RIES(1) RIES(1)
NAME
ries - find algebraic equations, given their solution
SYNOPSIS
ries [-ln] [-i[e]] [-s] [-x] [-Fn] [-Ssss] [-Nsss] [-Osss] [-Dxxx]
[-pfilename] [--extended-options [...]] value
ries --find-expression [expression [...]]
ries --eval-expression [expression [...]]
DESCRIPTION
Given a number, ries searches for algebraic equations in one variable
that have a solution (root) near that number. It avoids trivial or re-
ducible solutions like ``x/x = 1''. If value is an integer, ries 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 ries's
output will resemble the following:
$ ries 2.5063
Your target value: T = 2.5063 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}
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 expres-
sions with '=' in the middle), solution of equation (value of x
expressed as T plus a small error term), and total complexity score
(described below).
Each match is checked by solving for x using the Newton-Raphson method,
and the closeness of the match is judged by the difference between the
root (the value of x for which the two sides are equal) and your target
value T.
Options allow complete control over what symbols, constants and func-
tions are used in solutions, or to limit solutions to integer, ratio-
nal, constuctible, or algebraic values.
OPTIONS
Options must be separate: `ries -l1 -i -Ox 27', not `ries -l1iOx 27'.
-pname Profile (or Parameters): Load one or more options from file
name. -pname is equivalent to --include name, which is described
in the EXTENDED OPTIONS section below. -p alone (with no name)
has special meanings, also described under --include.
-ln Level: Specifies the level of the search (default 2). With each
increment of -l, ries will search about 10 times as many equa-
tions, 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, mea-
sured on a Core i7 at 3.2 GHz (using only one thread) invoked by
the command ries -ln 2.5063141592653589 for different values of
searchlevel n:
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
(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. ries 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 -l5.5) for more precise control
of how much memory ries will use before stopping its search.
When free memory is exhausted; performance will degrade signifi-
cantly and ries might exit, depending on your operating system.
Under Linux and Mac OS, ries keeps running but the system slows
to a crawl. If you don't know what your OS will do, be careful
before running ries 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 -S, -O and -N or when -i is
specified. -i in particular should allow about two more levels
in any given amount of memory. Large arguments tend to lengthen
runtime: for example, ries -l4 1058073667 takes about three
times as long as ries -l4 1.058073667.
Options to Select Symbols
Several options are used to choose which symbols (constants, opera-
tions, and functions) ries is allowed to use when searching for equa-
tions.
-Nsss Never use these: -N followed by one or more characters specifies
symbols (constants and operators) that ries should not use in
its equations. The symbols are as follows:
1-9 The integers 1 through 9. (ries constructs all larger
integers from combinations of these.)
p pi = 3.14159...
e e = 2.71828...
f phi = (1+sqrt(5))/2 = 1.61803...
n Negative
r Reciprocal
s Squared
q Square root
S C Sine, Cosine
T Tangent
A Two-argument Arctangent: arctan(s,c) is the angle of the
ray from the origin through the point (c,s). This func-
tion always gives a value in radians, and is unaffected
by the --trig-argument-scale option.
l ln (natural logarithm, also called log)
E e to the power of x
+ - Add, Subtract
* / Multiply, Divide
^ Power: 2 ^ 3 = 8
v Root (the ``v'' resembles part of the radical symbol): 3
v 27 = cube root of 27
L Logarithm to base A of B
W Lambert W function. Only available if using the stand-
alone maths library, described in the section ``STAND-
ALONE MATHS LIBRARY'' below. In addition, one must
explicitly choose it with -EW.
There are lots of potential uses for -N. For example, if you
invoke ries 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
-NlL to your command line, and all such solutions will be
skipped.
If you are checking an unknown number that you found in the con-
text 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 -N to rule out func-
tions you don't think are relevant.
Note that ries 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.
-Ssss permitted Symbol Set: Specifies a symbol set, as with -N, but
has the opposite effect: only these symbols will be used. A -S
also cancels any -E, -N or -O options that were given; meaning
that if you wish to combine these options, the -S should come
first.
-S can be used to solve those old problems of the sort ``How can
the number 27 be expressed using only the four basic operators,
powers and square roots, and the digit 4?'' The answer is given
by:
ries '-S+-*/^q4' 27 --one-sided
(The --one-sided option is described below). To solve the same
problem using the -N option, you'd need something like:
ries -Npef12356789rslLvE 27 --one-sided -l4
If you give the -S option with no symbols, ries will display a
table of all available symbols (as modified by any -E, -N, -O
and -S options that precede it) with their definition and
weights. This lone -S can be given along with a normal -S
option, but in any case ries will exit after showing the table.
-Esss Enable: Enables (or re-enables) the use of symbols that may have
been disabled by an earlier -S or -N option. This is mainly
intended for use in combination with the --include option. If
one include file disables some symbols, this option can be used
to re-enable some or all of them. It is also required for use of
the Lambert W function, which is disabled by default. See
``STAND-ALONE MATHS LIBRARY'' for instructions and use the
option -EW to request use of the Lambert W function in a ries
command.
-Osss Only One: Specifies symbols which should appear no more than
once on each side of the equation. This option can be combined
with -E or -N, in which case they augment each other. If used
with -S with the same symbol, the latter option takes effect.
One additional symbol is available with -O:
x The variable on the left-hand-side
Thus, you can use -Ox to limit ries's output to equations that
have only one 'x' in them and are therefore easy to solve for x
using only the most basic algebra techniques. This also makes
ries's output more like that of traditional expression-finders,
which search for expressions equal to x rather than equations in
x. Here's an example: ries -i 16 gives the answer ``x^2 - x =
3^5'' with x very close to 16, because 16^2-16 is close to 3^5.
ries -i 16 -Ox replaces that answer with ``5 x^2 = 6^4''.
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.
-i 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 x = 7^3''
where x=173. -i is interpreted as -r (described below) if the
supplied target value is not an integer.
-ie Integer, Exact. Like -i, but exits after reporting an exact
match (if found). This is equivalent to -i combined with --min-
match-distance 0. To not report any inexact matches at all, use
--max-match-distance 0 (described in more detail below).
-r Rational: Require that all equations have a single x and that
all subexpressions not involving x 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 -N and -E options to allow only addi-
tion, subtraction, multiplication and division, excluding irra-
tional constants and transcendental functions, etc.
-re Rational, ``Exact''. Like -r, but exits after reporting an
``exact'' match (if found). This is equivalent to -r combined
with --min-match-distance 0.
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 --max-match-distance with a very
small but nonzero epsilon (this option is described in detail
below).
-c Constructible: Require that all equations have a single x and
that all subexpressions are ``constructible'' in the sense of
Euclid's Elements, given a unit interval. This is like the -r
option except that squares, square roots and the golden ratio
phi are also allowed. All results will be easily solvable for x
and will use only addition, subtraction, multiplication, divi-
sion, and square roots.
If you add the option -Ex, more than one x may appear in
the solution. This gives answers that, when solved for x, are
not constructible from the unit interval, but both sides of the
(unsolved) equation are constructible given x and a unit inter-
val. For example ries 1.3263524026321 -c -Ex finds the solution
``x x^2 = 7/3''; x is the cube root of 7/3 which is not itself
constructible. ries's answer reflects the fact that 7/3 can be
constructed from its cube root (although the opposite construc-
tion is impossible).
-a Algebraic: Generate equations whose roots are ``algebraic num-
bers''. This is similar to the -c option, but also allows n^th
powers and roots. The forward trigonometric functions (sine,
cosine, and tangent) are allowed but their arguments will always
be rational multiples of pi; arctangent is not allowed because
for example tan^-1(1) is pi/4. More than one x is allowed
(unless you follow -a with the option -Ox) so the equations
might not be easy to solve.
-Ox Using the -O option (described above) with the symbol 'x' tells
ries to limit its search to solutions that can be expressed in
``closed form'' using the basic constants, elementary and tran-
scendental functions. This concept of ``closed-form number'' is
described by Timothy Chow in his 1998 paper What is a closed-
form number?.
-l Liouvillian: Generate equations whose roots are ``EL numbers''
(exponential-logarithmic), as described by Timothy Chow in his
1998 paper What is a closed-form number?.
Options Affecting Output Format
-s Sorta Solve, by Shifting to right-hand side: With this option,
ries will display equations with just a single ``x'' on the
left-hand side of the equal sign. It isn't ``solving'' the equa-
tions, but merely performing algebraic transformations to move
everything except one x to the right-hand side: ``x(x+1) = 7''
becomes ``x = 7/(x+1)''. You can combine this option with -Ox to
eliminate this issue, but with that option ries will no longer
find solutions that require more than one ``x'', like ``x^x =
2'' for 1.55961.
-x X Values: Print actual values of x (the roots of the equations
found) rather than expressing x as T plus/minus a small number,
where T is your target number. ``--absolute-roots'' is a syn-
onym for -x.
-Fn Format: Controls the way expressions are formatted in the main
output. If n is omitted it is 3 (``-F'' for ``FORTH Format'');
if -F is not specified at all, the format will be 2. The follow-
ing formats are available; each shows the output of ries
1.506591651 -Fn:
Format 0: Compressed FORTH-like postfix format: Each operator
and constant is just a single symbol. The symbols are as listed
above under the -N option. This is designed to be easy to parse,
and the parsing program pf2if.pl (available from the same place
you got the ries source code) shows how.
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}
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(x)) = p-1''
means ``sqrt(ln(x)) = pi - 1''.
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}
Format 2: Standard infix expression format (this is the
default).
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}
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 ries (to get the exact,
condensed format, use -F0). This is intended mainly for use by
scripts that use ries 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.
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}
Most of the symbols used by -F3 are self-explanatory. The nonob-
vious ones are: neg for negate, recip for reciprocal, dup* for
square, sqrt for square root, ** for power (A^B), root for Bth
root of A, logn for logarithm (to base B) of A. For these last
three, A is the first operand pushed on the stack and B is the
second.
The setting of -F does not affect expressions displayed by the
various -D diagnostic options (most of these use -F0, and -Ds
(``show your work'') uses -F2).
You may use the --symbol-names option (described below) to
redefine the appearance of formats 2 and 3.
-Dxx Display Diagnostic/Debugging Data: A detailed understanding of
the ries algorithms (described below) is assumed. -D is followed
by one or more letters specifying the messages you want to see.
Options A through L and a through l (except E and e) apply to
the LHS and RHS respectively. For each option, the number of
lines of output that you can expect from a command like ries -l2
2.5063141592653589 -Dx (with x replaced by a single letter) is
shown:
A,a [42836; 87770] show partial expressions that are
``pruned'' (ignored) because of arithmetic error (e.g.
divide by zero)
B,b [3173; 2714] show partial expressions pruned for being
zero, or derivative nearly zero; or outside range given
by --min-equate-value and --max-equate-value
C,c [81056; 697227] show partial expressions pruned for being
non-integer (and -i option was given); or irrational (and
-r option given); etc. (sample command is ries -l2 1047
-i -DC )
D,d [1751; 4350] show partial expressions pruned because of
overflow
E,e [102356; 272746] show expressions pruned because their
value matches one already in database
F,f [349368; 848882] show --canon-reduction operations on
expressions before adding to database (sample command is
ries -l2 2.50631415926 --canon-reduction nr25 -DF )
G,g [96112; 97337] show expressions added to database
H,h [409175; 816240] show attributes of each partial expres-
sion tested
I,i [3904331; 7759741] show each new symbol to be added
before complexity pruning
J,j [2579116; 5102516] show symbols skipped by complexity
pruning
K,k [257302; 558199] show symbols skipped by redundancy and
tautology rules
L,l [61994; 114453] show symbols skipped to obey -O option
(sample command is ries -l2 2.50631415926 '-O-+/^v*qsr-
lLeEpf' -DL )
m [10247603] show all metastack operations
M [46] show memory allocation benchmarks, and enable auto-
matic exit when memory gets slow (see --memory-abort-
threshold option)
n [136] show Newton iteration values and errors if any
N [461] show work in detail: operator/symbol, x and dx at
each step
o [539235] show match checks
p [112] show preprocessing transformations prior to conver-
sion to infix
Q [51] show manipulations to remove --canon-reduction from
equations before root-finding
q [140] show close matches dispatched to Newton and results
of test
r [1806085] show results (value and derivative of operands
and result) for each opcode executed
S [100] show solve-for-x work: displays all operations per-
formed by the --try-solve-for-x option to transform an
equation into ``solved'' form
s [277] show your work: displays values of each subexpres-
sion for every reported answer. Subexpressions are shown
in normal (infix) syntax, which is useful in combination
with -F0 to see the postfix format used with options like
--eval-expression
t [11017] show all abc-forms passed to expression genera-
tion
u [48895] show steps of min/max complexity ranging for each
abc-form
v [5525] show number of expressions generated by each abc-
form
w [32922] show details of abc-form generation (pruning,
weights, etc.)
x [91] show all rules used (varies with the -N, -O, and -S
options)
y [736] statistics and decisions made in main loop
z [55] initialization and other uncategorized messages
0 [1712490] list the entire expression database after every
pass through the main loop
Of these, -Ds is probably the most useful and fun to look at.
-Dy 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 grep; this can tell you why a certain subex-
pression is or is not appearing in results. -DG and -Dg can be
useful if you want to use ries to generate a massive list of
expressions for processing by another program; for this reason
its output uses infix notation. Most other -D options print
subexpressions in the -F0 terse postfix format.
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 --include files.
--include filename
-pfilename
-p Load one or more options from file filename. The options
``--include filename'' and ``-pfilename'' are equivalent; note
that one requires a space before filename and the other cannot
have a space (-p alone has a related function, described below).
ries will attempt to open the named file (which may be a simple
filename or a path), or the given name with ``.ries'' appended.
If either is found, ries will scan it for parameters and argu-
ments 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 ``hst.ries'' containing the following:
# 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"
then giving the option ``-phst'' is equivalent to giving the
options ``--trig-argument-scale 1.74532925199433e-2 -NLleEv -Ox
-x'', in that order.
A parameter file may additionally invoke another parameter
file with the --include option. When it encounters this option,
ries 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) ries will exit with an error.
It is an error for --include or the end of an included file
to come between an option and its arguments. For example, ``ries
1.2345 --eval-expression --include expressions.txt'' will pro-
duce an error regardless of the contents of ``expressions.txt'',
because --eval-expression must be followed immediately by its
argument(s). On the other hand, if ``expressions.txt'' contains
the --eval-expression option, like so:
# expressions.txt: Useful functions of one argument
--eval-expression
xsr # 1/(x^2)
1xq-r # 1/(1-sqrt(x))
2xl1+^ # 2^(ln(x)+1)
then the command ``ries 1.2345 --include expressions.txt''
works, and shows the values of the three expressions where x is
1.2345.
If you have a file called ``.ries_profile'' or ``ries_pro-
file.txt'' in your home directory, ries will load it as if you
specified it with a --include at the very beginning of the
parameters. If you have such a file and wish to prevent it from
being used, give a bare -p (without a filename) at the very
beginning of your ries options. If you wish to give some parame-
ters and have .ries_profile loaded after your parameters,
include -p again at the point where you want ries to use the
profile. For example, if your .ries_profile contains ``--trig-
argument-scale 1'' and you have a hst.ries with contents as
shown above, then giving the optiions ``-phst -p'' will use all
of the settings in hst.ries except the --trig-argument-scale.
Two more example profiles are the ``Latin'' and ``Mathemat-
ica'' settings files linked from the top of the main RIES web-
page.
--absolute-roots
This is a synonym for the -x option, described above.
--algebraic-subexpressions
This is a synonym for the -a option, described above.
--any-exponents
This option cancels any restrictions on subexpressions used as
an exponent, such as those set by the --algebraic-subexpressions
and --liouvillian-subexpressions options.
--any-subexpressions
This option cancels any restrictions on subexpressions, as
imposed by options such as --algebraic-subexpressions. This
might be useful if you are using one of the class selectors like
-a or -c as shorthand for all the restrictions of that particu-
lar class, and then re-enable a function like e^x using -EE.
--any-trig-args
This option cancels any restrictions on subexpressions used as
an argument to a trigonometric function, such as those set by
the --algebraic-subexpressions and --liouvillian-subexpressions
options.
--canon-reduction symbols
Apply simple transformations in an effort to make all expression
values fall in the range [1 ... 2). This option improves the
efficiency of the ries 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 ries will try to apply to expressions:
n Negate expressions when possible to make all values posi-
tive.
r Take the reciprocal when possible to make all expressions
fall outside the range (-1 ... 1).
2 Multiply by 2 when possible to increase the magnitude of
expressions in the range (-1 ... 1).
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).
In these descriptions, the words when possible refer to the fact
that --canon-reduction will respect any limits imposed by the
symbolset options -N, -O and -S. So if you use the option -On
together with --canon-reduction n, the negation operator will
still be used only once per expression.
Although it makes ries more efficient, this option also
causes the printed results to have greater complexity scores,
and complexity scores will increase somewhat more erratically.
ries will try to simplify its printed results by undoing
--canon-reduction transformations on both sides of the equal
sign. For example, ries 2.50618 --canon-reduction nr25 might
yield the result ``x^x/2 = (1+9)/2'', which simplifies to ``x^x
= 1+9''. But when only one side has a ``/2'', ries cannot fix
it, so the same example gives an overly complex ``1/(pi-x) =
pi/2''.
--canon-simplify
When reporting a match, remove common factors or terms from both
sides of the equation. This is the default.
--constructible-subexpressions
This is a synonym for the -c option, described above.
--derivative-margin value
Specify the limit to how small the derivative of any expression
or subexpression containing x can be in relation to the expres-
sion's value. By default this is 10^-6, so that an expression
containing x 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 ``x'' (with a derivative of
1.0) would be rejected. So if the magnitude of your target is
larger than 10^5, ries will set this limit correspondingly
lower. For example, if your target value is 10^8, ries automati-
cally sets a --derivative-margin 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 ries
12345 --derivative-margin 8e-5, derivatives of expressions can
be as small as 8x10^-5 times the expression's value. In calcu-
lating any possible answers, ries would allow ``x^2'' because
the ratio between d/dx x^2 and x^2 is about 1.62e-4, which is
big enough to surpass the margin. However, any answers involving
``sqrt(x)'' would be rejected because the ratio between d/dx
sqrt(x) and sqrt(x) is only 4.05e-5:
expression value d/dx(expr.) ratio
x^2 152399025 24690 1.62e-4
x 12345 1.0 8.1e-5
sqrt(x) 111.108 0.0045 4.05e-5
Indeed, the results of ries 12345 include the equation
``2(sqrt(x)-9) = (2 e)^pi'', but with the option --derivative-
margin 8e-5 that answer is left out.
--explicit-multiply
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 ries output into a calculator, com-
puter program or spreadsheet formula.
--integer-subexpressions
This is a synonym for the -i option, described above.
--match-all-digits
--mad Request that all reported matches should match all of the sup-
plied digits. This is equivalent to adding a '5' to the end of
your target value, along with a --max-match-distance value equal
to the magnitude of this appended '5' digit. It also selects the
--x option, unless the --wide option is also given.
For example, the command ries 2.5063 --mad is equivalent to
ries 2.50635 --max-match-distance 0.00005 -x, and the first
reported match is x^2+e = 9, which is true for x = 2.506335...
Without --mad it reports 2 x = 5 and a few other answers that do
not match all of the digits in 2.5063.
--max-equate-value value
--min-equate-value value
Specify the maximum and minimum values for the LHS and RHS of
any reported equations. For example, the command ries 2.50618
would normally give ``2 x = 5'' as the first solution; both
sides of that equation are about 5. But the command ries 2.50618
--max-equate-value 3 instead gives ``x-2 = 1/2'' as the first
answer: this is an equivalent solution, but expressed as an
equation in which both sides of the equal sign are less than 3.
Similarly, ries 2.50618 --min-equate-value 27 gives the answer
``(e^x)^2 = e^5''.
--max-match-distance value
Specify the maximum distance between your given target value T
and the roots x of any reported equations. This sets a minimum
level of accuracy, overriding the default, which is 1% of the
size of your target value. For example, the command ries 2.5063
will use a threshold that is 1% of 2.5063, or about 0.025. It
gives as its first answer the equation 2x = 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 ries 2.5063 --max-match-distance 0.001, then 2x
= 5 is not reported because 0.0063 is bigger than your threshold
0.001; instead the first match will be ``x^2 = 2 pi'' (which
comes within about 0.0003 of the target 2.5063).
Use a zero argument to specify that ries should only report
an ``exact'' match, if any (and note that this ``exact'' match
might be more complex than the obvious answer, because of round-
off errors; see UNEXPECTED BEHAVIOR and BUGS below). Note that
this is different from --min-match-distance 0, 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, --max-match-dis-
tance -0.001 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 --max-match-distance is so stringent that
the first match takes longer than 2 seconds, ries will display
progress messages until a match is found. Use the --no-slow-mes-
sages option to suppress these.
There is also a --min-match-distance option (described
below), which serves an entirely different purpose.
--max-matches N
-nN Limit the number of reported matches to a positive integer N.
This is particularly useful with certain options (such as --no-
refinement) that generate a lot of matches. The default N is
100.
--max-memory size
This option tells ries not to use more than the given amount of
memory (size specified in bytes). This is particularly useful in
combination with a high -l (search level) option. For example,
if you typically have about 2 gigabytes of free memory on your
machine, you could invoke ries with the option --max-memory
1.0e9, to ensure that it never uses more than 1 gigabyte of mem-
ory regardless of the search level.
ries also has an (experimental) feature that can automati-
cally detect when your system is slowing down; see the --memory-
abort-threshold option for details.
--max-trig-cycles value
Specify the maximum and minimum values for arguments to the
trigonometric functions, in multiples of one half-period. The
default is 1.0, which corresponds to an allowed range of -pi to
pi. The option --max-trig-cycles 4 would allow arguments from
-4pi to 4pi. In combination with --trig-argument-scale, this
argument is applied after the scaling, so that the number of
``cycles'', ``periods'' or ``waves'' remains the same. For exam-
ple, if the value pi/180 is used for --trig-argument-scale and
if --max-trig-cycles is set to 2, then the allowed range for
arguments to the trigonometric functions is -360 to +360 (which
is interpreted as being in units of degrees), corresponding to
two full cycles of the trigonometric functions. This option
does not affect the arctan2 function, which always returns
results from -pi to pi (1 full cycle).
--memory-abort-threshold N
This option is used with the -DM option, and overrides the
default slowness measurement after which ries will automatically
exit. With the -DM option, ries 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 --memory-abort-threshold for
more than 3 of the past 10 measurements, ries will exit. The
default --memory-abort-threshold 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.
NOTE: --memory-abort-threshold is an experimental ries fea-
ture and is likely to change in future versions of ries.
--min-match-distance value
Specify the minimum distance between your given target value T
and the roots x of any reported equations. This is useful for
finding approximate formulas for constants that have a known,
simple formula. For example, using the command ries
3.141592653589 -x --min-match-distance 1e-8 -NSCT one can dis-
cover the following approximate formulas for pi :
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
Among these results (after solving for x) are the ancient
approximations 19/6 and 22/7, and the more modern curiosity e^pi
= 20 + pi (which is called ``Gelfond's constant''). Other inter-
esting results can be found by omitting symbols with -N or by
using restricted classes. For example ries 3.1415926
-NASCTlLfEevp --min-match-distance 1e-8 (excluding most of the
scientific functions) gives the fraction approximation 355/113
in the form ``1/(x-3)-1 = 1/4^2+6''; and ries 3.1415926 -r
--min-match-distance 1e-8 (requesting only rational approxima-
tions) gives 355/113 in the form ``1/(x-3)-3 = 1/(4*4)+4''.
You will often get multiple equivalent results. In the
above example, the equations sqrt(1+pi) x = e^3/pi and x^2/e^3 =
1/sqrt(1+pi) can both be converted into the approximate rela-
tion:
pi = sqrt(sqrt(e^6/(pi+1)))
(which does not converge on the true value of pi if iterated).
If you give a value of 0: ``--min-match-distance 0'', and
ries finds an ``exact'' match, it will exit and report no fur-
ther results. Note that this is different from --max-match-dis-
tance 0, which will only print the ``exact'' match and will not
print any inexact matches.
There is also a --max-match-distance option (described
above), which serves an entirely different purpose.
--min-memory size
If ries is given the debug option -DM, 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 ries is
allowed to use all of your system's memory. (This feature is
only active with the -DM option because it is still being
tested).
When --min-memory is given in combination with -DM, it will
ensure that ries does not exit because of slow memory response
until at least size 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 com-
pute-intensive tasks, you could invoke ries with the options -DM
--min-memory 1.0e9, 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 ries' memory usage, use the
--max-memory option (without -DM) or use a suitably small -l
search level.
--no-canon-simplify
When reporting a match, do not remove common factors or terms
from both sides of the equation. This is useful mainly in combi-
nation with --max-equate-value and --min-equate-value. For exam-
ple, the command ries -Ox 2.6905 would normally give the answer
``x-2 = ln(2)'' in which both sides of the equation are about
0.693. Adding the options --no-canon-simplify and --min-equate-
value 1 reports the same answer as ``1/(x-2) = 1/ln(2)'' in
which both sides of the equation are about 1.443.
--no-refinement
After reporting a match, do not require that the next match come
closer to the target. This causes ries 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 ries 1.51301 --no-refinement
yields the solutions e^(x^2) = pi^2 and x/sqrt(2) =
sqrt(ln(pi)), both with the root 1.513096088... This option is
most effective in combination with --max-matches along with
--match-all-digits or --max-match-distance (using a stricter
argument than the default -0.01).
--no-slow-messages
Suppress the ``Still searching'' messages that ries would nor-
mally print if a search takes longer than 2 seconds without giv-
ing any results.
--no-solve-for-x
This option cancels the ``--try-solve-for-x'' option.
--numeric-anagram digits
Give a specific set of digits that can be used as constants in a
solution; this forces the --one-sided option. It will use only
as many of each digit as you specify. For example, if you give
``111223'' as the digits, ries 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:
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
--numeric-anagram can be used to set hard limits on the digits,
the constants e, phi, and pi, and the ``squared'' and ``recipro-
cal'' symbols. When you use --numeric-anagram, any of these sym-
bols that you do not list will be forbidden just as if you had
used the -N option. The last two examples here show the use of
``s'' to specify the squaring operation x^2 as distinguished
from any use of ``2''.
--one-sided
Force ries to ignore all LHS expressions except x. This results
in ``one-sided equations'' with x on the left-hand side. This
makes ries much slower, but all of its output will be ``solved
for x''.
This option is intended as a convenience for very special
problems (for example, --numeric-anagram automatically turns on
--one-sided), but it is not generally useful because the speed
advantage of the normal RIES bidirectional search algorithm is
lost. If you want ries to give answers that are solved for x,
use the --try-solve-for-x option possibly along with -Ox.
If you use this option, ries will be a lot slower and its
solutions for a given search level will not be nearly as accu-
rate. Whereas a normal ries search might quickly match your
target value to the first 10 decimal places, a search with the
--one-sided 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 -i, -N, -O, and -S.
--rational-exponents
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 -a or --alge-
braic-subexpressions option (described above).
--rational-subexpressions
This is a synonym for the -r option, described above.
--rational-trig-args
Require any argument to the forward trigonometric functions
(sine, cosine, tangent) to be a rational subexpression. The
restriction applies to the argument's value before multiplying
by the --trig-argument-scale (if any). This option exists mainly
to support the -a or --algebraic-subexpressions option
(described above).
--relative-roots
When printing each equation, show the root as T plus/minus a
small number (where T is your target number) rather than as the
actual value of the root. This is the default, so you'll only
need to use --relative-roots to cancel a -x or --absolute-roots
option in your .ries_profile or another --include file.
--ries-arguments-end
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 --include option above) ries will ignore
the rest of the contents of that file and continue with the next
option after the --include or -p option that invoked the pro-
file.
--show-work
This is a synonym for the -Ds option, described in the -D
(debugging/diagnostic) option above.
--significance-loss-margin digits
Specify the number of significant digits that may be lost in a
calculation. By default, ries tolerates a loss of 2 digits in
any calculation. For example, if x is 0.906402477... (the value
of Gamma[5/4]), ries would not use x+e^5 in any of its expres-
sions, because e^5 is more than 100 times as large as x. Due to
round-off, more than 2 digits of the value of x would be lost in
the addition. This restriction applies to constant expressions
too, so ries 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 ries behavior corresponds to a --significance-
loss-margin option with an argument of 2.0. Give a higher value
to allow more digits to be lost in calculations.
Conversely, if you suspect ries is generating meaningless
results due to round-off error, you can look at its calculations
in detail with the options -Ds and -F0, then evaluate specific
expressions with --eval-expression (described below). If it
seems appropriate, make ries more strict by giving --signifi-
cance-loss-margin with a lesser argument.
--symbol-names :sym:name [ :sym:name ... ]
This option allows you to set the ``names'' of individual sym-
bols. This affects how the expressions and equations are
printed, including in the -F2 and -F3 modes, and by special com-
mands such as --eval-expression. In addition to the symbols
listed above in the -N option, you may also define these sym-
bols:
( ) brackets to group sub-expressions
= equality symbol
Here are examples of the normal ries output, modified by chang-
ing the appearance of the exponentiation operator and parenthe-
ses. The single-quotes around each option are to avoid substitu-
tion by the shell:
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}
More examples of the use of --symbol-names are found in the
``Latin'' and ``Mathematica'' settings files linked from the top
of the main RIES webpage.
--symbol-weights N:sym [ N:sym ... ]
This option allows you to adjust the ``weights'', or complexity
ratings, of individual symbols. Use the option -S to see the
normal weights, then use this option to change one or more. Com-
pare these two examples; in the second one the cost of the sym-
bol x is reduced, and the costs of 2 and s (squared) are
increased.
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}
With the smaller weight of 5, x is considered ``less expen-
sive'', and ries uses x more often in its answers; and with the
number 2 and squaring more expensive, these show up less often
in the results. In many cases the new results are equivalent,
and ries has simply found a different way to get there.
Since the arguments of --symbol-weights 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.
NOTE: Changing the symbol weights can greatly reduce ries'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 experi-
ment with changing just one symbol-weight at a time to find
which is causing the problem.
--trig-argument-scale value
Specify a constant by which the argument of the sine, cosine and
tangent functions should be multiplied (however it does not
affect arctan2). By default this value is pi and the trig func-
tions are called sinpi, cospi and tanpi. sinpi(x) is the sine of
pi times x; so for example sinpi(1/3) is the sine of pi/3, which
is half the square root of 3. A full circle is 2 in these units:
sinpi(x) = -sinpi(x+1) = sinpi(x+2) for all x.
If you give this option, arguments will be multiplied by
the number you give instead of by pi. Useful values to give are:
6.2831853071795864769
This is ``tau'' (2 pi); use it to get units of 1 per
``full turn'': ``sin(1/16)'' will give 0.382683...
1 Use 1 to get natural units (radians): ``sin(pi/8)'' will
give 0.382683...
1.74532925199432957692e-2
This is pi/180, and is used for degrees: ``sin(22.5)''
will give 0.382683...
1.57079632679489661923e-2
This is pi/200, and is used for grads (also called gradi-
ans or gons): ``sin(25)'' will give 0.382683...
If you use this option, ries will call the functions sin, cos
and tan in its output, and the scale will be displayed after the
function definitions at the end.
--try-solve-for-x
-s --try-solve-for-x is equivalent to ``-s'', which is described in
the OPTIONS section above.
--version
Displays information about the version of ries, the calculation
precision and math library, any optional module(s), the cur-
rently enabled profile (see the --include option at the begin-
ning of the EXTENDED OPTIONS section above), and a brief copy-
right notice. The version is a date, such as ``2013 Jun 3''.
--wide-output
Use a wider (132-column) output format. This shows the roots of
equations (values of x) both in terms of the the actual value of
x, and as T plus/minus a delta; it also shows the ratio between
this delta and x as ``(1 part in N)'' where N is the delta
divided by x. For example, if x is 2.5 and target value is
2.501, the delta is 0.001, which is ``1 part in 2500''.
SPECIAL COMMANDS
ries provides some functions that supplement its main purpose. Com-
mands and their parameters must be separate: `ries 1.23 --trig-argu-
ment-scale 0.5', not `ries 1.23 --trig-argument-scale 0.5'. 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.
--eval-expression forth-expr [forth-expr ...]
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 syn-
tax that is displayed when you use the -F0 option. The symbols
are as listed above under the -N option. For example, xxq- is
the syntax for x-sqrt(x). Syntax errors and computation errors
such as overflow are reported; however successful execution by
--eval-expression does not guarantee that the expression will be
found in an actial ries search. For example, an expression con-
taining x only appears as the left-hand-side of an equation if
the x is the first symbol in the postfix form: ries will use x2+
(x+2) but will not use 2x+ (2+x).
--find-expression forth-expr [forth-expr ...]
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 -N option. For example, xxq- is the syntax for x-sqrt(x).
This command is useful for diagnostics; an example is given
below in the UNEXPECTED BEHAVIOR section.
STAND-ALONE MATHS LIBRARY
The ries source code includes an auxiliary file, msal_math64.c, which
can be downloaded from the same place as the main RIES source code and
this manual. It provides some of the standard trignonmetric functions,
whose implementation have been found to vary across different releases
of the standard libm. This is useful if you are running ries on a vari-
ety of newer or older systems and want to be able to rely on consistent
results.
msal_math64.c also provides the Lambert W function, defined to the sym-
bol 'W'.
To use msal_math64.c, compile ries in the normal way but with the addi-
tional compiler option -DRIES_USE_SA_M64. The resulting ries binary
will report ``mathlib: stand-alone'' when given the --version option.
Once ries has been compiled with the stand-alone maths library, the
Lambert W function is available by giving the option -EW on the ries
command-line. Its default weight is set to make it occur a little more
often than the two-argument exponential and root functions; use the
--symbol-weights option if you want to change this.
ALGORITHM
ries begins its search with small, simple equations and proceeds to
longer, more complex ones. It uses a set of complexity rules to compute
a measure (similar to Kolmogorov complexity), which determines the
order in which various candidate expressions are considered by ries.
For example:
1. If you add a symbol to an equation, the result is more complex:
x + 1 = 3 is more complex than x = 3
x + 1 = ln(3) is more complex than x + 1 = 3
x - 7 = 4^2 is more complex than x - 7 = 4
2. If two equations are the same except for one number, the equa-
tion with the higher number is more complex:
x + 1 = 5 is more complex than x + 1 = 3
x^3 + 1 = 3 is more complex than x^2 + 1 = 3
3. If two equations are the same except for one symbol, the equa-
tion with the "more exotic" symbol is more complex:
x ^ 5 = 3 is more complex than x + 5 = 3
As ries searches it finds solutions -- these are equations for which x
is close to being an exact answer. Each time it finds a solution it
prints it out. Then ries raises its standard for the next answer: The
next answer ries 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. ries 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, ries actually constructs half-
equations, called left-hand-side expressions and right-hand-side
expressions, 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 ries to find matches much faster. All LHS's contain
x 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 ries is able to check billions of equations in such a short
time.
The closeness of an LHS match depends on the value of x, and also on
the derivative with respect to x of the LHS expression. Because of
this, ries calculates derivatives of LHS's as well as their values.
There are dozens of optimization rules ries uses, like the following:
a+ Don't try ``K + K'' for any constant K because ``K * 2'' is
equivalent.
b+ Don't try ``3 + 4'' (or any two unequal integers from 1 to 5)
because another single integer (in this case ``7'') is shorter.
a* Don't try ``1 * K'' for any constant K because ``K'' is shorter.
b* Don't try both ``2 * 4'' and ``4 * 2'' because they are equiva-
lent.
c* Don't try ``K * K'' because ``K ^ 2'' is shorter.
ar Don't try ``1 / (1 / expr))'' for any expression expr because
``expr'' is shorter.
a^ Don't try ``2 ^ 2'' or ``2 squared'' because ``4'' is shorter.
b^ Don't try ``expr ^ 2'' for any expression expr because ``expr
squared'' is shorter.
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 -N, -O
or -S, some of these rules cannot be used. For each optimization rule
there are one or more symbolset exclude rules like the following:
Don't use rule a+ if either of the symbols '*' or '2' is dis-
abled.
In order to maintain maximum efficiency, ries checks each rule individ-
ually against the symbolset, and uses as many rules as it can. You can
see this process in action by trying a command like ries 1.4142136,
which gives the answer ``x^2 = 2''. If you disable the 's' (squared)
and '^' (power) symbols with ries 1.4142136 -Ns^, rule b^ goes away,
and the answer becomes ``x x = 2''. If you also disable '*' (multipli-
cation) the answer becomes ``x = sqrt(2)''. Disable 'q' (square root)
and you get ``log_2(x) = 1/2'' (the logarithm to base 2 of x is 1/2).
Disable 'L' and it becomes ``2/x = 2,/2'' (2/x is the square root of 2,
this time using the generalized root function). Disable 'v' and you get
``x/(1/x) = 2''. These are all trivially equivalent to the original
``x^2 = 2'', but as the easier operations like multiplication are
removed, exclude rules go away making it possible to get to the same
answer in an indirect way.
Continuing, we may disable '/' to get a trignometric identity involving
pi/4; disabling the trig functions as well, the command becomes ries
1.4142136 '-Ns^q*Lv/ASCT' and we finally get an answer that most people
probably would not guess: x-1/(x+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 gener-
ally increases as the solution becomes more and more obscure, and sim-
pler but poorly matching approximations like 1/cos(x) = 6 begin to
appear before the correct one.
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 ries 6.334119167042 --trig-argument-scale 1 -NT to report some-
thing like ``cos(sqrt(2)) x = sin(sqrt(2))''. Instead, ries gives
``sqrt(x^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 -S, -O and -N options often
causes unexpected changes in the output. For example, ries 2.2772
yields the solution ``1/(x-2) = 2+phi'' but ries 2.2772 '-N*/T' does
not give this solution in any form. This seems counterintuitive: there
was no *, /, or tan() in the ``1/(x-2) = 2+phi'' solution, so why did
ries decide not to report it?
In fact, the solution is still generated internally, but because
you have told it to exclude some operators, ries has to try other, more
exotic expressions sooner than it otherwise would. As it happens, the
next solution ``x+1/e = sqrt(7)'' (which matches the target value
2.2772 more closely than ``1/(x-2) = 2+phi'') ends up getting found
earlier.
The mysterious behavior results from the fact that ries always
tries to keep the number of LHS and RHS expressions equal as it per-
forms its search. Eliminating operators with the -N 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 complex-
ity grows a little more quickly than RHS complexity. The complexity of
``1/(x-2)'' is 40, while the complexity of ``x+1/e'' is only a little
more complex at 42. But the right-hand-side ``sqrt(7)'' is considered
less complex (27) than ``2+phi'' (35). Since both pieces of an equation
need to be found before an equation can be reported, ries is able to
locate both pieces of ``x+1/e = sqrt(7)'' sooner when the -N option is
given.
This is all made plain with the --find-expression option, giving
the expressions ``1/(x-2)'', ``2+phi'', ``x+1/e'' and ``sqrt(7)'' in
postfix form, to reveal the order in which they are generated by the
search algorithm:
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}
Then, repeating the same command with '-N*/T' shows that the [xer+] is
generated before [f2+].
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 ries only reports solutions that
are at least a 0.1% closer match than the previously-reported solution.
The -l option is meant to give control over the number of solutions
searched, but it actually controls the number of LHS and RHS expres-
sions 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 ries 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 ries to solve special prob-
lems, like the ``four fours problem'' exemplified by the command ries
--numeric-anagram 4444 -Ox 17. The current implementation represents
the author's attempt at a reasonable tradeoff.
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, ries will often print more than one
equivalent solution. It misses the fact that the multiple solutions are
equivalent because of roundoff error. For example, ries '-S4+-*/' -Ox
17 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).
This problem is common when the target is already known very pre-
cisely by the user. For example, ries 0.00088953230706449 gives the
(correct) answer ``ln(x)/pi = -sqrt(5)'', followed by redundant/equiva-
lent answers such as ``ln(x)/pi-2 = -(phi^3)'' (if ries was compiled
with regular precision) or ``sqrt(5),/x+4 = 1/e^pi+4'' (if compiled
with the -DRIES_WANT_LDBL option); but ries 0.000889532307 only gives
the first, simplest form of the answer.
Related to this, ries sometimes gives an overly-complex answer, again
because of roundoff error. For example, ries gets slightly different
values for ``2/3'' and ``1-1/3'', and stores both of these in its data-
base 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 --max-match-distance value, while giving an impre-
cise target value. For example, ries 1.9739208802178 --max-match-dis-
tance 0 might give ``cospi(1/(5 x)^2) = cospi(1/pi^4)'' whereas ries
1.9739208802178715 --max-match-distance 0 gives the expected ``5 x =
pi^2''. However, I've only gotten complaints about this from users who
give ries a problem to which they already know an answer.
If you know that 1.9739208802178... is pi^2/5, then you don't
really need ries to tell you that, do you? And if you're searching for
things that approximate pi^2/5, such as sqrt(sqrt(e^2+1)+1) =
1.9739267..., you can use the --min-match-distance option.
In deeper searches or with target values larger than 10^5, ries might
occasionally report ``solutions'' that are actually tautologies empty
in meaning. A typical example is ``x^(4/ln(sqrt(x))) = sqrt(e)^(4^2)''
(which is true for any value of x), but ries handles that case and most
others like it. If you suspect the solutions it gives, use the -Ds
option to show all calculations behind each proposed solution. The
options --derivative-margin, --min-match-distance, and --significance-
loss-margin may help avoid meaningless results.
You can also use -Ox to force ries to use only a single x in each
equation, which will prevent these tautologies entirely, but will also
prevent the discovery of interesting solutions like x^x = 10.
ACRONYM
ries (pronounced ``reese'' or ``reeze'') is an acronym for ``RILYBOT
Inverse Equation Solver''. The expansion of RILYBOT includes two more
acronyms whose combined length is greater than 11. The full expansion
of ries grows without limit and is well-defined but not primitive-
recursive. Contact the author for more information.
AUTHOR
Robert P. Munafo (contact information on mrob.com)
LICENSES
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with no
Invariant Sections, no Front-Cover Texts and no Back-Cover Texts.
ries itself is free software: you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the
Free Software Foundation, either version 3 of the License, or (at your
option) any later version.
ries and this document are distributed in the hope that they will be
useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Gen-
eral Public License for more details.
If you got ries.c from the website mrob.com, the GNU General Public
License may be retrieved at mrob.com/ries/COPYING.txt and the GNU Free
Documentation License may be found at mrob.com/ries/FDL-1.3.txt ; you
may also find copies of both licenses at www.gnu.org/licenses/
Manual version: 2024 May 20 RIES(1)