Oct

Introduction

Oct is a portable Lisp implementation of quad-double arithmetic. This gives about 65 digits of precision. Quad-double arithmetic uses four double-float numbers to represent an extended precision number.

The implementation is modeled on the quad-double package by Yozo Hida. This package is in C++, but we have translated parts of it and extended it to use Lisp. The intent is to provide all of the CL arithmetic functions with a quad-double implementation.

Further information will be provided at a later date. This is currently a work in progress, but the current code has the basic functionality implemented and includes all of the special functions specified by CL. There are, undoubtedly, many bugs.

Supported Lisps

This package has been tested with CMUCL, SBCL, Clisp, and Allegro. All tests pass. Note that development is primarily done using CMUCL.

What's New

2011-03-04

• Oct now has a wiki and ticket system. You can browse the sources there (or using gitweb).

2011-02-09

• For Lisp's that support signed zeros for floating point (like cmucl and sbcl), Oct now correctly handles signed quad-double zeroes.
• The CVS repository has been moved to Git. The CVS repository is still available but will not be updated. Use Git to obtain the latest versions. (See Oct gitweb for more links.)

2007-11-28

• Oct now passes all of the tests on Allegro/x86. The main change was not to compile with safety 0. This messes up Allegro's tracking of the precision.
• To help implementations (like Allegro or Clisp) that use arrays for storing %quad-double numbers, many functions now support an optional third arg which is where the result should be written. This is like how bit-and works. This should work correctly with both an array or (complex double-double-float) implementation, but perhaps some care is needed to make sure it works for both.
• CMUCL can use either an array or a (complex double-double-float) to store a %quad-double. However, simple timing measurements indicate that arrays are slightly slower to much slower than using (complex double-double-float).

2007-10-26

• A Pade approximation for exp has been added. This is currently not used, but tests show that it is 25% faster than the current method, but it loses about 3 bits of precision. This was contributed by Richard Fateman.

2007-10-15

• The OCT package has been renamed. The package is now NET.COMMON-LISP.OCT, with a nickname of OCT.
• QD is no longer a nickname for OCT
• QUAD-DOUBLE-INTERNAL has been renamed to OCT-INTERNAL, with a nickname of OCTI. The old nickname QDI has been removed
• Oct has a more accurate argument reduction for the trig functions. This allows Oct to compute cos(2^120) correctly (which is about -0.925879).

2007-10-10

• Oct now has print/read consistency, thanks to Richard Fateman, who provided a very nice method to convert bignums to quad-double.

Building Oct

1. Obtain a copy of Oct. Currently that means from the git repository http://common-lisp.net/r/projects/oct/oct.git, but also see the Oct gitweb for more links. The CVS repository is still available, but it is no longer used.
2. Get a copy of MK:DEFSYS from clocc, if you don't already have a copy. Or you can also use ASDF. Note that this might be out-of-date compared to the mk:defsys version.
3. (mk:oos "oct" :compile) will compile up everything. Or use (asdf:oos 'asdf:load-op :oct)
4. (mk:oos "oct-test" :compile) will compile up some simple tests. (qdi::all-tests) will run a series of tests. The accuracy of the results should be on the order of about 200 bits or more. There are a few cases where accuracy is much less than this, but that is due to the algorithm. If you have a copy of RT, you should run (rt:do-tests) to run the tests. All tests should pass.

Using Oct

Everything in Oct resides in the NET.COMMON-LISP.OCT package, with a nickname of OCT. The basic arithmetic operations of CL are shadowed in this package so you can use natural Lisp expressions to operate on quad-doubles. Special functions are included.

There are two types added:

QD-REAL

A quad-double type. This has about 65 digits of precision, or about 212 bits.

QD-COMPLEX

A complex type consisting of two QD-REAL values.

The reader is also modified to make it easier to enter quad-double values. #q is used to enter both QD-REAL and QD-COMPLEX types. For example #q1.25q5 is the QD-REAL with the value 125000. The exponent marker is q. To enter a QD-COMPLEX value, use #q(r i) where r and i are the real and imaginary parts. The parts will be coerced to QD-REAL type if necessary.

Examples

Here are a few examples:

  QD> (/ (sqrt #q3) 2)
  #q0.866025403784438646763723170752936183471402626905190314027903489696q0
  QD> (sin (/ +pi+ 3))
  #q0.86602540378443864676372317075293618347140262690519031402790348972q0
  QD> (sqrt #q-1)
  #q(#q0.0q0 #q1.0q0)
  QD> (coerce 7 'qd-complex)
  #q(#q7.0q0 #q0.0q0)
  QD> (integer-decode-float +pi+)
  165424160919322423196824703508232170249081435635340508251270944637
  -215
  1
  QD> +pi+
  #q3.1415926535897932384626433832795028841971693993751058209749445923q0
  QD> (* 4 (atan #q1))
  #q3.1415926535897932384626433832795028841971693993751058209749445923q0
 

Note that +pi+ is the QD-REAL value for pi.

Performance

Here are some simple benchmarks on the performance of this quad-double implementation. These benchmarks were run using CMUCL on a 1.42 GHz PPC. The columns are times relative to a double-float. The %quad-double represents the time using the internal implementaiton, without the overhead of CLOS. The QD-REAL column shows the effect of CLOS dispatch.

Here are some timing results using CMUCL on a 1.5 GHz UltraSparc IIIi

Here are some timing results using CMUCL with SSE2 support on a 3.06 GHz Core i3

Hida's QD package has a few timing tests. The lisp equivalent was written and here are the timing results. Note that the Lisp equivalent tried to be exactly the same as the QD reference, but no guarantees on that.

The second and third columns are microsec per operation. The last column is the relative time of Oct vs QD. All of these were run on a 1.5 GHz Ultrasparc III. Sun Studio 11 was used to compile the C code. CMUCL 2007-10 was used for the Lisp code.

It's surprising that Oct does as well as it does. To be fair, the times for Oct include the cost of CLOS dispatch since QD uses templates and classes in the tests. Except for add and mul, QD and Oct are within a few percent. The sin test is a bit slower in Oct. I don't know why, but the test did include the accurate argument reduction. The log test is quite a bit faster for Oct. This is probably due to using a different algorithm. QD uses a Newton iteration to compute the log. Oct uses Halley's iteration.

Mailing Lists

• Oct-devel
  for developers
• Oct-cvs
  CVS log feed.
• Oct-announce
  for announcements.

Download

Currently, there are no official releases. However, you may find occasional snapshots of the sources here.

Known Issues

• There is a known issue with Oct on x86 machines. Some Lisp's set up the FPU to use a rounding mode of 64-bits. That is, it sets the rounding assuming all arithmetic is using double extended floating-point registers (80-bits). However, since Oct is using double precision floats, this will cause bad things to happen. Rounding will be done incorrectly so incorrect result may be returned. The rounding mode must be set to 53 bits (double precision floats).

  It is known that CMUCL and SBCL have the desired rounding by default. Allegro works fine on ppc and x86 now.

• When the Oct system is loaded, it always adds the #q reader macro. This may interfere with other systems that want to use #q.

CVS

You can browse our CVS repository or download the development tree via anonymous cvs, as described here. Note that CVS is no longer up-to-date

Git

You can browse our git repository or clone it if desired.