REF TRANSARITH                                  James Anderson, Oct 2010

           COPYRIGHT James Anderson 2010. All rights reserved.

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This file describes  how to  install Poplog and  the transarith  package
which  implements  transreal  and   transcomplex  arithmetic.  It   also
describes the identifiers, such as procedures and constants, defined  by
the package. Both  the implementation and  documentation comprising  the
package  are  available  as  on-line  appendices  to  a  journal   paper
describing transcomplex arithmetic.

         CONTENTS - (Use <ENTER> g to access required sections)

 --  1  Installation
 --  1.1  Install Poplog
 --  1.2  Install transarith implementation
 --  1.3  Install transarith documentation
 --  1.4  Test the transarith package
 --  2  Introduction
 --  3  Naming Conventions
 --  4  Constants
 --  5  Predicates Relating to Numbers
 --  6  Type Checking
 --  7  Comparisons on Numbers
 --  8  Number Representation
 --  9  Arithmetical Operations
 --  10 Mathematical Functions
 --  11 Test Procedures
 --  12 Transcript


1  Installation
---------------

Take the following three  steps to install  the transarith package  on a
computer running a Microsoft  Windows or a  Unix operating system.  Then
take the fourth step to demonstrate that the package is working. You can
check the installation at every step.

If you are new to Poplog it might take some time to work through these
steps.


1.1  Install Poplog
-------------------

Install Poplog  from  the  following  web site.  This  will  create  the
directory $poplocal and the related directories needed for the remaining
installation steps.

    http://www.cs.bham.ac.uk/research/projects/poplog/freepoplog.html

Keep a note of where $poplocal is on your disk. If you forget, you can
recover this information by typing the following comma/quadrand
into Pop11.

    sysfileok('$poplocal') =>

It is a good idea to start Pop11 and type this command. This will allow
you to check that you have installed Poplog correctly.

Experts in  Poplog  will  have  more sophisticated  ways  of  doing  the
installation. If you  are confident that  you know what  you are  doing,
then install the transarith package any way you want to.


1.2  Install transarith implementation
--------------------------------------

Move the file transarith.p  to the directory $poplocal/../lib/lib/  This
completes the installation of the source code. If you want to, make the
file read only so that its contents cannot be changed by accident.

You can  check the  installation by  typing the  following command  into
pop11. It will  open the  contents of transarith.p  in Ppp11's  editor -
ved. Be careful not to change the contents of the file!

    showlib transarith


1.3  Install transarith documentation
-------------------------------------

Move the file,  transarith, to the  directory $poplocal/../ref/ and,  if
you want to,  make the file  read only  so that its  contents cannot  be
changed by accident.

Type the following two  commands into Pop11.  They will reconstruct  the
index of reference documents:

    lib mkrefindex;
    mkrefindex('$poplocal/../ref');

This completes  the installation  of the  documentation. The  transarith
package is now ready to use.

You can check that you have put the file transarith in the correct place
by typing the following command into Pop11. It will open the contents of
transarith in Pop11's editor.

    ref transarith

You can check  that you have  rebuilt the index  of reference  documents
correctly by typing the following command  into Pop11. It will open  the
reference document in Pop11's editor close to the definition of nullity.

    help nuly

A better way to access  the definition of the  constant nuly is to  type
the following command into ved's command line:

    ?? nuly

If all else fails, just print out the documentation file transarith  and
refer to it as needed!


1.4  Test the transarith package
--------------------------------

Load the transarith package by typing the following command into Pop11:

    lib transarith

Pop11 pints a message when it loads the package. Now type the following
command to raise zero to the power of zero.

    0 ##** 0 =>

Pop11 prints  two  asterisks, which  mark  the beginning  of  an  output
stream, followed by the  printable symbol for  nullity. Nullity is  zero
divided by zero.

    ** 0_/0



2  Introduction
---------------

The transreal and  transcomplex numbers  support a  total arithmetic  in
which  any  defined  arithmetical  operation  can  be  applied  to   any
transnumber(s) such that  the result  is a  transnumber. Hence,  neither
transreal nor transcomplex  arithmetic has any  undefined or  indefinite
values. In particular, division by zero is well defined.
    In principal, transreal and transcomplex arithmetic never suffer  an
arithmetical exception at run time, but the implementation supplied here
is layered on top of an ordinary implementation of Pop11 so it can  fail
on a floating-point error  or a memory fault.  This package is  provided
simply as a model of what might be achieved with transreal hardware.
    The  package  has  been  implemented  so  that  it  always   takes a
transcomplex path to a solution if  one is available. This provides  the
strongest test  of transcomplex  arithmetic, but  it degrades  numerical
accuracy. This implementation is not suitable for applications requiring
a high degree of numerical accuracy.
    The transreal numbers are all of Pop11's real number types and three
constants, the words pinf  = "1_/0", ninf =  "-1_/0" and nuly =  "0_/0".
These constants  can  be  used to  introduce  the  non-finite  transreal
numbers into a  calculation. Alternatively,  the arithmetical  operators
can be used to  divide a number  by zero so as  to produce a  non-finite
number. For example, 1 ##/ 0 = pinf.
    The  transcomplex   numbers  are   a  Pop11   class   implementing a
three-tuple (r,c,s) of a non-negative transreal radius, r, which is  the
Euclidean transmetric of a transcomplex vector;  and c and s which  are,
respectively, the cosine and sine  this vector makes with the  positive,
real axis.
    Pop11's arithmetical procedures suffer  complex contagion, which  is
to say that if any argument to a procedure is complex then the result is
complex. However,  the  arithmetical  procedures supplied  here  do  not
suffer contagion. The transcomplex procedures return a transreal  result
wherever this is possible. This  provides stronger diagnostics, but  its
impact on execution time and programming idioms is not known at present.
(It is unlikely that the current practice of contagion will long survive
the introduction of quaternions  and octonions into computer  languages,
though such introduction may well be a very long time away.)



3  Naming Conventions
---------------------

Procedures and constants may have a name which begins with a prefix,
followd by an underbar, as follows.

    "tr_"     means "transreal".
    "tc_"     means "transcomplex".
    "trans_"  means polymorphic "trans_real" or "trans_complex".
    "test_"   means the identifier is a testing procedure for use
              by system developers and readers of the companion paper.



4  Constants
------------

pinf = "1_/0"                                                 [constant]
    Constant denoting transreal positive infinity.


ninf = "-1_/0"                                                [constant]
    Constant denoting transreal negative infinity.


nuly = "0_/0"                                                 [constant]
    Constant denoting transreal nullity.


tc_pinf = constranscomplex(pinf,    1,    0)                  [constant]
    Constant denoting  transreal  positive infinity  as  a  transcomplex
    number.


tc_ninf = constranscomplex(pinf,   -1,    0)                  [constant]
    Constant denoting  transreal  negative infinity  as  a  transcomplex
    number.


tc_nuly = constranscomplex(nuly, nuly, nuly)                  [constant]
    Constant denoting transreal nullity as a transcomplex number.


tc_zero = constranscomplex(   0,    1,    0)                  [constant]
    Constant denoting real zero as a transcomplex number.


tc_i    = constranscomplex(   1,    0,    1)                  [constant]
    Constant denoting the complex  unit (i.e. the  square root of  minus
    one) as a transcomplex number.



5  Predicates Relating to Numbers
---------------------------------

istransreal(item) -> bool                                    [procedure]
    Returns true if item is of type transreal. Returns false  otherwise.
    All of Pop11's built in real types are transreal.


isstrictlytransreal(item) -> bool                            [procedure]
    Returns true if  item is  one of the  non-finite transreal  numbers:
    pinf, ninf, nuly. Returns false otherwise.


istranscomplex(item) -> bool                                 [procedure]
    Returns true if item is of type transcomplex. Pop11's complex  types
    are NOT transcomplex  and do NOT  generalise to transcomplex  types.
    However, the  whole  of  complex  arithmetic  can  be  performed  by
    transcomplex arithmetic. The difference is  in the structure of  the
    underlying numbers, not in the  arithmetical operations that can  be
    applied to the numbers.


isstrictlytranscomplex(item) -> bool                         [procedure]
    Returns true if item  is of type transcomplex  and has at least  one
    component which is non-finite.


istransnumber(item) -> bool                                  [procedure]
    Returns true if item is of type transreal or transcomplex.



6  Type Checking
----------------

checktransreal(item)                                         [procedure]
checkstrictlytransreal(item)                                 [procedure]
checktranscomplex(item)                                      [procedure]
checkstrictlytranscomplex(item)                              [procedure]
checktransnumber(item)                                       [procedure]
    These procedures check,  respectively, that item  returns true  with
    the   following   predicates:   istransreal,    isstrictlytransreal,
    istranscomplex, isstrictlytranscomplex, or istransnumber. If so,  no
    action is taken; otherwise a mishap message is printed and execution
    halts, unless trapped.



7  Comparisons on Numbers
-------------------------

The following  operators  operate  on transreal  numbers.  The  equality
operators have  the  same precedence  (7)  as Pop11's  builtin  equality
operators.  The  builtin  equality  operators   can  also  be  used   on
transnumbers. The remaining relational operators have precedence (6)  to
match the precedence of the corresponding relational operators built  in
to Pop11. Operators with a precedence closer to zero bind more  tightly.
All of the  operators provided  here can  be applied  to the  individual
components of a transnumber.

num1 ##=    num2 -> bool
num1 ##/=   num2 -> bool
    These operators compare their  transreal arguments for  mathematical
    equality and inequality respectively.


num1 ##>    num2 -> bool
num1 ##/>   num2 -> bool
num1 ##>=   num2 -> bool
num1 ##/>=  num2 -> bool
num1 ##<    num2 -> bool
num1 ##/<   num2 -> bool
num1 ##<=   num2 -> bool
num1 ##/<=  num2 -> bool
num1 ##<>   num2 -> bool
num1 ##/<>  num2 -> bool
num1 ##<=>  num2 -> bool
num1 ##/<=> num2 -> bool
    These operators  compare  their  first  transreal  argument  to  be,
    respectively, grater than, not greater than, greater than or  equal,
    not greater than or  equal, less than, not  less than, less than  or
    equal, not less than or equal,  less than or greater than, not  less
    than or greater than, less than  or equal or greater than, not  less
    than or equal or greater than their second transreal argument.

    Note that whereas real numbers obey the axiom of trichotomy - a real
    nember can  be less  than, equal  to,  or greater  than zero  -  the
    transreal numbers obey the axiom of quadrachotomy with a fourth case
    - a  number can  be equal  to nullity.  In particular  0 ##<=>  nuly
    returns false and 0 ##/<=> nuly returns true.



8  Number Representation
------------------------

stdrcs(num1, num2, num3) -> (num4, num5, num6)               [procedure]
    The arguments  num1,  num2,  num3 are  arbitrary  transreal  numbers
    corresponding, respectively, to  the radius, cosine,  and sine  of a
    transcomplex  number.  The  return  values  num4,  num5,  num6  are,
    respectively, the radius, cosine, and  sine in canonical form.  That
    is, with all of the mathematical constraints between the  components
    applied.


constransreal(num1, num2) -> num3                            [procedure]
    Constructs  a  transreal  number,  num3,  with  numerator  num1  and
    denominator num2. That is,  num3 = num1 ##/  num2. If the result  is
    irrational then the denominator is not stored explicitly, instead it
    is assumed to be unity.


desttransreal(num1) -> (num2, num3)                          [procedure]
    Destroys a transreal number num1, returning its numerator, num2, and
    denominator, num3.


tr_num(num1) -> num2                                         [procedure]
    Returns the numerator,  num2, of  the transreal  number, num1.  This
    procedure does not have an updater.


tr_den(num1) -> num2                                         [procedure]
    Returns the denominator, num2, of  the transreal number, num1.  This
    procedure does not have an updater.


transcomplex_key                                              [constant]
    Structure key for transcomplex numbers. No key for the transreals is
    supplied here, because transreals use builtin types and have builtin
    keys. See REF * KEYS.


constranscomplex(num1, num2, num3) -> num4                   [procedure]
    Constructs a  transcomplex number,  num4, with  radius num1,  cosine
    num2, and sine num3. This procedure creates num4 in canonical  form,
    respecting  all  of   the  mathematical   constraints  between   the
    components num1, num2, num3.


desttranscomplex(num1) -> (num2,num3,num4) [procedure]
    Destroys a  transcomplex or  transreal number,  num1, returning  its
    radius, cosine and sine.


tc_rad(num1) -> num2                                         [procedure]
num2 -> tc_rad(num1)                                           [updater]
    This  procedure  and  its  updater  access  the  radius  field  of a
    transcomplex number. The mathematical constraints between components
    of a transcomplex number are not enforced.


tc_cos(num1) -> num2                                         [procedure]
num2 -> tc_cos(num1)                                           [updater]
    This  procedure  and  its  updater  access  the  cosine  field  of a
    transcomplex number. The mathematical constraints between components
    of a transcomplex number are not enforced.


tc_sin(num1) -> num2                                         [procedure]
num2 -> tc_rad(num1)                                           [updater]
    This  procedure  and  its  updater   access  the  sine  field   of a
    transcomplex number. The mathematical constraints between components
    of a transcomplex number are not enforced.


casttransreal(num1) -> num2                                  [procedure]
    This procedure casts num1 to the equivalent transreal number,  num2,
    if there  is one,  otherwise it  returns num1  as num2.  Casting  is
    possible where the transcomplex number lies  on the real axis or  is
    nullity.


casttranscomplex(num1) -> num2                               [procedure]
    This procedure  casts num1  to the  equivalent transcomplex  number,
    num2.



9  Arithmetical Operations
--------------------------

The symbol # denotes a number, so num1 ##- num2 is an infix  subtraction
applying to two  numbers and  #- is a  prefix negation  applying to  one
number. It would  be possible  to overlay  Pop11's builtin  arithmetical
operators, without  introducing  any new  syntax,  but this  would  have
massive impact on the performance of the Poplog system, especially where
it interacts with external languages.

num1 ##+  num2 -> num3                                      [operator 5]
num1 ##-  num2 -> num3                                      [operator 5]
num1 ##*  num2 -> num3                                      [operator 4]
num1 ##/  num2 -> num3                                      [operator 4]
    These operators respectively  add, subtract,  multiply, and  divide,
    any transnumbers.


num1 ##** num2 -> num3                                      [operator 3]
    This operator raises  transnumber num1 to  the power of  transnumber
    num2.



10 Mathematical Functions
-------------------------

The prefix "tr_" means transreal, "tc_" means transcomplex, and "trans_"
means transnumber,  that  is, a  number  which is  either  transreal  or
transcomplex.  Some  of  the  functions  supplied  here  are  needed  to
implement transarithmetic, but most are provided only because they might
be helpful to users.

tr_abs(num1) -> num2                                         [procedure]
    Returns the absolute value of num1. If num1 is transreal then num2 =
    num1 when num1 ##>= 0, but num2 = ##- num1 when num1 ##< 0. If  num1
    is transcomplex then num2 is the  radius of num1. For any  transreal
    num1 it is the case that  num1 = tr_sgn(num1) ##* tr_abs(num1).  See
    also trans_abs.


tr_sgn(num1) -> num2                                         [procedure]
    Returns the sign of  num1. So num2 =  1 when num1 ##>  0; num2 =  -1
    when num1 ##< 0; num2 = 0 when num1 ##= 0; num2 = nuly when num1 ##=
    nuly. For any transreal num1 it is the case that num1 = tr_sgn(num1)
    ##* tr_abs(num1). See also trans_sgn which returns 2 at some angle
    for non-finite transreal or transcomplex numbers.


tr_signeddistance(num1, num2) -> num3                        [procedure]
    Returns the signed distance from num2 to num1. When num1 = num2  the
    signed distance num3 = 0, otherwise num3 = num1 ##- num2.


tr_distance(num1, num2) -> num3                              [procedure]
    Returns the absolute value  of the distance  between num1 and  num2.
    Consequently, tr_abs(x) is equivalent to tr_distance(x, 0).


tr_cos(num1) -> num2                                         [procedure]
    Returns the transreal cosine,  num2, corresponding to the  transreal
    angle, num1. Hence -1  <= num2 <=  1 or num2  = nuly. The  transreal
    cosine is identical to the ordinary cosine for all real angles.


tr_sin(num1) -> num2                                         [procedure]
    This procedure returns  the transreal sine,  num2, corresponding  to
    the transreal angle, num1. Hence -1 <= num2 <= 1 or num2 = nuly. The
    transreal sine  is  identical to  the  ordinary sine  for  all  real
    angles.


tr_exp(num1) -> num2                                         [procedure]
    Returns the transreal  exponential, num2, of  the transreal  number,
    num1. The transreal  exponential, tr_exp, is  identical to the  real
    exponential for  all finite  transreal numbers.  See also  the  more
    general function trans_exp.


tr_log(num1) -> num2                                         [procedure]
    Returns the  transreal logarithm,  num2,  of the  transreal  number,
    num1. The  transreal logarithm,  tr_log, is  identical to  the  real
    logarithm for  all  finite  transreal numbers.  See  also  the  more
    general function trans_log.


trans_exp(num1) -> num2                                      [procedure]
    Returns  the  transcomplex  exponential,  num2,  of  the   arbitrary
    transnumber, num1. The transexponential is identical to the real and
    complex exponentials  for  all  finite  transreal  and  transcomplex
    numbers.


trans_log(num1) -> num2                                      [procedure]
    Returns  the  transcomplex   logarithm,  num2,   of  the   arbitrary
    transnumber, num1. The translogarithm is  identical to the real  and
    complex  logarithms  for  all  finite  transreal  and   transcomplex
    numbers.


trans_sqrt(num1) -> num2                                     [procedure]
    Returns  the  transcomplex  square  root,  num2,  of  the  arbitrary
    transnumber, num1.


trans_abs(num1) -> num2                                      [procedure]
    The argument, num1, is a transnumber with the (implied) transcomplex
    components (r1,c1,s1). The  return value,  num2, is  a transreal  or
    transcomplex number with the  (implied) components (r1,1,0)  reduced
    to canonical form by stdrcs. In  other words, it is the radius  laid
    along the positive, real axis. For all transcomplex num1, it is  the
    case that  casttransreal(num1) =  casttransreal(trans_sgn(num1)  ##*
    trans_abs(num1)). See also: tr_abs.


trans_sgn(num1) -> num2                                      [procedure]
    The argument, num1, is a transnumber with the (implied) transcomplex
    components (r1,c1,s1). The  return value,  num2, is  a transreal  or
    transcomplex number with the (implied) components (1,c1,s1) when  r1
    in finite and (2,c1,s1) when r1 is non-finite. In each case, num2 is
    reduced to canonical form by stdrcs. In other words, it is a  radius
    laid along the vector (r1,c1,s1).  For all transcomplex num1, it  is
    the case  that casttransreal(num1)  =  casttransreal(trans_sgn(num1)
    ##* trans_abs(num1)).  It is  also the  case that  the sign  of  any
    transnumber is tr_sgn(c1). See also: tr_sgn.



11 Test Procedures
------------------

The prefix "test_" introduces a test procedure. The procedures "test_N",
where N is a number, are used to check numerical examples that appear in
the companion journal paper, excluding quotations within that paper. For
example, test_1() tests  the examples on  Page 6 of  the paper. But  the
page numbers given here are the page numbers of the text of the paper as
submitted for  review.  These  page  numbers may  be  different  in  the
published version of the paper.  For example, the published paper  might
not start on page one.

The output of the procedures is given in the next section.

    Page       Test
    ----       ----
       6          1
      11          2
      12          3
      13          4
      15          5
      16          6
      17          7
      18          8
      19          9
      20         10
      21         11
      22         12, 13
      23         14
      24         15
      34,35,36   16
      41         17
      43         18
      46         19
      51         20
      52         21
      53         22



12 Transcript
-------------

This is  a  transcript  of  the computation  of  various  transreal  and
transcomplex values,  including all  of the  numerical examples  in  the
companion journal paper. The transcript  has been composed from  several
programming  sessions  and  has  had   some  line  lengths  altered   to
accommodate the obligatory format of a reference (documentation) file.

The test procedures, described  in the section above,  print out a  page
number which is where the examples  occur in the paper as submitted  for
review. These page numbers might change during the publication  process.
For example, the published paper might not start on page one.

Some calculations appear on several pages  of the paper and appear  once
in each test procedure corresponding to a page.

: test_1();
** Page  6
** [nuly => 0_/0]
** [pinf => 1_/0]
** [ninf => -1_/0]
** [pinf ##/ pinf => 0_/0]
** [pinf ##- pinf => 0_/0]
** [0 ##** 0 => 0_/0]
** [trans_exp ( ninf ) => 0]
** [trans_log ( 0 ) => -1_/0]
** [nuly = 0 ##** 0 => <true>]
** [nuly = trans_exp ( trans_log ( 0 ##** 0 ) ) => <true>]
** [nuly = trans_exp ( 0 ##* trans_log ( 0 ) ) => <true>]

: test_2();
** Page 11
** [#- 0 = 0 => <true>]
** [ninf = ninf => <true>]
** [#- nuly = nuly => <true>]
** [trans_sqrt ( 0 ) = 0 => <true>]
** [trans_sqrt ( #- 0 ) = 0 => <true>]
** [0 ##/ 1 = 0 => <true>]
** [0 ##/ #- 1 = 0 => <true>]
** [1 ##/ 0 = pinf => <true>]
** [#- 1 ##/ 0 = ninf => <true>]
** [0 ##/ 0 = nuly => <true>]
** [pinf ##/ pinf = nuly => <true>]
** [ninf ##/ pinf = nuly => <true>]
** [pinf ##/ ninf = nuly => <true>]
** [ninf ##/ ninf = nuly => <true>]
** [pinf ##* 0 = nuly => <true>]
** [ninf ##* 0 = nuly => <true>]
** [0 ##* pinf = nuly => <true>]
** [0 ##* ninf = nuly => <true>]
** [pinf ##- pinf = nuly => <true>]

: test_3();
** Page 12
** [pinf ##+ 1 = pinf => <true>]

: test_4();
** Page 13
** [1 ##/ ( pinf ##+ 1 ) = 0 => <true>]
** [nuly = 0 ##* tr_sin ( pinf ) => <true>]

: test_5();
** Page 15
** [1 = 1 ##/ 1 => <true>]
** [#- 1 = #- 1 ##/ 1 => <true>]

: test_6();
** Page 16
** [0 = 0 ##/ 1 => <true>]
** [pinf = 1 ##/ 0 => <true>]
** [ninf = #- 1 ##/ 0 => <true>]
** [nuly = 0 ##/ 0 => <true>]
** [nuly = pinf ##/ pinf => <true>]
** [ninf = pinf ##/ #- 3 => <true>]

: test_7();
** Page 17
** [nuly = pinf ##- pinf => <true>]
** [nuly = nuly ##- pinf => <true>]

: test_8();
** Page 18
** [nuly = pinf ##* ( 2 ##- 2 ) => <true>]
** [nuly = pinf ##* 2 ##- pinf ##* 2 => <true>]
** [pinf = pinf ##* ( 2 ##- 1 ) => <true>]
** [nuly = pinf ##* 2 ##- pinf ##* 1 => <true>]
** [nuly = pinf ##* 0 => <true>]
** [nuly = pinf ##- pinf => <true>]

: test_9();
** Page 19

** Gravitational attraction between P_1 and P_2.
** [( ( 6.7 ##* 10 ##** #- 11 ) ##* ( 3.2 ##* 10 ##** #- 27 ) ##* ( 3.2
    ##* 10 ##** #- 27 ) ) ##/ ( 0 ##** 2 ) => 1_/0]

** Gravitational attraction between singularity and P_3.
** [( ( 6.7 ##* 10 ##** #- 11 ) ##* ( 2 ##* 3.2 ##* 10 ##** #- 27 ) ##*
    ( 3.2 ##* 10 ##** #- 27 ) ) ##/ ( ( 1.6 ##* 10 ##* #- 35 ) ##** 2 )
    => 5360000.0]

** Electrostatic repulsion between P_1 and P_2.
** [( ( 9 ##* 10 ##** 9 ) ##* ( 1.6 ##* 10 ##** #- 19 ) ##* ( 1.6 ##* 10
    ##** #- 19 ) ) ##/ ( 0 ##** 2 ) => 1_/0]

** Electrostatic repulsion between singularity and P_3.
** [( ( 9 ##* 10 ##** 9 ) ##* ( 2 ##* 1.6 ##* 10 ##** #- 19 ) ##* ( 1.6
    ##* 10 ##** #- 19 ) ) ##/ ( ( 1.6 #* 10 ##** #- 35 ) ##** 2 ) => 180
    0000000000000800000000000000000000000000.0]

: test_10();
** Page 20
** [nuly = pinf ##- pinf => <true>]
** [5.4 ##* 10 ##** 6 ##- 1.8 ##* 10 ##** 42 => -18000000000000190000000
   00000000000000000000.0]

: test_11();
** Page 21
** [pinf = pinf ##/ 0 => <true>]
** [nuly = pinf ##/ pinf => <true>]

: test_12();
** Page 22
** Table 3: F = ma with k_m = 2, k_a = 3
** [m = 0 a = -1_/0 F = 0_/0]
** [m = 0 a = -3 F = 0]
** [m = 0 a = 0 F = 0]
** [m = 0 a = 3 F = 0]
** [m = 0 a = 1_/0 F = 0_/0]
** [m = 0 a = 0_/0 F = 0_/0]
** [m = 2 a = -1_/0 F = -1_/0]
** [m = 2 a = -3 F = -6]
** [m = 2 a = 0 F = 0]
** [m = 2 a = 3 F = 6]
** [m = 2 a = 1_/0 F = 1_/0]
** [m = 2 a = 0_/0 F = 0_/0]
** [m = 1_/0 a = -1_/0 F = -1_/0]
** [m = 1_/0 a = -3 F = -1_/0]
** [m = 1_/0 a = 0 F = 0_/0]
** [m = 1_/0 a = 3 F = 1_/0]
** [m = 1_/0 a = 1_/0 F = 1_/0]
** [m = 1_/0 a = 0_/0 F = 0_/0]
** [m = 0_/0 a = -1_/0 F = 0_/0]
** [m = 0_/0 a = -3 F = 0_/0]
** [m = 0_/0 a = 0 F = 0_/0]
** [m = 0_/0 a = 3 F = 0_/0]
** [m = 0_/0 a = 1_/0 F = 0_/0]
** [m = 0_/0 a = 0_/0 F = 0_/0]

: test_13();
** Page 22
** Table 4: a = F/m with k_F = 2, k_m = 3
** [F = -1_/0 m = 0 a = -1_/0]
** [F = -1_/0 m = 3 a = -1_/0]
** [F = -1_/0 m = 1_/0 a = 0_/0]
** [F = -1_/0 m = 0_/0 a = 0_/0]
** [F = -2 m = 0 a = -1_/0]
** [F = -2 m = 3 a = -2_/3]
** [F = -2 m = 1_/0 a = 0]
** [F = -2 m = 0_/0 a = 0_/0]
** [F = 0 m = 0 a = 0_/0]
** [F = 0 m = 3 a = 0]
** [F = 0 m = 1_/0 a = 0]
** [F = 0 m = 0_/0 a = 0_/0]
** [F = 2 m = 0 a = 1_/0]
** [F = 2 m = 3 a = 2_/3]
** [F = 2 m = 1_/0 a = 0]
** [F = 2 m = 0_/0 a = 0_/0]
** [F = 1_/0 m = 0 a = 1_/0]
** [F = 1_/0 m = 3 a = 1_/0]
** [F = 1_/0 m = 1_/0 a = 0_/0]
** [F = 1_/0 m = 0_/0 a = 0_/0]
** [F = 0_/0 m = 0 a = 0_/0]
** [F = 0_/0 m = 3 a = 0_/0]
** [F = 0_/0 m = 1_/0 a = 0_/0]
** [F = 0_/0 m = 0_/0 a = 0_/0]

: test_14();
** Page 23
** Table 5: m = F/a with k_F = 2, k_a = 3
** [F = -1_/0 a = -1_/0 m = 0_/0]
** [F = -1_/0 a = -3 m = 1_/0]
** [F = -1_/0 a = 0 m = -1_/0]
** [F = -1_/0 a = 3 m = -1_/0]
** [F = -1_/0 a = 1_/0 m = 0_/0]
** [F = -1_/0 a = 0_/0 m = 0_/0]
** [F = -2 a = -1_/0 m = 0]
** [F = -2 a = -3 m = 2_/3]
** [F = -2 a = 0 m = -1_/0]
** [F = -2 a = 3 m = -2_/3]
** [F = -2 a = 1_/0 m = 0]
** [F = -2 a = 0_/0 m = 0_/0]
** [F = 0 a = -1_/0 m = 0]
** [F = 0 a = -3 m = 0]
** [F = 0 a = 0 m = 0_/0]
** [F = 0 a = 3 m = 0]
** [F = 0 a = 1_/0 m = 0]
** [F = 0 a = 0_/0 m = 0_/0]
** [F = 2 a = -1_/0 m = 0]
** [F = 2 a = -3 m = -2_/3]
** [F = 2 a = 0 m = 1_/0]
** [F = 2 a = 3 m = 2_/3]
** [F = 2 a = 1_/0 m = 0]
** [F = 2 a = 0_/0 m = 0_/0]
** [F = 1_/0 a = -1_/0 m = 0_/0]
** [F = 1_/0 a = -3 m = -1_/0]
** [F = 1_/0 a = 0 m = 1_/0]
** [F = 1_/0 a = 3 m = 1_/0]
** [F = 1_/0 a = 1_/0 m = 0_/0]
** [F = 1_/0 a = 0_/0 m = 0_/0]
** [F = 0_/0 a = -1_/0 m = 0_/0]
** [F = 0_/0 a = -3 m = 0_/0]
** [F = 0_/0 a = 0 m = 0_/0]
** [F = 0_/0 a = 3 m = 0_/0]
** [F = 0_/0 a = 1_/0 m = 0_/0]
** [F = 0_/0 a = 0_/0 m = 0_/0]

: test_15();
** Page 25

** Exact integer computation:
** [2 ** ( 10 + 1 ) - 2 => 2046]
** [2 ** ( 23 + 1 ) - 2 => 16777214]
** [2 ** ( 52 + 1 ) - 2 => 9007199254740990]
** [2 ** ( 112 + 1 ) - 2 => 10384593717069655257060992658440190]

** Floating-point computation following a transcomplex path:
** [2 ##** ( 10 ##+ 1 ) ##- 2 => 2046.0]
** [2 ##** ( 23 ##+ 1 ) ##- 2 => <transcomplex 16777213.99999998 0.99999
   99999999998 0.0>]
** [2 ##** ( 52 ##+ 1 ) ##- 2 => 9007199254740984.0]
** [2 ##** (112 ##+ 1 ) ##- 2 => 10384593717069630000000000000000000.0]

** No signed zero:
** [0 = #- 0 and 1 ##/ 0 = 1 ##/ #- 0 => <true>]

: test_16();
** Page 34
** [<transcomplex 1_/0 1.0 0.0> + <transcomplex 1_/0 0.0 1.0> =>
    <transcomplex 1_/0 0.707107 0.707107>]

: test_17();
** Page 41
** [1_/0 ##/ 0 = 1_/0 => <true>]

: test_18();
** Page 43
** [nuly ##/ tr_sqrt ( nuly ##** 2 ##+ nuly ##** 2 ) = nuly => <true>]

: test_19();
** Page 46
** [pinf ##- pinf = nuly => <true>]
** [ninf ##+ pinf = nuly => <true>]
** [nuly ##- nuly = nuly => <true>]
** [#- nuly ##+ nuly = nuly => <true>]
** [tr_cos ( ninf ) = nuly and tr_sin ( ninf ) = nuly => <true>]
** [tr_cos ( pinf ) = nuly and tr_sin ( pinf ) = nuly => <true>]
** [tr_cos ( nuly ) = nuly and tr_sin ( nuly ) = nuly => <true>]

: test_20();
** Page 51
** This is not a numerical example in the paper, but it is important

** [z1 = <transcomplex 1 0.9092974268256818 -0.4161468365471425>]
** [z2 = <transcomplex 1 -0.9092974268256818 0.4161468365471423>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 -1.0>]

** [z1 = <transcomplex 1 0.9540857816096938 -0.2995335061895742>]
** [z2 = <transcomplex 1 -0.9540857816096938 0.2995335061895741>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 -1.0>]

** [z1 = <transcomplex 1 0.983985946873937 -0.1782460556494921>]
** [z2 = <transcomplex 1 -0.983985946873937 0.178246055649492>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 -1.0>]

** [z1 = <transcomplex 1 0.9985313405398315 -0.0541771350269364>]
** [z2 = <transcomplex 1 -0.9985313405398315 0.0541771350269363>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 -1.0>]

** [z1 = <transcomplex 1 0.9974949866040545 0.0707372016677029>]
** [z2 = <transcomplex 1 -0.9974949866040545 -0.070737201667703>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 -1.0>]

** [z1 = <transcomplex 1 0.9808930570231556 0.1945477079889871>]
** [z2 = <transcomplex 1 -0.9808930570231556 -0.1945477079889872>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 -1.0>]

** [z1 = <transcomplex 1 0.9489846193555863 0.3153223623952686>]
** [z2 = <transcomplex 1 -0.9489846193555863 -0.3153223623952687>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 -1.0>]

** [z1 = <transcomplex 1 0.9022675940990952 0.4311765167986661>]
** [z2 = <transcomplex 1 -0.9022675940990952 -0.4311765167986662>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 -1.0>]

** [z1 = <transcomplex 1 0.8414709848078965 0.5403023058681397>]
** [z2 = <transcomplex 1 -0.8414709848078965 -0.5403023058681398>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 -1.0>]

** [z1 = <transcomplex 1 0.767543502236027 0.6409968581633251>]
** [z2 = <transcomplex 1 -0.7675435022360269 -0.6409968581633252>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000002 0.7071067811865475
    -0.7071067811865475>]

** [z1 = <transcomplex 1 0.6816387600233343 0.7316888688738209>]
** [z2 = <transcomplex 1 -0.6816387600233341 -0.7316888688738209>]
** [z1 ##+ z2 => 0.0000000000000001]

** [z1 = <transcomplex 1 0.5850972729404623 0.8109631195052177>]
** [z2 = <transcomplex 1 -0.5850972729404621 -0.8109631195052179>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000002 0.7071067811865475
    -0.7071067811865475>]

** [z1 = <transcomplex 1 0.4794255386042031 0.8775825618903726>]
** [z2 = <transcomplex 1 -0.4794255386042029 -0.8775825618903728>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000002 0.7071067811865475
    -0.7071067811865475>]

** [z1 = <transcomplex 1 0.3662725290860476 0.9305076219123142>]
** [z2 = <transcomplex 1 -0.3662725290860475 -0.9305076219123142>]
** [z1 ##+ z2 => 0.0000000000000001]

** [z1 = <transcomplex 1 0.247403959254523 0.9689124217106448>]
** [z2 = <transcomplex 1 -0.2474039592545229 -0.9689124217106448>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000002 0.7071067811865475
    -0.7071067811865475>]

** [z1 = <transcomplex 1 0.1246747333852277 0.992197667229329>]
** [z2 = <transcomplex 1 -0.1246747333852276 -0.992197667229329>]
** [z1 ##+ z2 => 0.0000000000000001]

** [z1 = <transcomplex 1 0.0000000000000001 1.0>]
** [z2 = <transcomplex 1 0.0000000000000001 -1.0>]
** [z1 ##+ z2 => 0.0000000000000001]

** [z1 = <transcomplex 1 -0.1246747333852276 0.992197667229329>]
** [z2 = <transcomplex 1 0.1246747333852277 -0.992197667229329>]
** [z1 ##+ z2 => 0.0000000000000001]

** [z1 = <transcomplex 1 -0.2474039592545229 0.9689124217106448>]
** [z2 = <transcomplex 1 0.247403959254523 -0.9689124217106448>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000002 0.7071067811865475
    0.7071067811865475>]

** [z1 = <transcomplex 1 -0.3662725290860475 0.9305076219123142>]
** [z2 = <transcomplex 1 0.3662725290860476 -0.9305076219123142>]
** [z1 ##+ z2 => 0.0000000000000001]

** [z1 = <transcomplex 1 -0.4794255386042029 0.8775825618903728>]
** [z2 = <transcomplex 1 0.4794255386042031 -0.8775825618903726>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000002 0.7071067811865475
    0.7071067811865475>]

** [z1 = <transcomplex 1 -0.5850972729404621 0.8109631195052179>]
** [z2 = <transcomplex 1 0.5850972729404623 -0.8109631195052177>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000002 0.7071067811865475
    0.7071067811865475>]

** [z1 = <transcomplex 1 -0.6816387600233341 0.7316888688738209>]
** [z2 = <transcomplex 1 0.6816387600233343 -0.7316888688738209>]
** [z1 ##+ z2 => 0.0000000000000001]

** [z1 = <transcomplex 1 -0.7675435022360269 0.6409968581633252>]
** [z2 = <transcomplex 1 0.767543502236027 -0.6409968581633251>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000002 0.7071067811865475
    0.7071067811865475>]

** [z1 = <transcomplex 1 -0.8414709848078965 0.5403023058681398>]
** [z2 = <transcomplex 1 0.8414709848078965 -0.5403023058681397>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 1.0>]

** [z1 = <transcomplex 1 -0.9022675940990952 0.4311765167986662>]
** [z2 = <transcomplex 1 0.9022675940990952 -0.4311765167986661>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 1.0>]

** [z1 = <transcomplex 1 -0.9489846193555863 0.3153223623952687>]
** [z2 = <transcomplex 1 0.9489846193555863 -0.3153223623952686>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 1.0>]

** [z1 = <transcomplex 1 -0.9808930570231556 0.1945477079889872>]
** [z2 = <transcomplex 1 0.9808930570231556 -0.1945477079889871>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 1.0>]

** [z1 = <transcomplex 1 -0.9974949866040545 0.070737201667703>]
** [z2 = <transcomplex 1 0.9974949866040545 -0.0707372016677029>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 1.0>]

** [z1 = <transcomplex 1 -0.9985313405398315 -0.0541771350269363>]
** [z2 = <transcomplex 1 0.9985313405398315 0.0541771350269364>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 1.0>]

** [z1 = <transcomplex 1 -0.983985946873937 -0.178246055649492>]
** [z2 = <transcomplex 1 0.983985946873937 0.1782460556494921>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 1.0>]

** [z1 = <transcomplex 1 -0.9540857816096938 -0.2995335061895741>]
** [z2 = <transcomplex 1 0.9540857816096938 0.2995335061895742>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 1.0>]

** [z1 = <transcomplex 1 -0.9092974268256818 -0.4161468365471423>]
** [z2 = <transcomplex 1 0.9092974268256818 0.4161468365471425>]
** [z1 ##+ z2 => <transcomplex 0.0000000000000001 0.0 1.0>]

: test_21();
** Page 52
** [<transcomplex 1_/0 0_/0 0_/0> ##* <transcomplex 1_/0 0_/0 0_/0> =
    <transcomplex 1_/0 0_/0 0_/0> => <true>]
** [istranscomplex ( <transcomplex 1_/0 0.707107 0.707107> ) => <true>]
** [istransreal ( <transcomplex 1_/0 0.707107 0.707107> ) => <false>]
** [<transcomplex 1_/0 1.0 0.0> ##* <transcomplex 1_/0 1.0 0.0> = 1_/0
    => <true>]

: test_22();
** Page 52
** [<transcomplex 1_/0 1.0 0.0> ##+ <transcomplex 1_/0 1.0 0.0> = 1_/0
    => <true>]
** [<transcomplex 1_/0 1.0 0.0> ##- <transcomplex 1_/0 1.0 0.0> = 0_/0
    => <true>]
** [<transcomplex 1_/0 1.0 0.0> ##/ <transcomplex 1_/0 1.0 0.0> = 0_/0
    => <true>]
** [0 ##/ 0 = 0_/0 => <true>]
** [<transcomplex 1_/0 1.0 0.0> ##* 0 = 0_/0 => <true>]


--- Copyright James Anderson 2010. All rights reserved. ---------------