1 Introduction

Ratmath

Ratmath is a collection of utilities for working with rational numbers, approximations, and intervals in Common Lisp. Functions are included to calculate the closest single approximant given a maximum numerator/denominator (rat) or the smallest bounded rational interval (infsup). The infsup function can find the floating point constant pi with a denominator limit of, e.g. 256, to be within the interval 688/219 to 355/113, and the with-interval-math function can use this interval instead of exact values to carry out further interval arithmetic to yield interval results.

A Better (rational NUM) and (rationalize NUM) Function

In its simplest use case, it is an extension of Common Lisp's rationlize that can take keyword arguments :limd and :limn representing the maximum numerator/denominator:

CL-USER> (ratmath:rat pi :limd 255)
355/113

Calculating Rational Intervals

Interval arithmetic is a powerful and convenient way of doing calculations in engineering but is seldom accomodated in the base functionality of computer languages. Wikipedia

There are other libraries for interval math using floats, but this library instead uses rationals, a numerical data type not commonly appreciated as a base numerical type amongst the other languages. With Common Lisp's built-in infinite size integers, there is no limit to the precision of rational numbers which simplifies a lot of the inherent complexity of writing floating point interval libraries which have to worry about round off on every operation.

The disadvantage of using rationals rather than floats is that some calculations can end up generating extremely large numerators and denominators quickly. Conveniently, these large numbers can be rationally approximated into intervals of smaller numbers at any time in the middle or at the end of a calculation. As as example, the number representing e * pi using high precision rational constants on e and pi yields a rational result which has 72-bit numerator (which would cause overflow in anything other than Common Lisp). This bignum rational could be reduced to the following 16-bit rational interval with ratmath:

CL-USER> (* (rational pi) (rational (exp 1.0)))
2520484590068807905375/295147905179352825856
CL-USER> (ratmath:infsup * :limd 65535 :limn 65535)
(23749/2781 . 50077/5864)
32894771015794204823/36893488147419103232

You can usually ignore the second value returned from infsup. It represents an alpha, or a number between 0 and 1 representing the location of the original number inside the returned interval. The alpha calculation returns a rational that may be of higher order than even the original rational. For most cases, it would be recommended to coerce this rational into a float or use the rat function to reduce its precision.

Approximations From Rational Intervals

There are routines to work in reverse as well. That is, to go from an interval to the simplest rational number within that interval. The Common Lisp builtin rational differs only from its cousin rationalize by calculating a rational approximation from an interval about the original floating point number equal to the epsilon of the floating point representation. Here's how you do something similar, but with an explicitly bounded interval with ratmath:

CL-USER> (ratmath:rat '(110/333 . 221/333))
1/2

Trivial Interval Arithmetic

Intervals can be useful outside of rational approximation and used in the general arithmetic of non-exact measurements and numbers. For instance, on a calculation returning voltage measurements of an analog input, one usually needs to plug in numbers representing the voltage reference and any input resistor divider ratios. Using exact numbers for these is suboptimal as the truth is that these specifications are fuzzy and have non-zero tolerances. Interval math can be used instead to output an interval representing where the analog input true value might actually be between.

An interval at its core is simply a cons of two numbers. e.g. '(99/20 . 101/20). To facilitate creation, it is also possible to create the same interval with alternative notation:

CL-USER> (ratmath:infsup 5 :abstol 5/100)             ;; 5, +/-.05
(99/20 . 101/20)
CL-USER> (ratmath:infsup 99/20 :abstol '(0 . 1/10))   ;; 99/20, -0/+.1
(99/20 . 101/20)
CL-USER> (ratmath:infsup 101/20 :abstol '(1/10 . 0))  ;; 101/20, -.1/+0
(99/20 . 101/20)
CL-USER> (ratmath:infsup 5 :tol 1/100)                ;; 5, +/-1%
(99/20 . 101/20)
CL-USER> (ratmath:infsup 5 :tol '(0 . 1/100))         ;; 5, -0%/+1%
(5 . 101/20)

The gist is that :tol takes either an interval or an exact representing the percent tolerance and :abstol uses absolute values +/- rather than percentages. Refer to the documentation of the infsup function for further details.

with-interval-math Blocks

From within a ratmath:with-interval-math block, some of the basic arithmetic operations are replaced with interval aware variants. For instance:

CL-USER> (ratmath:with-interval-math
           (* '(1 . 2) 5 (ratmath:infsup 3.3 :tol .01)))
(16.335 . 33.329998)

Notice that the usual rules of floating point contagion apply. Since the infsup was given arguments in floats, the result is a float. We can turn this float back into rationals or exacts with:

CL-USER> (ratmath:infsup '(16.335 . 33.329998) :limd 100)
(49/3 . 3333/100)
CL-USER> (ratmath:rat *)
17

The rat function converts the interval '(16.335 . 33.329998) into the simplest number within. For this interval, it is 17. Simplest may not be the best term here, the full truth is its first number on the infinite Stern-Brocot fraction tree that is recursively traversed in the process of analyzing the best convergents and semi-convergents of a given continued fraction. What I mean to illustrate here is that, under this definition, this number will rarely be the midpoint of the interval, and also that mathematically correct rational approximation is not as trivial as it may seem at first glance. There is some really interesting algorithms and concepts (to math geeks, anyway) this library is abstracting that many programs might gloss over. In fact, in the beginning of 2020, the Linux kernel doesn't even have a fully correct implementation which it uses (amongst other things) to calculate correct PLL dividers for CPU clock rates. One that was close was published and posted, but abandoned for non-obvious reasons probably having to do with Linux kernel politics and style (or perhaps some microbenchmark suffered).

Syntactic Sugar

Within a with-interval-math block, the full power of the Common Lisp macro system is utilized to realize a cute feature regarding the parsing of numeric literals. This can be illustrated most succinctly in the following single-line piece of code:

CL-USER> (ratmath:with-interval-math (* 2 ~5))
(9 . 11)

The with-interval-math turns the numeric literal ~5 automatically into the interval '(4.5 . 5.5) which is then multiplied by exactly 2 to return the interval '(9 . 11). The rules of conversion for ~ literals should not be surprising. It should be noted that the interval will always result in rationals even when normal numeric literal parsing rules would have it be a float. The other key to remember regarding the tilde literal is they are parsed according to the rules of significant figures. e.g. ~5 is '(4.5 . 5.5) but ~5.00 is '(4.995 . 5.005) Exponential notation is honored as one might predict, ~1e3 is '(500 . 1500), always as rationals.

These numeric literals can also be parsed outside of a with-interval-math block by passing a string to parse-interval. In parse-interval the tilde character ornamentation is not necessary.

The tilde '~' character is also a function, but it behaves differently depending on the context. Inside a with-interval-math block it turns an exact into an interval and behaves the same way as the infsup function. Outside of this block, it turns intervals into exacts according to some &key argument controls. Refer to the documentation of the ~ function for full details, but here are a few examples:

Random numbers for (e.g.) Monte Carlo analysis using a uniform probability distribution within the interval:

CL-USER> (ratmath:~ '(1 . 2) :random t)  ;; this will always be a float
1.3231988

The :discrete boolean means to only return upper or lower limit. Assumes a 50/50 chance to return either, unless the :alpha option is supplied.

CL-USER> (ratmath:~ '(1 . 2) :random t :discrete t) ;; random select upper/lower
1
CL-USER> (ratmath:~ '(1 . 2) :random t :discrete t :alpha 3/4) ;; 75% chance of upper
2

Several types of calculations are only interested in worst-case/best-case which will be either the upper or lower limit of the interval, e.g.:

CL-USER> (ratmath:~ '(1/14 . 3/7) :upper t)
3/7
CL-USER> (ratmath:~ '(1/14 . 3/7) :lower t)
1/14

Other Tricks and Functions (useful in the REPL)

A common application of these algorithms are to print out a progression of rational approximations of a given number as the size of the numerator and denominator terms is allowed to increase. The following prints out the best convergents and semi-convergents and their error in PPM or PPB (parts-per-million/billion) on the way to system's pi constant:

CL-USER> (ratmath:fractions pi :order 10)
2'd3/1'd1 45070.341 PPM
5'd22/3'd7 -402.499 PPM
8'd179/6'd57 395.270 PPM
9'd355/7'd113 -8.491D+1 PPB
16'd52163/15'd16604 8.474D+1 PPB
19'd312689/17'd99532 -9.277D-3 PPB
20'd833719/19'd265381 2.774D-3 PPB
21'd1146408/19'd364913 -5.128D-4 PPB
23'd5419351/21'd1725033 -7.088D-6 PPB
27'd80143857/25'd25510582 1.453D-7 PPB
28'd245850922/27'd78256779 -1.410D-8 PPB
30'd817696623/28'd260280919 1.531D-9 PPB
32'd2698940791/30'd859099536 1.074D-10 PPB
34'd9978066541/32'd3176117225 -9.265D-12 PPB
35'd32633140414/34'd10387451211 3.833D-13 PPB
38'd238410049439/37'd75888275702 -2.047D-14 PPB
40'd747863288731/38'd238052278317 -2.855D-15 PPB
41'd1257316528023/39'd400216280932 4.860D-16 PPB
43'd5777129400823/41'd1838917402045 5.350D-17 PPB
44'd10296942273623/42'd3277618523158 6.901D-19 PPB
50'd884279719003555/49'd281474976710656 0.000D-1 PPB

The :order 10 argument says to only print a best convergent/semi-convergent at most once per decimal order of magnitude. This avoids flooding the output with semi-convergents. If not specified, the default order is 2, which prints the best approximation for every bit size width in the numerator and denominator.

The ##'d syntax at the beginning of each number is Verilog syntax to describe the bit width of the constant. I wrote these routines originally to facilitate writing Verilog in FPGAs for fractional clock dividers and PWM controllers. The Verilog syntax for size is more convenient than manually counting digits to figure how big of a register I have to instantiate.

It is worth noting that over half of the above convergents to pi will vary from implementation to implementation. The pi constant is defined as a floating point number in Common Lisp and the default behavior of ratmath is to convert floats into rationals using the rational function rather than rationalize, which takes into consideration the host machines limits of floating point precision. (i.e. epsilon) If we instead prefer to stop the approximation at the end of the floating point constant's precision, we can use:

CL-USER> (ratmath:fractions (rationalize pi) :order 10)
2'd3/1'd1 45070.341 PPM
5'd22/3'd7 -402.499 PPM
8'd179/6'd57 395.270 PPM
9'd355/7'd113 -8.491D+1 PPB
16'd52163/15'd16604 8.474D+1 PPB
19'd312689/17'd99532 -9.277D-3 PPB
20'd833719/19'd265381 2.774D-3 PPB
21'd1146408/19'd364913 -5.127D-4 PPB
23'd5419351/21'd1725033 -7.074D-6 PPB
27'd80143857/25'd25510582 1.594D-7 PPB
28'd245850922/27'd78256779 0.000D-1 PPB

Pipe functions

A few public ratmath functions utilize a pattern unique to Common Lisp and return whats described in chapter 25, "Streams and Delayed Evaluation", in the book "Lisp, 3rd Edition" by Winston/Horn. The chapter describes a stream data structure (that I have renamed to "pipe" to avoid name clash confusion), that utilizes Lisp's lexical closures to create something akin to an infinite list. I have modeled much of the internal implementation of this library around the simple concepts described therein. Utilities to manipulate, map, filter, and transform these data structures are included in pipe.lisp and are made public in the ratmath package, though not specifically relating to the ratmath primary interfaces. These routines are superficially documented in pipe.lisp, though depending on familiarity with the above book or Lisp's lexical enclosures and functional programming paradigms, might either seem extremely trivial or extremely obtuse. This library was coded partly as an athletic challenge/experiment in avoiding all usage of iteration/loops/assignments in favor of recursion and functional programming. This (arbitrary) implementation constraint was simultaneously pleasing and frustrating and the pipe construct was an invaluable aid in realizing this.

The most likely useful pipe function is ratmath:rat-pipe, which takes as argument an input number and optional exponent base number which defaults to 2. This optional arg n filters out semi-convergents from the same power-of-n and only outputs the best approximation for each power-of-n increase in denominator/numerator.

CL-USER> (ratmath:rat-pipe pi 10)
(3 #<COMPILED-LEXICAL-CLOSURE (:INTERNAL RATMATH::PIPE-TRANSFORM) #x15B1D73E>)

Seeing the REPL output for this function shows that function returns a pipe. In the above, the number 3 is the first element and the second element is actually not an element at all, but a lexical closure that upon funcalling, will calculate and expose another rational + lexical closure, ad infinitum.

Since we know that this pipe is not infinite and has an end, we can safely convert the pipe into an ordinary list with (ratmath:pipe-to-list *) Note that the real pi, as an irrational number, has an infinite continued fraction and therefore an infinite pipe of rational approximations, but we are doing a rational approximation on Common Lisp's floating point constant which has finite precision and, therefore, a finite continued fraction.

CL-USER> (ratmath:pipe-to-list *)
(3 22/7 179/57 355/113 52163/16604 312689/99532 833719/265381 1146408/364913 
5419351/1725033 80143857/25510582 245850922/78256779 817696623/260280919
2698940791/859099536 9978066541/3176117225 32633140414/10387451211
238410049439/75888275702 747863288731/238052278317 1257316528023/400216280932
5777129400823/1838917402045 10296942273623/3277618523158
884279719003555/281474976710656)

The rat-pipe does allow for irrational number approximation, you simply have to create an infinite pipe of continued fraction terms to send as the first argument instead of an atomic real number. An example generator is in fractions.lisp in the function ratmath:napiers-constant-generator which returns the infinite terms of the continued fraction expansion of Euler's number e

CL-USER> (ratmath:rat-pipe (ratmath:napiers-constant-generator))
(3 #<COMPILED-LEXICAL-CLOSURE (:INTERNAL RATMATH::PIPE-TRANSFORM) #x15C8111E>)
CL-USER> (ratmath::pipe-sink-until 
          (lambda (x) (> (integer-length (denominator x)) 4096)) 
          *)
(41720850604724578859 .... ;; cut to not flood output with a 4096bit rational

Generating 8kbit integer rationals like this makes a good benchmark of Common Lisp and how well it handles its bignums. Letting your rationals grow in size to bignum territory likely will hurt performance significantly. Indeed, I would predict large gains in speed sprinking declare forms throughout the implementation of ratmath signifying intent to only ever use fixnums inputs and outputs.

Another useful pipe function generates lists of ordered rational fractions called Farey sequences. Wikipedia. The function ratmath:farey-pipe is used mostly for the implementation of infsup, but it is possible to use on its own, e.g. to list out a farey sequence of order 7 (no fraction denominator greater than 7):

CL-USER> (ratmath:pipe-to-list (ratmath:farey-pipe 7))
(1/7 1/6 1/5 1/4 2/7 1/3 2/5 3/7 1/2 4/7 3/5 2/3 5/7 3/4 4/5 5/6 6/7 1 7/6 
6/5 5/4 4/3 7/5 3/2 5/3 7/4 2 7/3 5/2 3 7/2 4 5 6 7)

Todo

• My attempt at allowing compile-time collapsing and constant folding of interval/ratmath needs further testing/review. Perhaps utilize numcl's constantfold package?
• Test routines. Esp. for the interval ops.
• Maybe further distill and refine the public interfaces for working with Stern-Brocot fraction trees and Farey sequences.
• Incorporate any other useful algorithms people are thinking of when they hear of or google for a library named ratmath.
• More interval aware math functions in with-interval-math blocks
• To assist in calculating operations where the same interval appears more than once, create a function that loops on the block with the permutations of min/max on each interval and collects the hull of results. (This can avoid intervals unnecessarily widening)
• Maybe allow for gaussian or truncated gaussian probability distributions in the random conversion of intervals back to exacts.

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• ratmath

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• Lisp

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• ratmath

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• Public Interface
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