant propagation) but using a broader scope that may include multiple procedures, multiple files, or the entire program. Intel's compiler has been criticized for applying, by default, floating-point optimizations not allowed by the C standard and that require special flags with other compilers such as gcc. Except for the Intel Bi-Endian C++ Compiler, Intel C++ compilers are not available in standalone form. They are available in suites: The suites include other build tools, such as libraries, and tools for threading

neg: Type punning effectively the same storage space. This is often seen when two structures are used in mutual exclusivity for optimization. Not all examples of type punning involve structures, as the previous example did. Suppose we want to determine whether a floating-point number is negative. We could write: bool is_negative(float x) { However, supposing that floating-point comparisons are expensive, and also supposing that codice_12 is represented according to the IEEE floating-point standard, and integers are 32 bits wide, we could engage in type punning to extract the sign bit of the floating-point number using only integer operations: bool is_negative(float x) { Note

neg: RISC-V are 64-bit (i.e. double-width), and the "F" subset is coordinated with the "D" set. A quad-precision 128-bit floating-point ISA ("Q") is also defined. RISC-V computers without floating-point can use a floating-point software library. RISC-V does not cause exceptions on arithmetic errors, including overflow, underflow, subnormal and divide by zero. Instead, both integer and floating-point arithmetic produce reasonable default values and set status bits. Divide-by-zero can be discovered by a single branch after the division. The status bits can be tested by an operating system or periodic interrupt. RISC-V supports computers that share memory between multiple CPUs and threads. RISC-V's standard

neg: Fast inverse square root the original constant was chosen through derivation or trial and error. Lomont pointed out that the magic number for 64-bit IEEE754 size type double is 0x5FE6EC85E7DE30DA, but it was later shown by Matthew Robertson to be exactly 0x5FE6EB50C7B537A9. Fast inverse square root Fast inverse square root, sometimes referred to as Fast InvSqrt() or by the hexadecimal constant 0x5F3759DF, is an algorithm that estimates , the reciprocal (or multiplicative inverse) of the square root of a 32-bit floating-point number in . This operation is used in digital signal processing to normalize a vector, i.e., scale it to length 1. For example,

neg: Type conversion as unintended consequences can arise. Data can be lost when converting representations from floating-point to integer, as the fractional components of the floating-point values will be truncated (rounded toward zero). Conversely, precision can be lost when converting representations from integer to floating-point, since a floating-point type may be unable to exactly represent an integer type. For example, might be an IEEE 754 single precision type, which cannot represent the integer 16777217 exactly, while a 32-bit integer type can. This can lead to unintuitive behavior, as demonstrated by the following code: On compilers that implement floats as IEEE single precision, and

neg: Strictfp in the earlier versions; that is, only with IEEE single and double precision types used. Using strictfp guarantees that results of floating-point calculations are identical on all platforms. It can be used on classes, interfaces and non-abstract methods. When applied to a method, it causes all calculations inside the method to use strict floating-point math. When applied to a class, all calculations inside the class use strict floating-point math. Compile-time constant expressions must always use strict floating-point behavior. Examples Strictfp strictfp is a keyword in the Java programming language that restricts floating-point calculations to ensure portability. The strictfp command was

neg: Constant maturity swap Constant maturity swap A constant maturity swap, also known as a CMS, is a swap that allows the purchaser to fix the duration of received flows on a swap. The floating leg of an interest rate swap typically resets against a published index. The floating leg of a constant maturity swap fixes against a point on the swap curve on a periodic basis. A constant maturity swap is an interest rate swap where the interest rate on one leg is reset periodically, but with reference to a market swap rate rather than LIBOR. The other leg of the swap is

neg: C data types that codice_16 is not smaller than codice_17, which is not smaller than codice_18. Usually, the 32-bit and 64-bit IEEE 754 binary floating-point formats are used, if supported by hardware. The C99 standard includes new real floating-point types codice_19 and codice_20, defined in codice_21. They correspond to the types used for the intermediate results of floating-point expressions when codice_22 is 0, 1, or 2. These types may be wider than codice_16. C99 also added complex types: codice_24, codice_25, codice_26. C99 added a boolean (true/false) type codice_27. Additionally, the new codice_28 header defines codice_29 as a convenient alias for this type, and

neg: Octuple-precision floating-point format one bit. The minimum positive normal value is 2 ≈ 2.4824 × 10. The maximum representable value is 2 − 2 ≈ 1.6113 × 10. These examples are given in bit "representation", in hexadecimal, of the floating-point value. This includes the sign, (biased) exponent, and significand. By default, 1/3 rounds down like double precision, because of the odd number of bits in the significand. So the bits beyond the rounding point are codice_1 which is less than 1/2 of a unit in the last place. Octuple precision is rarely implemented since usage of it is extremely rare. Apple Inc. had

neg: Sizeof cast. A cast is a data type enclosed in parenthesis. Data types may not only be primitive types, such as integer and floating-point types, but also pointer types, and compound datatypes (unions, structs, and C++ classes). Many programs must know the storage size of a particular datatype. Though for any given implementation of C or C++ the size of a particular datatype is constant, the sizes of even primitive types in C and C++ may be defined differently for different platforms of implementation. For example, runtime allocation of array space may use the following code, in which the sizeof operator

neg: Endianness there have been many floating-point formats with no "network" standard representation for them, the XDR standard uses big-endian IEEE 754 as its representation. It may therefore appear strange that the widespread IEEE 754 floating-point standard does not specify endianness. Theoretically, this means that even standard IEEE floating-point data written by one machine might not be readable by another. However, on modern standard computers (i.e., implementing IEEE 754), one may in practice safely assume that the endianness is the same for floating-point numbers as for integers, making the conversion straightforward regardless of data type. (Small embedded systems using special floating-point formats

neg: Complex data type sequence of floating-point machine instructions or into library calls. Those languages may also provide support for other operations, such as formatting, equality testing, etc. As in mathematics, those languages often interpret a floating-point value as equivalent to a complex value with a zero imaginary part. The codice_9 data type was provided in FORTRAN IV. Complex data type Some programming languages provide a complex data type for complex number storage and arithmetic as a built-in (primitive) data type. In some programming environments the term "complex data type" (in contrast to primitive data types) is a synonym for the composite data type.

neg: Double-precision floating-point format indicate a signaling NaN. By default, / rounds down, instead of up like single precision, because of the odd number of bits in the significand. In more detail: Using double-precision floating-point variables and mathematical functions (e.g., sin, cos, atan2, log, exp and sqrt) are slower than working with their single precision counterparts. One area of computing where this is a particular issue is for parallel code running on GPUs. For example, when using NVIDIA's CUDA platform, calculations with double precision take approximately 2 times longer to complete compared to those done using single precision. Doubles are implemented in many programming

neg: Signal (IPC) generate signals (here, SIGFPE "floating point exception" and SIGSEGV "segmentation violation" respectively, which both by default cause a core dump and a program exit). The kernel can generate signals to notify processes of events. For example, SIGPIPE will be generated when a process writes to a pipe which has been closed by the reader; by default, this causes the process to terminate, which is convenient when constructing shell pipelines. Typing certain key combinations at the controlling terminal of a running process causes the system to send it certain signals: These default key combinations with modern operating systems can be changed

neg: Quadruple-precision floating-point format 10 and has a precision of only one bit. The minimum positive normal value is 2 ≈ and has a precision of 113 bits, i.e. ±2 as well. The maximum representable value is ≈ . These examples are given in bit "representation", in hexadecimal, of the floating-point value. This includes the sign, (biased) exponent, and significand. By default, 1/3 rounds down like double precision, because of the odd number of bits in the significand. So the bits beyond the rounding point are codice_1 which is less than 1/2 of a unit in the last place. A common software technique to

neg: Type system numbers. They will thus use floating-point-specific microprocessor operations on those values (floating-point addition, multiplication, etc.). The depth of type constraints and the manner of their evaluation affect the "typing" of the language. A programming language may further associate an operation with various resolutions for each type, in the case of type polymorphism. Type theory is the study of type systems. The concrete types of some programming languages, such as integers and strings, depend on practical issues of computer architecture, compiler implementation, and language design. Formally, type theory studies type systems. A programming language must have occurrence to type check using

neg: Long double at a lower precision (i.e. FLT_EVAL_METHOD == 2). With gcc on Linux, 80-bit extended precision is the default; on several BSD operating systems (FreeBSD and OpenBSD), double-precision mode is the default, and codice_1 operations are effectively reduced to double precision. (NetBSD 7.0 and later, however, defaults to 80-bit extended precision ). However, it is possible to override this within an individual program via the FLDCW "floating-point load control-word" instruction. On x86_64 the BSDs default to 80-bit extended precision. Microsoft Windows with Visual C++ also sets the processor in double-precision mode by default, but this can again be overridden within an

neg: Literal (computer programming) Literal (computer programming) In computer science, a literal is a notation for representing a fixed value in source code. Almost all programming languages have notations for atomic values such as integers, floating-point numbers, and strings, and usually for booleans and characters; some also have notations for elements of enumerated types and compound values such as arrays, records, and objects. An anonymous function is a literal for the function type. In contrast to literals, variables or constants are symbols that can take on one of a class of fixed values, the constant being constrained not to change. Literals are often used

neg: Q (number format) a number from floating point to Q"m"."n" format: To convert a number from Q"m"."n" format to floating point: Q numbers are a ratio of two integers: the numerator is kept in storage, the denominator is equal to 2. Consider the following example: If the Q number's base is to be maintained ("n" remains constant) the Q number math operations must keep the denominator constant. The following formulas show math operations on the general Q numbers formula_5 and formula_6. formula_7 Because the denominator is a power of two the multiplication can be implemented as an arithmetic shift to the left and

neg: Double-precision floating-point format double in IEEE 754-1985. IEEE 754 specifies additional floating-point formats, including 32-bit base-2 "single precision" and, more recently, base-10 representations. One of the first programming languages to provide single- and double-precision floating-point data types was Fortran. Before the widespread adoption of IEEE 754-1985, the representation and properties of floating-point data types depended on the computer manufacturer and computer model, and upon decisions made by programming-language implementers. E.g., GW-BASIC's double-precision data type was the 64-bit MBF floating-point format. Double-precision binary floating-point is a commonly used format on PCs, due to its wider range over single-precision floating point, in spite of its

neg: Primitive data type Literals for floating point numbers include a decimal point, and typically use e or E to denote scientific notation. Examples of floating-point literals are: Some languages (e.g., Fortran, Python, D) also have a complex number type comprising two floating-point numbers: a real part and an imaginary part. A fixed-point number represents a limited-precision rational number that may have a fractional part. These numbers are stored internally in a scaled-integer form, typically in binary but sometimes in decimal. Because fixed-point numbers have limited precision, only a subset of real or rational numbers are exactly representable; other numbers can be represented only

neg: Type inference returns one integer. (This is how it works in, for example, OCaml.) From this, the type inferencer can infer that the type of codice_12 is an integer, which means codice_4 is an integer and thus the return value of codice_2 is an integer. Similarly, since codice_11 requires both of its arguments be of the same type, codice_6 must be an integer, and thus, codice_2 accepts one integer as an argument. However, in the subsequent line, "result2" is calculated by adding a decimal codice_18 with floating-point arithmetic, causing a conflict in the use of codice_6 for both integer and floating-point expressions.

neg: Fixed-point arithmetic The Ada programming language includes built-in support for both fixed-point (binary and decimal) and floating-point. JOVIAL and Coral 66 also provide both floating- and fixed-point types. ISO/IEC TR 18037 specifies fixed-point data types for the C programming language; vendors are expected to implement the language extensions for fixed point arithmetic in coming years. Fixed-point support is implemented in GCC. Fixed-point should not be confused with Decimal floating point in programming languages like C# and Python. Almost all relational databases, and the SQL, support fixed-point decimal arithmetic and storage of numbers. PostgreSQL has a special numeric type for exact storage of

neg: Atari BASIC math routines on the Atari were very slow, notably the BCD to binary conversion function, and this affected all programs. In the "Byte" benchmark, the constant conversion of the array indexes greatly slows the program down. In Ahl benchmark, a single exponent operation, which loops over a poorly optimized multiplication function, was responsible for much of the machine's poor showing. Atari BASIC does not support clauses with statements and (e.g. ) is not recognized; the user must type or line number. Atari BASIC uses 48-bit floating point arithmetic. While this has the advantage of greater numerical precision than the 32

neg: Exception handling IEEE 754 floating point hardware standard refers in general to exceptional conditions and defines an exception as "an event that occurs when an operation on some particular operands has no outcome suitable for every reasonable application. That operation might signal one or more exceptions by invoking the default or, if explicitly requested, a language-defined alternate handling." By default, an IEEE 754 exception is resumable and is handled by substituting a predefined value for different exceptions, e.g. infinity for a divide by zero exception, and providing status flags for later checking of whether the exception occurred (see C99 programming language for

neg: Kahan summation algorithm codice_3, eliminating the error compensation. In practice, many compilers do not use associativity rules (which are only approximate in floating-point arithmetic) in simplifications unless explicitly directed to do so by compiler options enabling "unsafe" optimizations, although the Intel C++ Compiler is one example that allows associativity-based transformations by default. The original K&R C version of the C programming language allowed the compiler to re-order floating-point expressions according to real-arithmetic associativity rules, but the subsequent ANSI C standard prohibited re-ordering in order to make C better suited for numerical applications (and more similar to Fortran, which also prohibits re-ordering), although in

neg: Decimal data type operator overloading facilities provided by the language. These operations may be translated by the compiler into a sequence of integer machine instructions, or into library calls. Support may also extend to other operations, such as formatting, rounding to an integer or floating point value, etc.. An example of this is 123.456 IEEE 754 specifies three standard floating-point decimal data types of different precision: Decimal data type Fractional numbers are supported on most programming languages as floating-point numbers or fixed-point numbers. However, such representations typically restrict the denominator to a power of two. Most decimal fractions (or most fractions in general)

neg: Mathematical software Mathematical software Mathematical software is software used to model, analyze or calculate numeric, symbolic or geometric data. It is a type of application software which is used for solving mathematical problems or mathematical study. There are various views to what is the mathematics, so there is various views of the category of mathematical software which used for them, over from narrow to wide sense. A type of mathematical software (math library) also used by built in the part of another scientific software. A most primary them (for example, to calculate elementary function by floating point arithmetic) may be in the

neg: Denormal number and Fortran compilers enable the denormals-are-zero (DAZ) and flush-to-zero (FTZ) flags for SSE by default for optimization levels higher than -O0. The effect of DAZ is to treat denormal input arguments to floating point operations as zero, and the effect of FTZ is to return zero instead of a denormal float for operations which would result in a denormal float, even if the input arguments are not themselves denormal. clang and gcc have varying default states depending on platform and optimization level. A non-C99-compliant method of enabling the DAZ and FTZ flags on targets supporting SSE is given below, but

neg: Single-precision floating-point format 41460000 Note: consider converting 68.123 into IEEE 754 binary32 format: Using the above procedure you expect to get 42883EF9 with the last 4 bits being 1001. However, due to the default rounding behaviour of IEEE 754 format, what you get is 42883EFA, whose last 4 bits are 1010. "Ex 1:" Consider decimal 1. We can see that: formula_16 From which we deduce: From these we can form the resulting 32 bit IEEE 754 binary32 format representation of real number 1 as: 0-01111111-00000000000000000000000 = 3f800000 "Ex 2:" Consider a value 0.25. We can see that: formula_17 From which we deduce: From

neg: Floating-point arithmetic data type of the C# and Python programming languages, and the decimal formats of the IEEE 754-2008 standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal. Expectations from mathematics may not be realized in the field of floating-point computation. For example, it is known that formula_47, and that formula_48, however these facts cannot be relied on when the quantities involved are the result of floating-point computation. The use of the equality test (codice_2) requires care when dealing

neg: Type punning verify portability expectations, helps to keep the code maintainable. For a practical example popularized by Quake III, see fast inverse square root. In addition to the assumption about bit-representation of floating-point numbers, the previous floating-point type-punning example also violates the C language's constraints on how objects are accessed:<ref name="ISOs6.5/7">ISO/IEC 9899:1999 s6.5/7</ref> the declared type of codice_13 is codice_12 but it is read through an expression of type codice_18. On many common platforms, this use of pointer punning can create problems if different pointers are aligned in machine-specific ways. Furthermore, pointers of different sizes can alias accesses to the same memory,

neg: S-algol (record) type for a binary tree of cstrings ! the pntr data type can point to a structure of any type, type checking is done at runtime structure tree.node(cstring name ; pntr left, right) ! inserts a new string into the binary tree head procedure insert.tree(cpntr head ; cstring new -> pntr) ! the case clause ends with a mandatory default option, use default : {} if it is not needed case true of procedure print.tree(cpntr head) if head ~= nil do ! ~= is the not equals operator begin end let fruit := nil fruit := insert.tree(fruit, "banana") fruit

neg: Type inference The correct type-inference algorithm for such a situation has been known since 1958 and has been known to be correct since 1982. It revisits the prior inferences and uses the most general type from the outset: in this case floating-point. This can however have detrimental implications, for instance using a floating-point from the outset can introduce precision issues that would have not been there with an integer type. Frequently, however, degenerate type-inference algorithms are used that cannot backtrack and instead generate an error message in such a situation. This behavior may be preferable as type inference may not always be

neg: Long double Long double In C and related programming languages, codice_1 refers to a floating-point data type that is often more precise than double-precision. As with C's other floating-point types, it may not necessarily map to an IEEE format. The codice_1 type was present in the original 1989 C standard, but support was improved by the 1999 revision of the C standard, or C99, which extended the standard library to include functions operating on codice_1 such as codice_5 and codice_6. Long double constants are floating-point constants suffixed with "L" or "l" (lower-case L), e.g., 0.333333333333333333L. Without a suffix, the evaluation depends on

neg: Rounding a specified number of digits. In other words, should be a multiple of a number that depends on the magnitude of . The number is a power of the base (usually 2 or 10) of the floating-point representation. Apart from this detail, all the variants of rounding discussed above apply to the rounding of floating-point numbers as well. The algorithm for such rounding is presented in the Scaled rounding section above, but with a constant scaling factor = 1, and an integer base > 1. Where the rounded result would overflow the result for a directed rounding is either the

neg: Long double to be IEEE 754-2008 binary128 a.k.a. quadruple precision without using that name. Long double In C and related programming languages, codice_1 refers to a floating-point data type that is often more precise than double-precision. As with C's other floating-point types, it may not necessarily map to an IEEE format. The codice_1 type was present in the original 1989 C standard, but support was improved by the 1999 revision of the C standard, or C99, which extended the standard library to include functions operating on codice_1 such as codice_5 and codice_6. Long double constants are floating-point constants suffixed with "L" or

neg: C++11 null terminator. An alternative mechanism for processing integer and floating point raw literals is via a variadic template: This instantiates the literal processing function as codice_161. In this form, there is no null character terminating the string. The main purpose for doing this is to use C++11's codice_17 keyword to ensure that the compiler will transform the literal entirely at compile time, assuming codice_163 is a constexpr-constructible and copyable type, and the literal processing function is a codice_17 function. For numeric literals, the type of the cooked literal is either codice_165 for integral literals or codice_166 for floating point literals.

neg: Decimal64 floating-point format Decimal64 floating-point format In computing, decimal64 is a decimal floating-point computer numbering format that occupies 8 bytes (64 bits) in computer memory. It is intended for applications where it is necessary to emulate decimal rounding exactly, such as financial and tax computations. Decimal64 supports 16 decimal digits of significand and an exponent range of −383 to +384, i.e. to . (Equivalently, to .) In contrast, the corresponding binary format, which is the most commonly used type, has an approximate range of to . Because the significand is not normalized, most values with less than 16 significant digits have multiple possible

neg: Fast inverse square root proper way to accomplish the fast inverse square root, in a more standard conforming way, is to type-pun floating point values and integers through a union type. Type-punning with a union type is considered undefined behavior in C++ standards. As an example, the number can be used to calculate . The first steps of the algorithm are illustrated below: Using IEEE 32-bit representation: Reinterpreting this last bit pattern as a floating point number gives the approximation , which has an error of about 3.4%. After one single iteration of Newton's method, the final result is , an error of only

neg: MIPS architecture (PS), which consisted of two single-precision (32-bit) floating-point numbers stored in the existing 64-bit floating-point registers. Variants of existing floating-point instructions for arithmetic, compare and conditional move were added to operate on this data type in a SIMD fashion. New instructions were added for loading, rearranging and converting PS data. It was the first instruction set to exploit floating-point SIMD with existing resources. When MIPS Technologies was spun-out of Silicon Graphics in 1998, it refocused on the embedded market. Up to MIPS V, each successive version was a strict superset of the previous version, but this property was found to

neg: Extended precision point sizes. On the x86 design most C compilers now support 80-bit extended precision via the long double type, and this was specified in the C99 / C11 standards (IEC 60559 floating-point arithmetic (Annex F)). Compilers on x86 for other languages often support extended precision as well, sometimes via nonstandard extensions: for example, Turbo Pascal offers an codice_1 type, and several Fortran compilers have a codice_2 type (analogous to codice_3 and codice_4). Such compilers also typically include extended-precision mathematical subroutines, such as square root and trigonometric functions, in their standard libraries. The 80-bit floating point format has a range (including

neg: Theta criterion For this type of verb, the potential object is usually semantically limited and therefore can be inferred from the verb at a default value . For instance, for (4a), the listener/reader automatically assumes that John ate "something." What necessitates the object in (4c) is the distinction from the default meaning achieved by specifying what John ate . As a result, this type of verbs can be treated the same as transitive verbs. The theta-roles of "agent" and "theme" can be assigned: In summary, by assigning the correct theta-roles, theta-criterion is able to tell the real intransitive verbs, such as "arrive"

neg: NaN NaN In computing, NaN, standing for not a number, is a numeric data type value representing an undefined or unrepresentable value, especially in floating-point calculations. Systematic use of NaNs was introduced by the IEEE 754 floating-point standard in 1985, along with the representation of other non-finite quantities like infinities. Two separate kinds of NaNs are provided, termed "quiet NaNs" and "signaling NaNs". Quiet NaNs are used to propagate errors resulting from invalid operations or values, whereas signaling NaNs can support advanced features such as mixing numerical and symbolic computation or other extensions to basic floating-point arithmetic. For example, 0/0 is

neg: Fixed-point arithmetic numbers with up to 1000 digits. Fixed-point arithmetic In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after (and sometimes also before) the radix point (after the decimal point '.' in English decimal notation). Fixed-point number representation can be compared to the more complicated (and more computationally demanding) floating-point number representation. Fixed-point numbers are useful for representing fractional values, usually in base 2 or base 10, when the executing processor has no floating point unit (FPU) or if fixed-point provides improved performance or accuracy for the application at

neg: Fixed-point arithmetic Fixed-point arithmetic In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits after (and sometimes also before) the radix point (after the decimal point '.' in English decimal notation). Fixed-point number representation can be compared to the more complicated (and more computationally demanding) floating-point number representation. Fixed-point numbers are useful for representing fractional values, usually in base 2 or base 10, when the executing processor has no floating point unit (FPU) or if fixed-point provides improved performance or accuracy for the application at hand. Older or low-cost embedded microprocessors

neg: Constant proportion debt obligation Constant proportion debt obligation A Constant proportion debt obligation (or CPDO) is a type of credit derivative sold to investors looking for exposure to credit risk. A CPDO is normally embedded in a note rated by a credit rating agency. CPDOs employ dynamic leveraging in a similar (but opposite) way to Credit CPPI trades. CPDOs are formed first by creating a SPV that issues a debt note. The SPV invests in an index of debt securities, commonly credit default swap indices such as CDX and iTraxx (in theory, this could be deal-specific, such as a bespoke portfolio of sovereign debt),

neg: Constant proportion debt obligation although not all agencies have rated them and some agencies had provided the market with early pre-crisis warnings. Constant proportion debt obligation A Constant proportion debt obligation (or CPDO) is a type of credit derivative sold to investors looking for exposure to credit risk. A CPDO is normally embedded in a note rated by a credit rating agency. CPDOs employ dynamic leveraging in a similar (but opposite) way to Credit CPPI trades. CPDOs are formed first by creating a SPV that issues a debt note. The SPV invests in an index of debt securities, commonly credit default swap indices such

neg: Constant maturity credit default swap maturity and the present value of the premium leg of the constant maturity CDS. CMCDS may be combined with CDS on the same entity to take only spread risk and not default risk on an entity. Indeed, as the default leg is the same, buying a CDS and selling a CMCDS or vice versa will offset the default legs and leave only the difference in the premium legs, that are driven by spread risk. Valuation of CMCDS has been explored by Damiano Brigo (2004) and Anlong Li (2006) Constant maturity credit default swap A constant maturity credit default swap (CMCDS)

neg: Fixed-point arithmetic and microcontrollers do not have an FPU. A value of a fixed-point data type is essentially an integer that is scaled by an implicit specific factor determined by the type. For example, the value 1.23 can be represented as 1230 in a fixed-point data type with scaling factor of 1/1000, and the value 1,230,000 can be represented as 1230 with a scaling factor of 1000. Unlike floating-point data types, the scaling factor is the same for all values of the same type, and does not change during the entire computation. The scaling factor is usually a power of 10 (for

neg: C syntax scientific notation may be used by adding e or E followed by a decimal exponent, also known as E notation, e.g. 1.23e2 (which has the value 1.23 × 10 = 123.0). Either a decimal point or an exponent is required (otherwise, the number is parsed as an integer constant). Hexadecimal floating-point constants follow similar rules, except that they must be prefixed by 0x and use p or P to specify a binary exponent, e.g. 0xAp-2 (which has the value 2.5, since A × 2 = 10 × 2 = 10 ÷ 4). Both decimal and hexadecimal floating-point constants may be

neg: Criticism of Java After a serialized object has been written into a file, it can be read from the file and deserialized that is, the type information and bytes that represent the object and its data can be used to recreate the object in memory This poses some problems.. Although Java's floating point arithmetic is largely based on IEEE 754 ("Standard for Binary Floating-Point Arithmetic"), certain features are not supported even when using the codice_1 modifier, such as Exception Flags and Directed Roundings — capabilities mandated by IEEE Standard 754. Additionally, the extended precision floating-point types permitted in 754 and present in many

neg: Type conversion an "implicit" conversion, either during compilation or during run time. For example, in an expression mixing integer and floating point numbers (like 5 + 0.1), the compiler will automatically convert integer representation into floating point representation so fractions are not lost. Explicit type conversions are either indicated by writing additional code (e.g. adding type identifiers or calling built-in routines) or by coding conversion routines for the compiler to use when it otherwise would halt with a type mismatch. In most ALGOL-like languages, such as Pascal, Modula-2, Ada and Delphi, "conversion" and "casting" are distinctly different concepts. In these languages, "conversion"

neg: Half-precision floating-point format half datatype in the Cg language, released in early 2002, and implemented it in silicon in the GeForce FX, released in late 2002. ILM was searching for an image format that could handle a wide dynamic range, but without the hard drive and memory cost of floating-point representations that are commonly used for floating-point computation (single and double precision). The hardware-accelerated programmable shading group led by John Airey at SGI (Silicon Graphics) invented the s10e5 data type in 1997 as part of the 'bali' design effort. This is described in a SIGGRAPH 2000 paper (see section 4.3) and further documented

neg: Chemical potential function, Fermi energy, and Fermi level. For example, n-type silicon has a higher internal chemical potential of electrons than p-type silicon. In a p–n junction diode at equilibrium the chemical potential ("internal" chemical potential) varies from the p-type to the n-type side, while the total chemical potential (electrochemical potential, or, Fermi level) is constant throughout the diode. As described above, when describing chemical potential, one has to say "relative to what". In the case of electrons in semiconductors, internal chemical potential is often specified relative to some convenient point in the band structure, e.g., to the bottom of the conduction

neg: Floating charge in bankruptcy), whereas the floating charge can be granted only by corporate entities. When the Quebec Civil Code came into force in 1994 and superseded the Civil Code of Lower Canada, it abolished the "charge flottante" "floating charge" and created and introduced an analogous security device into Quebec law under the name "hypothèque ouverte", or "floating mortgage". As a mortgage, it The floating mortgage can be specific or general with respect to immovables and movables, separately or together. The mortgage is not perfected until it crystallises. Crystallisation occurs upon default of the mortgagor and registration of a notice of default,

neg: Fast inverse square root is not known how the constant was originally derived, though investigation has shed some light on possible methods. The inverse square root of a floating point number is used in calculating a normalized vector. Programs can use normalized vectors to determine angles of incidence and reflection. 3D graphics programs must perform millions of these calculations every second to simulate lighting. When the code was developed in the early 1990s, most floating-point processing power lagged behind the speed of integer processing. This was troublesome for 3D graphics programs before the advent of specialized hardware to handle transform and lighting. The length

neg: Microsoft Binary Format Not long afterwards the Z80 ports, such as Level II BASIC for the TRS-80 (1978), introduced the 64-bit, double-precision format as a separate data type from 32-bit, single-precision. Microsoft used the same floating-point formats in their implementation of Fortran and for their macro assembler MASM, although their spreadsheet Multiplan and their COBOL implementation used binary-coded decimal (BCD) floating point. Even so, for a while MBF became the de facto floating-point format on home computers, to the point where people still occasionally encounter legacy files and file formats using it. As early as in 1976, Intel was starting the development of

neg: Decimal data type In the fixed-point case, the denominator would be set to a fixed power of ten. In the floating-point case, a variable exponent would represent the power of ten to which the mantissa of the number is multiplied. Languages that support a rational data type usually allow the construction of such a value from a string, instead of a base-2 floating-point number, due to the loss of exactness the latter would cause. Usually the basic arithmetic operations ('+', '−', '×', '/', integer powers) and comparisons ('=', '<', '>', '≤') would be extended to act on them — either natively or through

neg: Single-precision floating-point format these we can form the resulting 32 bit IEEE 754 binary32 format representation of real number 0.25 as: 0-01111101-00000000000000000000000 = 3e800000 "Ex 3:" Consider a value of 0.375. We saw that formula_18 Hence after determining a representation of 0.375 as formula_19 we can proceed as above: From these we can form the resulting 32 bit IEEE 754 binary32 format representation of real number 0.375 as: 0-01111101-10000000000000000000000 = 3ec00000 These examples are given in bit "representation", in hexadecimal and binary, of the floating-point value. This includes the sign, (biased) exponent, and significand. By default, 1/3 rounds up, instead of down like

neg: Strictfp single or double precisions. This made it expensive on common x87-based hardware to ensure that overflows would occur where required. Since JVM 1.2, intermediate computations are, by default, allowed to exceed the standard exponent ranges associated with IEEE 32 bit and 64 bit formats. They may instead be represented as a member of the "extended-exponent" value set. On platforms like x87, overflows and underflows may not occur where expected, producing possibly more meaningful, but less repeatable, results instead. Since X87 floating point is no longer widely used by Java implementations, there is an active proposal to again make all floating

neg: Type safety point to a valid object of correct type. In the example, this leads to garbage value being printed. The problem could have been avoided by replacing codice_12 with codice_13 that throws an exception on invalid casts. Type safety In computer science, type safety is the extent to which a programming language discourages or prevents type errors. A type error is erroneous or undesirable program behaviour caused by a discrepancy between differing data types for the program's constants, variables, and methods (functions), e.g., treating an integer (int) as a floating-point number (float). Type safety is sometimes alternatively considered to be a

neg: Quadruple-precision floating-point format switch) or simply as being synonymous with double precision (e.g. Microsoft Visual C++), rather than as quadruple precision. On a few other architectures, some C/C++ compilers implement codice_8 as quadruple precision, e.g. gcc on PowerPC (as double-double) and SPARC, or the Sun Studio compilers on SPARC. Even if codice_8 is not quadruple precision, however, some C/C++ compilers provide a nonstandard quadruple-precision type as an extension. For example, gcc provides a quadruple-precision type called codice_14 for x86, x86-64 and Itanium CPUs, and on PowerPC as IEEE 128-bit floating-point using the -mfloat128-hardware or -mfloat128 options; and some versions of Intel's C/C++ compiler

neg: Constant folding host architecture matches that on the target architecture, as otherwise enabling constant folding will change the behaviour of the program. This is of particular importance in the case of floating point operations, whose precise implementation may vary widely. Constant propagation is the process of substituting the values of known constants in expressions at compile time. Such constants include those defined above, as well as intrinsic functions applied to constant values. Consider the following pseudocode: Propagating x yields: Continuing to propagate yields the following (which would likely be further optimized by dead code elimination of both x and y.) Constant propagation

neg: Dyadic rational a data type used by computers, floating-point numbers are often defined as integers multiplied by positive or negative powers of two, and thus all numbers that can be represented for instance by binary IEEE floating-point datatypes are dyadic rationals. The same is true for the majority of fixed-point datatypes, which also uses powers of two implicitly in the majority of cases. Dyadic rational In mathematics, a dyadic fraction or dyadic rational is a rational number whose denominator, when the ratio is in minimal (coprime) terms, is a power of two, i.e., a number of the form formula_1 where "a" is

neg: CDC 6000 series type of operation. The function units provided were: branch, Boolean, shift, long integer add, floating-point add, floating-point divide, two floating-point multipliers, and two "increment" (18-bit integer add) units. Functional unit latencies were between a very fast three clock cycles (increment add) and 29 clock cycles (floating-point divide). The 6600 processor could issue a new instruction every clock cycle, assuming that various processor (functional unit, register) resources were available. These resources were kept track of by a "scoreboard" mechanism. Also contributing to keeping the issue rate high was an "instruction stack", which cached the contents of several instruction words. Small loops

neg: C syntax value is allowed), and values other than the constants may be assigned to paint_color, or any other variable of type enum colors. The floating-point form is used to represent numbers with a fractional component. They do not, however, represent most rational numbers exactly; they are instead a close approximation. There are three types of real values, denoted by their specifiers: single precision (float), double precision (double), and double extended precision (long double). Each of these may represent values in a different form, often one of the IEEE floating-point formats. Floating-point constants may be written in decimal notation, e.g. 1.23. Decimal

neg: Intelligent code completion supported by the Visual Studio editors for C++, C#, J#, Visual Basic, XML, HTML and XSLT among others. As of Visual Studio 2005, IntelliSense is now activated by default when the user begins to type, instead of requiring marker characters (though this behavior can be turned off). The IDE has the capability of inferring a greater amount of context based on what the developer is typing, to the point that basic language constructs such as for and while are also included in the choice list. In 2017 Microsoft announced Intellicode , which uses machine learning to infer exactly which language

neg: Partial function In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. The IEEE floating point standard defines a not-a-number value which is returned when a floating point operation is undefined and exceptions are suppressed, e.g. when the square root of a negative number is requested. In a programming language where function parameters are statically typed, a function may be defined as a partial function because the language's type system cannot express the exact domain of the function, so the programmer instead gives it the smallest domain which is expressible as a type and contains

neg: Random Fibonacci sequence of |"f"| converges to a constant value "almost surely", or with probability one: An explicit expression for this constant was found by Divakar Viswanath in 1999. It uses Furstenberg's formula for the Lyapunov exponent of a random matrix product and integration over a certain fractal measure on the Stern–Brocot tree. Moreover, Viswanath computed the numerical value above using floating point arithmetics validated by an analysis of the rounding error. The Embree–Trefethen constant describes the qualitative behavior of the random sequence with the recurrence relation for different values of β. Random Fibonacci sequence In mathematics, the random Fibonacci sequence is a

neg: Coprocessor for the 8086/8088 central processor was the 8089 input/output coprocessor. It used the same programming technique as 8087 for input/output operations, such as transfer of data from memory to a peripheral device, and so reducing the load on the CPU. But IBM didn't use it in IBM PC design and Intel stopped development of this type of coprocessor. The Intel 80386 microprocessor used an optional "math" coprocessor (the 80387) to perform floating point operations directly in hardware. The Intel 80486DX processor included floating-point hardware on the chip. Intel released a cost-reduced processor, the 80486SX, that had no floating point hardware,

neg: Primitive data type "true". C99 adds a distinct boolean type that can be included with stdbool.h, and C++ supports codice_14 as a built-in type and "true" and "false" as reserved words. A floating-point number represents a limited-precision rational number that may have a fractional part. These numbers are stored internally in a format equivalent to scientific notation, typically in binary but sometimes in decimal. Because floating-point numbers have limited precision, only a subset of real or rational numbers are exactly representable; other numbers can be represented only approximately. Many languages have both a single precision (often called "float") and a double precision type.

neg: Exception handling removable singularities. The default IEEE 754 exception handling behaviour of resumption following pre-substitution of a default value avoids the risks inherent in changing flow of program control on numerical exceptions. For example, in 1996 the maiden flight of the Ariane 5 (Flight 501) ended in a catastrophic explosion due in part to the Ada programming language exception handling policy of aborting computation on arithmetic error, which in this case was a 64-bit floating point to 16-bit integer conversion overflow. In the Ariane Flight 501 case, the programmers protected only four out of seven critical variables against overflow due to concerns

neg: Discrete space any function from a discrete metric space to another metric space bounded by 1 is short. Going the other direction, a function "f" from a topological space "Y" to a discrete space "X" is continuous if and only if it is "locally constant" in the sense that every point in "Y" has a neighborhood on which "f" is constant. A discrete structure is often used as the "default structure" on a set that doesn't carry any other natural topology, uniformity, or metric; discrete structures can often be used as "extreme" examples to test particular suppositions. For example, any group can

neg: C syntax to create named constants. Such constants are generally safer to use than macros, since they reside within a specific identifier namespace. An enumerated type is declared with the enum specifier and an optional name (or "tag") for the enum, followed by a list of one or more constants contained within curly braces and separated by commas, and an optional list of variable names. Subsequent references to a specific enumerated type use the enum keyword and the name of the enum. By default, the first constant in an enumeration is assigned the value zero, and each subsequent value is incremented by

neg: Saffron Type System an ADFPath (which represents a glyph); it generates implicit ADF cells from the preprocessed ADFPath and renders each implicit ADF cell by first reconstructing the distance field within the implicit ADF cell using its reconstruction method and then mapping the reconstructed distances to density values. The Saffron library offers both floating-point and fixed-point implementations of the renderer with identical image quality. The fixed-point implementation is ideal for embedded systems and mobile devices, which often lack floating-point hardware. The program supports traditional outline-based fonts, uniform-width stroke-based fonts, and Stylized Stroke Fonts (SSFs). Stroke-based fonts are useful for memory-constrained applications (e.g., embedded

neg: Ship mill Ship mill A ship mill is a type of watermill. The milling and grinding technology and the drive (waterwheel) are built on a floating platform on this type of mill. Its first recorded use dates back to mid-6th century AD Italy. Between barge and boat well is the undershot water wheel, which is driven by the flowing water of the current. There is also evidence of water mills for which both sides had a narrower water wheel, similar to an old paddle steamer. The floating platform is anchored at the most intense point in the current, to the bridge piers

neg: Single-precision floating-point format and properties of floating-point data types depended on the computer manufacturer and computer model, and upon decisions made by programming-language designers. E.g., GW-BASIC's single-precision data type was the 32-bit MBF floating-point format. Single precision is termed "REAL" in Fortran, "SINGLE-FLOAT" in Common Lisp, "float" in C, C++, C#, Java, "Float" in Haskell, and "Single" in Object Pascal (Delphi), Visual Basic, and MATLAB. However, "float" in Python, Ruby, PHP, and OCaml and "single" in versions of Octave before 3.2 refer to double-precision numbers. In most implementations of PostScript, and some embedded systems, the only supported precision is single. The IEEE 754