Documentation

## Range and Precision

The range of a number gives the limits of the representation, while the precision gives the distance between successive numbers in the representation. The range and precision of a fixed-point number depend on the length of the word and the scaling.

### Range

The following figure illustrates the range of representable numbers for an unsigned fixed-point number of size ws, scaling S, and bias B. The following figure illustrates the range of representable numbers for a two's complement fixed-point number of size ws, scaling S, and bias B where the values of ws, scaling S, and bias B allow for both negative and positive numbers. For both the signed and unsigned fixed-point numbers of any data type, the number of different bit patterns is 2ws.

For example, if the fixed-point data type is an integer with scaling defined as $S=1$ and B = 0, then the maximum unsigned value is ${2}^{ws-1}$, because zero must be represented. In two's complement, negative numbers must be represented as well as zero, so the maximum value is ${2}^{ws-1}-1$. Additionally, since there is only one representation for zero, there must be an unequal number of positive and negative numbers. This means there is a representation for $-{2}^{ws-1}$ but not for ${2}^{ws-1}$.

### Precision

The precision of a data type is given by the slope. In this usage, precision means the difference between neighboring representable values.

### Fixed-Point Data Type Parameters

The low limit, high limit, and default binary-point-only scaling for the supported fixed-point data types discussed in Binary-Point-Only Scaling are given in the following table. SeeLimitations on Precision and Limitations on Range for more information.

Fixed-Point Data Type Range and Default Scaling

Name

Data Type

Low Limit

High Limit

Default Scaling (~Precision)

Unsigned Integer

`fixdt(0,ws,0)`

0

${2}^{ws}-1$

`1`

Signed Integer

`fixdt(1,ws,0)`

$-{2}^{ws-1}$

${2}^{ws-1}-1$

`1`

Unsigned Binary Point

`fixdt(0,ws,fl)`

0

$\left({2}^{ws}-1\right){2}^{-fl}$

${2}^{-fl}$

Signed Binary Point

`fixdt(1,ws,fl)`

$-{2}^{ws-1-fl}$

$\left({2}^{ws-1}-1\right){2}^{-fl}$

${2}^{-fl}$

Unsigned Slope Bias

`fixdt(0,ws,s,b)`

`b`

$s\left({2}^{ws}-1\right)+b$

s

Signed Slope Bias

`fixdt(1,ws,s,b)`

$-s\left({2}^{ws-1}\right)+b$

$s\left({2}^{ws-1}-1\right)+b$

s

s = Slope, b = Bias, ws = WordLength, fl = FractionLength

### Range of an 8-Bit Fixed-Point Data Type — Binary-Point-Only Scaling

The precisions, range of signed values, and range of unsigned values for an 8-bit generalized fixed-point data type with binary-point-only scaling are listed in the follow table. Note that the first scaling value (21) represents a binary point that is not contiguous with the word.

Scaling

Precision

Range of Signed Values (Low, High)

Range of Unsigned Values (Low, High)

21

2.0

-256, 254

0, 510

20

1.0

-128, 127

0, 255

2-1

0.5

-64, 63.5

0, 127.5

2-2

0.25

-32, 31.75

0, 63.75

2-3

0.125

-16, 15.875

0, 31.875

2-4

0.0625

-8, 7.9375

0, 15.9375

2-5

0.03125

-4, 3.96875

0, 7.96875

2-6

0.015625

-2, 1.984375

0, 3.984375

2-7

0.0078125

-1, 0.9921875

0, 1.9921875

2-8

0.00390625

-0.5, 0.49609375

0, 0.99609375

### Range of an 8-Bit Fixed-Point Data Type — Slope and Bias Scaling

The precision and ranges of signed and unsigned values for an 8-bit fixed-point data type using slope and bias scaling are listed in the following table. The slope starts at a value of `1.25` with a bias of `1.0` for all slopes. Note that the slope is the same as the precision.

Bias

Slope/Precision

Range of Signed Values (low, high)

Range of Unsigned Values (low, high)

1

1.25

-159, 159.75

1, 319.75

1

0.625

-79, 80.375

1, 160.375

1

0.3125

-39, 40.6875

1, 80.6875

1

0.15625

-19, 20.84375

1, 40.84375

1

0.078125

-9, 10.921875

1, 20.921875

1

0.0390625

-4, 5.9609375

1, 10.9609375

1

0.01953125

-1.5, 3.48046875

1, 5.98046875

1

0.009765625

-0.25, 2.240234375

1, 3.490234375

1

0.0048828125

0.375, 1.6201171875

1, 2.2451171875

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