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Hexadecimal floating point (now called HFP by IBM) is a format for encoding floating-point numbers first introduced on the IBM System/360 computers, and supported on subsequent machines based on that architecture, [1] [2] [3] as well as machines which were intended to be application-compatible with System/360.
The "decimal" 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.
The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −42 10, −B01D9 to represent −721369 10 and so on. Hexadecimal can also be used to express the exact bit patterns used in the processor, so a sequence of hexadecimal digits may represent a signed or even a floating-point value.
To approximate the greater range and precision of real numbers, we have to abandon signed integers and fixed-point numbers and go to a "floating-point" format. In the decimal system, we are familiar with floating-point numbers of the form (scientific notation): 1.1030402 × 10 5 = 1.1030402 × 100000 = 110304.02. or, more compactly: 1.1030402E5
A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum value of 2 31 − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2 −23) × 2 127 ≈ 3.4028235 ...
The standard recommends providing conversions to and from external hexadecimal-significand character sequences, based on C99's hexadecimal floating point literals. Such a literal consists of an optional sign ( + or - ), the indicator "0x", a hexadecimal number with or without a period, an exponent indicator "p", and a decimal exponent with ...
Hexadecimal floating-point arithmetic in the Interdata 8/32 computer in the 1970s [1] Hexadecimal floating-point arithmetic in the Manchester MU5 computer in 1972 [ 2 ] Hexadecimal floating-point arithmetic in the Data General Eclipse S/200 computer in ca. 1974 [ 1 ]
Swift introduced half-precision floating point numbers in Swift 5.3 with the Float16 type. [20] OpenCL also supports half-precision floating point numbers with the half datatype on IEEE 754-2008 half-precision storage format. [21] As of 2024, Rust is currently working on adding a new f16 type for IEEE half-precision 16-bit floats. [22]