# Ulps Plots Reveal Math Function Accuracy

"ULP" stands for "unit in the last place." An *ulps plot* samples a fundamental math function such as $\sin{x}$, or a more esoteric function like a Bessel function. The samples are compared with more accurate values obtained from a higher precision computation. A plot of the accuracy, measured in ulps, reveals valuable information about the underlying algorithms.

### Contents

#### fdlibm

`libm` is the library of elementary math functions that supports the C compiler. `fdlibm` is "freely distributable" source for `libm` developed and put into the public domain over 25 years ago by K. C. Ng and perhaps a few others at Sun Microsystems. I wrote about `fdlibm` in our newsletter in 2002, The Tetragamma Function and Numerical Craftsmanship.

Mathematically `fdlibm` shows immaculate craftsmanship. We still use it today for our elementary transcendental functions. And I suspect all other mathematical software projects do as well. If they don't, they should.

#### ulps plots

`ulps(x)` is the distance from `x` to the next larger floating point number. It's the same as `eps(x)`.

To assess the accuracy of a computed value

y = f(x)

compare it with the more accurate value obtained from the Symbolic Math Toolbox

```
Y = f(sym(x,'f'))
```

The `'f'` flag says to convert `x` to a `sym` exactly, without trying to guess that it is an inverse power of 10 or the `sqrt` of something. The relative error in `y`, measured in units in the last place, is then

u = (y - Y)/eps(abs(Y))

Since this is *relative* error, it is a stringent measure near the zeros of `f(x)`.

If `y` is the floating point number obtained by correctly rounding `Y` to double precision, then

-0.5 <= u <= 0.5

This is the best that can be hoped for. Compute the exact mathematical value of `f(x)` and make a single rounding error to obtain the final result. Half-ulp accuracy is difficult to obtain algorithmically, and too expensive in execution time. All of the functions in MATLAB that are derived from `fdlibm` have better than one-ulp accuracy.

-1.0 < u < 1.0

Each of the following plots involves 100,000 random arguments `x`, uniformly distributed in an appropriate interval.

#### sin

We see about 0.8 ulp accuracy from this sample. That's typical.

Argument reduction is the first step in computing $\sin{x}$. An integer multiple $n$ of $\pi/2$ is subtracted from the argument to bring it into the interval

$$ -\frac{\pi}{4} \le x - n \frac{\pi}{2} \le \frac{\pi}{4} $$

Then, depending upon whether $n$ is odd or even, a polynomial approximation of degree 13 to either $\sin$ or $\cos$ gives the nearly correctly rounded result for the reduced argument, and hence for the original argument. The ulps plot shows a slight degradation in accuracy at odd multiples of $\pi/4$, which are the extreme points for the polynomial approximations.

It is important to note that the accuracy is better than one ulp even near the end-points of the sample interval, $0$ and $2\pi$. This is where $\sin{x}$ approaches $0$ and the approximation must follow carefully so that the relative error remains bounded.

#### tan

Again, roughly 0.8 ulp accuracy.

Similar argument reduction results in similar behavior near odd multiples of $\pi/4$. In between these points, at $\pi/2$ and $3\pi/2$, $\tan{x}$ has a pole and the approximation must follow suit. The algorithm uses a reciprocal and the identity

$$ \tan x = -1/\tan{(x+\frac{\pi}{2})} $$

This comes close to dividing by zero as you approach a pole, but the resulting approximation remains better than one ulp.

#### atan

Good to within slightly more than 0.8 ulp. The underlying approximation is a piecewise polynomial with breakpoints at a few multiples of 1/16 that are evident in the plot and marked on the axis.

#### exp

About the same accuracy as the previous plots.

The argument reduction involves the key value

$$ r = \ln{2} \approx 0.6931 $$

and the identity

$$ \exp{x} = 2^n \exp{(x-n r)} $$

The resulting ulps plot shows the extremes of the error at odd multiples of $r/2$.

#### Lambert W

Now for two functions that are not in `fdlibm`. If you follow this blog, you might have noticed that I am a big fan of the Lambert W function. I blogged about it a couple of years ago, The Lambert W Function. The Wikipedia article is excellent, Lambert W Function. And, you cn just Google "lambert w function" for many more interesting links.

The Lambert W function is not available in MATLAB itself. The Symbolic Math Toolbox has an @double method that accesses the symbolic code for type double arguments. Or, my blog post mentioned above has some simple (but elegant, if I do say so myself) code.

Neither code is one-ulp accurate. The primary branch of the function has a zero at the origin. As we get near that zero, the relative error measured in *ulps* is unbounded. The absolute accuracy is OK, but the relative accuracy is not. In fact, you might see a billion ulps error. That's a *gigaulp*, or *gulp* for short.

As `lambertw(x)` crosses a power of two, the unit in the last place, `eps(lambertw(x))`, jumps by a factor of two. Three of these points are marked by ticks on the x-axis in the ulps plot.

for a = [1/8 1/4 1/2] z = fzero(@(x) lambertw(x)-a,.5) end

z = 0.1416 z = 0.3210 z = 0.8244

#### Bessel functions

Our codes for Bessel functions have fairly good, although not one-ulp, absolute accuracy. But they do not have high relative accuracy near the zeros. Here are the first five zeros, and an ulps plot covering the first two, for $J_0(x)$, the zero-th order Bessel function of the first kind.

for a = (1:5)*pi z = fzero(@(x) besselj(0,x), a) end

z = 2.4048 z = 5.5201 z = 8.6537 z = 11.7915 z = 14.9309

#### erfinv

Here is the inverse of the error function. It looks very interesting, but I haven't investigated it yet.

#### Code

I have recently updated Cleve's Laboratory in the MATLAB Central file exchange. The latest version includes `ulpsapp.m`, which generates the ulps plots in this blog.

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