## ULP Considerations of Native Floating-Point Operators

The representation of infinitely real numbers with a finite number of bits requires an approximation. This approximation can result in rounding errors in floating-point computation. To measure the rounding errors, the floating-point standard uses relative error and ULP (Units in the Last Place) error. To learn about relative error, see Relative Accuracy and ULP Considerations.

If the exponent range is not upper-bounded, Units in Last Place (ULP) of a floating-point number x is the distance between two closest straddling floating-point numbers a and b nearest to x. The IEEE-754 standard requires that the result of an elementary arithmetic operation such as addition, multiplication, and division is correctly round. A correctly rounded result means that the rounded result is within 0.5 ULP of the exact result.

### Adherence of Native Floating Point Operators to IEEE-754 Standard

Native floating point technology in HDL Coder™ follows IEEE standard of floating-point arithmetic. Basic arithmetic
operations such as addition, subtraction, multiplication, division, and reciprocal are
mandated by IEEE to have zero ULP error. When you perform these operations in native
floating-point mode, the numerical results obtained from the generated HDL code match the
original Simulink^{®} model.

Certain advanced math operations such as exponential, logarithm, and trigonometric operators have machine-specific implementation behaviors because these operators use recurring taylor series and remez expression based implementations. When you use these operators in native floating-point mode, there can be relatively small differences in numerical results between the Simulink model and the generated HDL code.

You can measure the difference in numerical results as a relative error or ULP. A
nonzero ULP for these operators does not mean noncompliance with the IEEE standard. A ULP of
one is equivalent to a relative error of `10^-7`

. You can ignore such
relatively small errors by specifying a custom tolerance value for the ULP when generating a
HDL test bench. For example, you can specify a custom floating-point tolerance of one ULP to
ignore the error when verifying the generated code. For more information, see Floating-Point Tolerance Parameters.

### ULP Values of Floating Point Operators

The table enumerates the ULP of floating-point operators that have a nonzero ULP. In addition to these operators, the HDL Reciprocal block has a ULP of five.

### Considerations

For certain floating-point input values, some blocks can produce simulation results that
vary from the MATLAB^{®} simulation results. To see the difference in results, before you generate
code, enable generation of the validation model. In the Configuration Parameters dialog box,
on the **HDL Code Generation** pane, select the **Generate
validation model** check box.

If you perform computations that involve complex numbers and an exception such as

`Inf`

or`NaN`

, the HDL simulation result with native floating point can potentially vary from the Simulink simulation result. For example, if you multiply a complex input with`Inf`

, the Simulink simulation result is`Infi`

whereas the HDL simulation result is`NaN+Infi`

.HDL Coder does not generate a mismatch error between reference and native floating-point values if both values are

`NaN`

.If you compute the square root or logarithm of a negative number, the HDL simulation result with native floating point is

`0`

. This result matches the simulation result when you verify the design with a SystemVerilog DPI test bench. In Simulink, the result obtained is`NaN`

. According to the IEEE-754 standard, if you compute the square root or logarithm of a negative number, the result is that number itself.If the input to the Direct Lookup Table (n-D) is of floating-point data type, but the elements of the table use a smaller data type, the generated HDL code can be potentially incorrect. For example, the input is of

`single`

type and the elements use`uint8`

type. To obtain accurate HDL simulation results, use the same data type for the input signal and the elements of the lookup table.If you use the Cosine block with the inputs

`-7.729179E28`

or`7.729179E28`

, the generated HDL code has a ULP of 4. For all other inputs, the ULP is 2.When you use a Math Function block to compute

`mod(a,b)`

or`rem(a,b)`

, where`a`

is the dividend and`b`

is the divisor, the simulation result in native floating-point point mode varies from the MATLAB simulation result in these cases:If $$b\text{isintegerand}\frac{a}{b}{2}^{32}$$, the simulation result in native floating-point mode is zero. For such significant difference in magnitude between the numbers

`a`

and`b`

, this implementation saves area on the target FPGA device.If $$\frac{a}{b}\text{iscloseto}{2}^{23}$$, the simulation result in native floating-point mode can potentially vary from the MATLAB simulation results.