.TH RIES 1L "Manual version: 2014 Jul 29" \" -*- nroff -*-
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.SH NAME
ries \- find algebraic equations, given their solution
.SH SYNOPSIS
\fBries\fR
[\fB-l\fIn\fR] [\fB-i[e]\fR] [\fB-s\fR] [\fB-x\fR] [\fB-F\fIn\fR]
[\fB-S\fIsss\fR] [\fB-N\fIsss\fR] [\fB-O\fIsss\fR]
[\fB-D\fIxxx\fR] [\fB-p\fIfilename\fR]
[\fB--extended-options\fR [...]] \fIvalue\fR
.B ries --find-expression \fR[\fIexpression\fR [...]]
.B ries --eval-expression \fR[\fIexpression\fR [...]]
.SH DESCRIPTION
Given a number, \fBries\fR searches for algebraic equations in one
variable that have a solution (root) near that number. It avoids
trivial or reducible solutions like ``\fIx\fR/\fIx\fR = 1''. If
\fIvalue\fR is an integer, \fBries\fR can find an exact solution
expressed in terms of single-digit integers.
For example, if you supply the value 2.5063, the first part of
\fBries\fR's output will resemble the following:
.fam C
.ps -2
$ ries 2.5063
Your target value: T = 2.5063 www.mrob.com/ries
2 x = 5 for x = T - 0.0063 {49}
8 x = e^3 for x = T + 0.00439212 {66}
x^2 = 2 pi for x = T + 0.000328275 {55}
x^x = 1+9 for x = T - 0.000115854 {69}
x^2+e = 9 for x = T + 3.56063e-05 {63}
ln(6) x = sqrt(pi)+e for x = T + 2.73037e-05 {93}
x/4+1 = 4,/7 for x = T + 6.24679e-06 {91}
sinpi(ln(x))^2 = 1/(5 pi) for x = T + 2.75665e-06 {92}
.ps +2
.fam T
The output gives progressively ``more complex'' solutions (as described
below) that come progressively closer to matching your number. There
are four columns: equations in symbolic form (two columns of
expressions with '=' in the middle), solution of equation (value of
\fIx\fR expressed as \fIT\fR plus a small error term), and total complexity
score (described below).
Each match is checked by solving for \fIx\fR using the Newton-Raphson
method, and the closeness of the match is judged by the difference
between the root (the value of \fIx\fR for which the two sides are
equal) and your target value \fIT\fR.
Options allow complete control over what symbols, constants
and functions are used in solutions, or to limit solutions to
integer, rational, constuctible, or algebraic values.
.SH OPTIONS
Options must be separate: `\FCries -l1 -i -Ox 27\FT',
not `\FCries -l1iOx 27\FT'.
.IP \fB-p\fIname\fR
Profile (or Parameters): Load one or more options from file
\fIname\fR. \fB-p\fIname\fR is equivalent to \fB--include\fR
\fIname\fR, which is described in the EXTENDED OPTIONS section below.
\fB-p\fR alone (with no \fIname\fR) has special meanings, also
described under \fB--include\fR.
.IP \fB-l\fIn\fR
Level: Specifies the level of the search (default 2).
With each increment of \fB-l\fR, \fBries\fR
will search about 10 times as many equations, use
3.5 times as much memory and take at least 4 times as long.
Use higher levels to add more factors of 10.
The level can be fractional or negative.
Here are typical figures, measured on a Core i7 at 3.2 GHz
(using only one thread) invoked by the command
\fBries -l\fR\fIn\fR\fB 2.5063141592653589\fR
for different values of searchlevel \fIn\fR:
.fam C
.ps -2
memory equations tested digits run time
-l0 1.2M 89,400,000 6+ 0.025 sec
-l1 4.0M 932,000,000 7+ 0.08 sec
-l2 14 M 11,400,000,000 8+ 0.33 sec
-l3 45 M 134,000,000,000 9+ 1.8 sec
-l4 158M 1,600,000,000,000 11+ 8.8 sec
-l5 490M 15,000,000,000,000 12+ 37.1 sec
-l6 1.7G 184,000,000,000,000 13+ 190 sec
.ps +2
.fam T
(these times are a little quicker than a 2.33-GHz Core 2 Duo; on a
733-MHz Pentium 3, the times were about 5 times longer. If compiled
for an environment with 32-bit pointers, memory usage figures are about 20% lower.
\fBries\fR also works on much older and smaller systems, and can test
billions of equations in less than a minute on 1990's hardware)
Use a fractional argument (like \fB-l5.5\fR) for more precise control
of how much memory \fBries\fR will use before stopping its search.
When free memory is exhausted; performance will degrade significantly
and \fBries\fR might exit, depending on your operating system. Under
Linux and Mac OS, \fBries\fR keeps running but the system slows to a
crawl.
If you don't know what your OS will do, be careful before running
\fBries\fR with higher levels. In extreme cases your computer's
response might slow down so much that you are unable to save your work in
other applications.
The memory limits are not reached nearly as quickly when the symbolset
is greatly limited with \fB-S\fR, \fB-O\fR and \fB-N\fR or when
\fB-i\fR is specified. \fB-i\fR in particular should allow about two
more levels in any given amount of memory. Large arguments tend to
lengthen runtime: for example, \FCries -l4 1058073667\FT takes about
three times as long as \FCries -l4 1.058073667\FT.
.SH Options to Select Symbols
Several options are used to choose which symbols (constants,
operations, and functions) \fBries\fR is allowed to use when searching
for equations.
.IP \fB-N\fIsss\fR
Never use these: \fB-N\fR followed by one or more characters specifies
symbols (constants and operators) that \fBries\fR
should not use in its equations. The symbols are as follows:
.RS 0.7i
.IP 1-9
The integers 1 through 9. (\fBries\fR constructs all larger
integers from combinations of these.)
.IP p
pi = 3.14159...
.IP e
e = 2.71828...
.IP f
phi = (1+sqrt(5))/2 = 1.61803...
.IP n
Negative
.IP r
Reciprocal
.IP s
Squared
.IP q
Square root
.IP "S C"
Sine, Cosine
.IP T
Tangent
.IP l
ln (natural logarithm, also called log)
.IP E
e to the power of x
.IP "+ -"
Add, Subtract
.IP "* /"
Multiply, Divide
.IP ^
Power: 2 ^ 3 = 8
.IP v
Root (the ``v'' resembles a radical symbol): 3 v 27 = cube root of 27
.IP L
Logarithm to base A of B
.RE
.IP
There are lots of potential uses for \fB\-N\fR. For example,
if you invoke \fBries\fR on a small irrational number,
you might get several solutions that involve
the unary and binary logarithm operators 'ln' (natural logarithm) and 'log_'
(log to base A of B). If you decide
you aren't interested in such solutions you can just
add \fB-NlL\fR to your command line, and all such solutions
will be skipped.
If you are checking an unknown number that you found in the context of
some larger problem, you probably have some idea what
constants and operators may be involved, or not involved,
in the phenomenon that produced your number. Use \fB-N\fR to
rule out functions you don't think are relevant.
Note that \fBries\fR will often run considerably slower when you limit
it to a very small set of symbols, mainly because it cannot
use its optimization rules (described below under
ALGORITHM). Also, with fewer symbols the average length of
expressions is longer, and that makes the search slower.
.IP \fB-S\fIsss\fR
permitted Symbol Set: Specifies a symbol set, as with \fB\-N\fR, but has the
opposite effect: \fIonly\fR these symbols will be used. A \fB-S\fR also
cancels any \fB\-E\fR, \fB\-N\fR or \fB\-O\fR options that were given;
meaning that if you wish to combine these options, the \fB\-S\fR
should come first.
\fB\-S\fR can be used to solve those old problems of the
sort ``How can the number 27 be expressed using only the
four basic operators and the digit 4?'' The answer is
given by:
\FCries '-S4+-*/' 27 -Ox\FT
(The \fB-Ox\fR option is described next).
To solve the same problem using the \fB-N\fR
option, you'd have to type:
\FCries -Npef12356789rsqlL^v 27 -Ox\FT
If you give the \fB\-S\fR option with no symbols, \fBries\fR will
display a table of all available symbols (as modified by any \fB-E\fR,
\fB-N\fR, \fB-O\fR and \fB-S\fR options) with their definition and
weights. This lone \fB\-S\fR can be given along with a normal
\fB\-S\fR option, but in any case \fBries\fR will exit after showing
the table.
.IP \fB-E\fIsss\fR
Enable: Enables (or re-enables) the use of symbols that may have been
disabled by an earlier \fB-S\fR or \fB-N\fR option. This is mainly
intended for use in combination with the \fB--include\fR option. If
one include file disables some symbols, this option can be used to
re-enable some or all of them.
.IP \fB-O\fIsss\fR
Only One: Specifies symbols which should appear no more than once
on \fIeach side\fR of the equation. This option can be combined with \fB-E\fR
or \fB\-N\fR, in which case they augment each other.
If used with \fB-S\fR with the same symbol, the latter option
takes effect.
One additional symbol is available with \fB-O\fR:
.RS 1.2i
.IP x
The variable on the left-hand-side
.RE
.IP
Thus, you can use \fB-Ox\fR to limit \fBries\fR's output to equations
that have only one '\fIx\fR' in them and are therefore easy to
solve for \fIx\fR using only the most basic algebra techniques.
This also makes \fBries\fR's output more like that of
traditional expression-finders, which search for
expressions equal to \fIx\fR rather than equations in \fIx\fR. Here's an
example: \FCries -i 16\FT gives the answer
``\fIx\fR^2 - \fIx\fR = 3^5'' with \fIx\fR very close to 16,
because 16^2-16 is close to 3^5. \FCries -i 16 -Ox\FT replaces
that answer with ``5 \fIx\fR^2 = 6^4''.
.SH Options Limiting the Type of Solutions
Several options are used to choose what types of expression can be
used in the equations that are presented as solutions.
.IP \fB-i\fR
Integer: Require that all expressions, and all subexpressions, must
have integer values. This is primarily useful if you are searching for
an exact solution for a large integer. Note that inexact solutions
will still be given, but both sides of the equation will be integers.
An example of this is ``2 \fIx\fR = 7^3'' where \fIx\fR=173.
\fB-i\fR is interpreted as \fB-r\fR (described below) if the supplied
target value is not an integer.
.IP \fB-ie\fR
Integer, Exact. Like \fB-i\fR, but exits after reporting an exact
match (if found). This equivalent to \fB-i\fR combined with
\fB--min-match-distance 0\fR. To not report any inexact matches at
all, use \fB--max-match-distance 0\fR (described in more detail
below).
.IP \fB-r\fR
Rational: Require that all equations have a single \fIx\fR and that
all subexpressions not involving \fIx\fR are rational (an exact ratio
of two integers). This is primarily useful if you are searching for
rational approximations. This option is essentially just shorthand for
using the \fB-N\fR and \fB-E\fR options to allow only addition,
subtraction, multiplication and division.
.IP \fB-re\fR
Rational, ``Exact''. Like \fB-r\fR, but exits after reporting an
``exact'' match (if found). This equivalent to \fB-r\fR combined with
\fB--min-match-distance 0\fR.
Note that computers are (famously) unable to make exact
calculations with fractions as simple as 1/3. To exit on a match
within some ``epsilon'', use \fB--max-match-distance\fR with a nonzero
epsilon; to reject inexact matches use \fB--max-match-distance\fR
(these options are described in detail below).
.IP \fB-c\fR
Constructible: Require that all equations have a single \fIx\fR and
that all subexpressions are ``constructible'' in the sense of
Euclid's \fIElements\fR, given a unit interval. This is
like the \fB-r\fR option except that squares, square roots and the
golden ratio \fIphi\fR are also allowed. All results will be easily
solvable for \fIx\fR and will use only addition, subtraction,
multiplication, division, and square roots.
If you add the option \fB-Ex\fR, more than one \fIx\fR may appear
in the solution. This gives answers that, when solved for \fIx\fR, are
not constructible from the unit interval, but both sides of the
(unsolved) equation \fIare\fR constructible given \fIx\fR and a unit
interval. For example \FCries 1.3263524026321 -c -Ex\FT finds the
solution ``\fIx\fR \fIx\fR^2 = 7/3''; \fIx\fR is the cube root of
7/3 which is not itself constructible. \fBries\fR's answer reflects
the fact that 7/3 can be constructed from its cube root (although the
opposite construction is impossible).
.IP \fB-a\fR
Algebraic: Generate equations whose roots are ``algebraic numbers''.
This is similar to the \fB-c\fR option, but also allows \fIn\fR^th
powers and roots. The trigonometric functions (sine, cosine, and
tangent) are allowed but their arguments will always be rational
multiples of \fIpi\fR. More than one \fIx\fR is allowed (unless you
follow \fB-a\fR with the option \fB-Ox\fR) so the equations might not
be easy to solve.
.IP \fB-Ox\fR
Using the \fB-O\fR option (described above) with the symbol 'x' tells
\fBries\fR to limit its search to solutions that can be expressed in
``closed form'' using the basic constants, elementary and
transcendental functions. This concept of ``closed-form number'' is
described by Timothy Chow in his 1998 paper \fIWhat is a closed-form
number?\fR.
.IP \fB-l\fR
Liouvillian: Generate equations whose roots are ``Liouvillian
numbers'', as described by Timothy Chow in his 1998 paper \fIWhat is a
closed-form number?\fR.
.SH Options Affecting Output Format
.IP \fB-s\fR
Sorta Solve, by Shifting to right-hand side: With this option,
\fBries\fR will display equations with just a single ``\fIx\fR'' on
the left-hand side of the equal sign. It isn't ``solving'' the
equations, but merely performing algebraic transformations to move
everything except one \fIx\fR to the right-hand side:
``\fIx\fR(\fIx\fR+1) = 7'' becomes ``\fIx\fR = 7/(\fIx\fR+1)''. You
can combine this option with \fB-Ox\fR to eliminate this issue, but
with that option \fBries\fR will no longer find solutions that require
more than one ``\fIx\fR'', like ``\fIx\fR^\fIx\fR = 2'' for 1.55961.
.IP \fB-x\fR
X Values: Print actual values of \fIx\fR (the roots of the equations
found) rather than expressing \fIx\fR as \fIT\fR plus/minus a small
number, where \fIT\fR is your target number.
``\fB--absolute-roots\fR'' is a synonym for \fB-x\fR.
.IP \fB-F\fIn\fR
Format: Controls the way expressions are formatted in the main output.
If \fIn\fR is omitted it is 3 (``-F'' for ``FORTH Format''); if
\fB-F\fR is not specified at all, the format will be 2. The following
formats are available; each shows the output of \FCries 1.506591651
-F\FT\fIn\fR:
Format 0: Compressed FORTH-like postfix format: Each operator and
constant is just a single symbol. The symbols are as listed above under
the \fB-N\fR option.
.fam C
.ps -2
x1- = 2r for x = T - 0.00659165 {50}
xlr = 6q for x = T - 0.00241106 {62}
x4^ = p2+ for x = T - 0.000766951 {68}
x1+s = p2* for x = T + 3.66236e-05 {69}
.ps +2
.fam T
Format 1: Infix format, but with single-letter symbols. If this format
is specified, a table of symbols will be printed after the main table
of results. The rest of the expression syntax is the same as the normal
format. For example, ``q(l(\fIx\fR)) = p-1'' means ``sqrt(ln(\fIx\fR))
= pi - 1''.
.fam C
.ps -2
x-1 = 1/2 for x = T - 0.00659165 {50}
1/l(x) = q(6) for x = T - 0.00241106 {62}
x^4 = 2+p for x = T - 0.000766951 {68}
(x+1)^2 = 2.p for x = T + 3.66236e-05 {69}
.ps +2
.fam T
Format 2: Standard infix expression format (this is the default).
.fam C
.ps -2
x-1 = 1/2 for x = T - 0.00659165 {50}
1/ln(x) = sqrt(6) for x = T - 0.00241106 {62}
x^4 = 2+pi for x = T - 0.000766951 {68}
(x+1)^2 = 2 pi for x = T + 3.66236e-05 {69}
.ps +2
.fam T
Format 3: Print solutions in postfix format, similar to that used
in FORTH and on certain old pocket calculators. This is close to the
format used internally by \fBries\fR (to get the exact, condensed format,
use \fI-F0\fR). This is intended mainly for
use by scripts that use \fBries\fR as an engine to generate
equations and then perform further manipulation on them. However,
this option will also help you distinguish
what symbols were actually used internally to generate an answer.
For example, 'squared' and 'to the power of 2' both show up as '^2' in
the normal output, but in postfix they appear as "dup*" and "2 **"
respectively.
.fam C
.ps -2
x 1 - = 2 recip for x = T - 0.00659165 {50}
x ln recip = 6 sqrt for x = T - 0.00241106 {62}
x 4 ** = pi 2 + for x = T - 0.000766951 {68}
x 1 + dup* = pi 2 * for x = T + 3.66236e-05 {69}
.ps +2
.fam T
Most of the symbols used by \fB-F3\fR are self-explanatory. The
nonobvious ones are:
\fBneg\fR for negate,
\fBrecip\fR for reciprocal,
\fBdup*\fR for square,
\fBsqrt\fR for square root,
\fB**\fR for power (A^B),
\fBroot\fR for Bth root of A,
\fBlogn\fR for logarithm (to base B) of A.
For these last three, \fIA\fR is the first
operand pushed on the stack and \fIB\fR is the second.
The setting of \fB-F\fR does not affect expressions displayed by the
various \fB-D\fR diagnostic options (most of these use \fB-F0\fR,
and \fB-Ds\fR (``show your work'') uses \fB-F2\fR).
You may use the \fB--symbol-names\fR option (described below)
to redefine the appearance of formats 2 and 3.
.IP \fB-D\fIxx\fR
Display Diagnostic/Debugging Data: A detailed understanding of the \fBries\fR
algorithms (described below) is assumed. \fB-D\fR is followed by one
or more letters specifying the messages you want to see. Options
\fBA\fR through \fBL\fR and \fBa\fR through \fBl\fR (except \fBE\fR
and \fBe\fR) apply to the LHS and RHS respectively. For each option,
the number of lines of output that you can expect from a command like
\FCries -l2 2.5063141592653589 -D\FT\fIx\fR\FC\FT is shown:
.RS 0.7i
.IP A,a
\fB[61927; 130510]\fR partial expression error (e.g. divide by zero); pruned
.IP B,b
\fB[3357; 2666]\fR partial expression is zero, or derivative nearly zero; pruned
.IP C,c
\fB[79245; 671255]\fR partial expression is non-integer (and -i option given); pruned
.IP D,d
\fB[1948; 4321]\fR partial expression overflowed; pruned
.IP E,e
\fB[105526; 281232]\fR expression value matches one already in database; pruned
.IP F,f
\fB[366634; 884213]\fR show \fB--canon-reduction\fR operations on expressions before adding to database
.IP G,g
\fB[102307; 103587]\fR expression added to database
.IP H,h
\fB[418287; 624260]\fR show attributes of each partial expression tested
.IP I,i
\fB[3985434; 5935446]\fR show each new symbol to be added before complexity pruning
.IP J,j
\fB[2596074; 3839185]\fR show symbols skipped by complexity pruning
.IP K,k
\fB[251052; 417411]\fR show symbols skipped by redundancy and tautology rules
.IP L,l
\fB[72560; 95743]\fR show symbols skipped to obey -O option
.IP m
\fB[9173410]\fR show all metastack operations
.IP M
\fB[46]\fR show memory allocation benchmarks, and enable automatic exit when memory gets slow (see \fB--memory-abort-threshold\fR option)
.IP n
\fB[174]\fR show Newton iteration values and errors if any
.IP N
\fB[153]\fR show work in detail: operator/symbol, x and dx at each step
.IP o
\fB[562749]\fR show match checks
.IP p
\fB[118]\fR show preprocessing transformations prior to conversion to infix
.IP Q
\fB[53]\fR show manipulations to remove \fB--canon-reduction\fR from equations before root-finding
.IP q
\fB[125]\fR show close matches dispatched to Newton and results of test
.IP r
\fB[1605525]\fR show results (value and derivative of operands and result) for each opcode executed
.IP S
\fB[51]\fR show solve-for-x work: displays all operations performed by the \fB--try-solve-for-x\fR option to transform an equation into ``solved'' form
.IP s
\fB[272]\fR show your work: displays values of each subexpression for every reported answer. Subexpressions are shown in normal (infix) syntax, which is useful in combination with \fB-F0\fR to see the postfix format used with options like \fB--eval-expression\fR
.IP t
\fB[10438]\fR show all abc-forms passed to expression generation
.IP u
\fB[46724]\fR show steps of min/max complexity ranging for each abc-form
.IP v
\fB[5236]\fR show number of expressions generated by each abc-form
.IP w
\fB[31662]\fR show details of abc-form generation (pruning, weights, etc.)
.IP x
\fB[90]\fR show all rules used (varies with the -N -O and -S options)
.IP y
\fB[732]\fR statistics and decisions made in main loop
.IP z
\fB[52]\fR initialization and other uncategorized messages
.IP 0
\fB[1811037]\fR list the entire expression database after every pass
through the main loop
.RE
.IP
Of these, \fB-Ds\fR is probably the most useful and fun to look at.
\fB-Dy\fR gives a nice top-level view of the statistics of the search.
Most of the options that generate lots of output are useful if
filtered through \FCgrep\FT; this can tell you why a certain
subexpression is or is not appearing in results. \fB-DG\fR and
\fB-Dg\fR can be useful if you want to use \fBries\fR to generate a
massive list of expressions for processing by another program; for
this reason its output uses infix notation. Most other \fB-D\fR
options print subexpressions in the \fB-F0\fR terse postfix format.
.SH EXTENDED OPTIONS
Longer names are used for options that are thought to be less commonly
wanted, or are more likely to be used only within \fB--include\fR files.
.IP \fB--include\fR\ \fIfilename\fR
.IP \fB-p\fIfilename\fR
.IP \fB-p\fR
Load one or more options from file \fIfilename\fR. The options
``\fB--include\fR \fIfilename\fR'' and ``\fB-p\fIfilename\fR'' are
equivalent; note that one requires a space before \fIfilename\fR and
the other cannot have a space (\fB-p\fR alone has a related function,
described below). \fBries\fR will attempt to open the named file
(which may be a simple filename or a path), or the given name with
``\FC.ries\FT'' appended. If either is found, \fBries\fR will scan it
for parameters and arguments separated by whitespace. Any control
characters count as whitespace. Any '#' character that comes at the
beginning of a line or immediately after blank space denotes a comment
and the rest of the line will be ignored. For example, if there is a
file ``\FChst.ries\FT'' containing the following:
.fam C
.ps -2
# hst.ries: High School Trigonometry settings
--trig-argument-scale 1.74532925199433e-2 # pi/180
-NLleEv # No log, ln, e, e^x or arbitrary roots
-Ox # Only allow one 'x' on the left-hand-side
-x # Show equation roots as "x = 123.456"
# rather than "x = T + 1.23e-4"
.ps +2
.fam T
then giving the option ``\fB-phst\fR'' is equivalent to giving the
options ``\fB--trig-argument-scale 1.74532925199433e-2 -NLleEv -Ox -x\fR'',
in that order.
A parameter file may additionally invoke another parameter file
with the \fB--include\fR option. When it encounters this option,
\fBries\fR will apply the options in the included file, then continue
with the rest of the first file. These may be nested up to 25 levels
deep. If a file includes itself recursively (either directly or
indirectly) \fBries\fR will exit with an error.
It is an error for \fB--include\fR or the end of an included file
to come between an option and its arguments. For example, ``\fBries
1.2345 --eval-expression --include expressions.txt\fR'' will produce
an error regardless of the contents of ``\FCexpressions.txt\FT'',
because \fB--eval-expression\fR must be followed immediately by its
argument(s). On the other hand, if ``\FCexpressions.txt\FT'' contains
the \fB--eval-expression\fR option, like so:
.fam C
.ps -2
# expressions.txt: Useful functions of one argument
--eval-expression
xsr # 1/(x^2)
1xq-r # 1/(1-sqrt(x))
2xl1+^ # 2^(ln(x)+1)
.ps +2
.fam T
then the command ``\fBries 1.2345 --include expressions.txt\fR''
works, and shows the values of the three expressions where \fIx\fR is
1.2345.
If you have a file called ``\FC.ries_profile\FT'' or
``\FCries_profile.txt\FT'' in your home directory, \fBries\fR will
load it as if you specified it with a \fB--include\fR at the very
beginning of the parameters. If you have such a file and wish to
prevent it from being used, give a bare \fB-p\fR (without a filename)
at the very beginning of your \fBries\fR options. If you wish to give
some parameters and have \FC.ries_profile\FT loaded \fIafter\fR your
parameters, include \fB-p\fR again at the point where you want
\fBries\fR to use the profile. For example, if your
\FC.ries_profile\FT contains ``\fB--trig-argument-scale 1\fR'' and you
have a \FChst.ries\FT with contents as shown above, then giving the
optiions ``\fB-phst\fR \fB-p\fR'' will use all of the settings in
\FChst.ries\FT except the \fB--trig-argument-scale\fR.
Two more example profiles are the ``Latin'' and ``Mathematica''
settings files linked from the top of the main RIES webpage.
.IP \fB--absolute-roots\fR
This is a synonym for the \fB-x\fR option, described above.
.IP \fB--algebraic-subexpressions\fR
This is a synonym for the \fB-a\fR option, described above.
.IP \fB--any-exponents\fR
This option cancels any restrictions on subexpressions used as an
exponent, such as those set by the \fB--algebraic-subexpressions\fR
and \fB--liouvillian-subexpressions\fR options.
.IP \fB--any-trig-args\fR
This option cancels any restrictions on subexpressions used as an
argument to one of the trigonometric function, such as those set by
the \fB--algebraic-subexpressions\fR and
\fB--liouvillian-subexpressions\fR options.
.IP \fB--canon-reduction\fR\ \fIsymbols\fR
Apply simple transformations in an effort to make all expression
values fall in the range [1 ... 2). This option improves the
efficiency of the \fBries\fR algorithm (described in the ``ALGORITHM''
section below) by increasing the chances of two expressions forming a
match. This allows it to use less memory and time to achieve any given
amount of precision.
This option should be followed by one or more symbols which
represent the operations \fBries\fR will try to apply to expressions:
.RS 0.7i
.IP n
Negate expressions when possible to make all values positive.
.IP r
Take the reciprocal when possible to make all expressions fall outside
the range (-1 ... 1).
.IP 2
Multiply by 2 when possible to increase the magnitude of
expressions in the range (-1 ... 1).
.IP 5
Divide by 2 (i.e. multiply by 0.5) when possible to decrease the
magnitude of expressions that fall outside the range (-2 ... 2).
.RE
.IP
In these descriptions, the words \fIwhen possible\fR refer to the fact
that \fB--canon-reduction\fR will respect any limits imposed by the
symbolset options \fB-N\fR, \fB-O\fR and \fB-S\fR. So if you use the
option \fB-On\fR together with \fB--canon-reduction n\fR, the negation
operator will still be used only once per expression.
Although it makes \fBries\fR more efficient, this option also
causes the printed results to have greater complexity scores, and
complexity scores will increase somewhat more erratically. \fBries\fR
will try to simplify its printed results by undoing
\fB--canon-reduction\fR transformations on both sides of the equal
sign. For example, \FCries 2.50618 --canon-reduction nr25\FT might
yield the result ``\fIx\fR^\fIx\fR/2 = (1+9)/2'', which simplifies
to ``\fIx\fR^\fIx\fR = 1+9''. But when only one side has a ``/2'',
\fBries\fR cannot fix it, so the same example gives an overly complex
``1/(\fIpi\fR-\fIx\fR) = \fIpi\fR/2''.
.IP \fB--constructible-subexpressions\fR
This is a synonym for the \fB-c\fR option, described above.
.IP \fB--derivative-margin\fR\ \fIvalue\fR
Specify the limit to how small the derivative of any expression or
subexpression containing \fIx\fR can be in relation to the
expression's value. By default this is 10^-6, so that an expression
containing \fIx\fR is rejected if its value is more than a million
times its derivative. For really large target values, this doesn't
work because the expression ``\fIx\fR'' (with a derivative of 1.0)
would be rejected. So if the magnitude of your target is larger than
10^5, \fBries\fR will set this limit correspondingly lower. For
example, if your target value is 10^8, \fBries\fR automatically sets
a \fB--derivative-margin\fR value of 10^-9.
If you do not consider these defaults suitable, use this option
to pick your own value. For example in the command \FCries 12345
--derivative-margin 8e-5\FT, derivatives of expressions can be as
small as 8x10^-5 times the expression's value. In calculating any
possible answers, \fBries\fR would allow ``\fIx\fR^2'' because the
ratio between d/d\fIx\fR \fIx\fR^2 and \fIx\fR^2 is about 1.62e-4,
which is big enough to surpass the margin. However, any answers
involving ``sqrt(\fIx\fR)'' would be rejected beause the ratio between
d/d\fIx\fR sqrt(\fIx\fR) and sqrt(\fIx\fR) is only 4.05e-5:
.ta 1.2iC 2.2iC 3.4iC 4.4iC
expression value d/d\fIx\fR(expr.) ratio
.br
\fIx\fR^2 152399025 24690 1.62e-4
.br
\fIx\fR 12345 1.0 8.1e-5
.br
sqrt(\fIx\fR) 111.108 0.0045 4.05e-5
Indeed, the results of \FCries 12345\FT include the equation
``2(sqrt(\fIx\fR)-9) = (2 e)^pi'', but with the option
\fB--derivative-margin 8e-5\fR that answer is left out.
.IP \fB--explicit-multiply\fR
Always use the ``*'' symbol when displaying results, rather than the
default behavior of omitting ``*'' when multiplicaton can be implied
by writing the multiplicands next to each other. This is useful if you
need to copy \fBries\fR output into a calculator, computer program or
spreadsheet formula.
.IP \fB--integer-subexpressions\fR
This is a synonym for the \fB-i\fR option, described above.
.IP \fB--match-all-digits\fR
.IP \fB--mad\fR
Request that all reported matches should match all of the supplied
digits. This is equivalent to adding a '5' to the end of your target
value, along with a \fB--max-match-distance\fR value equal to the
magnitude of this appended '5' digit. It also selects the \fB--x\fR
option, unless the \fB--wide\fR option is also given.
For example, the command \FCries 2.5063 --mad\FT is equivalent to
\FCries 2.50635 --max-match-distance 0.00005 -x\FT, and the first
reported match is \fIx\fR^2+e = 9, which is true for \fIx\fR =
2.506335... Without \fB--mad\fR it reports 2 \fIx\fR = 5 and a few
other answers that do not match all of the digits in 2.5063.
.IP \fB--max-match-distance\fR\ \fIvalue\fR
Specify the maximum distance between your given target value \fIT\fR
and the roots \fIx\fR of any reported equations. This sets a minumum
level of accuracy, overriding the default, which is 1% of the
size of your target value. For example, the command \FCries 2.5063\FT
will use a threshold that is 1% of 2.5063, or about 0.025. It
gives as its first answer the equation 2\fIx\fR = 5, an equation whose
root (solution) is 2.5. This differs from the target value 2.5063 by
0.0063. If you specify an initial threshold of 0.001 with the command
\FCries 2.5063 --max-match-distance 0.001\FT, then 2\fIx\fR = 5
is not reported because 0.0063 is bigger than your threshold 0.001;
instead the first match will be ``\fIx\fR^2 = 2 \fIpi\fR''
(which comes within about 0.0003 of the target 2.5063).
Use a zero argument to specify that \fBries\fR should only report
an ``exact'' match, if any (and note that this ``exact'' match might
be more complex than the obvious answer, because of roundoff errors;
see UNEXPECTED BEHAVIOR and BUGS below). Note that this is different
from \fB--min-match-distance 0\fR, which prints inexact matches and
stops after the ``exact'' match.
Use a negative argument to specify a match threshold in
proportion to your target value. For example, \FC--max-match-distance
-0.001\FT specifies that the first match must be within 1 part in
1000 (or 1/10 of one percent) of the magnitude of the target.
If your choice of \fB--max-match-distance\fR is so stringent that
the first match takes longer than 2 seconds, \fBries\fR will display
progress messages until a match is found. Use the \fB--no-slow-messages\fR
option to suppress these.
There is also a \fB--min-match-distance\fR option (described
below), which serves an entirely different purpose.
.IP \fB--max-matches\fR\ \fIN\fR
.IP \fB-n\fIN\fR
Limit the number of reported matches to a positive integer \fIN\fR.
This is particularly useful with certain options (such as
\fB--no-refinement\fR) that generate a lot of matches. The default
\fIN\fR is 100.
.IP \fB--max-memory\fR\ \fIsize\fR
This option tells \fBries\fR not to use more than the given amount of
memory (size specified in bytes). This is particularly useful in
combination with a
high \fB-l\fR (search level) option. For example, if you typically
have about 2 gigabytes of free memory on your machine, you could
invoke \fBries\fR with the option \fB--max-memory\fR \fB1.0e9\fR,
to ensure that it never uses more than 1 gigabyte of memory regardless
of the search level.
\fBries\fR also has an (experimental) feature that can
automatically detect when your system is slowing down; see the
\fB--memory-abort-threshold\fR option for details.
.IP \fB--memory-abort-threshold\fR\ \fIN\fR
This option is used with the \fB-DM\fR option, and overrides the
default slowness measurement after which \fBries\fR will automatically
exit. With the \fB-DM\fR option, \fBries\fR measures how fast it is
running, as compared to an estimate of how fast it ``should'' be
running. If this ratio is greater than the
\fI--memory-abort-threshold\fR for more than 3 of the past 10
measurements, \fBries\fR will exit. The default
\fI--memory-abort-threshold\fR is 2.0. The value must be at least 1.0,
and values less than about 1.5 are unlikely to be of much use.
\fINOTE\fR: \fB--memory-abort-threshold\fR is an experimental
\fBries\fR feature and is likely to change in future versions of
\fBries\fR.
.IP \fB--min-match-distance\fR\ \fIvalue\fR
Specify the minimum distance between your given target value \fIT\fR
and the roots \fIx\fR of any reported equations. This is useful for
finding approximate formulas for constants that have a known, simple
formula. For example, using the command \FCries 3.141592653589 -x
--min-match-distance 1e-8 -NSCT\FT one can discover the following approximate
formulas for \fIpi\fR :
.fam C
.ps -2
x-3 = 1/6 for x = 3.16666666666667
x-3 = 1/7 for x = 3.14285714285714
ln(ln(x)) = 1/e^2 for x = 3.14219183392327
x^2+1 = 4 e for x = 3.14215329254258
e^x+2 = 8 pi for x = 3.14124898321672
x/phi^2 = 1/5+1 for x = 3.14164078649987
e^x-pi = 4*5 for x = 3.14163154625921
e^(x^2)+1 = e^(pi^2) for x = 3.14158442136535
pi-x = 1/e^(4^2) for x = 3.14159254105462
sqrt(1+pi) x = e^3/pi for x = 3.14159272240341
x^2/e^3 = 1/sqrt(1+pi) for x = 3.1415926879966
9(x-pi) = 1/-(e^(4^2)) for x = 3.14159264108588
.ps +2
.fam T
Among these results (after solving for \fIx\fR) are the ancient
approximations 19/6 and 22/7, and the more modern curiosity
\fIe\fR^\fIpi\fR \o"~=" 20 + \fIpi\fR (which is called ``\fIGelfond's
constant\fR''). Other interesting results can be found by omitting
symbols with \fB-N\fR, for example \fBries 3.1415926 -NSCTlLfEevp\fR
(excluding most of the scientific functions) gives the fraction
approximation 355/113 in the form ``1/(\fIx\fR-3)-1 = 1/4^2+6''.
You will often get multiple equivalent results. In the above
example, the equations sqrt(1+\fIpi\fR) \fIx\fR = e^3/\fIpi\fR and
\fIx\fR^2/e^3 = 1/sqrt(1+\fIpi\fR) can both be converted into the
approximate relation:
.RS 1.2i
\fIpi\fR \o"~=" sqrt(sqrt(\fIe\fR^6/(\fIpi\fR+1)))
.RE
.IP
(which does \fInot\fR converge on the true value of \fIpi\fR if iterated).
If you give a value of 0: ``\FC--min-match-distance 0\FT'', and
\fBries\fR finds an ``exact'' match, it will exit and report no further
results. Note that this is different from \fB--max-match-distance 0\fR,
which will only print the ``exact'' match and will not print any inexact
matches.
There is also a \fB--max-match-distance\fR option (described
above), which serves an entirely different purpose.
.IP \fB--min-memory\fR\ \fIsize\fR
If \fBries\fR is given the debug option \fB-DM\fR, it will try to
measure the responsiveness of the system and automatically exit if it
gets very slow. This is intended as an automatic safeguard against
virtual memory ``thrashing'' that will happen if \fBries\fR is allowed
to use all of your system's memory. (This feature is only active with
the \fB-DM\fR option because it is still being tested).
When \fB--min-memory\fR is given in combination with \fB-DM\fR, it
will ensure that \fBries\fR does not exit because of slow memory
response until at least \fIsize\fR bytes of memory have been used. For
example, if you know that you always have about 1 gigabyte of free
memory on your machine, and your machine often gets slow for other
compute-intensive tasks, you could invoke \fBries\fR with the options
\fB-DM\fR \fB--min-memory\fR \fB1.0e9\fR, and slow system detection
would be enabled but would not trigger (if at all) until a gigabyte of
memory has been used.
For more direct control over \fBries\fR' memory usage, use the
\fB--max-memory\fR option (without \fB-DM\fR) or use a suitably small
\fB-l\fR search level.
.IP \fB--no-refinement\fR
After reporting a match, do \fInot\fR require that the next match come
closer to the target. This causes \fBries\fR to emit many more matches
than it normally would. The matches will not be given in order of
closeness, but they will still be (roughly) ordered by increasing
``complexity''. Many will be equivalent to one another, for example
the command \FCries 1.51301 --no-refinement\FT yields the solutions
\fIe\fR^(\fIx\fR^2) = pi^2 and \fIx\fR/sqrt(2) = sqrt(ln(pi)),
both with the root 1.513096088... This option is most effective in
combination with \fB--max-matches\fR along with
\fB--match-all-digits\fR or \fB--max-match-distance\fR (using a
stricter argument than the default -0.01).
.IP \fB--no-slow-messages\fR
Suppress the ``Still searching'' messages that \fBries\fR would normally
print if a search takes longer than 2 seconds without giving any results.
.IP \fB--no-solve-for-x\fR
This option cancels the ``\fB--try-solve-for-x\fR'' option.
.IP \fB--numeric-anagram\fR\ \fIdigits\fR
Give a specific set of digits that can be used as constants in a
solution; this forces the \fB--one-sided\fR option. It will use only
as many of each digit as you specify. For example, if you give
``111223'' as the digits, \fBries\fR will use up to three 1's, two
2's, and/or a single 3. This is meant to aid in solving certain
puzzles of the ``four fours'' variety:
.fam C
.ps -2
ries 17 -ie --numeric-anagram 4444
x = 4*4+4/4 ('exact' match) {92}
ries 17 -ie --numeric-anagram 111223
x = 2^(1+3)+1 ('exact' match) {77}
ries 12 --numeric-anagram 44s
x = 4^2-4
ries 12 --numeric-anagram 442
x = 2*4+4
.ps +2
.fam T
\fB--numeric-anagram\fR can be used to set hard limits on the digits,
the constants \fIe\fR, \fIphi\fR, and \fIpi\fR, and the ``squared''
and ``reciprocal'' symbols. When you use \fB--numeric-anagram\fR, any
of these symbols that you do not list will be forbidden just as if you
had used the \fB-N\fR option. The last two examples here show the use
of ``s'' to specify the squaring operation \fIx\fR^2 as
distinguished from any use of ``2''.
.IP \fB--one-sided\fR
Force \fBries\fR to ignore all LHS expressions except \fIx\fR. This
results in ``one-sided equations'' with \fIx\fR on the left-hand side.
This makes \fBries\fR much slower, but all of its output will be
``solved for \fIx\fR''.
This option is intended as a convenience for very special problems
(for example, \fB--numeric-anagram\fR automatically turns on
\fB--one-sided\fR), but it is not generally useful because the speed
advantage of the normal RIES bidirectional search algorithm is lost.
If you want \fBries\fR to give answers that are solved for \fIx\fR,
use the \fB--try-solve-for-x\fR option possibly along with \fB-Ox\fR.
If you use this option, \fBries\fR will be a lot slower and its
solutions for a given search level will not be nearly as accurate.
Whereas a normal \fBries\fR search might quickly match your target
value to the first 10 decimal places, a search with the
\fB--one-sided\fR option, taking the same amount of computation time,
will only match the first 5 decimal places. Sometimes this is
acceptable, particularly when used with other options that restrict
the search, such as \fB-i\fR, \fB-N\fR, \fB-O\fR, and \fB-S\fR.
.IP \fB--rational-exponents\fR
Require any exponent to be a rational subexpression. For example
sqrt(2) is allowed because it is 2 to the power of 1/2, but 2 to the
power of sqrt(2) is not allowed because sqrt(2) is not rational. This
option exists mainly to support the \fB-a\fR or
\fB--algebraic-subexpressions\fR option (described above).
.IP \fB--rational-subexpressions\fR
This is a synonym for the \fB-r\fR option, described above.
.IP \fB--rational-trig-args\fR
Require any argument to a trigonometric function be a rational
subexpression. The restriction applies to the argument's value before
multiplying by the \fB--trig-argument-scale\fR (if any). This
option exists mainly to support the \fB-a\fR or
\fB--algebraic-subexpressions\fR option (described above).
.IP \fB--relative-roots\fR
When printing each equation, show the root as \fIT\fR plus/minus a
small number (where \fIT\fR is your target number) rather than as the
actual value of the root. This is the refault, so you'll only need to
use \fB--relative-roots\fR to cancel a \fB-x\fR or
\fB--absolute-roots\fR option in your \FC.ries_profile\FT or another
\fB--include\fR file.
.IP \fB--ries-arguments-end\fR
This `option' signals the end of options and arguments; any that come
after it will be ignored. If it occurs within a profile (described under
the \fB--include\fR option above) \fBries\fR will ignore the rest of the
contents of that file and continue with the next option after the
\fB--include\fR or \fB--p\fR option that invoked the profile.
.IP \fB--significance-loss-margin\fR\ \fIdigits\fR
Specify the number of significant digits that may be lost in a
calculation. By default, \fBries\fR tolerates a loss of 2 digits in
any calculation. For example, if \fIx\fR is 0.906402477... (the value
of Gamma[5/4]), \fBries\fR would not use \fIx\fR+e^5 in any of its
expressions, because \fIe\fR^5 is more than 100 times as large as
\fIx\fR. Due to round-off, more than 2 digits of the value of \fIx\fR
would be lost in the addition. This restriction applies to constant
expressions too, so \fBries\fR also avoids 1+e^5 and 1+e^-5.
Similar restrictions apply to any function that can cause precision to
be lost (if evaluated at a point where the function's derivative is
very low).
The normal \fBries\fR behavior corresponds to a
\fB--significance-loss-margin\fR option with an argument of 2.0. Give
a higher value to allow more digits to be lost in calculations.
Conversely, if you suspect \fBries\fR is generating meaningless
results due to round-off error, you can look at its calculations in
detail with the options \fB-Ds\fR and \fB-F0\fR, then evaluate specific
expressions with \fB--eval-expression\fR (described below). If
it seems appropriate, make \fBries\fR more strict by giving
\fB--significance-loss-margin\fR with a lesser argument.
.IP \fB--symbol-names\fR\ :\fIsym\fR:\fIname\fR\ [\ :\fIsym\fR:\fIname\fR\ ...\ ]
This option allows you to set the ``names'' of individual symbols.
This affects how the expressions and equations are printed in any of
the \fB-F\fR formatting modes, and by special commands such as
\fB--eval-expression\fR. In addition to the symbols listed above
in the \fB-N\fR option, you may also define these symbols:
.RS 0.7i
.IP (\ )
brackets to group sub-expressions
.IP =
equality symbol
.RE
.IP
Here are examples of the normal \fBries\fR output, modified by
changing the appearance of the exponentiation operator and
parentheses. The single-quotes around each option are to avoid
substitution by the shell:
.fam C
.ps -2
ries -l0 2.5063
2 x = 5 for x = T - 0.0063 {49}
8 x = e^3 for x = T + 0.00439212 {66}
x^2 = 2 pi for x = T + 0.000328275 {55}
x^x = 1+9 for x = T - 0.000115854 {69}
x^2+e = 9 for x = T + 3.56063e-05 {63}
ln(6) x = sqrt(pi)+e for x = T + 2.73037e-05 {93}
ries -l0 2.5063 --symbol-names ':^:**' ':(:[' ':):]'
2 x = 5 for x = T - 0.0063 {49}
8 x = e**3 for x = T + 0.00439212 {66}
x**2 = 2 pi for x = T + 0.000328275 {55}
x**x = 1+9 for x = T - 0.000115854 {69}
x**2+e = 9 for x = T + 3.56063e-05 {63}
ln[6] x = sqrt[pi]+e for x = T + 2.73037e-05 {93}
.ps +2
.fam T
More examples of the use of \fB--symbol-names\fR are found in the
``Latin'' and ``Mathematica'' settings files linked from the top
of the main RIES webpage.
.IP \fB--symbol-weights\fR\ \fIN\fR:\fIsym\fR\ [\ \fIN\fR:\fIsym\fR\ ...\ ]
This option allows you to adjust the ``weights'', or complexity
ratings, of individual symbols. Use the option \fB-S\fR to see the
normal weights, then use this option to change one or more. Compare
these two examples; in the second one the cost of the symbol \fIx\fR
is reduced, and the costs of \fB2\fR and \fBs\fR (squared) are
increased.
.fam C
.ps -2
ries -l0 2.5063
2 x = 5 for x = T - 0.0063 {49}
8 x = e^3 for x = T + 0.00439212 {66}
x^2 = 2 pi for x = T + 0.000328275 {55}
x^x = 1+9 for x = T - 0.000115854 {69}
x^2+e = 9 for x = T + 3.56063e-05 {63}
ries -l0 2.5063 --symbol-weights 5:x 25:2 15:s
x sqrt(x) = 4 for x = T + 0.0135421 {39}
-x-x = -5 for x = T - 0.0063 {46}
x/x^x = 1/4 for x = T + 0.00150933 {49}
x x^x = 5^2 for x = T - 0.00118185 {57}
x^x = 1+9 for x = T - 0.000115854 {49}
x (1/x-x) = e-8 for x = T + 3.56063e-05 {71}
.ps +2
.fam T
With the smaller weight of 5, \fIx\fR is considered ``less
expensive'', and \fBries\fR uses \fIx\fR more often in its answers;
and with the number \fB2\fR and squaring more expensive, these show up
less often in the results. In many cases the new results are
equivalent, and \fBries\fR has simply found a different way to get
there.
Since the arguments of \fB--symbol-weights\fR start with a digit, your
target value will be treated as a symbol-weight specifier unless you
place it somewhere else in the parameter list (as shown in the
example), or use a single dash ``-'' to signal the end of the
parameters.
\fINOTE\fR: Changing the symbol weights can greatly reduce
\fBries\fR's efficiency, causing it to run for a very long time and
giving little or no output. If this happens, it usually can be fixed
by using weights closer to the default values. You can also experiment
with changing just one symbol-weight at a time to find which is
causing the problem.
.IP \fB--trig-argument-scale\fR\ \fIvalue\fR
Specify a constant by which the argument of the sine, cosine and
tangent functions should be multiplied. By default this value is
\fIpi\fR and the trig functions are called \FCsinpi\FT, \FCcospi\FT
and \FCtanpi\FT. sinpi(\fIx\fR) is the sine of \fIpi\fR times \fIx\fR;
so for example sinpi(1/3) is the sine of \fIpi\fR/3, which is half the
square root of 3. A full circle is 2 in these units: sinpi(\fIx\fR) =
-sinpi(\fIx\fR+1) = sinpi(\fIx\fR+2) for all \fIx\fR.
If you give this option, arguments will be multiplied by the number
you give instead of by \fIpi\fR. Useful values to give are:
.RS 0.7i
.IP \fB1\fR
Use 1 to get natural units (radians): sin(1/3) = 0.327194...
.IP \fB1.74532925199433e-2\fR
This is \fIpi\fR/180, and is used for degrees: sin(22.5) = 0.382683...
.IP \fB1.570796326794897e-2\fR
This is \fIpi\fR/200, and is used for grads (gradians or gons):
sin(25) = 0.382683...
.RE
.IP
If you use this option, \fBries\fR will call the functions
\FCsin\FT, \FCcos\FT and \FCtan\FT in its output, and the scale will
be displayed after the function definitions at the end.
.IP \fB--try-solve-for-x\fR
.IP \fB-s\fR
\fB--try-solve-for-x\fR is equivalent to ``\fB-s\fR'', which is
described in the OPTIONS section above.
.IP \fB--version\fR
Displays information about the version of \fBries\fR, the calculation
precision and math library, any optional module(s), the currently
enabled profile (see the \fB--include\fR option at the beginning of
the EXTENDED OPTIONS section above), and a brief copyright notice. The
version is a date, such as ``2013 Jun 3''.
.IP \fB--wide-output\fR
Use a wider (132-column) output format. This shows the roots of
equations (values of \fIx\fR) both in terms of the the actual value of
\fIx\fR, and as \fIT\fR plus/minus a delta; it also shows the ratio
between this delta and \fIx\fR as ``(1 part in \fIN\fR)'' where \fIN\fR
is the delta divided by \fIx\fR. For example, if \fIx\fR is 2.5 and
target value is 2.501, the delta is 0.001, which is ``1 part in
2500''.
.SH SPECIAL COMMANDS
\fBries\fR provides some functions that supplement its main purpose.
Commands and their parameters must be separate:
`\FCries 1.23 --find-expression xxS+\FT',
not `\FCries 1.23 --find-expressionxxS+\FT'. Because the parameters
are given separately, your target value might be interpreted as a parameter
if you give it right after a special command. To avoid this, use a single
dash ``-'' to signal the end of the parameters.
.IP \fB--eval-expression\fR\ \fIforth-expr\fR\ [\fIforth-expr\fR\ ...]
Evaluate one or more expressions, showing intermediate values,
derivatives, and the complexity score of the full expression. The
expression(s) should be given in the FORTH-like postfix syntax that is
displayed when
you use the -F0 option. The symbols are as listed above under the
\fB-N\fR option. For example, \FCxxq-\FT is the syntax for
\fIx\fR-sqrt(\fIx\fR). Syntax errors and computation errors such as
overflow are reported; however successful execution by \fB--eval-expression\fR
does not guarantee that the expression will be found in an actial
\fBries\fR search. For example, an expression containing \fIx\fR
only appears as the left-hand-side of an equation if the \fIx\fR is the
first symbol in the postfix form: \fBries\fR will use \FCx2+\FT (\fIx\fR+2)
but will not use \FC2x+\FT (2+\fIx\fR).
.IP \fB--find-expression\fR\ \fIforth-expr\fR\ [\fIforth-expr\fR\ ...]
Perform the normal equation-finding search algorithm, and report
specific expressions when they are found, along with their value,
derivative, and complexity. The expression(s) should be given in the
FORTH-like postfix syntax that is displayed when you use the -F0
option. The symbols are as listed above under the \fB-N\fR option. For
example, \FCxxq-\FT is the syntax for \fIx\fR-sqrt(\fIx\fR). This
command is useful for diagnostics; an example is given below in the
UNEXPECTED BEHAVIOR section.
.SH ALGORITHM
\fBries\fR begins its search with small, simple equations and proceeds
to longer, more complex ones. It uses a set of \fIcomplexity rules\fR
to compute a measure (similar to Kolmogorov complexity), which
determines the order in which various candidate expressions are
considered by \fBries\fR. For example:
.IP 1.
If you add a symbol to an equation, the result is more
complex:
\fIx\fR + 1 = 3 is more complex than \fIx\fR = 3
\fIx\fR + 1 = ln(3) is more complex than \fIx\fR + 1 = 3
\fIx\fR - 7 = 4^2 is more complex than \fIx\fR - 7 = 4
.IP 2.
If two equations are the same except for one number, the
equation with the higher number is more complex:
\fIx\fR + 1 = 5 is more complex than \fIx\fR + 1 = 3
\fIx\fR^3 + 1 = 3 is more complex than \fIx\fR^2 + 1 = 3
.IP 3.
If two equations are the same except for one symbol, the
equation with the "more exotic" symbol is more complex:
\fIx\fR ^ 5 = 3 is more complex than \fIx\fR + 5 = 3
.P
As \fBries\fR searches it finds solutions -- these are equations for
which \fIx\fR is close to being an exact answer. Each time it finds a
solution it prints it out. Then \fBries\fR raises its standard for
the next answer: The next answer \fBries\fR prints must be a closer
match to your supplied value than all the answers it has given so
far. (The only exception to this rule is an 'exact' match, one for which
both sides match to within the limits of numerical precision. \fBries\fR
will print at most one of these, and will then continue to
print more inexact solutions. Sometimes the approximations are
of greater interest than the exact match.)
Instead of trying complete equations, \fBries\fR
actually constructs half-equations, called
\fIleft-hand-side expressions\fR
and \fIright-hand-side expressions\fR, abbreviated LHS's and RHS's.
It keeps a list of LHS's and a list of RHS's, and it keeps
these lists in numerical order at all times. This enables \fBries\fR
to find matches much faster. All LHS's contain \fIx\fR and all
RHS's do not. Thus, 1000 LHS's and 1000 RHS's make a total of
1000000 possible equations, and all 1000000 combinations can be quickly checked
just by scanning through the two lists in numerical order. This is why
\fBries\fR is able to check billions of equations in such a short time.
The closeness of an LHS match depends on the value of \fIx\fR,
and also on the derivative with respect to \fIx\fR of the LHS
expression. Because of this, \fBries\fR
calculates derivatives of LHS's as well as their values.
There are dozens of optimization rules \fBries\fR
uses, like the following:
.IP \fBa+\fR
Don't try ``K + K'' for any constant \fBK\fR because ``K * 2''
is equivalent.
.IP \fBb+\fR
Don't try ``3 + 4'' (or any two unequal integers from 1 to 5)
because another single integer (in this case ``7'') is shorter.
.IP \fBa*\fR
Don't try ``1 * K'' for any constant \fBK\fR because ``K'' is shorter.
.IP \fBb*\fR
Don't try both ``2 * 4'' and ``4 * 2'' because they are equivalent.
.IP \fBc*\fR
Don't try ``K * K'' because ``K ^ 2'' is shorter.
.IP \fBar\fR
Don't try ``1 / (1 / expr))'' for any expression \fBexpr\fR
because ``expr'' is shorter.
.IP \fBa^\fR
Don't try ``2 ^ 2'' or ``2 squared'' because ``4'' is shorter.
.IP \fBb^\fR
Don't try ``expr ^ 2'' for any expression \fBexpr\fR
because ``expr squared'' is shorter.
.RE
There are over 50 rules like this, and together they make the search about
10 times faster. However, if the symbol set is limited via \fB-N\fR,
\fB-O\fR or \fB-S\fR, some of these rules cannot be used. For each
optimization rule there are one or more symbolset exclude rules
like the following:
.RS 0.6i
Don't use rule \fBa+\fR if either of the symbols '*' or '2' is disabled.
.RE
In order to maintain maximum efficiency, \fBries\fR
checks each rule individually against the symbolset,
and uses as many rules as it can.
You can see this process in action by trying a command like
\FCries 1.4142135\FT, which gives the answer ``\fIx\fR^2 = 2''.
If you disable the 's' (squared) and '^' (power) symbols with
\FCries 1.4142135 -Ns^\FT, rule \fBb^\fR goes
away, and the answer becomes ``\fIx x\fR = 2''.
If you also disable '*' (multiplication) the answer
becomes ``\fIx\fR = sqrt(2)''.
Disable 'q' (square root) and you get ``log_2(\fIx\fR)
= 1/2'' (the logarithm to base 2 of \fIx\fR is 1/2).
Disable 'L' and it becomes ``\fIx\fR = 2,/2'' (square root of 2, this time
using the generalized root function). Disable 'v' and you
get ``\fIx\fR/(1/\fIx\fR) = 2''. Disabling '/', we get a trignometric
identity involving \fIpi\fR/4; disabling the trig functions as well, the
command becomes \FCries 1.4142135 '-Ns^q*Lv/SCT'\FT and we finally get an
answer that most people probably would not guess:
\fIx\fR-1/(\fIx\fR+1) = 1 (note that the '/' in this answer is
actually part of '1/', which is the reciprocal operator 'r'). Throughout
this progression the complexity score of the equation increases as
the solution becomes more and more obscure, and simpler but poorly
matching ``solutions'' like 1/cos(\fIx\fR) = 6 begin to appear.
.SH UNEXPECTED BEHAVIOR
Sometimes a more complex equation will be given before the (simpler)
equation that you expect. For example, tan(sqrt(2)) =
sin(sqrt(2))/cos(sqrt(2)) is 6.334119167042..., so you might expect
the command \FCries 6.334119167042 --trig-argument-scale 1 -NT\FT to
report something like ``cos(sqrt(2)) \fIx\fR = sin(sqrt(2))''.
Instead, \fBries\fR gives ``sqrt(\fIx\fR^2+1) = 1/cos(sqrt(2))'',
which is equivalent by
trigonometric identity, because it considers this equation to be
``more balanced'' (complexity score 45+38) than the other (which
has a score of 48+29). Please read the preceding section ``ALGORITHM''
for more details.
Adding or changing the symbolset with the \fB-S\fR, \fB-O\fR and
\fB-N\fR options often causes unexpected changes in the output. For
example, \FCries 2.2772\FT yields the solution ``1/(\fIx\fR-2) =
2+\fIphi\fR'' but \FCries 2.2772 '-N*/T'\FT does not give this
solution in any form. This seems counterintuitive: there was no *, /,
or tan() in the ``1/(\fIx\fR-2) = 2+\fIphi\fR'' solution, so why did
\fBries\fR decide not to report it?
In fact, the solution is still generated internally, but because
you have told it to exclude some operators, \fBries\fR has to try
other, more exotic expressions sooner than it otherwise would. As it
happens, the next solution ``\fIx\fR+1/e = sqrt(7)'' (which matches
the target value 2.2772 more closely than ``1/(\fIx\fR-2) =
2+\fIphi\fR'') ends up getting found earlier.
The mysterious behavior results from the fact that \fBries\fR
always tries to keep the number of LHS and RHS expressions equal as it
performs its search. Eliminating operators with the \fB-N\fR option
means that more complex expressions must be generated to reach the
``quota''. In this particular case, the symbolset restriction has a
greater effect on the LHS than on the RHS, so as the search is
progressing, LHS complexity grows a little more quickly than RHS
complexity. The complexity of ``1/(\fIx\fR-2)'' is 40, while the
complexity of ``\fIx\fR+1/e'' is only a little more complex at 42. But
the right-hand-side ``sqrt(7)'' is considered less complex (27) than
``2+\fIphi\fR'' (35). Since both pieces of an equation need to be
found before an equation can be reported, \fBries\fR is able to locate
both pieces of ``\fIx\fR+1/e = sqrt(7)'' sooner when the \FC-N\FT
option is given.
This is all made plain with the \fB--find-expression\fR option,
giving the expressions ``1/(\fIx\fR-2)'', ``2+\fIphi\fR'',
``\fIx\fR+1/e'' and ``sqrt(7)'' in postfix form, to reveal the order
in which they are generated by the search algorithm:
.fam C
.ps -2
ries 2.2772 --find-expression x2-r f2+ xer+ 7q
[7q] = 2.64575131106459, complexity = {27}
[x2-r] = 3.60750360750361, d/dx = -13.0141, complexity = {40}
[f2+] = 3.61803398874989, complexity = {35}
[xer+] = 2.64507944117144, d/dx = 1, complexity = {42}
.ps +2
.fam T
Then, repeating the same command with \fB'-N*/T'\fR shows that the
\FC[xer+]\FT is generated before \FC[f2+]\FT.
In the case of two equivalent solutions (like the 6.334119167042...
example earlier in this section), both equations come equally close to
the supplied value, but only one can be found first. Once the
first one is reported, the other is not, because \fBries\fR only
reports solutions that are at least a 0.1% closer match than the
previously-reported solution.
The \fB-l\fR option is meant to give control over the
number of solutions searched, but it actually controls the
number of LHS and RHS expressions generated. Because two RHS's
often have the same value, and only one (the first) gets kept,
the number of solutions checked (which is the RHS count
times the LHS count) depends on how often you get two LHS's or RHS's
with the same value. This happens particularly often when the
symbol set is severely restricted. If \fBries\fR tried to compensate
for this, the result would be that severely limited symbolsets would
take a very long time to run and would generate really long
equations. This is an important issue for those using \fBries\fR
to solve special problems, like the ``four fours problem''
exemplified by the command \FCries --numeric-anagram 4444 -Ox 17\FT.
The current implementation represents the author's
attempt at a reasonable tradeoff.
.SH BUGS
Performance does not degrade gracefully when the physical memory limit
is hit, because expression nodes are allocated in sequential order in
memory, without regard to where they will end up in the tree. This
could be improved in the future with percentile demographics and a
sort performed one time only, after the tree reaches a healthy (but
not excessive) size.
Although it tries to avoid it, \fBries\fR will often print more than
one equivalent solution. It misses the fact that the multiple
solutions are equivalent because of roundoff error. For example,
\FCries '-S4+-*/' -Ox 17\FT gives ``x-4*4 = 4/4'' and ``(4/(4-4*4))
x+4 = (4*4+4)/(4-4*4)'' (or a similar long form), only recognizing the
first as an exact match (the more complex one involves divisions by
multiples of 3, which require rounding).
Related to this, \fBries\fR sometimes gives an overly-complex answer,
again because of roundoff error. For example, \fBries\fR gets slightly
different values for ``2/3'' and ``1-1/3'', and stores both of these
in its database of RHS values. When reporting a solution in which both
sides of the equation equal 2/3, it might give ``1-1/3'' for
the right-hand side if it is closer to the (rounded) value of the
left-hand side.
This is particularly common if you request an exact match with a
zero or very small \fB--max-match-distance\fR value,
while giving an imprecise target value.
For example, \FCries 1.9739208802178 --max-match-distance 0\FT
might give ``cospi(1/(5 x)^2) = cospi(1/pi^4)'' whereas
\FCries 1.9739208802178715 --max-match-distance 0\FT gives the expected
``5 x = pi^2''. However, I've only gotten complaints about this from
users who give \fBries\fR a problem to which they already know an answer.
If you know that 1.9739208802178... is \fIpi\fR^2/5, then you don't
really need \fBries\fR to tell you that, do you? And if you're searching
for things that approximate \fIpi\fR^2/5, such as sqrt(sqrt(e^2+1)+1)
= 1.9739267..., you can use the \fB--min-match-distance\fR option.
In deeper searches or with target values larger than 10^5,
\fBries\fR might occasionally report ``solutions'' that are actually
tautologies empty in meaning. A typical example is
``\fIx\fR^(4/ln(sqrt(\fIx\fR))) = sqrt(e)^(4^2)'' (which is true
for any value of \fIx\fR), but \fBries\fR handles that case and most
others like it. If you suspect the solutions it gives, use the
\fB-Ds\fR option to show all calculations behind each proposed
solution. The options \fB--derivative-margin\fR,
\fB--min-match-distance\fR, and \fB--significance-loss-margin\fR may
help avoid meaningless results.
You can also use \fB-Ox\fR to force \fBries\fR to use only a
single \fIx\fR in each equation, which will prevent these tautologies
entirely, but will also prevent the discovery of interesting solutions
like \fIx\fR^\fIx\fR = 10.
.SH ACRONYM
\fBries\fR (pronounced ``reese'' or ``reeze'') is an acronym for
``RILYBOT Inverse Equation Solver''. The expansion of
\fIRILYBOT\fR includes two
more acronyms whose combined length is greater than 11. The full
expansion of \fBries\fR grows without limit and is well-defined but not
primitive-recursive. Contact the author for more information.
.SH AUTHOR
Robert P. Munafo (contact information on mrob.com)