= frac =

Frac is a C library for computing with http://crypto.stanford.edu/pbc/notes/contfrac/[continued fractions].

== Installation ==

The GMP library is required. Type:
 
  $ make

to compile.

Then try "pi 2000" to generate 2000 digits of pi using a continued
fraction. It takes longer than I had hoped, but it's faster than

  $ echo "scale = 2000; 4*a(1)" | bc -l

Also try "hakmem 1000" to compute 1000 digits of the example continued fraction
constant from HAKMEM, which compares favourably with:

  $ echo "scale=1000
          pi=4*a(1)
          x=2*sqrt(5)
          (sqrt(3/pi^2+e(1)))/((e(x)-1)/(e(x)+1)-s(69))" | bc -l

Unlike frac, bc can only output the final answer in one go after all
calculations are complete.

There's no API documentation; see the source.

== Continued fractions ==

The binary representation of 1/3 is infinite. Floating-point numbers, our
standard tool for working with reals, cannot even handle numbers a young child
could understand.

In contrast, every rational can be uniquely and exactly represented by a finite
continued fraction (provided we forbid those with last term 1). Continued
fractions are immune from oddities such as 1/3 = 0.333... and 1 = 0.999...,
living in a realm free from the tyranny of binary, or any other base.

Every irrational number can also be uniquely expressed as an infinite continued
fraction. Many frequently encountered algebraic and transcendental numbers have
easily computable continued fraction expansions which can be truncated to yield
approximations with built-in error bounds, unlike floating-point
approximations, which must be accompanied at all times with their error bounds.

Furthermore, suppose we use floating-point to compute some answer to 100
decimal places. What if we now want 200 decimal places? Even if we possess the
first 100 digits and carefully preserve the program state, we must start from
scratch and redo all our arithmetic with 200 digit precision. If we had used
continued fractions, we could arbitrarily increase the precision of our answer
without redoing any work, and without any tedious error analysis.

But we must pay a price for these advantages. We have a representation from
which we can instantly infer error bounds. A representation for which we may
arbitrarily increase precision without repeating any calculations. A
representation which in some sense is the closest we can hope to get to an
exact number using rationals. Is it any wonder therefore that binary operations
on continued fractions, whose output must also uphold these high standards, are
clumsy and convoluted?

These drawbacks are almost certainly why these fascinating creatures remain
obscure. Yet surely there are some problems that scream for continued
fractions. How about a screen saver that computes more and more digits of pi,
each time picking up where it left off?  Or imagine a real-time application
where all continued fractions simply output the last computed convergent on a
timer interrupt. Under heavy system load, approximations are coarser but the
show goes on.

=== Disclaimer ===

Lest my salesmanship backfire, let me temper my anti-floating-point rhetoric.
Increasing the precision of floating-point operations without redoing work is
in fact possible for many common cases. For example, Newton's method is
self-correcting, meaning we may use low precision at first, and increase it for
future iterations. Even pi enjoys such methods: there exists a formula
revealing any hexadecimal digit of pi without computing any previous digits,
though it requires about the same amount of work.

Moreover, several floating-point algorithms converge quadratically or faster,
thus good implementations will asymptotically outperform continued fractions
as these often converge linearly.

Nonetheless, for lower accuracies, the smaller overhead may give continued
fractions the edge in certain problems such as finding the square root of 2.
Additionally, precision decisions are automatic, for example, one simply needs
enough bits to hold the last computed convergents. Built-in error analysis
simplifies and hence accelerates development.

== Bugs and other issues ==

I intend to devote time to projects that operate on real numbers rather than
study real numbers for its own sake. But perhaps then I'll find a perfect
application for continued fractions, which will push me to fix deficiencies in
this code:

- I want to replace the condition variable with a semaphore to silence Helgrind
  (whose documentation states that condition variables almost unconditionally
  and invariably raise false alarms).

- Computing the continued fraction expansion of pi from the Chudnovsky
  brothers' Ramanujan formula would be much faster. More constants could
  benefit from using efficiently computable sequences of narrower intervals
  for their continued fraction expansions.

- The API could use cosmetic surgery.

- The Makefile is fragile and annoying to maintain and use.

- The testing infrastructure needs upgrading.

- Much more, e.g. see TODOs sprinkled throughout the code.

== Design ==

I was inspired by Doug McIlroy's paper,
``http://swtch.com/~rsc/thread/squint.pdf['Squinting at Power Series']'', a
captivating introduction to the communicating sequential processes (CSP) model
of concurrent programming: some processes generate coefficients of a power
series, while other processes read coefficients on input channels and write
their output on another channel as soon as possible. Each output coefficient is
written as soon as it is known, before reading more input. Like Unix pipes,
these processes are connected together to compute sums, products, and so on.

For example, a process that repeatedly spits out 1 on its output channel
represents the power series 1 + 'x' + 'x'^2^ + ... This is similar to
running

 $ yes 1

This process could be fed into another process that produces
the derivative of its input. A Unix equivalent might be:

 $ yes 1 | awk \'{ print (NR-1)*$0 }\'

Continued fractions likewise naturally fit the CSP model, as Gosper suggets.
The simplest are processes that output terms according to some
pattern while maintaining barely any internal state. Other processes can then
consume these terms to compute convergents, decimal expansions, sums, products,
square roots and so on.

Thus we spawn at least one thread per continued fraction or operation. These
threads communicate via crude demand channels, sending terms of continued
fractions back and forth.

For example, consider the
code for generating 'e' = 2.71828...

..............................................................................
// e = [2; 1, 2, 1, 1, 4, 1, ...]
static void *e_expansion(cf_t cf) {
  int even = 2;
  cf_put_int(cf, even);

  while(cf_wait(cf)) {
    cf_put_int(cf, 1);
    cf_put_int(cf, even);
    even += 2;
    cf_put_int(cf, 1);
  }

  return NULL;
}
..............................................................................

The function +cf_put+ places a term on the output channel, and +cf_wait+ is our
implementation of demand channels, a sort of cooperative multitasking. Without
it, not only would we have to destroy and clean up after the thread ourselves,
but more seriously, the thread might consume vast amounts of resources
computing unwanted terms. The +cf_wait+ functions instructs the thread to stay
idle. Our threads call this function often, and if it returns zero, our threads
clean themselves up and exit.

Threads are the future, if not already the present. Multicore systems are
already commonplace, and as time passes, the number of cores per system will
steadily march upward. Happily, this suits continued fractions.