## Create Lookup Tables for a Sine Function

### Introduction

The sections that follow explain how to use the function `fixpt_look1_func_approx` to create lookup tables. It gives examples that show how to create lookup tables for the function sin(2πx) on the interval from 0 to 0.25.

### Set Function Parameters for the Lookup Table

First, define parameter values for the `fixpt_look1_func_approx` function.

```% Required parameters funcstr = 'sin(2*pi*x)'; % Ideal function xmin = 0; % Minimum input of interest xmax = 0.25; % Maximum input of interest xdt = ufix(16); % x data type xscale = 2^-16; % x data scaling ydt = sfix(16); % y data type yscale = 2^-14; % y data scaling rndmeth = 'Floor'; % Rounding method % Optional parameters errmax = 2^-10; % Maximum allowed error of the lookup table nptsmax = 21; % Maximum number of points of the lookup table```

The parameters `errmax`, `nptsmax`, and `spacing` are optional. You must use at least one of the parameters `errmax` and `nptsmax`. If you use only the `errmax` parameter without `nptsmax`, the function creates a lookup table with the fewest points for which the worst-case error is at most `errmax`. If you use only the `nptsmax` parameter without `errmax`, the function creates a lookup table with at most `nptsmax` points which has the smallest worst-case error. You can use the optional `spacing` parameter to restrict the spacing between breakpoints of the lookup table.

Use `errmax` with Unrestricted Spacing

Create a lookup table that has the fewest data points for a specified worst-case error, with unrestricted spacing.

```[xdata,ydata,errworst] = fixpt_look1_func_approx(funcstr, ... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,[]);```

Note that the `nptsmax` and `spacing` parameters are not specified.

`length(xdata)`
```ans = 16 ```

16 points are required to approximate $\mathrm{sin}\left(2\pi \mathit{x}\right)$ to within the tolerance specified by `errmax`. The maximum error is:

`errworst`
```errworst = 9.7656e-04 ```

The value of `errworst` is less than or equal to the value of `errmax`.

Plot the results:

```figure(1) fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ... xscale,ydt,yscale,rndmeth);```

The upper plot shows the ideal function $\mathrm{sin}\left(2\pi \mathit{x}\right)$ and the fixed-point lookup table approximation between the breakpoints. In this example, the ideal function and the approximation are so close together that the two graphs appear to coincide. The lower plot displays the errors.

In this example, the Y data points, `ydata`, are equal to the ideal function applied to the point in `xdata`. However, you can define a different set of values for `ydata` after running `fixpt_look1_func_approx`. This can sometimes reduce the maximum error.

You can also change the values of `xmin` and `xmax` to evaluate the lookup table on a subset of the original interval.

To find the new maximum error after changing `ydata`, `xmin`, or `xmax`, type

`errworst = fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax, ...`

`xdt,xscale,ydt,yscale,rndmeth)`

Use `nptsmax` with Unrestricted Spacing

Create a lookup table that minimizes the worst-case error for a specified maximum number of data points, with unrestricted spacing.

```[xdata, ydata, errworst] = fixpt_look1_func_approx(funcstr, ... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,[],nptsmax);```

The empty brackets tell the function to ignore the parameter `errmax`. Omitting `errmax` causes the function to return a lookup table of size specified by `nptsmax`, with the smallest worst-case error.

`length(xdata)`
```ans = 21 ```

The function returns a vector `xdata` with 21 points. The maximum error is:

`errworst`
```errworst = 5.1139e-04 ```

The value of `errworst` is less than or equal to the value of `errmax`.

Plot the results:

```figure(2) fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ... xscale,ydt,yscale,rndmeth);```

Restricting the Spacing

In the previous two examples, the function `fixpt_look1_func_approx` creates lookup tables with unrestricted spacing between the points. You can restrict the spacing to improve the computational efficiency of the lookup table using the spacing parameter. Both power of two, `'pow2'`, and even spacing, `'even'`, increase the computational speed of the lookup table and use less command read-only memory (ROM). However, specifying either of the spacing restrictions along with `errmax` usually requires more data points in the lookup table than does unrestricted spacing to achieve the same degree of accuracy. See Effects of Spacing on Speed, Error, and Memory Usage for more information.

Use `errmax` with Even Spacing

Create a lookup table that has evenly spaced breakpoints and a specified worst-case error.

```spacing = 'even'; [xdata,ydata,errworst] = fixpt_look1_func_approx(funcstr, ... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,[],spacing); length(xdata)```
```ans = 20 ```
`errworst`
```errworst = 9.2109e-04 ```

Plot the results:

```figure(3) fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ... xscale,ydt,yscale,rndmeth);```

Use `nptsmax` with Even Spacing

Create a lookup table that has evenly spaced breakpoints and minimizes the worst-case error for a specified maximum number of points.

```spacing = 'even'; [xdata, ydata, errworst] = fixpt_look1_func_approx(funcstr, ... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,[],nptsmax,spacing); length(xdata)```
```ans = 21 ```
`errworst`
```errworst = 8.3793e-04 ```

The result requires 21 evenly spaced points to achieve a maximum absolute error of `2^-10.2209`.

Plot the results:

```figure(4) fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ... xscale,ydt,yscale,rndmeth);```

Use `errmax` with Power of Two Spacing

Create a lookup table that has power of two spacing and a specified worst-case error.

```spacing = 'pow2'; [xdata, ydata, errworst] = ... fixpt_look1_func_approx(funcstr,xmin, ... xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,[],spacing); length(xdata)```
```ans = 33 ```

33 points are required to achieve the worst-case error specified by `errmax`. To verify that these points are evenly spaced, type

`widths = diff(xdata)`
```widths = 32×1 0.0078 0.0078 0.0078 0.0078 0.0078 0.0078 0.0078 0.0078 0.0078 0.0078 ⋮ ```

This generates a vector whose entries are the differences between the consecutive points in `xdata`. Every entry of `widths` is `2^-7`.

The maximum error is:

`errworst`
```errworst = 3.7209e-04 ```

This is less than the value of `errmax`.

Plot the results:

```figure(5) fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ... xscale,ydt,yscale,rndmeth);```

Use `nptsmax` with Power of Two Spacing

Create a lookup table that has power of two spacing and minimizes the worst-case error for a specified number of points.

```spacing = 'pow2'; [xdata, ydata, errworst] = ... fixpt_look1_func_approx(funcstr,xmin, ... xmax,xdt,xscale,ydt,yscale,rndmeth,[],nptsmax,spacing); length(xdata)```
```ans = 17 ```
`errworst`
```errworst = 0.0013 ```

The result requires 17 points to achieve a maximum absolute error of `2^-9.6267`.

Plot the results:

```figure(6) fixpt_look1_func_plot(xdata,ydata,funcstr,xmin,xmax,xdt, ... xscale,ydt,yscale,rndmeth);```

### Specifying Both errmax and nptsmax

If you include both the `errmax` and the `nptsmax` parameters, the function `fixpt_look1_func_approx` tries to find a lookup table with at most `nptsmax` data points, whose worst-case error is at most `errmax`. If it can find a lookup table meeting both conditions, it uses the following order of priority for spacing:

1. Power of two

2. Even

3. Unrestricted

If the function cannot find any lookup table satisfying both conditions, it ignores `nptsmax` and returns a lookup table with unrestricted spacing, whose worst-case error is at most `errmax`. In this case, the function behaves the same as if the `nptsmax` parameter were omitted.

The following examples illustrate the results of using different values for `nptsmax` when you enter

```[xdata ydata errworst] = fixpt_look1_func_approx(funcstr, ... xmin,xmax,xdt,xscale,ydt,yscale,rndmeth,errmax,nptsmax);```

The results for three different settings for `nptsmax` are as follows:

• `nptsmax = 33;` — The function creates the lookup table with 33 points having power of two spacing, as in Example 3.

• `nptsmax = 21;` — Because the `errmax` and `nptsmax` conditions cannot be met with power of two spacing, the function creates the lookup table with 20 points having even spacing, as in Example 5.

• `nptsmax = 16;` — Because the `errmax` and `nptsmax` conditions cannot be met with either power of two or even spacing, the function creates the lookup table with 16 points having unrestricted spacing, as in Example 1.

### Comparison of Example Results

The following table summarizes the results for the examples. When you specify `errmax`, even spacing requires more data points than unrestricted, and power of two spacing requires more points than even spacing.

ExampleOptionsSpacingWorst-Case ErrorNumber of Points in Table

1

`errmax=2^-10`

`'unrestricted'`

`2^-10`

16

2

`nptsmax=21`

`'unrestricted'`

`2^-10.933`

21

3

`errmax=2^-10`

`'even'`

`2^-10.0844`

20

4

`nptsmax=21`

`'even'`

`2^-10.2209`

21

5

`errmax=2^-10`

`'pow2'`

`2^-11.3921`

33

6

`nptsmax=21`

`'pow2'`

`2^-9.627`

17

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