Single-precision floating-point format

Single-precision floating-point format is a computer number format, usually occupying 32 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. Single-precision floating-point format is a computer number format, usually occupying 32 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. 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 231 − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2−23) × 2127 ≈ 3.4028235 × 1038. All integers with 6 or fewer significant decimal digits, and any number that can be written as 2n such that n is a whole number from -126 to 127, can be converted into an IEEE 754 floating-point value without loss of precision. In the IEEE 754-2008 standard, the 32-bit base-2 format is officially referred to as binary32; it was called single in IEEE 754-1985. IEEE 754 specifies additional floating-point types, such as 64-bit base-2 double 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 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 standard specifies a binary32 as having: This gives from 6 to 9 significant decimal digits precision. If a decimal string with at most 6 significant digits is converted to IEEE 754 single-precision representation, and then converted back to a decimal string with the same number of digits, the final result should match the original string. If an IEEE 754 single-precision number is converted to a decimal string with at least 9 significant digits, and then converted back to single-precision representation, the final result must match the original number. Sign bit determines the sign of the number, which is the sign of the significand as well. Exponent is either an 8-bit signed integer from −128 to 127 (2's complement) or an 8-bit unsigned integer from 0 to 255, which is the accepted biased form in IEEE 754 binary32 definition. If the unsigned integer format is used, the exponent value used in the arithmetic is the exponent shifted by a bias – for the IEEE 754 binary32 case, an exponent value of 127 represents the actual zero (i.e. for 2e − 127 to be one, e must be 127). Exponents range from −126 to +127 because exponents of −127 (all 0s) and +128 (all 1s) are reserved for special numbers. The true significand includes 23 fraction bits to the right of the binary point and an implicit leading bit (to the left of the binary point) with value 1, unless the exponent is stored with all zeros. Thus only 23 fraction bits of the significand appear in the memory format, but the total precision is 24 bits (equivalent to log10(224) ≈ 7.225 decimal digits). The bits are laid out as follows: