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) as is the case for older or low-cost embedded microprocessors and microcontrollers, if fixed-point provides improved performance or accuracy for the application at hand, or if their use is more natural for the problem (such as for the representation of angles).
Representation.
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 human convenience) or a power of 2 (for computational efficiency). However, other scaling factors may be used occasionally, e.g. a time value in hours may be represented as a fixed-point type with a scale factor of 1/3600 to obtain values with one-second accuracy.
The maximum value of a fixed-point type is simply the largest value that can be represented in the underlying integer type multiplied by the scaling factor; and similarly for the minimum value.
Operations.
To convert a number from a fixed point type with scaling factor "R" to another type with scaling factor "S", the underlying integer must be multiplied by "R" and divided by "S"; that is, multiplied by the ratio "R"/"S". Thus, for example, to convert the value 1.23 = 123/100 from a type with scaling factor "R"=1/100 to one with scaling factor "S"=1/1000, the underlying integer 123 must be multiplied by (1/100)/(1/1000) = 10, yielding the representation 1230/1000. If "S" does not divide "R" (in particular, if the new scaling factor "S" is greater than the original "R"), the new integer will have to be rounded. The rounding rules and methods are usually part of the language's specification.
To add or subtract two values of the same fixed-point type, it is sufficient to add or subtract the underlying integers, and keep their common scaling factor. The result can be exactly represented in the same type, as long as no overflow occurs (i.e. provided that the sum of the two integers fits in the underlying integer type). If the numbers have different fixed-point types, with different scaling factors, then one of them must be converted to the other before the sum.
To multiply two fixed-point numbers, it suffices to multiply the two underlying integers, and assume that the scaling factor of the result is the product of their scaling factors. This operation involves no rounding. For example, multiplying the numbers 123 scaled by 1/1000 (0.123) and 25 scaled by 1/10 (2.5) yields the integer 123×25 = 3075 scaled by (1/1000)×(1/10) = 1/10000, that is 3075/10000 = 0.3075. If the two operands belong to the same fixed-point type, and the result is also to be represented in that type, then the product of the two integers must be explicitly multiplied by the common scaling factor; in this case the result may have to be rounded, and overflow may occur. For example, if the common scaling factor is 1/100, multiplying 1.23 by 0.25 entails multiplying 123 by 25 to yield 3075 with an intermediate scaling factor of 1/10000. This then must be multiplied by 1/100 to yield either 31 (0.31) or 30 (0.30), depending on the rounding method used, to result in a final scale factor of 1/100.
To divide two fixed-point numbers, one takes the integer quotient of their underlying integers, and assumes that the scaling factor is the quotient of their scaling factors. The first division involves rounding in general. For example, division of 3456 scaled by 1/100 (34.56) and 1234 scaled by 1/1000 (1.234) yields the integer 3456÷1234 = 3 (rounded) with scale factor (1/100)/(1/1000) = 10, that is, 30. One can obtain a more accurate result by first converting the dividend to a more precise type: in the same example, converting 3456 scaled by 1/100 (34.56) to 3,456,000 scaled by 1/100000, before dividing by 1234 scaled by 1/1000 (1.234), would yield 3456000÷1234 = 2801 (rounded) with scaling factor (1/100000)/(1/1000) = 1/100, that is 28.01 (instead of 30). If both operands and the desired result all have the same scaling factor, then the quotient of the two integers must be explicitly multiplied by that common scaling factor.
Binary vs. decimal.
The two most common classes of fixed-point types are decimal and binary. Decimal fixed-point types have a scaling factor that is a power of ten; for binary fixed-point types it is a power of two.
Binary fixed-point types are most commonly used, because the rescaling operations can be implemented as fast bit shifts. Binary fixed-point numbers can represent fractional powers of two exactly, but, like binary floating-point numbers, cannot exactly represent fractional powers of ten. If exact fractional powers of ten are desired, then a decimal format should be used. For example, one-tenth (0.1) and one-hundredth (0.01) can be represented only approximately by binary fixed-point or binary floating-point representations, while they can be represented exactly in decimal fixed-point or decimal floating-point representations. These representations may be encoded in many ways, including binary-coded decimal (BCD).
A worked Example.
Suppose there is the following multiplication with 2 fixed point 3 decimal place numbers.
=1*10.500 + 0.050*10.500 
= 10.500+0.525000=11.025000
Note how since there are 3 decimal places we show the trailing zeros. To re-characterize this as an integer multiplication we must first multiply by 1000 (10^3) moving all the decimal places in to integer places, then we will multiply by formula_1 (10^-3) to put them back the equation now looks like
This works equivalently if we choose a different base, notably base 2 for computing, since a bit shift is the same as a multiplication or division by an order of 2. Three decimal digits is equivalent to about 10 binary digits, so we should round 0.05 to 10 bits after the binary point. The closest approximation is then 0.0000110011.
Thus our multiplication becomes
This rounds to 11.023 with three digits after the decimal point.
Notation.
There are various notations used to represent word length and radix point in a binary fixed-point number. In the following list, "f" represents the number of fractional bits, "m" the number of magnitude or integer bits, "s" the number of sign bits, and "b" the total number of bits.
Precision loss and overflow.
Because fixed point operations can produce results that have more digits than the operands, information loss is possible. For instance, the result of fixed point multiplication could potentially have as many digits as the sum of the number of digits in the two operands. In order to fit the result into the same number of digits as the operands, the answer must be rounded or truncated. If this is the case, the choice of which digits to keep is very important. When multiplying two fixed point numbers with the same format, for instance with formula_2 integer digits, and formula_3 fractional digits, the answer could have up to formula_4 integer digits, and formula_5 fractional digits.
For simplicity, many fixed-point multiply procedures use the same result format as the operands. This has the effect of keeping the middle digits; the I-number of least significant integer digits, and the Q-number of most significant fractional digits. Fractional digits lost below this value represent a precision loss which is common in fractional multiplication. If any integer digits are lost, however, the value will be radically inaccurate. Some model-based fixed-point packages allow specifying a result format different from the input formats, enabling the user to maximize precision and avoid overflow.
Some operations, like divide, often have built-in result limiting so that any positive overflow results in the largest possible number that can be represented by the current format. Likewise, negative overflow results in the largest negative number represented by the current format. This built in limiting is often referred to as "saturation".
Some processors support a hardware overflow flag that can generate an exception on the occurrence of an overflow, but it is usually too late to salvage the proper result at this point.
Computer language implementations.
Very few computer languages include built-in support for fixed point values other than with the radix point immediately to the right of the least significant digit (i.e. integer), because for most applications, binary or decimal floating-point representations are usually simpler to use and accurate enough. Floating-point representations are easier to use than fixed-point representations, because they can handle a wider dynamic range and do not require programmers to specify the number of digits after the radix point. However, if they are needed, fixed-point numbers can be implemented even in programming languages like C and C++, which do not commonly include such support.
A common use of fixed-point BCD numbers is for storing monetary values, where the inexact values of binary floating-point numbers are often a liability. Historically, fixed-point representations were the norm for decimal data types; for example, in PL/I or COBOL. 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 <samp>numeric</samp> type for exact storage of numbers with up to 1000 digits.