IEEE 754 formats

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single, double and extended are FPC's data types implementing Pascal’s real. All of them are implemented according IEEE standard 754, where single is “single-precision”, double is “double-precision”, and extended has 80 bits.

single

value range 1.5E-45 .. 3.4E38
accuracy 6-9 significant decimal digits precision
memory requirement 4 bytes or 32 bits
property The single- data-type data field can hold floating-point values ​​and signed and unsigned integer values.

Assigning other values ​​will result in error messages from the compiler when the program is compiled, and the compile will be aborted. That is, the executable program is not created.

Definition of a data field of data type Single:

var
  s: single;

Examples of assigning valid values:

s := -123.45678;
// Note: '0' is an integer literal. 0.0 is an real literal.
// Here, FPC will make an implicit typecast from integer to single:
s := 0;
// a positive sign is optional
s := 123.45678;

Examples of assigning invalid values:

s := '-123.45678';
s := '0';
s := '123.45678';

The difference between the two examples is that the upper example is the assignment of Integer and Floating literals, while the assignment of the lower example is literals of the String type.

Binary floating-point format

Any value stored as a single requires 32 bits, formatted as shown in the table below:

Bits Usage
31 Sign (0 = positive, 1 = negative)
30 to 23 Exponent, biased by 127
22 to 0 Fraction f of the number 1.f

double

Any value stored as a double requires 64 bits, formatted as shown in the table below:

Bits Usage
63 Sign (0 = positive, 1 = negative)
62 to 52 Exponent, biased by 1023
51 to 0 Fraction f of the number 1.f

Example of converting from raw data to double (Data is array [0..7] of byte):

function ToDouble(const Data; IntelEndianness: Boolean = False):Double;inline;
var
  ADouble: Double absolute Data;
  AQWord: QWord absolute Result;
begin
  Result := ADouble;
  if not IntelEndianness then
    AQWord := SwapEndian(AQWord);
end;

extended

extended is a 80-bit wide floating-point data type. There are FPUs that internally use 80 bits for increased precision. FPC allows to use this gain in precision.

Note-icon.png

Note: If some platform does not support the extended data type, it will be mapped to largest available floating-point number data available, i. e. usually double.

Using the maximum precision for constants

In this example:

var
  f: double;
  n: integer = 1758;
  m: integer = 0;
begin
  f := n * 1.2E6 + (2*m+1) * 50E3; // 2109650048

FPC will interpret constant 1.2E6 as Single type, because it fits into Single type. This leads to some precision loss in the calculation: result is 2109650048 instead of 2109650000.

You can use FPC command-line flag -CF64 to force floating point constants to have at least 64 bits precision. Or you can use {$excessprecision on} to force the compiler to evaluate all floating point expressions always using the maximum precision supported by the FPU.


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