Minifloat

At these small sizes other bias values may be interesting, for instance a bias of -2 will make the numbers 0-16 have the same bit representation as the integers 0-16, with the loss that no non-integer values can be represented.

0 0000 000 = 0.0002 × 21 - (-2) = 0.0 × 23 = 0 (subnormal number)
0 0000 001 = 0.0012 × 21 - (-2) = 0.125 × 23 = 1 (subnormal number)
0 0000 111 = 0.1112 × 21 - (-2) = 0.875 × 23 = 7 (subnormal number)
0 0001 000 = 1.0002 × 21 - (-2) = 1.000 × 23 = 8 (normalized number)
0 0001 111 = 1.1112 × 21 - (-2) = 1.875 × 23 = 15 (normalized number)
0 0010 000 = 1.0002 × 22 - (-2) = 1.000 × 24 = 16 (normalized number)

Zero is represented as zero exponent with a zero mantissa. The zero exponent means zero is a subnormal number with a leading "0." prefix, and with the zero mantissa all bits after the decimal point are zero, meaning this value is interpreted as ${\displaystyle 0.000_{2}\times 2^{-6}=0}$. Floating point numbers use a signed zero, so ${\displaystyle -0}$ is also available and is equal to positive ${\displaystyle 0}$.

0 0000 000 = 0
1 0000 000 = −0

The significand is extended with "0." and the exponent value is treated as 1 higher like the least normalized number:

0 0000 001 = 0.0012 × 21 - 7 = 0.125 × 2-6 = 0.001953125 (least subnormal number)
...
0 0000 111 = 0.1112 × 21 - 7 = 0.875 × 2-6 = 0.013671875 (greatest subnormal number)

The significand is extended with "1.":

0 0001 000 = 1.0002 × 21 - 7 = 1 × 2-6 = 0.015625 (least normalized number)
0 0001 001 = 1.0012 × 21 - 7 = 1.125 × 2-6 = 0.017578125
...
0 0111 000 = 1.0002 × 27 - 7 = 1 × 20 = 1
0 0111 001 = 1.0012 × 27 - 7 = 1.125 × 20 = 1.125 (least value above 1)
...
0 1110 000 = 1.0002 × 214 - 7 =  1.000 × 27 =  128
0 1110 001 = 1.0012 × 214 - 7 =  1.125 × 27 =  144
...
0 1110 110 = 1.1102 × 214 - 7 =  1.750 × 27 = 224
0 1110 111 = 1.1112 × 214 - 7 =  1.875 × 27 = 240 (greatest normalized number)

Infinity values have the highest exponent, with the mantissa set to zero. The sign bit can be either positive or negative.

0 1111 000 = +infinity
1 1111 000 = −infinity

NaN values have the highest exponent, with a non-zero value for the mantissa. A float with 1-bit sign and 3-bit mantissa has ${\displaystyle 2\times (2^{3}-1)=14}$ NaN values.

s 1111 mmm = NaN (if mmm ≠ 000)

This is a chart of all possible values for this example 8-bit float.

There are only 242 different non-NaN values (if +0 and −0 are regarded as different), because 14 of the bit patterns represent NaNs.

Tables like the above can be generated for any combination of SEMB values using a script in Python or in GDScript.

Addition of (1.3.2.3)-minifloats

The graphic demonstrates the addition of even smaller (1.3.2.3)-minifloats with 6 bits. This floating-point system follows the rules of IEEE 754 exactly. NaN as operand produces always NaN results. Inf − Inf and (−Inf) + Inf results in NaN too (green area). Inf can be augmented and decremented by finite values without change. Sums with finite operands can give an infinite result (i.e. 14.0 + 3.0 = +Inf as a result is the cyan area, −Inf is the magenta area). The range of the finite operands is filled with the curves x + y = c, where c is always one of the representable float values (blue and red for positive and negative results respectively).

The other arithmetic operations can be illustrated similarly:

• 
  Subtraction
• 
  Multiplication
• 
  Division

The smallest possible float size that follows all IEEE principles, including normalized numbers, subnormal numbers, signed zero, signed infinity, and multiple NaN values, is a 4-bit float with 1-bit sign, 2-bit exponent, and 1-bit mantissa.[8] In the table below, the columns have different values for the sign and mantissa bits, and the rows are different values for the exponent bits.

If normalized numbers are not required, the size can be reduced to 3-bit by reducing the exponent down to 1.

In situations where the sign bit can be excluded, each of the above examples can be reduced by 1 bit further, keeping only the left half of the above tables. A 2-bit float with 1-bit exponent and 1-bit mantissa would only have 0, 1, Inf, NaN values.

If the mantissa is allowed to be 0-bit, a 1-bit float format would have a 1-bit exponent, and the only two values would be 0 and Inf. The exponent must be at least 1 bit or else it no longer makes sense as a float (it would just be a signed number).