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C++ Mathematical Expression Library

branch: master
readme.txt
C++ Mathematical Expression Toolkit Library

[00 - INTRODUCTION]
The C++ Mathematical Expression  Toolkit Library (ExprTk) is  a simple
to  use,  easy  to  integrate  and  extremely  efficient  mathematical
expression parsing and evaluation engine. The parsing engine  supports
numerous forms  of functional  and logic  processing semantics  and is
easily extendible.



[01 - CAPABILITIES]
The  ExprTk expression  evaluator supports  the following  fundamental
arithmetic operations, functions and processes:

 (0) Basic operators: +, -, *, /, %, ^

 (1) Functions:       abs, avg, ceil, clamp, equal, erf, erfc, exp,
                      expm1, floor, frac, log, log10, log1p, log2,
                      logn, max, min, mul, nequal, root, round,
                      roundn, sgn, sqrt, sum, trunc

 (2) Trigonometry:    acos, acosh, asin, asinh, atan, atanh, atan2,
                      cos, cosh, cot, csc, sec, sin, sinh, tan, tanh,
                      hypot, rad2deg,  deg2grad, deg2rad, grad2deg

 (3) Equalities &
     Inequalities:    =, ==, <>, !=, <, <=, >, >=

 (4) Boolean logic:   and, mand, mor, nand, nor, not, or, shl, shr,
                      xnor, xor, true, false

 (5) Conditional,
     Switch &
     Loop statements: if-then-else, ternary conditional, switch-case,
                      while, repeat-until

 (6) Assignment:      :=

 (7) String
     processing:      in, like, ilike

 (8) Optimisations:   constant-folding and simple strength reduction

 (9) Calculus:        numerical integration and differentiation



[02 - EXAMPLE EXPRESSIONS]
The  following  is  a  short listing  of  the types  of  mathematical
expressions that can be parsed and evaluated using the ExprTk library.

  (01) sqrt(1 - (3 / x^2))
  (02) clamp(-1, sin(2 * pi * x) + cos(y / 2 * pi), +1)
  (03) sin(2.34e-3 * x)
  (04) if(((x + 2) == 3) and ((y + 5) <= 9),1 + w, 2 / z)
  (05) inrange(-2,m,+2) == if(({-2 <= m} and [m <= +2]),1,0)
  (06) ({1/1}*[1/2]+(1/3))-{1/4}^[1/5]+(1/6)-({1/7}+[1/8]*(1/9))
  (07) a * exp(2.2 / 3.3 * t) + c
  (08) z := x + sin(2.567 * pi / y)
  (09) u := 2.123 * {pi * z} / (w := x + cos(y / pi))
  (10) 2x + 3y + 4z + 5w == 2 * x + 3 * y + 4 * z + 5 * w
  (11) 3(x + y) / 2.9 + 1.234e+12 == 3 * (x + y) / 2.9 + 1.234e+12
  (12) (x + y)3.3 + 1 / 4.5 == [x + y] * 3.3 + 1 / 4.5
  (13) (x + y)z + 1.1 / 2.7 == (x + y) * z + 1.1 / 2.7
  (14) (sin(x / pi) cos(2y) + 1) == (sin(x / pi) * cos(2 * y) + 1)
  (15) 75x^17 + 25.1x^5 - 35x^4 - 15.2x^3 + 40x^2 - 15.3x + 1
  (16) (avg(x,y) <= x + y ? x - y : x * y) + 2.345 * pi / x
  (17) fib_i := fib_i + (x := y + 0 * (fib_i := x + (y := fib_i)))
  (18) while (x <= 100) { x := x + 1 }
  (19) x <= 'abc123' and (y in 'AString') or ('1x2y3z' != z)
  (20) (x like '*123*') or ('a123b' ilike y)



[03 - COPYRIGHT NOTICE]
Free  use  of  the  C++  Mathematical  Expression  Toolkit  Library is
permitted under the guidelines and in accordance with the most current
version of the Common Public License.

http://www.opensource.org/licenses/cpl1.0.php



[04 - DOWNLOADS & UPDATES]
The most  recent version  of the C++ Mathematical  Expression  Toolkit
Library including all updates and tests can be found at the  following
locations:

  (1) Download:   http://www.partow.net/programming/exprtk/index.html
  (2) Repository: https://exprtk.googlecode.com/svn/



[05 - INSTALLATION]
The header  file exprtk.hpp  should be  placed in a project or  system
include path (e.g: /usr/include/).



[06 - COMPILATION]
 (a) For a complete build: make clean all
 (b) For a PGO build: make clean pgo
 (c) To strip executables: make strip_bin
 (d) Execute valgrind check: make valgrind_check



[07 - COMPILER COMPATIBILITY]
 (*) GNU Compiler Collection (4.1+)
 (*) Intel C++ Compiler (9.x+)
 (*) Clang/LLVM (1.1+)
 (*) PGI C++ (10.x+)
 (*) Microsoft Visual Studio C++ Compiler (8.1+)
 (*) Comeau C++ Compiler (4.3+)
 (*) IBM XL C/C++ (10.x+)
 (*) C++ Builder (XE4+)



[08 - BUILT-IN OPERATIONS & FUNCTIONS]

(0) Arithmetic Operators
+----------+---------------------------------------------------------+
| OPERATOR | DEFINITION                                              |
+----------+---------------------------------------------------------+
|  +       | Addition between x and y. (eg: x + y)                   |
+----------+---------------------------------------------------------+
|  -       | Subtraction between x and y. (eg: x - y)                |
+----------+---------------------------------------------------------+
|  *       | Multiplication between x and y. (eg: x * y)             |
+----------+---------------------------------------------------------+
|  /       | Division between x and y (eg: x / y)                    |
+----------+---------------------------------------------------------+
|  %       | Modulus of x with respect to y. (eg: x % y)             |
+----------+---------------------------------------------------------+
|  ^       | x to the power of y. (eg: x ^ y)                        |
+----------+---------------------------------------------------------+
|  :=      | Assign the value of x to y. (eg: y := x)                |
|          | Where y is a variable type.                             |
+----------+---------------------------------------------------------+

(1) Equalities & Inequalities
+----------+---------------------------------------------------------+
| OPERATOR | DEFINITION                                              |
+----------+---------------------------------------------------------+
| == or =  | True only if x is strictly equal to y. (eg: x == y)     |
+----------+---------------------------------------------------------+
| <> or != | True only if x does not equal y. (eg: x <> y or x != y) |
+----------+---------------------------------------------------------+
|  <       | True only if x is less than y. (eg: x < y)              |
+----------+---------------------------------------------------------+
|  <=      | True only if x is less than or equal to y. (eg: x <= y) |
+----------+---------------------------------------------------------+
|  >       | True only if x is greater than y. (eg: x > y)           |
+----------+---------------------------------------------------------+
|  >=      | True only if x greater than or equal to y. (eg: x >= y) |
+----------+---------------------------------------------------------+

(2) Boolean Operations
+----------+---------------------------------------------------------+
| OPERATOR | DEFINITION                                              |
+----------+---------------------------------------------------------+
| true     | True state or any value other than zero (typically 1).  |
+----------+---------------------------------------------------------+
| false    | False state, value of zero.                             |
+----------+---------------------------------------------------------+
| and      | Logical AND, True only if x and y are both true.        |
|          | (eg: x and y)                                           |
+----------+---------------------------------------------------------+
| mand     | Multi-input logical AND, True only if all inputs are    |
|          | true. Left to right short-circuiting of expressions.    |
|          | (eg: mand(x > y, z < w, u or v, w and x))               |
+----------+---------------------------------------------------------+
| mor      | Multi-input logical OR, True if at least one of the     |
|          | inputs are true. Left to right short-circuiting of      |
|          | expressions.  (eg: mor(x > y, z < w, u or v, w and x))  |
+----------+---------------------------------------------------------+
| nand     | Logical NAND, True only if either x or y is false.      |
|          | (eg: x nand y)                                          |
+----------+---------------------------------------------------------+
| nor      | Logical NOR, True only if the result of x or y is false |
|          | (eg: x nor y)                                           |
+----------+---------------------------------------------------------+
| not      | Logical NOT, Negate the logical sense of the input.     |
|          | (eg: not(x and y) == x nand y)                          |
+----------+---------------------------------------------------------+
| or       | Logical OR, True if either x or y is true. (eg: x or y) |
+----------+---------------------------------------------------------+
| xor      | Logical XOR, True only if the logical states of x and y |
|          | differ.  (eg: x xor y)                                  |
+----------+---------------------------------------------------------+
| xnor     | Logical XNOR, True iff the biconditional of x and y is  |
|          | satisfied.  (eg: x xnor y)                              |
+----------+---------------------------------------------------------+
| &        | Similar to AND but with left to right expression short  |
|          | circuiting optimisation.  (eg: (x & y) == (y and x))    |
+----------+---------------------------------------------------------+
| |        | Similar to OR but with left to right expression short   |
|          | circuiting optimisation.  (eg: (x | y) == (y or x))     |
+----------+---------------------------------------------------------+

(3) General Purpose Functions
+----------+---------------------------------------------------------+
| FUNCTION | DEFINITION                                              |
+----------+---------------------------------------------------------+
| abs      | Absolute value of x.  (eg: abs(x))                      |
+----------+---------------------------------------------------------+
| avg      | Average of all the inputs.                              |
|          | (eg: avg(x,y,z,w,u,v) == (x + y + z + w + u + v) / 6)   |
+----------+---------------------------------------------------------+
| ceil     | Smallest integer that is greater than or equal to x.    |
+----------+---------------------------------------------------------+
| clamp    | Clamp x in range between r0 and r1, where r0 < r1.      |
|          | (eg: clamp(r0,x,r1)                                     |
+----------+---------------------------------------------------------+
| equal    | Equality test between x and y using normalized epsilon  |
+----------+---------------------------------------------------------+
| erf      | Error function of x.  (eg: erf(x))                      |
+----------+---------------------------------------------------------+
| erfc     | Complimentary error function of x.  (eg: erfc(x))       |
+----------+---------------------------------------------------------+
| exp      | e to the power of x.  (eg: exp(x))                      |
+----------+---------------------------------------------------------+
| expm1    | e to the power of x minus 1, where x is very small.     |
|          | (eg: expm1(x))                                          |
+----------+---------------------------------------------------------+
| floor    | Largest integer that is less than or equal to x.        |
|          | (eg: floor(x))                                          |
+----------+---------------------------------------------------------+
| frac     | Fractional portion of x.  (eg: frac(x))                 |
+----------+---------------------------------------------------------+
| hypot    | Hypotenuse of x and y (eg: hypot(x,y) = sqrt(x*x + y*y))|
+----------+---------------------------------------------------------+
| iclamp   | Inverse-clamp x outside of the range r0 and r1. Where   |
|          | r0 < r1. If x is within the range it will snap to the   |
|          | closest bound. (eg: iclamp(r0,x,r1)                     |
+----------+---------------------------------------------------------+
| log      | Natural logarithm of x.  (eg: log(x))                   |
+----------+---------------------------------------------------------+
| log10    | Base 10 logarithm of x.  (eg: log10(x))                 |
+----------+---------------------------------------------------------+
| log1p    | Natural logarithm of 1 + x, where x is very small.      |
|          | (eg: log1p(x))                                          |
+----------+---------------------------------------------------------+
| log2     | Base 2 logarithm of x.  (eg: log2(x))                   |
+----------+---------------------------------------------------------+
| logn     | Base N logarithm of x. where n is a positive integer.   |
|          | (eg: logn(x,8))                                         |
+----------+---------------------------------------------------------+
| max      | Largest value of all the inputs. (eg: max(x,y,z,w,u,v)) |
+----------+---------------------------------------------------------+
| min      | Smallest value of all the inputs. (eg: min(x,y,z,w,u))  |
+----------+---------------------------------------------------------+
| mul      | Product of all the inputs.                              |
|          | (eg: mul(x,y,z,w,u,v,t) == (x * y * z * w * u * v * t)) |
+----------+---------------------------------------------------------+
| nequal   | Not-equal test between x and y using normalized epsilon |
+----------+---------------------------------------------------------+
| root     | Nth-Root of x. where n is a positive integer.           |
|          | (eg: root(x,3))                                         |
+----------+---------------------------------------------------------+
| round    | Round x to the nearest integer.  (eg: round(x))         |
+----------+---------------------------------------------------------+
| roundn   | Round x to n decimal places  (eg: roundn(x,3))          |
|          | where n > 0 and is an integer.                          |
|          | (eg: roundn(1.2345678,4) == 1.2346)                     |
+----------+---------------------------------------------------------+
| sgn      | Sign of x, -1 where x < 0, +1 where x > 0, else zero.   |
|          | (eg: sgn(x))                                            |
+----------+---------------------------------------------------------+
| sqrt     | Square root of x, where x > 0.  (eg: sqrt(x))           |
+----------+---------------------------------------------------------+
| sum      | Sum of all the inputs.                                  |
|          | (eg: sum(x,y,z,w,u,v,t) == (x + y + z + w + u + v + t)) |
+----------+---------------------------------------------------------+
| trunc    | Integer portion of x.  (eg: trunc(x))                   |
+----------+---------------------------------------------------------+

(4) Trigonometry Functions
+----------+---------------------------------------------------------+
| FUNCTION | DEFINITION                                              |
+----------+---------------------------------------------------------+
| acos     | Arc cosine of x expressed in radians. Interval [-1,+1]  |
|          | (eg: acos(x))                                           |
+----------+---------------------------------------------------------+
| acosh    | Inverse hyperbolic cosine of x expressed in radians.    |
|          | (eg: acosh(x))                                          |
+----------+---------------------------------------------------------+
| asin     | Arc sine of x expressed in radians. Interval [-1,+1]    |
|          | (eg: asin(x))                                           |
+----------+---------------------------------------------------------+
| asinh    | Inverse hyperbolic sine of x expressed in radians.      |
|          | (eg: asinh(x))                                          |
+----------+---------------------------------------------------------+
| atan     | Arc tangent of x expressed in radians. Interval [-1,+1] |
|          | (eg: atan(x))                                           |
+----------+---------------------------------------------------------+
| atan2    | Arc tangent of (x / y) expressed in radians. [-pi,+pi]  |
|          | eg: atan2(x,y)                                          |
+----------+---------------------------------------------------------+
| atanh    | Inverse hyperbolic tangent of x expressed in radians.   |
|          | (eg: atanh(x))                                          |
+----------+---------------------------------------------------------+
| cos      | Cosine of x.  (eg: cos(x))                              |
+----------+---------------------------------------------------------+
| cosh     | Hyperbolic cosine of x.  (eg: cosh(x))                  |
+----------+---------------------------------------------------------+
| cot      | Cotangent of x.  (eg: cot(x))                           |
+----------+---------------------------------------------------------+
| csc      | Cosecant of x.  (eg: csc(x))                            |
+----------+---------------------------------------------------------+
| sec      | Secant of x.  (eg: sec(x))                              |
+----------+---------------------------------------------------------+
| sin      | Sine of x.  (eg: sin(x))                                |
+----------+---------------------------------------------------------+
| sinh     | Hyperbolic sine of x.  (eg: sinh(x))                    |
+----------+---------------------------------------------------------+
| tan      | Tangent of x.  (eg: tan(x))                             |
+----------+---------------------------------------------------------+
| tanh     | Hyperbolic tangent of x.  (eg: tanh(x))                 |
+----------+---------------------------------------------------------+
| deg2rad  | Convert x from degrees to radians.  (eg: deg2rad(x))    |
+----------+---------------------------------------------------------+
| deg2grad | Convert x from degrees to gradians.  (eg: deg2grad(x))  |
+----------+---------------------------------------------------------+
| rad2deg  | Convert x from radians to degrees.  (eg: rad2deg(x))    |
+----------+---------------------------------------------------------+
| grad2deg | Convert x from gradians to degrees.  (eg: grad2deg(x))  |
+----------+---------------------------------------------------------+

(5) String Processing
+----------+---------------------------------------------------------+
| FUNCTION | DEFINITION                                              |
+----------+---------------------------------------------------------+
|  = , ==  | All common equality/inequality operators are applicable |
|  !=, <>  | to strings and are applied in a case sensitive manner.  |
|  <=, >=  | In the following example x, y and z are of type string. |
|  < , >   | (eg: not((x <= 'AbC') and ('1x2y3z' <> y)) or (z == x)  |
+----------+---------------------------------------------------------+
| in       | True only if x is a substring of y.                     |
|          | (eg: x in y or 'abc' in 'abcdefgh')                     |
+----------+---------------------------------------------------------+
| like     | True only if the string x matches the pattern y.        |
|          | Available wildcard characters are '*' and '?' denoting  |
|          | zero or more and zero or one matches respectively.      |
|          | (eg: x like y or 'abcdefgh' like 'a?d*h')               |
+----------+---------------------------------------------------------+
| ilike    | True only if the string x matches the pattern y in a    |
|          | case insensitive manner. Available wildcard characters  |
|          | are '*' and '?' denoting zero or more and zero or one   |
|          | matches respectively.                                   |
|          | (eg: x ilike y or 'a1B2c3D4e5F6g7H' ilike 'a?d*h')      |
+----------+---------------------------------------------------------+
| [r0:r1]  | The closed interval [r0,r1] of the specified string.    |
|          | eg: Given a string x with a value of 'abcdefgh' then:   |
|          | 1. x[1:4] == 'bcde'                                     |
|          | 2. x[ :5] == x[:5] == 'abcdef'                          |
|          | 3. x[3: ] == x[3:] =='cdefgh'                           |
|          | 4. x[ : ] == x[:] == 'abcdefgh'                         |
|          | 5. x[4/2:3+2] == x[2:5] == 'cdef'                       |
|          |                                                         |
|          | Note: Both r0 and r1 are assumed to be integers, where  |
|          | r0 <= r1. They may also be the result of an expression, |
|          | in the event they have fractional components truncation |
|          | will be performed. (eg: 1.67 --> 1)                     |
+----------+---------------------------------------------------------+

(6) Control Structures
+----------+---------------------------------------------------------+
|STRUCTURE | DEFINITION                                              |
+----------+---------------------------------------------------------+
| if       | If x is true then return y else return z.               |
|          | eg:                                                     |
|          | 1. if(x, y, z)                                          |
|          | 2. if((x + 1) > 2y, z + 1, w / v)                       |
+----------+---------------------------------------------------------+
| if-else  | The if-else/else-if statement. Subject to the condition |
|          | branch the statement will return either the value of the|
|          | consequent or the alternative branch.                   |
|          | eg:                                                     |
|          | 1. if (x > y) z; else w;                                |
|          | 2. if (x > y) z; else if (w != u) v;                    |
|          | 3. if (x < y) {z; w+1;} else u;                         |
|          | 4. if ((x != y) and (z > w))                            |
|          |    {                                                    |
|          |      y := sin(x) / u;                                   |
|          |      z := w+1;                                          |
|          |    }                                                    |
|          |    else if (x > (z + 1))                                |
|          |    {                                                    |
|          |      w := abs (x - y) + z;                              |
|          |      u := (x + 1) > 2y ? 2u : 3u;                       |
|          |    }                                                    |
+----------+---------------------------------------------------------+
| switch   | The first true case condition that is encountered will  |
|          | determine the result of the switch. If none of the case |
|          | conditions hold true, the default action is assumed as  |
|          | the final return value. This is sometimes also known as |
|          | a multi-way branch mechanism.                           |
|          | eg:                                                     |
|          | switch                                                  |
|          | {                                                       |
|          |   case x > (y + z) : 2 * x / abs(y - z);                |
|          |   case x < 3       : sin(x + y);                        |
|          |   default          : 1 + x;                             |
|          | }                                                       |
+----------+---------------------------------------------------------+
| while    | The structure will repeatedly evaluate the internal     |
|          | statement(s) 'while' the condition is true. The final   |
|          | statement in the final iteration will be used as the    |
|          | return value of the loop.                               |
|          | eg:                                                     |
|          | while ((x := (x - 1)) > 0)                              |
|          | {                                                       |
|          |   y := x + z;                                           |
|          |   w := z + y;                                           |
|          | }                                                       |
+----------+---------------------------------------------------------+
| repeat/  | The structure will repeatedly evaluate the internal     |
| until    | statement(s) 'until' the condition is true. The final   |
|          | statement in the final iteration will be used as the    |
|          | return value of the loop.                               |
|          | eg:                                                     |
|          | repeat                                                  |
|          |   y := x + z;                                           |
|          |   w := z + y;                                           |
|          | until ((x := (x - 1)) <= 0)                             |
+----------+---------------------------------------------------------+
| ?:       | Ternary conditional statement, similar to that of the   |
|          | above denoted if-statement.                             |
|          | eg:                                                     |
|          | 1. x ? y : z                                            |
|          | 2. x + 1 > 2y ? z + 1 : (w / v)                         |
|          | 3. min(x,y) > z ? (x < y + 1) ? x : y : (w * v)         |
+----------+---------------------------------------------------------+
| ~        | Evaluate each sub-expression, then return as the result |
|          | the value of the last sub-expression. This is sometimes |
|          | known as multiple sequence point evaluation.            |
|          | eg:                                                     |
|          | ~(i := x + 1, j := y / z, k := sin(w/u)) == (sin(w/u))) |
|          | ~{i := x + 1; j := y / z; k := sin(w/u)} == (sin(w/u))) |
+----------+---------------------------------------------------------+
| [*]      | Evaluate any consequent for which its case statement is |
|          | true. The return value will be either zero or the result|
|          | of the last consequent to have been evaluated.          |
|          | eg:                                                     |
|          | [*]                                                     |
|          | {                                                       |
|          |   case (x + 1) > (y - 2)   : x := z / 2 + sin(y / pi);  |
|          |   case (x + 2) < abs(y + 3): w / 4 + min(5y,9);         |
|          |   case (x + 3) = (y * 4)   : y := abs(z / 6) + 7y;      |
|          | }                                                       |
+----------+---------------------------------------------------------+

Note: In  the  above  tables, the  symbols x, y, z, w, u  and v  where
appropriate may represent any of one the following:

   1. Literal numeric/string value
   2. A variable
   3. An expression comprised of [1] or [2] (eg: 2 + x)



[09 - COMPONENTS]
There are three primary components, that are specialized upon a  given
numeric type, which make up the core of ExprTk. The components are  as
follows:

   1. Symbol Table  exprtk::symbol_table<NumericType>
   2. Expression    exprtk::expression<NumericType>
   3. Parser        exprtk::parser<NumericType>


(1) Symbol Table
A structure that is used  to store references to variables,  constants
and functions that are to  be used within expressions. Furthermore  in
the context  of composited  recursive functions  the symbol  table can
also be thought of as a simple representation of a stack specific  for
the expression(s) that reference it. The following is a list of the
types a symbol table can handle:

   (a) Numeric variables
   (b) Numeric constants
   (c) String variables
   (d) String constants
   (e) Functions
   (f) Vararg functions

During the compilation  process if an  expression is found  to require
any  of  the  elements   noted  above,  the  expression's   associated
symbol_table  will  be  queried  for  the  element  and  if  present a
reference to the element will be embedded within the expression's AST.
This allows for the original  element to be modified independently  of
the  expression  instance  and  to also  allow  the  expression  to be
evaluated using the current value of the element.

The  example  below demonstrates  the  relationship between variables,
symbol_table and expression. Note  the variables are modified  as they
normally would in a program, and when the expression is  evaluated the
current values assigned to the variables will be used.

   typedef exprtk::symbol_table<double> symbol_table_t;
   typedef exprtk::expression<double>     expression_t;
   typedef exprtk::parser<double>             parser_t;

   symbol_table_t symbol_table;
   expression_t   expression;
   parser_t       parser;

   double x = 0;
   double y = 0;

   std::string expression_string = "x * y + 3";
   symbol_table.add_variable("x",x);
   symbol_table.add_variable("y",y);

   expression.register_symbol_table(symbol_table);

   parser.compile(expression_string,expression);

   x = 1.0;
   y = 2.0;
   parser.value(); // 1 * 2 + 3
   x = 3.7;
   parser.value(); // 3.7 * 2 + 3
   y = -9.0;
   parser.value(); // 3.7 * -9 + 3


(2) Expression
A structure that holds an AST  for a specified expression and is  used
to evaluate said expression.  If a compiled Expression  uses variables
or user defined functions, it will then also have an associated Symbol
Table, which will contain  references to said variables,  functions et
al. An example AST structure for the denoted expression is as follows:

Expression:  z := (x + y^-2.345) * sin(pi / min(w - 7.3,v))

                  [Root]
                    |
               [Assignment]
        ________/        \_____
       /                       \
 Variable(z)            [Multiplication]
                ____________/      \___________
               /                               \
              /                          [Unary-Func(sin)]
         [Addition]                            |
      ____/      \____                    [Division]
     /                \                 ___/      \___
 Variable(x)   [Exponentiation]        /              \
              ______/   \______  Constant(pi)  [Binary-Func(min)]
             /                 \                ____/    \____
        Variable(y)        [Negation]          /              \
                               |              /           Variable(v)
                        Constant(2.345)      /
                                            /
                                     [Subtraction]
                                   ____/      \____
                                  /                \
                             Variable(w)      Constant(7.3)


(3) Parser
A  structure  which  takes  as input  a  string  representation  of an
expression and attempts to compile said input with the result being an
instance  of  Expression.  If  an  error  is  encountered  during  the
compilation  process, the  parser will  stop compiling  and return  an
error status code,  with a more  detailed description of  the error(s)
and  its  location  within  the  input  provided  by  the  'get_error'
interface.



[10 - COMPILATION OPTIONS]
The exprtk::parser  when being  instantiated takes  as input  a set of
options  to be  used during  the compilation  process of  expressions.
An  example instantiation  of exprtk::parser  where only  the  joiner,
commutative and strength reduction options are enabled is as  follows:

   typedef exprtk::parser<NumericType> parser_t;

   std::size_t compile_options = parser_t::e_joiner            +
                                 parser_t::e_commutative_check +
                                 parser_t::e_strength_reduction;

   parser_t parser(compile_options);


Currently  seven  types of  compile  time options  are  supported, and
enabled by default. The options and their explanations are as follows:

(1) Replacer (e_replacer)
Enable replacement of specific  tokens with other tokens.  For example
the token  "true" of  type symbol  will be  replaced with  the numeric
token of value one.

   (a) (x < y) == true   --->  (x < y) == 1
   (b) false == (x > y)  --->  0 == (x > y)


(2) Joiner (e_joiner)
Enable  joining  of  multi-character  operators  that  may  have  been
incorrectly  disjoint in the  string  representation  of the specified
expression. For example the consecutive tokens of ">" "=" will  become
">=" representing  the "greater  than or  equal to"  operator. If  not
properly resolved the  original form will  cause a compilation  error.
The  following is  a listing  of the  scenarios that  the joiner  can
handle:

   (a) ':' '='  --->  ':=' (assignment)
   (b) '>' '='  --->  '>=' (gte)
   (c) '<' '='  --->  '<=' (lte)
   (d) '=' '='  --->  '==' (equal)
   (e) '!' '='  --->  '!=' (not-equal)
   (f) '<' '>'  --->  '<>' (not-equal)

An example of the transformation that takes place is as follows:

   (a) (x > = y) and (z ! = w)  --->  (x >= y) and (z != w)


(3) Numeric Check (e_numeric_check)
Enable validation of tokens representing numeric types so as to  catch
any errors prior  to the costly  process of the  main compilation step
commencing.


(4) Bracket Check (e_bracket_check)
Enable  the  check for  validating  the ordering  of  brackets in  the
specified expression.


(5) Sequence Check (e_sequence_check)
Enable the  check for  validating that  sequences of  either pairs  or
triplets of tokens make sense.  For example the following sequence  of
tokens when encountered will raise an error:

   (a) (x + * 3)  --->  sequence error


(6) Commutative Check (e_commutative_check)
Enable the check that will transform sequences of pairs of tokens that
imply a multiplication operation.  The following are some  examples of
such transformations:

   (a) 2x             --->  2 * x
   (b) 25x^3          --->  25 * x^3
   (c) 3(x + 1)       --->  3 * (x + 1)
   (d) (x + 1)4       --->  (x + 1) * 4
   (e) 5foo(x,y)      --->  5 * foo(x,y)
   (f) foo(x,y)6 + 1  --->  foo(x,y) * 6 + 1
   (g) (4((2x)3))     --->  4 * ((2 * x) * 3)


(7) Strength Reduction Check (e_strength_reduction)
Enable  the  use  of  strength  reduction  optimisations  during   the
compilation  process.  In  ExprTk  strength  reduction   optimisations
predominantly involve  transforming sub-expressions  into other  forms
that  are algebraically  equivalent yet  less costly  to compute.  The
following are examples of the various transformations that can occur:

   (a) (x / y) / z        --->  x / (y * z)
   (b) (x / y) / (z / w)  --->  (x * w) / (y * z)
   (c) (2 * x) - (2 * y)  --->  2 * (x - y)
   (d) (2 / x) / (3 / y)  --->  (2 / 3) / (x * y)
   (e) (2 * x) * (3 * y)  --->  (2 * 3) * (x * y)

Note:
When using  strength reduction  in conjunction  with expressions whose
inputs or sub-expressions may result  in values nearing either of  the
bounds of the underlying numeric  type (eg: double), there may  be the
possibility of a decrease in the precision of results.

In  the following  example the  given expression  which represents  an
attempt at computing the average  between x and y will  be transformed
as follows:

   (x * 0.5) + (y * 0.5) ---> 0.5 * (x + y)

There  may be  situations where  the above  transformation will  cause
numerical overflows and  that the original  form of the  expression is
desired over the strength reduced form. In these situations it is best
to turn off strength reduction optimisations  or to use a type with  a
larger numerical bound.



[11 - SPECIAL FUNCTIONS]
The purpose  of special  functions in  ExprTk is  to provide  compiler
generated equivalents of common mathematical expressions which can  be
invoked by  using the  'special function'  syntax (eg:  $f12(x,y,z) or
$f82(x,y,z,w)).

Special functions dramatically decrease  the total evaluation time  of
expressions which would otherwise  have been written using  the common
form by reducing the total number  of nodes in the evaluation tree  of
an  expression  and  by  also  leveraging  the  compiler's  ability to
correctly optimize such expressions for a given architecture.

          3-Parameter                       4-Parameter
 +-------------+-------------+    +--------------+------------------+
 |  Prototype  |  Operation  |    |  Prototype   |    Operation     |
 +-------------+-------------+    +--------------+------------------+
   $f00(x,y,z) | (x + y) / z       $f48(x,y,z,w) | x + ((y + z) / w)
   $f01(x,y,z) | (x + y) * z       $f49(x,y,z,w) | x + ((y + z) * w)
   $f02(x,y,z) | (x + y) - z       $f50(x,y,z,w) | x + ((y - z) / w)
   $f03(x,y,z) | (x + y) + z       $f51(x,y,z,w) | x + ((y - z) * w)
   $f04(x,y,z) | (x - y) + z       $f52(x,y,z,w) | x + ((y * z) / w)
   $f05(x,y,z) | (x - y) / z       $f53(x,y,z,w) | x + ((y * z) * w)
   $f06(x,y,z) | (x - y) * z       $f54(x,y,z,w) | x + ((y / z) + w)
   $f07(x,y,z) | (x * y) + z       $f55(x,y,z,w) | x + ((y / z) / w)
   $f08(x,y,z) | (x * y) - z       $f56(x,y,z,w) | x + ((y / z) * w)
   $f09(x,y,z) | (x * y) / z       $f57(x,y,z,w) | x - ((y + z) / w)
   $f10(x,y,z) | (x * y) * z       $f58(x,y,z,w) | x - ((y + z) * w)
   $f11(x,y,z) | (x / y) + z       $f59(x,y,z,w) | x - ((y - z) / w)
   $f12(x,y,z) | (x / y) - z       $f60(x,y,z,w) | x - ((y - z) * w)
   $f13(x,y,z) | (x / y) / z       $f61(x,y,z,w) | x - ((y * z) / w)
   $f14(x,y,z) | (x / y) * z       $f62(x,y,z,w) | x - ((y * z) * w)
   $f15(x,y,z) | x / (y + z)       $f63(x,y,z,w) | x - ((y / z) / w)
   $f16(x,y,z) | x / (y - z)       $f64(x,y,z,w) | x - ((y / z) * w)
   $f17(x,y,z) | x / (y * z)       $f65(x,y,z,w) | ((x + y) * z) - w
   $f18(x,y,z) | x / (y / z)       $f66(x,y,z,w) | ((x - y) * z) - w
   $f19(x,y,z) | x * (y + z)       $f67(x,y,z,w) | ((x * y) * z) - w
   $f20(x,y,z) | x * (y - z)       $f68(x,y,z,w) | ((x / y) * z) - w
   $f21(x,y,z) | x * (y * z)       $f69(x,y,z,w) | ((x + y) / z) - w
   $f22(x,y,z) | x * (y / z)       $f70(x,y,z,w) | ((x - y) / z) - w
   $f23(x,y,z) | x - (y + z)       $f71(x,y,z,w) | ((x * y) / z) - w
   $f24(x,y,z) | x - (y - z)       $f72(x,y,z,w) | ((x / y) / z) - w
   $f25(x,y,z) | x - (y / z)       $f73(x,y,z,w) | (x * y) + (z * w)
   $f26(x,y,z) | x - (y * z)       $f74(x,y,z,w) | (x * y) - (z * w)
   $f27(x,y,z) | x + (y * z)       $f75(x,y,z,w) | (x * y) + (z / w)
   $f28(x,y,z) | x + (y / z)       $f76(x,y,z,w) | (x * y) - (z / w)
   $f29(x,y,z) | x + (y + z)       $f77(x,y,z,w) | (x / y) + (z / w)
   $f30(x,y,z) | x + (y - z)       $f78(x,y,z,w) | (x / y) - (z / w)
   $f31(x,y,z) | x * y^2 + z       $f79(x,y,z,w) | (x / y) - (z * w)
   $f32(x,y,z) | x * y^3 + z       $f80(x,y,z,w) | x / (y + (z * w))
   $f33(x,y,z) | x * y^4 + z       $f81(x,y,z,w) | x / (y - (z * w))
   $f34(x,y,z) | x * y^5 + z       $f82(x,y,z,w) | x * (y + (z * w))
   $f35(x,y,z) | x * y^6 + z       $f83(x,y,z,w) | x * (y - (z * w))
   $f36(x,y,z) | x * y^7 + z       $f84(x,y,z,w) | x*y^2 + z*w^2
   $f37(x,y,z) | x * y^8 + z       $f85(x,y,z,w) | x*y^3 + z*w^3
   $f38(x,y,z) | x * y^9 + z       $f86(x,y,z,w) | x*y^4 + z*w^4
   $f39(x,y,z) | x * log(y)+z      $f87(x,y,z,w) | x*y^5 + z*w^5
   $f40(x,y,z) | x * log(y)-z      $f88(x,y,z,w) | x*y^6 + z*w^6
   $f41(x,y,z) | x * log10(y)+z    $f89(x,y,z,w) | x*y^7 + z*w^7
   $f42(x,y,z) | x * log10(y)-z    $f90(x,y,z,w) | x*y^8 + z*w^8
   $f43(x,y,z) | x * sin(y)+z      $f91(x,y,z,w) | x*y^9 + z*w^9
   $f44(x,y,z) | x * sin(y)-z      $f92(x,y,z,w) | (x and y) ? z : w
   $f45(x,y,z) | x * cos(y)+z      $f93(x,y,z,w) | (x or  y) ? z : w
   $f46(x,y,z) | x * cos(y)-z      $f94(x,y,z,w) | (x <   y) ? z : w
   $f47(x,y,z) | x ? y : z         $f95(x,y,z,w) | (x <=  y) ? z : w
                                   $f96(x,y,z,w) | (x >   y) ? z : w
                                   $f97(x,y,z,w) | (x >=  y) ? z : w
                                   $f98(x,y,z,w) | (x ==  y) ? z : w
                                   $f99(x,y,z,w) | x*sin(y)+z*cos(w)



[12 - EXPRTK NOTES]
 (00) Precision and performance of expression evaluations are the
      dominant principles of the ExprTk library.

 (01) Supported types are float, double and long double.

 (02) Standard mathematical operator precedence is applied (BEDMAS).

 (03) Results of expressions that are deemed as being 'valid' are to
      exist within the set of Real numbers. All other results will be
      of the value: Not-A-Number (NaN).

 (04) Supported user defined types are numeric and string variables
      and functions.

 (05) All variable and function names are case-insensitive.

 (06) Variable and function names must begin with a letter
      (A-Z or a-z), then can be comprised of any combination of
      letters, digits and underscores. (eg: x, var1 or power_func99)

 (07) Expression lengths and sub-expression lists are limited only by
      storage capacity.

 (08) The life-time of objects registered with or created from a
      specific symbol-table must span at least the life-time of the
      compiled expressions which utilize objects, such as variables,
      of that symbol-table, otherwise the result will be undefined
      behavior.

 (09) Equal/Nequal are normalized equality routines, which use
      epsilons of 0.0000000001 and 0.000001 for double and float
      types respectively.

 (10) All trigonometric functions assume radian input unless
      stated otherwise.

 (11) Expressions may contain white-space characters such as
      space,  tabs, new-lines, control-feed et al.
      ('\n', '\r', '\t', '\b', '\v', '\f')

 (12) Strings may be constructed from any letters, digits or special
      characters such as (~!@#$%^&*()[]|=+ ,./?<>;:"`~_), and must
      be enclosed with single-quotes.
      eg: 'Frankly my dear, I do not give a damn!'

 (13) User defined normal functions can have up to 20 parameters,
      where as user defined vararg-functions can have an unlimited
      number of parameters.

 (14) The inbuilt polynomial functions can be at most of degree 12.

 (15) Where appropriate constant folding optimisations may be
      applied. (eg: The expression '2+(3-(x/y))' becomes '5-(x/y)')

 (16) If the strength reduction compilation option has been enabled,
      then where applicable strength reduction optimisations may be
      applied.

 (17) String processing capabilities are available by default.
      To turn them off, the following needs to be defined at
      compile time: exprtk_disable_string_capabilities

 (18) Composited functions can call themselves or any other functions
      that have been defined prior to their own definition.

 (19) Recursive calls made from within composited functions will have
      a stack size bound by the stack of the executing architecture.

 (20) The entity relationship between symbol_table and an expression
      is one-to-many. Hence the intended use case is to have a single
      symbol table manage the variable and function requirements of
      multiple expressions.

 (21) The common use-case for an expression is to have it compiled
      only once and then subsequently have it evaluated multiple
      times. An extremely inefficient approach would be to recompile
      an expression from its string form every time it requires
      evaluating.

 (22) The following are examples of compliant floating point value
      representations:
      (a) 12345        (b) -123.456
      (c) +123.456e+12 (d) 123.456E-12
      (e) +012.045e+07 (f) .1234
      (g) 123.456f     (h) -321.654E+3L

 (23) Expressions may contain any of the following comment styles:
      1. // .... \n
      2. #  .... \n
      3. /* .... */



[13 - SIMPLE EXPRTK EXAMPLE]
--- snip ---
#include <cstdio>
#include <string>

#include "exprtk.hpp"

template <typename T>
struct myfunc : public exprtk::ifunction<T>
{
   myfunc() : exprtk::ifunction<T>(2) {}

   T operator()(const T& v1, const T& v2)
   {
      return T(1) + (v1 * v2) / T(3);
   }
};

int main()
{
   typedef exprtk::symbol_table<double> symbol_table_t;
   typedef exprtk::expression<double>     expression_t;
   typedef exprtk::parser<double>             parser_t;
   typedef exprtk::parser_error::type          error_t;

   std::string expression_str =
                  "z := 2 myfunc([4 + sin(x / pi)^3],y ^ 2)";

   double x = 1.1;
   double y = 2.2;
   double z = 3.3;

   myfunc<double> mf;

   symbol_table_t symbol_table;
   symbol_table.add_constants();
   symbol_table.add_variable("x",x);
   symbol_table.add_variable("y",y);
   symbol_table.add_variable("z",z);
   symbol_table.add_function("myfunc",mf);

   expression_t expression;
   expression.register_symbol_table(symbol_table);

   parser_t parser;

   if (!parser.compile(expression_str,expression))
   {
      // A compilation error has occured. Attempt to
      // print all errors to the stdout.

      printf("Error: %s\tExpression: %s\n",
             parser.error().c_str(),
             expression_str.c_str());

      for (std::size_t i = 0; i < parser.error_count(); ++i)
      {
         // Include the specific nature of each error
         // and its position in the expression string.

         error_t error = parser.get_error(i);

         printf("Error: %02d Position: %02d "
                "Type: [%s] "
                "Message: %s "
                "Expression: %s\n",
                static_cast<int>(i),
                static_cast<int>(error.token.position),
                exprtk::parser_error::to_str(error.mode).c_str(),
                error.diagnostic.c_str(),
                expression_str.c_str());
      }

      return 1;
   }

   // Evaluate the expression and obtain its result.

   double result = expression.value();

   printf("Result: %10.5f\n",result);

   return 0;
}
--- snip ---



[14 - FILES]
(00) Makefile
(01) readme.txt
(02) exprtk.hpp
(03) exprtk_test.cpp
(04) exprtk_benchmark.cpp
(05) exprtk_simple_example_01.cpp
(06) exprtk_simple_example_02.cpp
(07) exprtk_simple_example_03.cpp
(08) exprtk_simple_example_04.cpp
(09) exprtk_simple_example_05.cpp
(10) exprtk_simple_example_06.cpp
(11) exprtk_simple_example_07.cpp
(12) exprtk_simple_example_08.cpp
(13) exprtk_simple_example_09.cpp
(14) exprtk_simple_example_10.cpp
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