# Arctan approximation example

Started by July 30, 2003
```>> On 30 Jul 2003 05:02:11 -0700, jomfrusti@image.dk
>> (=?ISO-8859-1?Q?Ren=E9?=) wrote:
>>
>> >Is it possible to get a fast fix point Arctan routine written
>> >in C using the principle with a lookup table or perhaps
>> >another principle?
>> >
>> >Regards,
>> >Ren&#2013265929;

An interesting way to compute arctan has appeared two days ago on
sci.math. But the OP wasn't after speed, and the formula has a whole
bunch of divisions. It is intended for single precision.

Martin

Robert Israel wrote in message <bg4dkq\$dll\$1@nntp.itservices.ubc.ca>:

> It suffices to be able to compute arctan(x) for x in [0,1].
> Try
> .0318159928972*y+.950551425796+3.86835495723/(y+8.05475522951+
>   39.4241153441/(y-2.08140771798-.277672591210/(y-8.27402153865+
>   95.3157060344/(y+10.5910515515))))
>
> where y = 2*x-1.  The maximum error is about 8*10^(-10).
```
```Martin Eisenberg wrote:

...
> An interesting way to compute arctan has appeared two days ago on
> sci.math. But the OP wasn't after speed, and the formula has a whole
> bunch of divisions. It is intended for single precision.

...
> Robert Israel wrote:
>
> > It suffices to be able to compute arctan(x) for x in [0,1].
> > Try
> > .0318159928972*y+.950551425796+3.86835495723/(y+8.05475522951+
> >   39.4241153441/(y-2.08140771798-.277672591210/(y-8.27402153865+
> >   95.3157060344/(y+10.5910515515))))
> >
> > where y = 2*x-1.  The maximum error is about 8*10^(-10).

That's a minmax rational polynomial approximation for
arccos(1/Sqrt(1+x^2)) = arctan(x) for x >= 0 with degree 5 in the
numerator and degree 4 in the denominator (just multiply the continued
fraction representation out). This uses only one division.

The accuracy is a bit high for 32-bit floating-point, a minmax
rational polynomial function with degree 4 / degree 2 is enough for
that:

0.05030176425872175099 (-6.9888366207752135 +
x)(3.14559995508649281e-7 + x)(2.84446368839622429 +
0.826399783297673451 x + x^2) /
(1 + 0.1471039133652469065841349249 x +  0.644464067689154755092299698
x^2)

The max absolute error in [0,1] is about 3.2e-7 which is still
slightly better than 32bit floating-point representation of arctan(x)
in [0,1].

Regards,
Andor
```
```Rene,

What is "fast" often depends on the low level instruction set of the
DSP/computer.

You could take a look at the Cordic algorithm for an atan2() or atan(). The
underlying idea is that you are going to rotate an (x,y) vector to align
with the positive x-axis through a series of known shifts that are
computationally efficient to apply. The final result will be the negative of
the sum of shifts applied to get as close to the x-axis as you wanted.
Generally, each additional phase bit output costs a sign check, 2 shifts and
There is a known scaling required (multiply) that can be at the end,
beginning, or could be distributed.

There is a discussion of it in an appendix in Frerking. Besides a typo, he
needs to extend his index range to one lower to be able to compute atan2 on
any angle in the [0,90] degree range he claims. You need to rotate the
vector into the  [0,90] degree range using x,y exchanges and sign
adjustments, accumulating the corresponding phase rotations in the process,
before you launch into his algorithm.

Dirk

Dirk A. Bell
DSP Consultant

"Ren&#2013265929;" <jomfrusti@image.dk> wrote in message
> Is it possible to get a fast fix point Arctan routine written in C
> using the principle with a lookup table or perhaps another principle?
>
> Regards,
> Ren&#2013265929;

```
```Is it possible to get a fast fix point Arctan routine written in C
using the principle with a lookup table or perhaps another principle?

Regards,
Ren&#2013265929;
```