
Single-precision floating-point format

Single-precision floating-point format (sometimes called FP32 or float32) 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 7 or fewer decimal digits, and any 2"n" for a whole number −149 ≤ "n" ≤ 127, can be converted exactly into an IEEE 754 single-precision floating-point value.
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.
IEEE 754 single-precision binary floating-point format: binary32.
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.
The sign bit determines the sign of the number, which is the sign of the significand as well. The exponent is an 8-bit unsigned integer from 0 to 255, in biased form: an exponent value of 127 represents the actual zero. 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:
The real value assumed by a given 32-bit "binary32" data with a given "sign", biased exponent "e" (the 8-bit unsigned integer), and a "23-bit fraction" is
which yields
In this example:
thus:
Note:
Exponent encoding.
The single-precision binary floating-point exponent is encoded using an offset-binary representation, with the zero offset being 127; also known as exponent bias in the IEEE 754 standard.
Thus, in order to get the true exponent as defined by the offset-binary representation, the offset of 127 has to be subtracted from the stored exponent.
The stored exponents 00H and FFH are interpreted specially.
The minimum positive normal value is formula_13 and the minimum positive (subnormal) value is formula_14.
Converting from decimal representation to binary32 format.
In general, refer to the IEEE 754 standard itself for the strict conversion (including the rounding behaviour) of a real number into its equivalent binary32 format.
Here we can show how to convert a base-10 real number into an IEEE 754 binary32 format using the following outline:
Conversion of the fractional part:
Consider 0.375, the fractional part of 12.375. To convert it into a binary fraction, multiply the fraction by 2, take the integer part and repeat with the new fraction by 2 until a fraction of zero is found or until the precision limit is reached which is 23 fraction digits for IEEE 754 binary32 format.
We see that formula_18 can be exactly represented in binary as formula_19. Not all decimal fractions can be represented in a finite digit binary fraction. For example, decimal 0.1 cannot be represented in binary exactly, only approximated. Therefore:
Since IEEE 754 binary32 format requires real values to be represented in formula_21 format (see Normalized number, Denormalized number), 1100.011 is shifted to the right by 3 digits to become formula_22
Finally we can see that: formula_23
From which we deduce:
From these we can form the resulting 32-bit IEEE 754 binary32 format representation of 12.375:
Note: consider converting 68.123 into IEEE 754 binary32 format: Using the above procedure you expect to get formula_26 with the last 4 bits being 1001. However, due to the default rounding behaviour of IEEE 754 format, what you get is formula_27, whose last 4 bits are 1010.
Example 1:
Consider decimal 1. We can see that: formula_28
From which we deduce:
From these we can form the resulting 32-bit IEEE 754 binary32 format representation of real number 1:
Example 2:
Consider a value 0.25. We can see that: formula_32
From which we deduce:
From these we can form the resulting 32-bit IEEE 754 binary32 format representation of real number 0.25:
Example 3:
Consider a value of 0.375. We saw that formula_35
Hence after determining a representation of 0.375 as formula_36 we can proceed as above:
From these we can form the resulting 32-bit IEEE 754 binary32 format representation of real number 0.375:
Single-precision examples.
These examples are given in bit "representation", in hexadecimal and binary, of the floating-point value. This includes the sign, (biased) exponent, and significand.
 0 00000000 000000000000000000000012 = 0000 000116 = 2−126 × 2−23 = 2−149 ≈ 1.4012984643 × 10−45
 (smallest positive subnormal number)
 0 00000000 111111111111111111111112 = 007f ffff16 = 2−126 × (1 − 2−23) ≈ 1.1754942107 ×10−38
 (largest subnormal number)
 0 00000001 000000000000000000000002 = 0080 000016 = 2−126 ≈ 1.1754943508 × 10−38
 (smallest positive normal number)
 0 11111110 111111111111111111111112 = 7f7f ffff16 = 2127 × (2 − 2−23) ≈ 3.4028234664 × 1038
 (largest normal number)
 0 01111110 111111111111111111111112 = 3f7f ffff16 = 1 − 2−24 ≈ 0.999999940395355225
 (largest number less than one)
 0 01111111 000000000000000000000002 = 3f80 000016 = 1 (one)
 0 01111111 000000000000000000000012 = 3f80 000116 = 1 + 2−23 ≈ 1.00000011920928955
 (smallest number larger than one)
 1 10000000 000000000000000000000002 = c000 000016 = −2
 0 00000000 000000000000000000000002 = 0000 000016 = 0
 1 00000000 000000000000000000000002 = 8000 000016 = −0
 0 11111111 000000000000000000000002 = 7f80 000016 = infinity
 1 11111111 000000000000000000000002 = ff80 000016 = −infinity
 0 10000000 100100100001111110110112 = 4049 0fdb16 ≈ 3.14159274101257324 ≈ π ( pi )
 0 01111101 010101010101010101010112 = 3eaa aaab16 ≈ 0.333333343267440796 ≈ 1/3
 x 11111111 100000000000000000000012 = ffc0 000116 = qNaN (on x86 and ARM processors)
 x 11111111 000000000000000000000012 = ff80 000116 = sNaN (on x86 and ARM processors)
By default, 1/3 rounds up, instead of down like double precision, because of the even number of bits in the significand. The bits of 1/3 beyond the rounding point are codice_1 which is more than 1/2 of a unit in the last place.
Encodings of qNaN and sNaN are not specified in IEEE 754 and implemented differently on different processors. The x86 family and the ARM family processors use the most significant bit of the significand field to indicate a quiet NaN. The PA-RISC processors use the bit to indicate a signalling NaN.
Converting from single-precision binary to decimal.
We start with the hexadecimal representation of the value, , in this example, and convert it to binary:
then we break it down into three parts: sign bit, exponent, and significand.
We then add the implicit 24th bit to the significand:
and decode the exponent value by subtracting 127:
Each of the 24 bits of the significand (including the implicit 24th bit), bit 23 to bit 0, represents a value, starting at 1 and halves for each bit, as follows:
 bit 23 = 1
 bit 22 = 0.5
 bit 21 = 0.25
 bit 20 = 0.125
 bit 19 = 0.0625
 bit 18 = 0.03125
 .
 bit 0 = 0.00000011920928955078125
The significand in this example has three bits set: bit 23, bit 22, and bit 19. We can now decode the significand by adding the values represented by these bits.
Then we need to multiply with the base, 2, to the power of the exponent, to get the final result:
Thus
This is equivalent to:
where is the sign bit, is the exponent, and is the significand.
Optimizations.
The design of floating-point format allows various optimisations, resulting from the easy generation of a base-2 logarithm approximation from an integer view of the raw bit pattern. Integer arithmetic and bit-shifting can yield an approximation to reciprocal square root (fast inverse square root), commonly required in computer graphics.