C++ Mathematical Expression Toolkit Library Documentation
Section 00 - Introduction
Section 01 - Capabilities
Section 02 - Example Expressions
Section 03 - Copyright Notice
Section 04 - Downloads & Updates
Section 05 - Installation
Section 06 - Compilation
Section 07 - Compiler Compatibility
Section 08 - Built-In Operations & Functions
Section 09 - Fundamental Types
Section 10 - Components
Section 11 - Compilation Options
Section 12 - Expression Structures
Section 13 - Variable, Vector & String Definition
Section 14 - Vector Processing
Section 15 - User Defined Functions
Section 16 - Expression Dependents
Section 17 - Hierarchies Of Symbol Tables
Section 18 - Unknown Unknowns
Section 19 - Enabling & Disabling Features
Section 20 - Expression Return Values
Section 21 - Compilation Errors
Section 22 - Runtime Library Packages
Section 23 - Helpers & Utils
Section 24 - Benchmarking
Section 25 - Exprtk Notes
Section 26 - Simple Exprtk Example
Section 27 - Build Options
Section 28 - Files
Section 29 - Language Structure
[SECTION 00 - INTRODUCTION]
The C++ Mathematical Expression Toolkit Library (ExprTk) is a simple
to use, easy to integrate and extremely efficient run-time
mathematical expression parsing and evaluation engine. The parsing
engine supports numerous forms of functional and logic processing
semantics and is easily extensible.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 01 - CAPABILITIES]
The ExprTk expression evaluator supports the following fundamental
arithmetic operations, functions and processes:
(00) Types: Scalar, Vector, String
(01) Basic operators: +, -, *, /, %, ^
(02) Assignment: :=, +=, -=, *=, /=, %=
(03) Equalities &
Inequalities: =, ==, <>, !=, <, <=, >, >=
(04) Logic operators: and, mand, mor, nand, nor, not, or, shl, shr,
xnor, xor, true, false
(05) Functions: abs, avg, ceil, clamp, equal, erf, erfc, exp,
expm1, floor, frac, log, log10, log1p, log2,
logn, max, min, mul, ncdf, nequal, root,
round, roundn, sgn, sqrt, sum, swap, trunc
(06) Trigonometry: acos, acosh, asin, asinh, atan, atanh, atan2,
cos, cosh, cot, csc, sec, sin, sinc, sinh,
tan, tanh, hypot, rad2deg, deg2grad, deg2rad,
grad2deg
(07) Control
structures: if-then-else, ternary conditional, switch-case,
return-statement
(08) Loop statements: while, for, repeat-until, break, continue
(09) String
processing: in, like, ilike, concatenation
(10) Optimisations: constant-folding, simple strength reduction and
dead code elimination
(11) Calculus: numerical integration and differentiation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 02 - EXAMPLE EXPRESSIONS]
The following is a short listing of infix format based 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] + 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[i])z + 1.1 / 2.7 == (x + y[i]) * 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) while (x <= 100) { x -= 1; }
(18) x <= 'abc123' and (y in 'AString') or ('1x2y3z' != z)
(19) ((x + 'abc') like '*123*') or ('a123b' ilike y)
(20) sgn(+1.2^3.4z / -5.6y) <= {-7.8^9 / -10.11x }
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 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 MIT License.
http://www.opensource.org/licenses/MIT
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 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:
(a) Download: http://www.partow.net/programming/exprtk/index.html
(b) Repository: https://github.com/ArashPartow/exprtk
https://github.com/ArashPartow/exprtk-extras
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 05 - INSTALLATION]
The header file exprtk.hpp should be placed in a project or system
include path (e.g: /usr/include/).
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 07 - COMPILER COMPATIBILITY]
ExprTk has been built error and warning free using the following set
of C++ compilers:
(*) GNU Compiler Collection (3.5+)
(*) Intel C++ Compiler (8.x+)
(*) Clang/LLVM (1.1+)
(*) PGI C++ (10.x+)
(*) Microsoft Visual Studio C++ Compiler (8.1+)
(*) IBM XL C/C++ (9.x+)
(*) C++ Builder (XE4+)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 08 - BUILT-IN OPERATIONS & FUNCTIONS]
(0) Arithmetic & Assignment 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. Where y is either a variable|
| | or vector type. (eg: y := x) |
+----------+---------------------------------------------------------+
| += | Increment x by the value of the expression on the right |
| | hand side. Where x is either a variable or vector type. |
| | (eg: x += abs(y - z)) |
+----------+---------------------------------------------------------+
| -= | Decrement x by the value of the expression on the right |
| | hand side. Where x is either a variable or vector type. |
| | (eg: x[i] -= abs(y + z)) |
+----------+---------------------------------------------------------+
| *= | Assign the multiplication of x by the value of the |
| | expression on the righthand side to x. Where x is either|
| | a variable or vector type. |
| | (eg: x *= abs(y / z)) |
+----------+---------------------------------------------------------+
| /= | Assign the division of x by the value of the expression |
| | on the right-hand side to x. Where x is either a |
| | variable or vector type. (eg: x[i + j] /= abs(y * z)) |
+----------+---------------------------------------------------------+
| %= | Assign x modulo the value of the expression on the right|
| | hand side to x. Where x is either a variable or vector |
| | type. (eg: x[2] %= y ^ 2) |
+----------+---------------------------------------------------------+
(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 exactly 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 normalised 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) |
+----------+---------------------------------------------------------+
| inrange | In-range returns 'true' when x is within the range r0 |
| | and r1. Where r0 < r1. (eg: inrange(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)) |
+----------+---------------------------------------------------------+
| ncdf | Normal cumulative distribution function. (eg: ncdf(x)) |
+----------+---------------------------------------------------------+
| nequal | Not-equal test between x and y using normalised epsilon |
+----------+---------------------------------------------------------+
| pow | x to the power of y. (eg: pow(x,y) == x ^ y) |
+----------+---------------------------------------------------------+
| root | Nth-Root of x. where n is a positive integer. |
| | (eg: root(x,3) == x^(1/3)) |
+----------+---------------------------------------------------------+
| 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)) |
+----------+---------------------------------------------------------+
| swap | Swap the values of the variables x and y and return the |
| <=> | current value of y. (eg: swap(x,y) or x <=> y) |
+----------+---------------------------------------------------------+
| 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)) |
+----------+---------------------------------------------------------+
| sinc | Sine cardinal of x. (eg: sinc(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) |
+----------+---------------------------------------------------------+
| := | Assign the value of x to y. Where y is a mutable string |
| | or string range and x is either a string or a string |
| | range. eg: |
| | 1. y := x |
| | 2. y := 'abc' |
| | 3. y := x[:i + j] |
| | 4. y := '0123456789'[2:7] |
| | 5. y := '0123456789'[2i + 1:7] |
| | 6. y := (x := '0123456789'[2:7]) |
| | 7. y[i:j] := x |
| | 8. y[i:j] := (x + 'abcdefg'[8 / 4:5])[m:n] |
| | |
| | Note: For options 7 and 8 the shorter of the two ranges |
| | will denote the number characters that are to be copied.|
+----------+---------------------------------------------------------+
| + | Concatenation of x and y. Where x and y are strings or |
| | string ranges. eg |
| | 1. x + y |
| | 2. x + 'abc' |
| | 3. x + y[:i + j] |
| | 4. x[i:j] + y[2:3] + '0123456789'[2:7] |
| | 5. 'abc' + x + y |
| | 6. 'abc' + '1234567' |
| | 7. (x + 'a1B2c3D4' + y)[i:2j] |
+----------+---------------------------------------------------------+
| += | Append to x the value of y. Where x is a mutable string |
| | and y is either a string or a string range. eg: |
| | 1. x += y |
| | 2. x += 'abc' |
| | 3. x += y[:i + j] + 'abc' |
| | 4. x += '0123456789'[2:7] |
+----------+---------------------------------------------------------+
| <=> | Swap the values of x and y. Where x and y are mutable |
| | strings. (eg: x <=> y) |
+----------+---------------------------------------------------------+
| [] | The string size operator returns the size of the string |
| | being actioned. |
| | eg: |
| | 1. 'abc'[] == 3 |
| | 2. var max_str_length := max(s0[],s1[],s2[],s3[]) |
| | 3. ('abc' + 'xyz')[] == 6 |
| | 4. (('abc' + 'xyz')[1:4])[] == 4 |
+----------+---------------------------------------------------------+
(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) |
| | 3. if (x > y) z; |
| | 4. if (x <= 2*y) { z + w }; |
+----------+---------------------------------------------------------+
| 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 -= 1) > 0) |
| | { |
| | y := x + z; |
| | w := u + 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 := u + y; |
| | until ((x += 1) > 100) |
+----------+---------------------------------------------------------+
| for | The structure will repeatedly evaluate the internal |
| | statement(s) while the condition is true. On each loop |
| | iteration, an 'incrementing' expression is evaluated. |
| | The conditional is mandatory whereas the initialiser |
| | and incrementing expressions are optional. |
| | eg: |
| | for (var x := 0; (x < n) and (x != y); x += 1) |
| | { |
| | y := y + x / 2 - z; |
| | w := u + y; |
| | } |
+----------+---------------------------------------------------------+
| break | Break terminates the execution of the nearest enclosed |
| break[] | loop, allowing for the execution to continue on external|
| | to the loop. The default break statement will set the |
| | return value of the loop to NaN, where as the return |
| | based form will set the value to that of the break |
| | expression. |
| | eg: |
| | while ((i += 1) < 10) |
| | { |
| | if (i < 5) |
| | j -= i + 2; |
| | else if (i % 2 == 0) |
| | break; |
| | else |
| | break[2i + 3]; |
| | } |
+----------+---------------------------------------------------------+
| continue | Continue results in the remaining portion of the nearest|
| | enclosing loop body to be skipped. |
| | eg: |
| | for (var i := 0; i < 10; i += 1) |
| | { |
| | if (i < 5) |
| | continue; |
| | j -= i + 2; |
| | } |
+----------+---------------------------------------------------------+
| return | Return immediately from within the current expression. |
| | With the option of passing back a variable number of |
| | values (scalar, vector or string). eg: |
| | 1. return [1]; |
| | 2. return [x, 'abx']; |
| | 3. return [x, x + y,'abx']; |
| | 4. return []; |
| | 5. if (x < y) |
| | return [x, x - y, 'result-set1', 123.456]; |
| | else |
| | return [y, x + y, 'result-set2']; |
+----------+---------------------------------------------------------+
| ?: | 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; |
| | } |
+----------+---------------------------------------------------------+
| [] | The vector size operator returns the size of the vector |
| | being actioned. |
| | eg: |
| | 1. v[] |
| | 2. max_size := max(v0[],v1[],v2[],v3[]) |
+----------+---------------------------------------------------------+
Note: In the tables above, 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. A vector element
4. A vector
5. A string
6. An expression comprised of [1], [2] or [3] (eg: 2 + x / vec[3])
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 09 - FUNDAMENTAL TYPES]
ExprTk supports three fundamental types which can be used freely in
expressions. The types are as follows:
(1) Scalar
(2) Vector
(3) String
(1) Scalar Type
The scalar type is a singular numeric value. The underlying type is
that used to specialise the ExprTk components (float, double, long
double, MPFR et al).
(2) Vector Type
The vector type is a fixed size sequence of contiguous scalar values.
A vector can be indexed resulting in a scalar value. Operations
between a vector and scalar will result in a vector with a size equal
to that of the original vector, whereas operations between vectors
will result in a vector of size equal to that of the smaller of the
two. In both mentioned cases, the operations will occur element-wise.
(3) String Type
The string type is a variable length sequence of 8-bit chars. Strings
can be assigned and concatenated to one another, they can also be
manipulated via sub-ranges using the range definition syntax. Strings
however can not interact with scalar or vector types.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 10 - COMPONENTS]
There are three primary components, that are specialised 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) Numeric vector elements
(d) String variables
(e) String constants
(f) Functions
(g) 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.
Note: Any variable reference provided to a given symbol_table
instance, must have a life time at least as long as the life-time of
the symbol_table instance. In the event the variable reference is
invalidated before the symbol_table or any dependent expression
instances have been destructed, then any associated expression
evaluations or variable referencing via the symbol_table instance will
result in undefined behaviour.
The following bit of code instantiates a symbol_table and expression
instance, then proceeds to demonstrate various ways in which
references to variables can be added to the symbol_table, and how
those references are subsequently invalidated resulting in various
forms of undefined behaviour.
typedef exprtk::symbol_table<double> symbol_table_t;
symbol_table_t symbol_table;
expression_t expression;
{
double x = 123.4567;
symbol_table.add_variable("x", x);
} // Reference to variable x has been invalidated
std::deque<double> y {1.1, 2.2, 3.3};
symbol_table.add_variable("y", y.back());
y.pop_back(); // Reference to variable y has been invalidated
std::vector<double> z {4.4, 5.5, 6.6};
symbol_table.add_variable("z", z.front());
z.erase(z.begin());
// Reference to variable z has been invalidated
double* w = new double(123.456);
symbol_table.add_variable("w", *w);
delete w; // Reference to variable w has been invalidated
const std::string expression_str = "x + y / z * w";
// Compilation of expression will succeed
parser.compile(expression_str,expression);
expression.value();
// Evaluation will result in undefined behaviour
symbol_table.get_variable("x")->ref() = 135.791;
// Assignment will result in undefined behaviour
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;
expression.value(); // 1 * 2 + 3
x = 3.7;
expression.value(); // 3.7 * 2 + 3
y = -9.0;
expression.value(); // 3.7 * -9 + 3
// 'x * -9 + 3' for x in range of [0,100) in steps of 0.0001
for (x = 0.0; x < 100.0; x += 0.0001)
{
expression.value(); // x * -9 + 3
}
Note: It is possible to register multiple symbol_tables with a single
expression object. In the event an expression has multiple symbol
tables, and where there exists conflicts between symbols, the
compilation stage will resolve the conflicts based on the order of
registration of the symbol_tables to the expression. For a more
expansive discussion please review section [17 - Hierarchies Of
Symbol Tables]
typedef exprtk::symbol_table<double> symbol_table_t;
typedef exprtk::expression<double> expression_t;
typedef exprtk::parser<double> parser_t;
symbol_table_t symbol_table0;
symbol_table_t symbol_table1;
expression_t expression;
parser_t parser;
double x0 = 123.0;
double x1 = 678.0;
std::string expression_string = "x + 1";
symbol_table0.add_variable("x",x0);
symbol_table1.add_variable("x",x1);
expression.register_symbol_table(symbol_table0);
expression.register_symbol_table(symbol_table1);
parser.compile(expression_string,expression);
expression.value(); // 123 + 1
The symbol table supports adding references to external instances of
types that can be accessed within expressions via the following
methods:
1. bool add_variable (const std::string& name, scalar_t&)
2. bool add_constant (const std::string& name, const scalar_t&)
3. bool add_stringvar(const std::string& name, std::string&)
4. bool add_vector (const std::string& name, vector_type&)
Note: The 'vector' type must be comprised from a contiguous array of
scalars with a size that is larger than zero. The vector type itself
can be any one of the following:
1. std::vector<scalar_t>
2. scalar_t(&v)[N]
3. scalar_t* and array size
4. exprtk::vector_view<scalar_t>
When registering a variable, vector, string or function with an
instance of a symbol_table, the call to 'add_...' may fail and return
a false result due to one or more of the following reasons:
1. Variable name contains invalid characters or is ill-formed
2. Variable name conflicts with a reserved word (eg: 'while')
3. Variable name conflicts with a previously registered variable
4. A vector of size (length) zero is being registered
5. A free function exceeding fifteen parameters is being registered
6. The symbol_table instance is in an invalid state
(2) Expression
A structure that holds an abstract syntax tree or AST for a specified
expression and is used to evaluate said expression. Evaluation of the
expression is accomplished by performing a post-order traversal of the
AST. If a compiled Expression uses variables or user defined
functions, it will have an associated Symbol Table, which will contain
references to said variables, functions or strings. 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-Function(sin)]
[Addition] |
____/ \____ [Division]
/ \ ___/ \___
Variable(x) [Exponentiation] / \
______/ \______ Constant(pi) [Binary-Function(min)]
/ \ ____/ \____
Variable(y) [Negation] / \
| / Variable(v)
Constant(2.345) /
/
[Subtraction]
____/ \____
/ \
Variable(w) Constant(7.3)
The above denoted AST will be evaluated in the following order:
(01) Load Variable (z) (10) Load Constant (7.3)
(02) Load Variable (x) (11) Subtraction (09 & 10)
(03) Load Variable (y) (12) Load Variable (v)
(04) Load Constant (2.345) (13) Min (11 & 12)
(05) Negation (04) (14) Division (08 & 13)
(06) Exponentiation (03 & 05) (15) Sin (14)
(07) Addition (02 & 06) (16) Multiplication (07 & 15)
(08) Load Constant (pi) (17) Assignment (01 & 16)
(09) Load Variable (w)
Generally an expression in ExprTk can be thought of as a free function
similar to those found in imperative languages. This form of pseudo
function will have a name, it may have a set of one or more inputs and
will return at least one value as its result. Futhermore the function
when invoked, may cause a side-effect that changes the state of the
host program.
As an example the following is a pseudo-code definition of a free
function that performs a computation taking four inputs, modifying one
of them and returning a value based on some arbitrary calculation:
ResultType foo(InputType x, InputType y, InputType z, InputType w)
{
w = 2 * x^y + z; // Side-Effect
return abs(x - y) / z; // Return Result
}
Given the above definition the following is a functionally equivalent
version using ExprTk:
const std::string foo_str = " w := 2 * x^y + z; "
" abs(x - y) / z; ";
T x, y, z, w;
symbol_table_t symbol_table;
symbol_table.add_variable("x",x);
symbol_table.add_variable("y",y);
symbol_table.add_variable("z",z);
symbol_table.add_variable("w",w);
expression_t foo;
foo.register_symbol_table(symbol_table);
parser_t parser;
if (!parser.compile(foo_str,foo))
{
// Error in expression...
return;
}
T result = foo.value();
(3) Parser
A component 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.
Note: The exprtk::expression and exprtk::symbol_table components are
reference counted entities. Copy constructing or assigning to or from
either component will result in a shallow copy and a reference count
increment, rather than a complete replication. Furthermore the
expression and symbol_table components being Default-Constructible,
Copy-Constructible and Copy-Assignable make them compatible with
various C++ standard library containers and adaptors such as
std::vector, std::map, std::stack etc.
The following is an example of two unique expressions, after having
being instantiated and compiled, one expression is assigned to the
other. The diagrams depict their initial and post assignment states,
including which control block each expression references and their
associated reference counts.
exprtk::expression e0; // constructed expression, eg: x + 1
exprtk::expression e1; // constructed expression, eg: 2z + y
+-----[ e0 cntrl block]----+ +-----[ e1 cntrl block]-----+
| 1. Expression Node 'x+1' | | 1. Expression Node '2z+y' |
| 2. Ref Count: 1 |<-+ | 2. Ref Count: 1 |<-+
+--------------------------+ | +---------------------------+ |
| |
+--[ e0 expression]--+ | +--[ e1 expression]--+ |
| 1. Reference to ]------+ | 1. Reference to ]-------+
| e0 Control Block | | e1 Control Block |
+--------------------+ +--------------------+
e0 = e1; // e0 and e1 are now 2z+y
+-----[ e1 cntrl block]-----+
| 1. Expression Node '2z+y' |
+----------->| 2. Ref Count: 2 |<----------+
| +---------------------------+ |
| |
| +--[ e0 expression]--+ +--[ e1 expression]--+ |
+---[ 1. Reference to | | 1. Reference to ]---+
| e1 Control Block | | e1 Control Block |
+--------------------+ +--------------------+
The reason for the above complexity and restrictions of deep copies
for the expression and symbol_table components is because expressions
may include user defined variables or functions. These are embedded as
references into the expression's AST. When copying an expression, said
references need to also be copied. If the references are blindly
copied, it will then result in two or more identical expressions
utilizing the exact same references for variables. This obviously is
not the default assumed scenario and will give rise to non-obvious
behaviours when using the expressions in various contexts such as
muli-threading et al.
The prescribed method for cloning an expression is to compile it from
its string form. Doing so will allow the 'user' to properly consider
the exact source of user defined variables and functions.
Note: The exprtk::parser is a non-copyable and non-thread safe
component, and should only be shared via either a reference, a shared
pointer or a std::ref mechanism, and considerations relating to
synchronisation taken into account where appropriate. The parser
represents an object factory, specifically a factory of expressions,
and generally should not be instantiated solely on a per expression
compilation basis.
The following diagram and example depicts the flow of data and
operations for compiling multiple expressions via the parser and
inserting the newly minted exprtk::expression instances into a
std::vector.
+----[exprtk::parser]---+
| Expression Factory |
| parser_t::compile(...)|
+--> ~.~.~.~.~.~.~.~.~.~ ->--+
| +-----------------------+ |
Expressions in | | Expressions as
string form A V exprtk::expression
| | instances
[s0:'x+1']--->--+ | | +-[e0: x+1]
| | | |
[s1:'2z+y']-->--+--+ +->+-[e1: 2z+y]
| |
[s2:'sin(k+w)']-+ +-[e2: sin(k+w)]
const std::string expression_str[3]
= { "x + 1", "2x + y", "sin(k + w)" };
std::vector<expression_t> expression_list;
parser_t parser;
expression_t expression;
symbol_table_t symbol_table;
expression.register_symbol_table(symbol_table);
for (std::size_t i = 0; i < 3; ++i)
{
if (parser.compile(expression_str[i],expression))
{
expression_list.push_back(expression);
}
else
std::cout << "Error in " << expression_str[i] << "\n";
}
for (auto& e : expression_list)
{
e.value();
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 11 - 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>::settings_t settings_t;
std::size_t compile_options = settings_t::e_joiner +
settings_t::e_commutative_check +
settings_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
(2) Joiner
(3) Numeric Check
(4) Bracket Check
(5) Sequence Check
(6) Commutative Check
(7) Strength Reduction Check
(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) '>' '=' ---> '>=' (gte)
(b) '<' '=' ---> '<=' (lte)
(c) '=' '=' ---> '==' (equal)
(d) '!' '=' ---> '!=' (not-equal)
(e) '<' '>' ---> '<>' (not-equal)
(f) ':' '=' ---> ':=' (assignment)
(g) '+' '=' ---> '+=' (addition assignment)
(h) '-' '=' ---> '-=' (subtraction assignment)
(i) '*' '=' ---> '*=' (multiplication assignment)
(j) '/' '=' ---> '/=' (division assignment)
(k) '%' '=' ---> '%=' (modulo assignment)
(l) '+' '-' ---> '-' (subtraction)
(m) '-' '+' ---> '-' (subtraction)
(n) '-' '-' ---> '+' (addition)
(o) '<=' '>' ---> '<=>' (swap)
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)
(h) w(x) + (y)z ---> w * x + y * z
(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:
(0.5 * x) + (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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 12 - EXPRESSION STRUCTURES]
Exprtk supports mathematical expressions in numerous forms based on a
simple imperative programming model. This section will cover the
following topics related to general structure and programming of
expression using ExprTk:
(1) Multi-Statement Expressions
(2) Statements And Side-Effects
(3) Conditional Statements
(4) Special Functions
(1) Multi-Statement Expressions
Expressions in ExprTk can be comprised of one more statements, which
may sometimes be called sub-expressions. The following are two
examples of expressions stored in std::string variables, the first a
single statement and the second a multi-statement expression:
std::string single_statement = " z := x + y ";
std::string multi_statement = " var temp := x; "
" x := y + z; "
" y := temp; ";
In a multi-statement expression, the final statement will determine
the overall result of the expression. In the following multi-statement
expression, the result of the expression when evaluated will be '2.3',
which will also be the value stored in the 'y' variable.
z := x + y;
y := 2.3;
As demonstrated in the expression above, statements within an
expression are separated using the semi-colon ';' operator. In the
event two statements are not separated by a semi-colon, and the
implied multiplication feature is active (enabled by default), the
compiler will assume a multiplication operation between the two
statements.
In the following example we have a multi-statement expression composed
of two variable definitions and initialisations for variables x and y
and two seemingly separate mathematical operations.
var x:= 2;
var y:= 3;
x + 1
y * 2
However the result of the expression will not be 6 as may have been
assumed based on the calculation of 'y * 2', but rather the result
will be 8. This is because the compiler will have conjoined the two
mathematical statements into one via a multiplication operation. The
expression when compiled will actually evaluate as the following:
var x:= 2;
var y:= 3;
x + 1 * y * 2; // 2 + 1 * 3 * 2 == 8
In ExprTk any valid statement will itself return a value. This value
can further be used in conjunction with other statements. This
includes language structures such as if-statements, loops (for, while)
and the switch statement. Typically the last statement executed in the
given construct (conditional, loop etc), will be the value that is
returned.
In the following example, the return value of the expression will be
11, which is the sum of the variable 'x' and the final value computed
within the loop body on its last iteration:
var x := 1;
x + for (var i := x; i < 10; i += 1)
{
i / 2;
i + 1;
}
(2) Statements And Side-Effects
Statements themselves may have side effects, which in-turn effect the
proceeding statements in multi-statement expressions.
A statement is said to have a side-effect if it causes the state of
the expression to change in some way - this includes but is not
limited to the modification of the state of external variables used
within the expression. Currently the following actions being present
in a statement will cause it to have a side-effect:
(a) Assignment operation (explicit or potentially)
(b) Invoking a user-defined function that has side-effects
The following are examples of expressions where the side-effect status
of the statements (or sub-exressions) within the expressions have been
noted:
+-+----------------------+------------------------------+
|#| Expression | Side Effect Status |
+-+----------------------+------------------------------+
|0| x + y | False |
+-+----------------------+------------------------------+
|1| z := x + y | True - Due to assignment |
+-+----------------------+------------------------------+
|2| abs(x - y) | False |
+-+----------------------+------------------------------+
|3| abs(x - y); | False |
| | z := (x += y); | True - Due to assignments |
+-+----------------------+------------------------------+
|4| abs(x - y); | False |
| | z := (x += y); | True - Due to assignments |
+-+----------------------+------------------------------+
|5| var t := abs(x - y); | True - Due to initialisation |
| | t + x; | False |
| | z := (x += y); | True - Due to assignments |
+-+----------------------+------------------------------+
|6| foo(x - y) | True - user defined function |
+-+----------------------+------------------------------+
Note: In example 6 from the above set, it is assumed the user defined
function foo has been registered as having a side_effect. By default
all user defined functions are assumed to have side-effects, unless
they are configured in their constructors to not have side-effects
using the 'disable_has_side_effects' free function. For more
information review Section 15 - User Defined Functions sub-section 7
Function Side-Effects.
At this point we can see that there will be expressions composed of
certain kinds of statements that when executed will not effect the
nature of the expression's result. These statements are typically
called 'dead code'. These statements though not effecting the final
result will still be executed and as such they will consume processing
time that could otherwise be saved. As such ExprTk attempts to detect
and remove such statements from expressions.
The 'Dead Code Elimination' (DCE) optimisation process, which is
enabled by default, will remove any statements that are determined to
not have a side effect in a multi-statement expression, excluding the
final or last statement.
By default the final statement in an expression will always be present
regardless of its side-effect status, as it is the statement whose
value will be used as the result of the expression.
In order to further explain the actions taken during the DCE process,
lets review the following expression:
var x := 2; // Statement 1
var y := x + 2; // Statement 2
x + y; // Statement 3
y := x + 3y; // Statement 4
x - y; // Statement 5
The above expression has five statements. Three of them (1, 2 and 4)
actively have side-effects. The first two are variable declaration and
initialisations, where as the third is due to an assignment operation.
There are two statements (3 and 5), that do not explicitly have
side-effects, however the latter, statement 5, is the final statement
in the expression and hence will be assumed to have a side-effect.
During compilation when the DCE optimisation is applied to the above
expression, statement 3 will be removed from the expression, as it has
no bearing on the final result of expression, the rest of the
statements will all remain. The optimised form of the expression is as
follows:
var x := 2; // Statement 1
var y := x + 2; // Statement 2
y := x + 3y; // Statement 3
x - y; // Statement 4
(3) Conditional Statements (If-Then-Else)
ExprTk support two forms of conditional branching or otherwise known
as if-statements. The first form, is a simple function based
conditional statement, that takes exactly three input expressions:
condition, consequent and alternative. The following is an example
expression that utilises the function based if-statement.
x := if (y < z, y + 1, 2 * z)
In the example above, if the condition 'y < z' is true, then the
consequent 'y + 1' will be evaluated, its value will be returned and
subsequently assigned to the variable 'x'. Otherwise the alternative
'2 * z' will be evaluated and its value will be returned. This is
essentially the simplest form of an if-then-else statement. A simple
variation of the expression where the value of the if-statement is
used within another statement is as follows:
x := 3 * if (y < z, y + 1, 2 * z) / 2
The second form of if-statement resembles the standard syntax found in
most imperative languages. There are two variations of the statement:
(a) If-Statement
(b) If-Then-Else Statement
(a) If-Statement
This version of the conditional statement returns the value of the
consequent expression when the condition expression is true, else it
will return a quiet NaN value as its result.
Example 1:
x := if (y < z) y + 3;
Example 2:
x := if (y < z)
{
y + 3
}
The two example expressions above are equivalent. If the condition
'y < z' is true, the 'x' variable will be assigned the value of the
consequent 'y + 3', otherwise it will be assigned the value of quiet
NaN. As previously discussed, if-statements are value returning
constructs, and if not properly terminated using a semi-colon, will
end-up combining with the next statement via a multiplication
operation. The following example will NOT result in the expected value
of 'w + x' being returned:
x := if (y < z) y + 3 // missing semi-colon ';'
w + x
When the above supposed multi-statement expression is compiled, the
expression will have a multiplication inserted between the two
'intended' statements resulting in the unanticipated expression:
x := (if (y < z) y + 3) * w + x
The solution to the above situation is to simply terminate the
conditional statement with a semi-colon as follows:
x := if (y < z) y + 3;
w + x
(b) If-Then-Else Statement
The second variation of the if-statement is to allow for the use of
Else and If-Else cascading statements. Examples of such statements are
as follows:
Example 1: Example 2: Example 3:
if (x < y) if (x < y) if (x > y + 1)
z := x + 3; { y := abs(x - z);
else y := z + x; else
y := x - z; z := x + 3; {
} y := z + x;
else z := x + 3;
y := x - z; };
Example 4: Example 5: Example 6:
if (2 * x < max(y,3)) if (x < y) if (x < y or (x + z) > y)
{ z := x + 3; {
y := z + x; else if (2y != z) z := x + 3;
z := x + 3; { y := x - z;
} z := x + 3; }
else if (2y - z) y := x - z; else if (abs(2y - z) >= 3)
y := x - z; } y := x - z;
else else
x * x; {
z := abs(x * x);
x * y * z;
};
In the case where there is no final else statement and the flow
through the conditional arrives at this final point, the same rules
apply to this form of if-statement as to the previous. That is a quiet
NaN will be returned as the result of the if-statement. Furthermore
the same requirements of terminating the statement with a semi-colon
apply.
(4) 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 optimise 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)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 13 - VARIABLE, VECTOR & STRING DEFINITION]
ExprTk supports the definition of expression local variables, vectors
and strings. The definitions must be unique as shadowing is not
allowed and object life-times are based on scope. Definitions use the
following general form:
var <name> := <initialiser>;
(1) Variable Definition
Variables are of numeric type denoting a single value. They can be
explicitly initialised to a value, otherwise they will be defaulted to
zero. The following are examples of variable definitions:
(a) Initialise x to zero
var x;
(b) Initialise y to three
var y := 3;
(c) Initialise z to the expression
var z := if (max(1, x + y) > 2, w, v);
(2) Vector Definition
Vectors are arrays of a common numeric type. The elements in a vector
can be explicitly initialised, otherwise they will all be defaulted to
zero. The following are examples of vector definitions:
(a) Initialise all values to zero
var x[3];
(b) Initialise all values to zero
var x[3] := {};
(c) Initialise all values to given expression
var x[3] := [123 + 3y + sin(w / z)];
(d) Initialise the first two values, all other elements to zero
var x[3] := { 1 + x[2], sin(y[0] / x[]) + 3 };
(e) Initialise the first three (all) values
var x[3] := { 1, 2, 3 };
(f) Initialise vector from a vector
var x[4] := { 1, 2, 3, 4 };
var y[3] := x;
(g) Initialise vector from a smaller vector
var x[3] := { 1, 2, 3 };
var y[5] := x; // 1, 2, 3, ??, ??
(h) Non-initialised vector
var x[3] := null; // ?? ?? ??
(i) Error as there are too many initialisers
var x[3] := { 1, 2, 3, 4 };
(j) Error as a vector of size zero is not allowed.
var x[0];
(3) String Definition
Strings are sequences comprised of 8-bit characters. They can only be
defined with an explicit initialisation value. The following are
examples of string variable definitions:
(a) Initialise to a string
var x := 'abc';
(b) Initialise to an empty string
var x := '';
(c) Initialise to a string expression
var x := 'abc' + '123';
(d) Initialise to a string range
var x := 'abc123'[2:4];
(e) Initialise to another string variable
var x := 'abc';
var y := x;
(f) Initialise to another string variable range
var x := 'abc123';
var y := x[2:4];
(g) Initialise to a string expression
var x := 'abc';
var y := x + '123';
(h) Initialise to a string expression range
var x := 'abc';
var y := (x + '123')[1:3];
(4) Return Value
Variable and vector definitions have a return value. In the case of
variable definitions, the value to which the variable is initialised
will be returned. Where as for vectors, the value of the first element
(eg: v[0]) will be returned.
8 == ((var x := 7;) + 1)
4 == (var y[3] := {4, 5, 6};)
(5) Variable/Vector Assignment
The value of a variable can be assigned to a vector and a vector or a
vector expression can be assigned to a variable.
(a) Variable To Vector:
Every element of the vector is assigned the value of the variable
or expression.
var x := 3;
var y[3] := { 1, 2, 3 };
y := x + 1;
(b) Vector To Variable:
The variable is assigned the value of the first element of the
vector (aka vec[0])
var x := 3;
var y[3] := { 1, 2, 3 };
x := y + 1;
Note: During the expression compilation phase, tokens are classified
based on the following priorities:
(a) Reserved keywords or operators (+, -, and, or, etc)
(b) Base functions (abs, sin, cos, min, max etc)
(c) Symbol table variables
(d) Expression local defined variables
(e) Symbol table functions
(f) Unknown symbol resolver based variables
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 14 - VECTOR PROCESSING]
ExprTk provides support for various forms of vector oriented
arithmetic, inequalities and processing. The various supported pairs
are as follows:
(a) vector and vector (eg: v0 + v1)
(b) vector and scalar (eg: v + 33)
(c) scalar and vector (eg: 22 * v)
The following is a list of operations that can be used in conjunction
with vectors:
(a) Arithmetic: +, -, *, /, %
(b) Exponentiation: vector ^ scalar
(c) Assignment: :=, +=, -=, *=, /=, %=, <=>
(d) Inequalities: <, <=, >, >=, ==, =, equal
(e) Boolean logic: and, nand, nor, or, xnor, xor
(f) Unary operations:
abs, acos, acosh, asin, asinh, atan, atanh, ceil, cos, cosh,
cot, csc, deg2grad, deg2rad, erf, erfc, exp, expm1, floor,
frac, grad2deg, log, log10, log1p, log2, rad2deg, round, sec,
sgn, sin, sinc, sinh, sqrt, swap, tan, tanh, trunc
(g) Aggregate and Reduce operations:
avg, max, min, mul, dot, dotk, sum, sumk, count, all_true,
all_false, any_true, any_false
(h) Transformation operations:
copy, rotate-left/right, shift-left/right, sort, nth_element
(i) BLAS-L1:
axpy, axpby, axpyz, axpbyz, axpbz
Note: When one of the above described operations is being performed
between two vectors, the operation will only span the size of the
smallest vector. The elements of the larger vector outside of the
range will not be included. The operation itself will be processed
element-wise over values the smaller of the two ranges.
The following simple example demonstrates the vector processing
capabilities by computing the dot-product of the vectors v0 and v1 and
then assigning it to the variable v0dotv1:
var v0[3] := { 1, 2, 3 };
var v1[3] := { 4, 5, 6 };
var v0dotv1 := sum(v0 * v1);
The following is a for-loop based implementation that is equivalent to
the previously mentioned dot-product computation expression:
var v0[3] := { 1, 2, 3 };
var v1[3] := { 4, 5, 6 };
var v0dotv1;
for (var i := 0; i < min(v0[],v1[]); i += 1)
{
v0dotv1 += (v0[i] * v1[i]);
}
Note: When the aggregate or reduction operations denoted above are
used in conjunction with a vector or vector expression, the return
value is not a vector but rather a single value.
var x[3] := { 1, 2, 3 };
sum(x) == 6
sum(1 + 2x) == 15
avg(3x + 1) == 7
min(1 / x) == (1 / 3)
max(x / 2) == (3 / 2)
sum(x > 0 and x < 5) == x[]
When utilizing external user defined vectors via the symbol table as
opposed to expression local defined vectors, the typical 'add_vector'
method from the symbol table will register the entirety of the vector
that is passed. The following example attempts to evaluate the sum of
elements of the external user defined vector within a typical yet
trivial expression:
std::string reduce_program = " sum(2 * v + 1) ";
std::vector<T> v0 { T(1.1), T(2.2), ..... , T(99.99) };
symbol_table_t symbol_table;
symbol_table.add_vector("v",v);
expression_t expression;
expression.register_symbol_table(symbol_table);
parser_t parser;
parser.compile(reduce_program,expression);
T sum = expression.value();
For the most part, this is a very common use-case. However there may
be situations where one may want to evaluate the same vector oriented
expression many times over, but using different vectors or sub ranges
of the same vector of the same size to that of the original upon every
evaluation.
The usual solution is to either recompile the expression for the new
vector instance, or to copy the contents from the new vector to the
symbol table registered vector and then perform the evaluation. When
the vectors are large or the re-evaluation attempts are numerous,
these solutions can become rather time consuming and generally
inefficient.
std::vector<T> v1 { T(2.2), T(2.2), ..... , T(2.2) };
std::vector<T> v2 { T(3.3), T(3.3), ..... , T(3.3) };
std::vector<T> v3 { T(4.4), T(4.4), ..... , T(4.4) };
std::vector<std::vector<T>> vv { v1, v2, v3 };
...
T sum = T(0);
for (auto& new_vec : vv)
{
v = new_vec; // update vector
sum += expression.value();
}
A solution to the above 'efficiency' problem, is to use the
exprtk::vector_view object. The vector_view is instantiated with a
size and backing based upon a vector. Upon evaluations if the backing
needs to be 'updated' to either another vector or sub-range, the
vector_view instance can be efficiently rebased, and the expression
evaluated as normal.
exprtk::vector_view<T> view = exprtk::make_vector_view(v,v.size());
symbol_table_t symbol_table;
symbol_table.add_vector("v",view);
...
T sum = T(0);
for (auto& new_vec : vv)
{
view.rebase(new_vec.data()); // update vector
sum += expression.value();
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 15 - USER DEFINED FUNCTIONS]
ExprTk provides a means whereby custom functions can be defined and
utilised within expressions. The concept requires the user to
provide a reference to the function coupled with an associated name
that will be invoked within expressions. Functions may take numerous
inputs but will always return a single value of the underlying numeric
type.
During expression compilation when required the reference to the
function will be obtained from the associated symbol_table and be
embedded into the expression.
There are five types of function interface:
+---+----------------------+-------------+----------------------+
| # | Name | Return Type | Input Types |
+---+----------------------+-------------+----------------------+
| 1 | ifunction | Scalar | Scalar |
| 2 | ivararg_function | Scalar | Scalar |
| 3 | igeneric_function | Scalar | Scalar,Vector,String |
| 4 | igeneric_function II | String | Scalar,Vector,String |
| 5 | function_compositor | Scalar | Scalar |
+---+----------------------+-------------+----------------------+
(1) ifunction
This interface supports zero to 20 input parameters of only the scalar
type (numbers). The usage requires a custom function be derived from
ifunction and to override one of the 21 function operators. As part of
the constructor the custom function will define how many parameters it
expects to handle. The following example defines a 3 parameter
function called 'foo':
template <typename T>
struct foo : public exprtk::ifunction<T>
{
foo() : exprtk::ifunction<T>(3)
{}
T operator()(const T& v1, const T& v2, const T& v3)
{
return T(1) + (v1 * v2) / T(v3);
}
};
(2) ivararg_function
This interface supports a variable number of scalar arguments as input
into the function. The function operator interface uses a std::vector
specialised upon type T to facilitate parameter passing. The following
example defines a vararg function called 'boo':
template <typename T>
struct boo : public exprtk::ivararg_function<T>
{
inline T operator()(const std::vector<T>& arglist)
{
T result = T(0);
for (std::size_t i = 0; i < arglist.size(); ++i)
{
result += arglist[i] / arglist[i > 0 ? (i - 1) : 0];
}
return result;
}
};
(3) igeneric_function
This interface supports a variable number of arguments and types as
input into the function. The function operator interface uses a
std::vector specialised upon the type_store type to facilitate
parameter passing.
Scalar <-- function(i_0, i_1, i_2....., i_N)
The fundamental types that can be passed into the function as
parameters and their views are as follows:
(1) Scalar - scalar_view
(2) Vector - vector_view
(3) String - string_view
The above denoted type views provide non-const reference-like access
to each parameter, as such modifications made to the input parameters
will persist after the function call has completed. The following
example defines a generic function called 'too':
template <typename T>
struct too : public exprtk::igeneric_function<T>
{
typedef typename exprtk::igeneric_function<T>::parameter_list_t
parameter_list_t;
too()
{}
inline T operator()(parameter_list_t parameters)
{
for (std::size_t i = 0; i < parameters.size(); ++i)
{
...
}
return T(0);
}
};
In the example above, the input 'parameters' to the function operator,
parameter_list_t, is a type of std::vector of type_store. Each
type_store instance has a member called 'type' which holds the
enumeration pertaining to the underlying type of the type_store. There
are three type enumerations:
(1) e_scalar - literals, variables, vector elements, expressions
eg: 123.456, x, vec[3x + 1], 2x + 3
(2) e_vector - vectors, vector expressions
eg: vec1, 2 * vec1 + vec2 / 3
(3) e_string - strings, string literals and range variants of both
eg: 'AString', s0, 'AString'[x:y], s1[1 + x:] + 'AString'
Each of the parameters can be accessed using its designated view. A
typical loop for processing the parameters is as follows:
inline T operator()(parameter_list_t parameters)
{
typedef typename exprtk::igeneric_function<T>::generic_type
generic_type;
typedef typename generic_type::scalar_view scalar_t;
typedef typename generic_type::vector_view vector_t;
typedef typename generic_type::string_view string_t;
for (std::size_t i = 0; i < parameters.size(); ++i)
{
generic_type& gt = parameters[i];
if (generic_type::e_scalar == gt.type)
{
scalar_t x(gt);
...
}
else if (generic_type::e_vector == gt.type)
{
vector_t vector(gt);
...
}
else if (generic_type::e_string == gt.type)
{
string_t string(gt);
...
}
}
return T(0);
}
Most often than not a custom generic function will require a specific
sequence of parameters, rather than some arbitrary sequence of types.
In those situations, ExprTk can perform compile-time type checking to
validate that function invocations are carried out using the correct
sequence of parameters. Furthermore performing the checks at compile
-time rather than at run-time (aka every time the function is invoked)
will result in expression evaluation performance gains.
Compile-time type checking of input parameters can be requested by
passing a string to the constructor of the igeneric_function that
represents the required sequence of parameter types. When no parameter
sequence is provided, it is implied the function can accept a variable
number of parameters comprised of any of the fundamental types.
Each fundamental type has an associated character. The following is a
listing of said characters and their meanings:
(1) T - Scalar
(2) V - Vector
(3) S - String
(4) Z - Zero or no parameters
(5) ? - Any type (Scalar, Vector or String)
(6) * - Wildcard operator
(7) | - Parameter sequence delimiter
No other characters other than the seven denoted above may be included
in the parameter sequence definition. If any such invalid characters
do exist, registration of the associated generic function to a symbol
table ('add_function' method) will fail. If the parameter sequence is
modified resulting in it becoming invalid after having been added to
the symbol table but before the compilation step, a compilation error
will be incurred.
The following example demonstrates a simple generic function
implementation with a user specified parameter sequence:
template <typename T>
struct moo : public exprtk::igeneric_function<T>
{
typedef typename exprtk::igeneric_function<T>::parameter_list_t
parameter_list_t;
moo()
: exprtk::igeneric_function<T>("SVTT")
{}
inline T operator()(parameter_list_t parameters)
{
...
}
};
In the example above the generic function 'moo' expects exactly four
parameters in the following sequence:
(1) String
(2) Vector
(3) Scalar
(4) Scalar
Note: The 'Z' or no parameter option may not be used in conjunction
with any other type option in a parameter sequence. When incorporated
in the parameter sequence list, the no parameter option indicates that
the function may be invoked without any parameters being passed. For
more information refer to the section: 'Zero Parameter Functions'
(4) igeneric_function II
This interface is identical to the igeneric_function, in that in can
consume an arbitrary number of parameters of varying type, but the
difference being that the function returns a string and as such is
treated as a string when invoked within expressions. As a result the
function call can alias a string and interact with other strings in
situations such as concatenation and equality operations.
String <-- function(i_0, i_1, i_2....., i_N)
The following example defines a generic function named 'toupper' with
the string return type function operator being explicitly overridden:
template <typename T>
struct toupper : public exprtk::igeneric_function<T>
{
typedef exprtk::igeneric_function<T> igenfunct_t
typedef typename igenfunct_t::generic_type generic_t;
typedef typename igenfunct_t::parameter_list_t parameter_list_t;
typedef typename generic_t::string_view string_t;
toupper()
: exprtk::igeneric_function<T>("S",igenfunct_t::e_rtrn_string)
{}
inline T operator()(std::string& result,
parameter_list_t parameters)
{
result.clear();
string_t string(params[0]);
for (std::size_t i = 0; i < string.size(); ++i)
{
result += std::toupper(string[i]);
}
return T(0);
}
};
In the example above the generic function 'toupper' expects only one
input parameter of type string, as noted by the parameter sequence
string passed during the constructor. Furthermore a second parameter
is passed to the constructor indicating that it should be treated as a
string returning function - by default it is assumed to be a scalar
returning function.
When executed, the function will return as a result a copy of the
input string converted to uppercase form. An example expression using
the toupper function registered as the symbol 'toupper' is as follows:
"'ABCDEF' == toupper('aBc') + toupper('DeF')"
Note: When adding a string type returning generic function to a symbol
table the 'add_function' is invoked. The example below demonstrates
how this can be done:
toupper<T> tu;
exprtk::symbol_table<T> symbol_table;
symbol_table.add_function("toupper",tu);
Note: Two further refinements to the type checking facility are the
possibilities of a variable number of common types which can be
accomplished by using a wildcard '*' and a special 'any type' which is
done using the '?' character. It should be noted that the wildcard
operator is associated with the previous type in the sequence and
implies one or more of that type.
template <typename T>
struct zoo : public exprtk::igeneric_function<T>
{
typedef typename exprtk::igeneric_function<T>::parameter_list_t
parameter_list_t;
zoo()
: exprtk::igeneric_function<T>("SVT*V?")
{}
inline T operator()(parameter_list_t parameters)
{
...
}
};
In the example above the generic function 'zoo' expects at least five
parameters in the following sequence:
(1) String
(2) Vector
(3) One or more Scalars
(4) Vector
(5) Any type (one type of either a scalar, vector or string)
A final piece of type checking functionality is available for the
scenarios where a single function name is intended to be used for
multiple distinct parameter sequences, another name for this feature
is function overloading. The parameter sequences are passed to the
constructor as a single string delimited by the pipe '|' character.
Two specific overrides of the function operator are provided one for
standard generic functions and one for string returning functions. The
overrides are as follows:
// Scalar <-- function(psi,i_0,i_1,....,i_N)
inline T operator()(const std::size_t& ps_index,
parameter_list_t parameters)
{
...
}
// String <-- function(psi,i_0,i_1,....,i_N)
inline T operator()(const std::size_t& ps_index,
std::string& result,
parameter_list_t parameters)
{
...
}
When the function operator is invoked the 'ps_index' parameter will
have as its value the index of the parameter sequence that matches the
specific invocation. This way complex and time consuming type checking
conditions need not be executed in the function itself but rather a
simple and efficient dispatch to a specific implementation for that
particular parameter sequence can be performed.
template <typename T>
struct roo : public exprtk::igeneric_function<T>
{
typedef typename exprtk::igeneric_function<T>::parameter_list_t
parameter_list_t;
moo()
: exprtk::igeneric_function<T>("SVTT|SS|TTV|S?V*S")
{}
inline T operator()(const std::size_t& ps_index,
parameter_list_t parameters)
{
...
}
};
In the example above there are four distinct parameter sequences that
can be processed by the generic function 'roo'. Any other parameter
sequences will cause a compilation error. The four valid sequences are
as follows:
Sequence-0 Sequence-1 Sequence-2 Sequence-3
'SVTT' 'SS' 'TTV' 'S?V*S'
(1) String (1) String (1) Scalar (1) String
(2) Vector (2) String (2) Scalar (2) Any Type
(3) Scalar (3) Vector (3) One or more Vectors
(4) Scalar (4) String
(5) function_compositor
The function compositor is a factory that allows one to define and
construct a function using ExprTk syntax. The functions are limited to
returning a single scalar value and consuming up to six parameters as
input.
All composited functions are registered with a symbol table, allowing
them to call other functions that have been registered with the symbol
table instance. Furthermore the functions can be recursive in nature
due to the inherent function prototype forwarding that occurs during
construction. The following example defines, by using two different
methods, composited functions and implicitly registering the functions
with the denoted symbol table.
typedef exprtk::symbol_table<T> symbol_table_t;
typedef exprtk::function_compositor<T> compositor_t;
typedef typename compositor_t::function function_t;
symbol_table_t symbol_table;
compositor_t compositor(symbol_table);
// define function koo0(v1,v2) { ... }
compositor
.add(
function_t(
"koo0",
" 1 + cos(v1 * v2) / 3;",
"v1","v2"));
// define function koo1(x,y,z) { ... }
compositor
.add(function_t()
.name("koo1")
.var("x").var("y").var("z")
.expression("1 + cos(x * y) / z;"));
(6) Using Functions In Expressions
For the above denoted custom and composited functions to be used in an
expression, an instance of each function needs to be registered with a
symbol_table that has been associated with the expression instance.
The following demonstrates how all the pieces are put together:
typedef exprtk::symbol_table<double> symbol_table_t;
typedef exprtk::expression<double> expression_t;
typedef exprtk::parser<double> parser_t;
typedef exprtk::function_compositor<double> compositor_t;
typedef typename compositor_t::function function_t;
foo<double> f;
boo<double> b;
too<double> t;
toupper<double> tu;
symbol_table_t symbol_table;
compositor_t compositor(symbol_table);
symbol_table.add_function("foo",f);
symbol_table.add_function("boo",b);
symbol_table.add_function("too",t);
symbol_table.add_function("toupper",
tu,
symbol_table_t::e_ft_strfunc);
compositor
.add(function_t()
.name("koo")
.var("v1")
.var("v2")
.expression("1 + cos(v1 * v2) / 3;"));
expression_t expression;
expression.register_symbol_table(symbol_table);
std::string expression_str =
" if (foo(1,2,3) + boo(1) > boo(1/2,2/3,3/4,4/5)) "
" koo(3,4); "
" else "
" too(2 * v1 + v2 / 3, 'abcdef'[2:4], 3.3); "
" ";
parser_t parser;
parser.compile(expression_str,expression);
expression.value();
(7) Function Side-Effects
All function calls are assumed to have side-effects by default. This
assumption implicitly disables constant folding optimisations when all
parameters being passed to the function are deduced as being constants
at compile time.
If it is certain that the function being registered does not have any
side effects and can be correctly constant folded where appropriate,
then during the construction of the function the side-effect trait of
the function can be disabled.
template <typename T>
struct foo : public exprtk::ifunction<T>
{
foo() : exprtk::ifunction<T>(3)
{
exprtk::disable_has_side_effects(*this);
}
T operator()(const T& v1, const T& v2, const T& v3)
{ ... }
};
(8) Zero Parameter Functions
When either an ifunction, ivararg_function or igeneric_function
derived type is defined with zero number of parameters, there are two
calling conventions within expressions that are allowed. For a
function named 'foo' with zero input parameters the calling styles are
as follows:
(1) x + sin(foo()- 2) / y
(2) x + sin(foo - 2) / y
By default the zero parameter trait is disabled. In order to enable
it, a process similar to that of enabling of the side effect trait is
carried out:
template <typename T>
struct foo : public exprtk::ivararg_function<T>
{
foo()
{
exprtk::enable_zero_parameters(*this);
}
inline T operator()(const std::vector<T>& arglist)
{ ... }
};
Note: For the igeneric_function type, there also needs to be a 'Z'
parameter sequence defined in order for the zero parameter trait to
properly take effect otherwise a compilation error will occur.
(9) Free Functions
The ExprTk symbol table supports the registration of free functions
and lambdas (anonymous functors) for use in expressions. The basic
requirements are similar to those found in ifunction derived user
defined functions. This includes support for free functions using
anywhere from zero up to fifteen input parameters of scalar type, with
a return type that is also scalar. Furthermore such functions will by
default be assumed to have side-effects and hence will not participate
in constant folding optimisations.
In the following example, a two input parameter free function named
'compute1', and a three input parameter lambda named 'compute2' will
be registered with the given symbol_table instance:
double compute1(double v0, double v1)
{
return 2.0 * v0 + v1 / 3.0;
}
.
.
.
typedef exprtk::symbol_table<double> symbol_table_t;
symbol_table_t symbol_table;
symbol_table.add_function("compute1", compute1);
symbol_table.add_function(
"compute2",
[](double v0, double v1, double v2) -> double
{ return v0 / v1 + v2; });
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 16 - EXPRESSION DEPENDENTS]
Any expression that is not a literal (aka constant) will have
dependencies. The types of 'dependencies' an expression can have are
as follows:
(a) Variables
(b) Vectors
(c) Strings
(d) Functions
(e) Assignments
In the following example the denoted expression has its various
dependencies listed:
z := abs(x + sin(2 * pi / y))
(a) Variables: x, y, z and pi
(b) Functions: abs, sin
(c) Assignments: z
ExprTk allows for the derivation of expression dependencies via the
'dependent_entity_collector' (DEC). When activated either through
'compile_options' at the construction of the parser or through calls
to enabler methods just prior to compilation, the DEC will proceed to
collect any of the relevant types that are encountered during the
parsing phase. Once the compilation process has successfully
completed, the caller can then obtain a list of symbols and their
associated types from the DEC.
The kinds of questions one can ask regarding the dependent entities
within an expression are as follows:
* What user defined variables, vectors or strings are used?
* What functions or custom user functions are used?
* Which variables, vectors or strings have values assigned to them?
The following example demonstrates usage of the DEC in determining the
dependents of the given expression:
typedef typename parser_t::
dependent_entity_collector::symbol_t symbol_t;
std::string expression_string =
"z := abs(x + sin(2 * pi / y))";
T x,y,z;
parser_t parser;
symbol_table_t symbol_table;
symbol_table.add_variable("x",x);
symbol_table.add_variable("y",y);
symbol_table.add_variable("z",z);
expression_t expression;
expression.register_symbol_table(symbol_table);
//Collect only variable and function symbols
parser.dec().collect_variables() = true;
parser.dec().collect_functions() = true;
if (!parser.compile(expression_string,expression))
{
// error....
}
std::deque<symbol_t> symbol_list;
parser.dec().symbols(symbol_list);
for (std::size_t i = 0; i < symbol_list.size(); ++i)
{
symbol_t& symbol = symbol_list[i];
switch (symbol.second)
{
case parser_t::e_st_variable : ... break;
case parser_t::e_st_vector : ... break;
case parser_t::e_st_string : ... break;
case parser_t::e_st_function : ... break;
}
}
Note: The 'symbol_t' type is a std::pair comprising of the symbol name
(std::string) and the associated type of the symbol as denoted by the
cases in the switch statement.
Having particular symbols (variable or function) present in an
expression is one form of dependency. Another and just as interesting
and important type of dependency is that of assignments. Assignments
are the set of dependent symbols that 'may' have their values modified
within an expression. The following are example expressions and their
associated assignments:
Assignments Expression
(1) x x := y + z
(2) x, y x += y += z
(3) x, y, z x := y += sin(z := w + 2)
(4) w, z if (x > y, z := x + 2, w := 'A String')
(5) None x + y + z
Note: In expression 4, both variables 'w' and 'z' are denoted as being
assignments even though only one of them can ever be modified at the
time of evaluation. Furthermore the determination of which of the two
variables the modification will occur upon can only be known with
certainty at evaluation time and not beforehand, hence both are listed
as being candidates for assignment.
The following builds upon the previous example demonstrating the usage
of the DEC in determining the 'assignments' of the given expression:
//Collect assignments
parser.dec().collect_assignments() = true;
if (!parser.compile(expression_string,expression))
{
// error....
}
std::deque<symbol_t> symbol_list;
parser.dec().assignment_symbols(symbol_list);
for (std::size_t i = 0; i < symbol_list.size(); ++i)
{
symbol_t& symbol = symbol_list[i];
switch (symbol.second)
{
case parser_t::e_st_variable : ... break;
case parser_t::e_st_vector : ... break;
case parser_t::e_st_string : ... break;
}
}
Note: The assignments will only consist of variable types and as such
will not contain symbols denoting functions.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 17 - HIERARCHIES OF SYMBOL TABLES]
Most situations will only require a single symbol_table instance to be
associated with a given expression instance.
However as an expression can have more than one symbol table instance
associated with itself, when building more complex systems that
utilise many expressions where each can in turn utilise one or more
variables from a large set of potential variables, functions or
constants, it becomes evident that grouping variables into layers of
symbol_tables will simplify and streamline the overall process.
A suggested hierarchy of symbol tables is as follows:
(a) Global constant value symbol table
(b) Global non side-effect functions symbol table
(c) Global variable symbol table
(d) Expression specific variable symbol table
(a) Global constant value symbol table
This symbol table will contain constant variables denoting immutable
values. These variables can be made available to all expressions, and
in turn expressions will assume the values themselves will never be
modified for the duration of the process run-time. Examples of such
variables are:
(1) pi or e
(2) speed_of_light
(3) avogadro_number
(4) num_cpus
(b) Global non side-effect functions symbol table
This symbol table will contain only user defined functions that will
not incur any side-effects that are visible to any of the expressions
that invoke them. These functions will be thread-safe or threading
invariant and will not maintain any form of state between invocations.
Examples of such functions are:
(1) calc_volume_of_sphere(r)
(2) distance(x0,y0,x1,y1)
(c) Global variable symbol table
This symbol table will contain variables that will be accessible to
all associated expressions and will not be specific or exclusive to
any one expression. This variant differs from (a) in that the values
of the variables can change (or be updated) between evaluations of
expressions - but through properly scheduled evaluations are
guaranteed to never change during the evaluation of any dependent
expressions. Furthermore it is assumed that these variables will be
used in a read-only context and that no expressions will attempt to
modify these variables via assignments or other means.
(1) price_of_stock_xyz
(2) outside_temperature or inside_temperature
(3) fuel_in_tank
(4) num_customers_in_store
(5) num_items_on_shelf
(d) Expression specific variable symbol table
This symbol_table is the most common form, and is used to store
variables that are specific and exclusive to a particular expression.
That is to say references to variables in this symbol_table will not
be part of another expression. Though it may be possible to have
expressions that contain the variables with the same name, in that
case those variables will be distinctly different. Which would mean if
a particular expression were to be compiled twice, each expression
would have its own unique symbol_table which in turn would have its
own instances of those variables. Examples of such variables could be:
(1) x or y
(2) customer_name
The following is a diagram depicting the possible version of the
denoted symbol table hierarchies. In the diagram there are two unique
expressions, each of which have a reference to the Global constant,
functions and variables symbol tables and an exclusive reference to a
local symbol table.
+-------------------------+ +-------------------------+
| Global Constants | | Global Functions |
| Symbol Table | | Symbol Table |
+----o--o-----------------+ +--------------------o----+
| | |
| | +-------+
| +------------------->----------------------------+ |
| +----------------------------+ | |
| | Global Variables | | |
| +------o Symbol Table o-----+ | V
| | +----------------------------+ | | |
| | | | |
| | +----------------+ +----------------+ | | |
| | | Symbol Table 0 | | Symbol Table 1 | | V |
| | +--o-------------+ +--o-------------+ | | |
| | | | | | |
| | | | | | |
+--V--V----V---------+ +-V---------------V--+ | |
| Expression 0 | | Expression 1 |<--+--+
| '2 * sin(x) - y' | | 'k + abs(x - y)' |
+--------------------+ +--------------------+
Bringing all of the above together, in the following example the
hierarchy of symbol tables are instantiated and initialised. An
expression that makes use of various elements of each symbol table is
then compiled and later on evaluated:
typedef exprtk::symbol_table<double> symbol_table_t;
typedef exprtk::expression<double> expression_t;
// Setup global constants symbol table
symbol_table_t glbl_const_symbol_table;
glbl_const_symbtab.add_constants(); // pi, epsilon and inf
glbl_const_symbtab.add_constant("speed_of_light",299e6);
glbl_const_symbtab.add_constant("avogadro_number",6e23);
// Setup global function symbol table
symbol_table_t glbl_funcs_symbol_table;
glbl_func_symbtab.add_function('distance',distance);
glbl_func_symbtab.add_function('calc_spherevol',calc_sphrvol);
......
// Setup global variable symbol table
symbol_table_t glbl_variable_symbol_table;
glbl_variable_symbtab.add_variable('temp_outside',thermo.outside);
glbl_variable_symbtab.add_variable('temp_inside' ,thermo.inside );
glbl_variable_symbtab.add_variable('num_cstmrs',store.num_cstmrs);
......
double x,y,z;
// Setup expression specific symbol table
symbol_table_t symbol_table;
symbol_table.add_variable('x',x);
symbol_table.add_variable('y',y);
symbol_table.add_variable('z',z);
expression_t expression;
// Register the various symbol tables
expression
.register_symbol_table(symbol_table);
expression
.register_symbol_table(glbl_funcs_symbol_table);
expression
.register_symbol_table(glbl_const_symbol_table);
expression
.register_symbol_table(glbl_variable_symbol_table);
std::string expression_str =
"abs(temp_inside - temp_outside) + 2 * speed_of_light / x";
parser_t parser;
parser.compile(expression_str,expression);
......
while (keep_evaluating)
{
....
T result = expression.value();
....
}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 18 - UNKNOWN UNKNOWNS]
In this section we will discuss the process of handling expressions
with a mix of known and unknown variables. Initially a discussion into
the types of expressions that exist will be provided, then a series of
possible solutions will be presented for each scenario.
When parsing an expression, there may be situations where one is not
fully aware of what if any variables will be used prior to the
expression being compiled.
This can become problematic, as in the default scenario it is assumed
the symbol_table that is registered with the expression instance will
already possess the externally available variables, functions and
constants needed during the compilation of the expression.
In the event there are symbols in the expression that can't be mapped
to either a reserved word, or located in the associated
symbol_table(s), an "Undefined symbol" error will be raised and the
compilation process will fail.
The numerous scenarios that can occur when compiling an expression
with ExprTk generally fall into one of the following three categories:
(a) No external variables
(b) Predetermined set of external variables
(c) Unknown set of variables
(a) No external variables
These are expressions that contain no external variables but may
contain local variables. As local variables cannot be accessed
externally from the expression, it is assumed that such expressions
will not have a need for a symbol_table and furthermore expressions
which don't make use of functions that have side-effects will be
evaluated completely at compile time resulting in a constant return
value. The following are examples of such expressions:
(1) 1 + 2
(2) var x := 3; 2 * x - 3
(3) var x := 3; var y := abs(x - 8); x - y / 7
(b) Predetermined set of external variables
These are expressions that are comprised of externally available
variables and functions and will only compile successfully if the
symbols that correspond to the variables and functions are already
defined in their associated symbol_table(s). This is by far the most
common scenario when using ExprTk.
As an example, one may have three external variables: x, y and z which
have been registered with the associated symbol_table, and will then
need to compile and evaluate expressions comprised of any subset of
these three variables. The following are a few examples of such
expressions:
(1) 1 + x
(2) x / y
(3) 2 * x * y / z
In this scenario one can use the 'dependent_entity_collector'
component as described in [Section 16] to further determine which of
the registered variables were actually used in the given expression.
As an example once the set of utilised variables are known, any
further 'attention' can be restricted to only those variables when
evaluating the expression. This can be quite useful when dealing with
expressions that can draw from a set of hundreds or even thousands of
variables.
(c) Unknown set of variables
These are expressions that are comprised of symbols other than the
standard ExprTk reserved words or what has been registered with their
associated symbol_table, and will normally fail compilation due to the
associated symbol_table not having a reference to them. As such this
scenario can be seen as a combination of scenario B, where one may
have a symbol_table with registered variables, but would also like to
handle the situation of variables that aren't present in said
symbol_table.
When dealing with expressions of category (c), one must perform all of
the following:
(1) Determine the variables used in the expression
(2) Populate a symbol_table(s) with the entities from (1)
(3) Compile the expression
(4) Provide a means by which the entities from (1) can be modified
Depending on the nature of processing, steps (1) and (2) can be done
either independently of each other or combined into one. The following
example will initially look at solving the problem of unknown
variables with the latter method using the 'unknown_symbol_resolver'
component.
typedef exprtk::symbol_table<T> symbol_table_t;
typedef exprtk::expression<T> expression_t;
typedef exprtk::parser<T> parser_t;
symbol_table_t unknown_var_symbol_table;
symbol_table_t symbol_table;
symbol_table.add_variable("x",x);
symbol_table.add_variable("y",y);
expression_t expression;
expression.register_symbol_table(unknown_var_symbol_table);
expression.register_symbol_table(symbol_table);
parser_t parser;
parser.enable_unknown_symbol_resolver();
std::string expression_str = "x + abs(y / 3k) * z + 2";
parser.compile(expression_str,expression);
In the example above, the symbols 'k' and 'z' will be treated as
unknown symbols. The parser in the example is set to handle unknown
symbols using the built-in default unknown_symbol_resolver (USR). The
default USR will automatically resolve any unknown symbols as a
variable (scalar type). The new variables will be added to the primary
symbol_table, which in this case is the 'unknown_var_symbol_table'
instance. Once the compilation has completed successfully, the
variables that were resolved during compilation can be accessed from
the primary symbol_table using the 'get_variable_list' and
'variable_ref' methods and then if needed can be modified accordingly
after which the expression itself can be evaluated.
std::vector<std::string> variable_list;
unknown_var_symbol_table.get_variable_list(variable_list);
for (auto& var_name : variable_list)
{
T& v = unknown_var_symbol_table.variable_ref(var_name);
v = ...;
}
...
expression.value();
Note: As previously mentioned the default USR will automatically
assume any unknown symbol to be a valid scalar variable, and will then
proceed to add said symbol as a variable to the primary symbol_table
of the associated expression during the compilation process. However a
problem that may arise, is that expressions that are parsed with the
USR enabled, but contain 'typos' or otherwise syntactic errors may
inadvertently compile successfully due to the simplistic nature of the
default USR. The following are some example expressions:
(1) 1 + abz(x + 1)
(2) sine(y / 2) - coz(3x)
The two expressions above contain misspelt symbols (abz, sine, coz)
which if implied multiplications and default USR are enabled during
compilation will result in them being assumed to be valid 'variables',
which obviously is not the intended outcome by the user. A possible
solution to this problem is for one to implement their own specific
USR that will perform a user defined business logic in determining if
an encountered unknown symbol should be treated as a variable or if it
should raise a compilation error. The following example demonstrates a
simple user defined USR:
typedef exprtk::symbol_table<T> symbol_table_t;
typedef exprtk::expression<T> expression_t;
typedef exprtk::parser<T> parser_t;
template <typename T>
struct my_usr : public parser_t::unknown_symbol_resolver
{
typedef typename parser_t::unknown_symbol_resolver usr_t;
bool process(const std::string& unknown_symbol,
typename usr_t::usr_symbol_type& st,
T& default_value,
std::string& error_message)
{
if (0 != unknown_symbol.find("var_"))
{
error_message = "Invalid symbol: " + unknown_symbol;
return false;
}
st = usr_t::e_usr_variable_type;
default_value = T(123.123);
return true;
}
};
...
symbol_table_t unknown_var_symbol_table;
symbol_table_t symbol_table;
symbol_table.add_variable("x",x);
symbol_table.add_variable("y",y);
expression_t expression;
expression.register_symbol_table(unknown_var_symbol_table);
expression.register_symbol_table(symbol_table);
my_usr<T> musr;
parser_t parser;
parser.enable_unknown_symbol_resolver(&musr);
std::string expression_str = "var_x + abs(var_y - 3) * var_z";
parser.compile(expression_str,expression);
In the example above, a user specified USR is defined, and is
registered with the parser enabling the USR functionality. Then when
an unknown symbol is encountered during the compilation process, the
USR's process method will be invoked. The USR in the example will only
'accept' unknown symbols that have a prefix of 'var_' as being valid
variables, all other unknown symbols will result in a compilation
error being raised.
In the example above the callback of the USR that is invoked during
the unknown symbol resolution process only allows for scalar variables
to be defined and resolved - as that is the simplest and most common
form.
There is also an extended version of the callback that can be
overridden that will allow for more control and choice over the type
of symbol being resolved. The following is an example definition of
said extended callback:
template <typename T>
struct my_usr : public parser_t::unknown_symbol_resolver
{
typedef typename parser_t::unknown_symbol_resolver usr_t;
my_usr()
: usr_t(usr_t::e_usrmode_extended)
{}
virtual bool process(const std::string& unknown_symbol,
symbol_table_t& symbol_table,
std::string& error_message)
{
bool result = false;
if (0 == unknown_symbol.find("var_"))
{
// Default value of zero
result = symbol_table.create_variable(unknown_symbol,0);
if (!result)
{
error_message = "Failed to create variable...";
}
}
else if (0 == unknown_symbol.find("str_"))
{
// Default value of empty string
result = symbol_table.create_stringvar(unknown_symbol,"");
if (!result)
{
error_message = "Failed to create string variable...";
}
}
else
error_message = "Indeterminable symbol type.";
return result;
}
};
In the example above, the USR callback when invoked will pass the
primary symbol table associated with the expression being parsed. The
symbol resolution business logic can then determine under what
conditions a symbol will be resolved including its type (scalar,
string, vector etc) and default value. When the callback successfully
returns the symbol parsing and resolution process will again be
executed by the parser. The idea here is that given the primary symbol
table will now have the previously detected unknown symbol registered,
it will be correctly resolved and the general parsing processing can
then resume as per normal.
Note: In order to have the USR's extended mode callback be invoked It
is necessary to pass the e_usrmode_extended enum value during the
constructor of the user defined USR.
Note: The primary symbol table for an expression is the first symbol
table to be registered with that instance of the expression.
Note: For a successful symbol resolution using the normal USR all of
the following are required:
(1) Only if successful shall the process method return TRUE
(2) The default_value parameter will have been set
(3) The error_message parameter will be empty
(4) usr_symbol_type input parameter field will be set to either:
(*) e_usr_variable_type
(*) e_usr_constant_type
Note: For a successful symbol resolution using the extended USR all of
the following are required:
(1) Only if successful shall the process method return TRUE
(2) symbol_table parameter will have had the newly resolved
variable or string added to it
(3) error_message parameter will be empty
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 19 - ENABLING & DISABLING FEATURES]
The parser can be configured via its settings instance to either allow
or disallow certain features that are available within the ExprTk
grammar. The features fall into one of the following six categories:
(1) Base Functions
(2) Control Flow Structures
(3) Logical Operators
(4) Arithmetic Operators
(5) Inequality Operators
(6) Assignment Operators
(1) Base Functions
The list of available base functions is as follows:
abs, acos, acosh, asin, asinh, atan, atanh, atan2, avg, ceil,
clamp, cos, cosh, cot, csc, equal, erf, erfc, exp, expm1,
floor, frac, hypot, iclamp, like, log, log10, log2, logn,
log1p, mand, max, min, mod, mor, mul, ncdf, pow, root, round,
roundn, sec, sgn, sin, sinc, sinh, sqrt, sum, swap, tan, tanh,
trunc, not_equal, inrange, deg2grad, deg2rad, rad2deg, grad2deg
The above mentioned base functions can be either enabled or disabled
'all' at once, as is demonstrated below:
parser_t parser;
expression_t expression;
parser.settings().disable_all_base_functions();
parser
.compile("2 * abs(2 - 3)",expression); // compilation failure
parser.settings().enable_all_base_functions();
parser
.compile("2 * abs(2 - 3)",expression); // compilation success
One can also enable or disable specific base functions. The following
example demonstrates the disabling of the trigonometric functions
'sin' and 'cos':
parser_t parser;
expression_t expression;
parser.settings()
.disable_base_function(settings_t::e_bf_sin)
.disable_base_function(settings_t::e_bf_cos);
parser
.compile("(sin(x) / cos(x)) == tan(x)",expression); // failure
parser.settings()
.enable_base_function(settings_t::e_bf_sin)
.enable_base_function(settings_t::e_bf_cos);
parser
.compile("(sin(x) / cos(x)) == tan(x)",expression); // success
(2) Control Flow Structures
The list of available control flow structures is as follows:
(a) If or If-Else
(b) Switch statement
(c) For Loop
(d) While Loop
(e) Repeat Loop
The above mentioned control flow structures can be either enabled
or disabled 'all' at once, as is demonstrated below:
parser_t parser;
expression_t expression;
std::string program =
" var x := 0; "
" for (var i := 0; i < 10; i += 1) "
" { "
" x += i; "
" } ";
parser.settings().disable_all_control_structures();
parser
.compile(program,expression); // compilation failure
parser.settings().enable_all_control_structures();
parser
.compile(program,expression); // compilation success
One can also enable or disable specific control flow structures. The
following example demonstrates the disabling of the for-loop control
flow structure:
parser_t parser;
expression_t expression;
std::string program =
" var x := 0; "
" for (var i := 0; i < 10; i += 1) "
" { "
" x += i; "
" } ";
parser.settings()
.disable_control_structure(settings_t::e_ctrl_for_loop);
parser
.compile(program,expression); // failure
parser.settings()
.enable_control_structure(settings_t::e_ctrl_for_loop);
parser
.compile(program,expression); // success
(3) Logical Operators
The list of available logical operators is as follows:
and, nand, nor, not, or, xnor, xor, &, |
The above mentioned logical operators can be either enabled or
disabled 'all' at once, as is demonstrated below:
parser_t parser;
expression_t expression;
parser.settings().disable_all_logic_ops();
parser
.compile("1 or not(0 and 1)",expression); // compilation failure
parser.settings().enable_all_logic_ops();
parser
.compile("1 or not(0 and 1)",expression); // compilation success
One can also enable or disable specific logical operators. The
following example demonstrates the disabling of the 'and' logical
operator:
parser_t parser;
expression_t expression;
parser.settings()
.disable_logic_operation(settings_t::e_logic_and);
parser
.compile("1 or not(0 and 1)",expression); // failure
parser.settings()
.enable_logic_operation(settings_t::e_logic_and);
parser
.compile("1 or not(0 and 1)",expression); // success
(4) Arithmetic Operators
The list of available arithmetic operators is as follows:
+, -, *, /, %, ^
The above mentioned arithmetic operators can be either enabled or
disabled 'all' at once, as is demonstrated below:
parser_t parser;
expression_t expression;
parser.settings().disable_all_arithmetic_ops();
parser
.compile("1 + 2 / 3",expression); // compilation failure
parser.settings().enable_all_arithmetic_ops();
parser
.compile("1 + 2 / 3",expression); // compilation success
One can also enable or disable specific arithmetic operators. The
following example demonstrates the disabling of the addition '+'
arithmetic operator:
parser_t parser;
expression_t expression;
parser.settings()
.disable_arithmetic_operation(settings_t::e_arith_add);
parser
.compile("1 + 2 / 3",expression); // failure
parser.settings()
.enable_arithmetic_operation(settings_t::e_arith_add);
parser
.compile("1 + 2 / 3",expression); // success
(5) Inequality Operators
The list of available inequality operators is as follows:
<, <=, >, >=, ==, =, != <>
The above mentioned inequality operators can be either enabled or
disabled 'all' at once, as is demonstrated below:
parser_t parser;
expression_t expression;
parser.settings().disable_all_inequality_ops();
parser
.compile("1 < 3",expression); // compilation failure
parser.settings().enable_all_inequality_ops();
parser
.compile("1 < 3",expression); // compilation success
One can also enable or disable specific inequality operators. The
following example demonstrates the disabling of the less-than '<'
inequality operator:
parser_t parser;
expression_t expression;
parser.settings()
.disable_inequality_operation(settings_t::e_ineq_lt);
parser
.compile("1 < 3",expression); // failure
parser.settings()
.enable_inequality_operation(settings_t::e_ineq_lt);
parser
.compile("1 < 3",expression); // success
(6) Assignment Operators
The list of available assignment operators is as follows:
:=, +=, -=, *=, /=, %=
The above mentioned assignment operators can be either enabled or
disabled 'all' at once, as is demonstrated below:
parser_t parser;
expression_t expression;
symbol_table_t symbol_table;
T x = T(0);
symbol_table.add_variable("x",x);
expression.register_symbol_table(symbol_table);
parser.settings().disable_all_assignment_ops();
parser
.compile("x := 3",expression); // compilation failure
parser.settings().enable_all_assignment_ops();
parser
.compile("x := 3",expression); // compilation success
One can also enable or disable specific assignment operators. The
following example demonstrates the disabling of the '+=' addition
assignment operator:
parser_t parser;
expression_t expression;
symbol_table_t symbol_table;
T x = T(0);
symbol_table.add_variable("x",x);
expression.register_symbol_table(symbol_table);
parser.settings()
.disable_assignment_operation(settings_t::e_assign_addass);
parser
.compile("x += 3",expression); // failure
parser.settings()
.enable_assignment_operation(settings_t::e_assign_addass);
parser
.compile("x += 3",expression); // success
Note: In the event of a base function being disabled, one can redefine
the base function using the standard custom function definition
process. In the following example the 'sin' function is disabled then
redefined as a function taking degree input.
template <typename T>
struct sine_deg : public exprtk::ifunction<T>
{
sine_deg() : exprtk::ifunction<T>(1) {}
inline T operator()(const T& v)
{
const T pi = exprtk::details::numeric::constant::pi;
return std::sin((v * T(pi)) / T(180));
}
};
...
typedef exprtk::symbol_table<T> symbol_table_t;
typedef exprtk::expression<T> expression_t;
typedef exprtk::parser<T> parser_t;
typedef typename parser_t::settings_store settings_t;
sine_deg<T> sine;
symbol_table.add_reserved_function("sin",sine);
expression_t expression;
expression.register_symbol_table(symbol_table);
parser_t parser;
parser.settings()
.disable_base_function(settings_t::e_bf_sin);
parser.compile("1 + sin(30)",expression);
In the example above, the custom 'sin' function is registered with the
symbol_table using the method 'add_reserved_function'. This is done so
as to bypass the checks for reserved words that are carried out on the
provided symbol names when calling the standard 'add_function' method.
Normally if a user specified symbol name conflicts with any of the
ExprTk reserved words, the add_function call will fail.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 20 - EXPRESSION RETURN VALUES]
ExprTk expressions can return immediately from any point by utilizing
the return call. Furthermore the return call can be used to transfer
out multiple return values from within the expression.
If an expression evaluation exits using a return point, the result of
the call to the 'value' method will be NaN, and it's expected that the
return values will be available from the results_context.
In the following example there are three return points in the
expression. If neither of the return points are hit, then the
expression will return normally.
std::string expression_string =
" if (x < y) "
" return [x + 1,'return-call 1']; "
" else if (x > y) "
" return [y / 2, y + 1, 'return-call 2']; "
" else if (equal(x,y)) "
" x + y; "
" return [x, y, x + y, x - y, 'return-call 3'] ";
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;
symbol_table.add_variable("x",x);
symbol_table.add_variable("y",y);
expression.register_symbol_table(symbol_table);
parser.compile(expression_string,expression);
T result = expression.value();
if (expression.results().count())
{
typedef exprtk::results_context<T> results_context_t;
typedef typename results_context_t::type_store_t type_t;
typedef typename type_t::scalar_view scalar_t;
typedef typename type_t::vector_view vector_t;
typedef typename type_t::string_view string_t;
const results_context_t& results = expression.results();
for (std::size_t i = 0; i < results.count(); ++i)
{
type_t t = results[i];
switch (t.type)
{
case type_t::e_scalar : ...
break;
case type_t::e_vector : ...
break;
case type_t::e_string : ...
break;
default : continue;
}
}
Note: Processing of the return results is similar to that of the
generic function call parameters.
It is however recommended that if there is to be only a single flow of
execution through the expression, that the simpler approach of
registering external variables of appropriate type be used.
This method simply requires the variables that are to hold the various
results that are to be computed within the expression to be registered
with an associated symbol_table instance. Then within the expression
itself to have the result variables be assigned the appropriate
values.
typedef exprtk::symbol_table<double> symbol_table_t;
typedef exprtk::expression<double> expression_t;
typedef exprtk::parser<double> parser_t;
std::string expression_string =
" var x := 123.456; "
" var s := 'ijk'; "
" result0 := x + 78.90; "
" result1 := s + '123' ";
double result0;
std::string result1;
symbol_table_t symbol_table;
symbol_table.add_variable ("result0",result0);
symbol_table.add_stringvar("result1",result1);
expression_t expression;
expression.register_symbol_table(symbol_table);
parser_t parser;
parser.compile(expression_string,expression);
expression.value();
printf("Result0: %15.5f\n", result0 );
printf("Result1: %s\n" , result1.c_str());
In the example above, the expression will compute two results. As such
two result variables are defined to hold the values named result0 and
result1 respectively. The first is of scalar type (double), the second
is of string type. Once the expression has been evaluated, the two
variables will have been updated with the new result values, and can
then be further utilised from within the calling program.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 21 - COMPILATION ERRORS]
When attempting to compile a malformed or otherwise erroneous ExprTk
expression, the compilation process will result in an error, as is
indicated by the 'compile' method returning a false value. A
diagnostic indicating the first error encountered and its cause can be
obtained by invoking the 'error' method, as is demonstrated in the
following example:
if (!parser.compile(expression_string,expression))
{
printf("Error: %s\n", parser.error().c_str());
return false;
}
Any error(s) resulting from a failed compilation will be stored in the
parser instance until the next time a compilation is performed. Before
then errors can be enumerated in the order they occurred by invoking
the 'get_error' method which itself will return a 'parser_error' type.
A parser_error object will contain an error diagnostic, an error mode
(or class), and the character position of the error in the expression
string. The following example demonstrates the enumeration of error(s)
in the event of a failed compilation.
if (!parser.compile(expression_string,expression))
{
for (std::size_t i = 0; i < parser.error_count(); ++i)
{
typedef exprtk::parser_error::type error_t;
error_t error = parser.get_error(i);
printf("Error[%02d] Position: %02d Type: [%14s] Msg: %s\n",
i,
error.token.position,
exprtk::parser_error::to_str(error.mode).c_str(),
error.diagnostic.c_str());
}
return false;
}
Assuming the following expression '2 + (3 / log(1 + x))' which uses a
variable named 'x' that has not been registered with the appropriate
symbol_table instance and is not a locally defined variable, once
compiled the above denoted post compilation error handling code shall
produce the following output:
Error: ERR184 - Undefined symbol: 'x'
Error[00] Pos:17 Type:[Syntax] Msg: ERR184 - Undefined symbol: 'x'
For expressions comprised of multiple lines, the error position
provided in the parser_error object can be converted into a pair of
line and column numbers by invoking the 'update_error' function as is
demonstrated by the following example:
if (!parser.compile(program_str,expression))
{
for (std::size_t i = 0; i < parser.error_count(); ++i)
{
typedef exprtk::parser_error::type error_t;
error_t error = parser.get_error(i);
exprtk::parser_error::update_error(error,program_str);
printf("Error[%02d] at line: %d column: %d\n",
i,
error.line_no,
error.column_no);
}
return false;
}
Note: There are five distinct error modes in ExprTk which denote the
class of an error. These classes are as follows:
(a) Syntax
(b) Token
(c) Numeric
(d) Symbol Table
(e) Lexer
(a) Syntax Errors
These are errors related to invalid syntax found within the denoted
expression. Examples are invalid sequences of operators and variables,
incorrect number of parameters to functions, invalid conditional or
loop structures and invalid use of keywords.
eg: 'for := sin(x,y,z) + 2 * equal > until[2 - x,3]'
(b) Token Errors
Errors in this class relate to token level errors detected by one or
more of the following checkers:
(1) Bracket Checker
(2) Numeric Checker
(3) Sequence Checker
(c) Numeric Errors
This class of error is related to conversion of numeric values from
their string form to the underlying numerical type (float, double
etc).
(d) Symbol Table Errors
This is the class of errors related to failures when interacting with
the registered symbol_table instance. Errors such as not being able to
find, within the symbol_table, symbols representing variables or
functions, to being unable to create new variables in the symbol_table
via the 'unknown symbol resolver' mechanism.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 22 - RUNTIME LIBRARY PACKAGES]
ExprTk contains a set of simple extensions, that provide
functionalities beyond basic numerical calculations. Currently the
available packages are:
+---+--------------------+-----------------------------------+
| # | Package Name | Namespace/Type |
+---+--------------------+-----------------------------------+
| 1 | Basic I/O | exprtk::rtl::io::package<T> |
| 2 | File I/O | exprtk::rtl::io::file::package<T> |
| 3 | Vector Operations | exprtk::rtl::vecops::package<T> |
+---+--------------------+-----------------------------------+
In order to make the features of a specific package available within
an expression, an instance of the package must be added to the
expression's associated symbol table. In the following example, the
file I/O package is made available for the given expression:
typedef exprtk::symbol_table<T> symbol_table_t;
typedef exprtk::expression<T> expression_t;
typedef exprtk::parser<T> parser_t;
exprtk::rtl::io::file::package<T> fileio_package;
std::string expression_string =
" var file_name := 'file.txt'; "
" var stream := null; "
" "
" stream := open(file_name,'w'); "
" "
" write(stream,'Hello world....\n'); "
" "
" close(stream); "
" ";
symbol_table_t symbol_table;
symbol_table.add_package(fileio_package);
expression_t expression;
expression.register_symbol_table(symbol_table);
parser_t parser;
parser.compile(expression_string,expression);
expression.value();
(1) Basic I/O functions:
(a) print
(b) println
(2) File I/O functions:
(a) open (b) close
(c) write (d) read
(e) getline (f) eof
(3) Vector Operations functions:
(a) all_true (b) all_false
(c) any_true (d) any_false
(e) count (f) copy
(g) rotate-left (h) rotate-right
(i) shift-left (j) shift-right
(k) sort (l) nth_element
(m) iota (n) sumk
(o) axpy (p) axpby
(q) axpyz (r) axpbyz
(s) axpbz (t) dot
(u) dotk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 23 - HELPERS & UTILS]
The ExprTk library provides a series of usage simplifications via
helper routines that combine various processes into a single 'function
call' making certain actions easier to carry out though not
necessarily in the most efficient way possible. A list of the routines
are as follows:
(a) collect_variables
(b) collect_functions
(c) compute
(d) integrate
(e) derivative
(f) second_derivative
(g) third_derivative
(a) collect_variables
This function will collect all the variable symbols in a given string
representation of an expression and return them in an STL compatible
sequence data structure (eg: std::vector, dequeue etc) specialised
upon a std::string type. If an error occurs during the parsing of the
expression then the return value of the function will be false,
otherwise it will be true. An example use of the given routine is as
follows:
std::string expression = "x + abs(y / z)";
std::vector<std::string> variable_list;
if (exprtk::collect_variables(expression, variable_list))
{
for (const auto& var : variable_list)
{
...
}
}
else
printf("An error occured.");
(b) collect_functions
This function will collect all the function symbols in a given string
representation of an expression and return them in an STL compatible
sequence data structure (eg: std::vector, dequeue etc) specialised
upon a std::string type. If an error occurs during the parsing of the
expression then the return value of the function will be false,
otherwise it will be true. An example use of the given routine is as
follows:
std::string expression = "x + abs(y / cos(1 + z))";
std::deque<std::string> function_list;
if (exprtk::collect_functions(expression, function_list))
{
for (const auto& func : function_list)
{
...
}
}
else
printf("An error occured.");
Note: When either the 'collect_variables' or 'collect_functions' free
functions return true - that does not necessarily indicate the
expression itself is valid. It is still possible that when compiled
the expression may have certain 'type' related errors - though it is
highly likely that no semantic errors will occur if either return
true.
Note: The default interface provided for both the collect_variables
and collect_functions free_functions, assumes that expressions will
only be utilising the ExprTk reserved funnctions (eg: abs, cos, min
etc). When user defined functions are to be used in an expression, a
symbol_table instance containing said functions can be passed to
either routine, and will be incorparated during the compilation and
Dependent Entity Collection processes. In the following example, a
user defined free function named 'foo' is registered with a
symbol_table. Finally the symbol_table instance and associated
expression string are passed to the exprtk::collect_functions routine.
template <typename T>
T foo(T v)
{
return std::abs(v + T(2)) / T(3);
}
......
exprtk::symbol_table<T> sym_tab;
symbol_table.add_function("foo",foo);
std::string expression = "x + foo(y / cos(1 + z))";
std::deque<std::string> function_list;
if (exprtk::collect_functions(expression, sym_tab, function_list))
{
for (const auto& func : function_list)
{
...
}
}
else
printf("An error occured.");
(c) compute
This free function will compute the value of an expression from its
string form. If an invalid expression is passed, the result of the
function will be false indicating an error, otherwise the return value
will be true indicating success. The compute function has three
overloads, the definitions of which are:
(1) No variables
(2) One variable called x
(3) Two variables called x and y
(3) Three variables called x, y and z
Example uses of each of the three overloads for the compute routine
are as follows:
T result = T(0);
// No variables overload
std::string no_vars = "abs(1 - (3 / pi)) * 5";
if (!exprtk::compute(no_vars,result))
printf("Failed to compute: %s",no_vars.c_str());
else
printf("Result: %15.5f\n",result);
// One variable 'x' overload
T x = 123.456;
std::string one_var = "abs(x - (3 / pi)) * 5";
if (!exprtk::compute(one_var, x, result))
printf("Failed to compute: %s",one_var.c_str());
else
printf("Result: %15.5f\n",result);
// Two variables 'x' and 'y' overload
T y = 789.012;
std::string two_var = "abs(x - (y / pi)) * 5";
if (!exprtk::compute(two_var, x, y, result))
printf("Failed to compute: %s",two_var.c_str());
else
printf("Result: %15.5f\n",result);
// Three variables 'x', 'y' and 'z' overload
T z = 345.678;
std::string three_var = "abs(x - (y / pi)) * z";
if (!exprtk::compute(three_var, x, y, z, result))
printf("Failed to compute: %s",three_var.c_str());
else
printf("Result: %15.5f\n",result);
(d) integrate
This free function will attempt to perform a numerical integration of
a single variable compiled expression over a specified range and step
size. The numerical integration is based on the three point form of
Simpson's rule. The integrate function has two overloads, where the
variable of integration can either be passed as a reference or as a
name in string form. Example usage of the function is as follows:
typedef exprtk::parser<T> parser_t;
typedef exprtk::expression<T> expression_t;
typedef exprtk::symbol_table<T> symbol_table_t;
std::string expression_string = "sqrt(1 - (x^2))";
T x = T(0);
symbol_table_t symbol_table;
symbol_table.add_variable("x",x);
expression_t expression;
expression.register_symbol_table(symbol_table);
parser_t parser;
parser.compile(expression_string,expression);
....
// Integrate in domain [-1,1] using a reference to x variable
T area1 = exprtk::integrate(expression, x, T(-1), T(1));
// Integrate in domain [-1,1] using name of x variable
T area2 = exprtk::integrate(expression, "x", T(-1), T(1));
(e) derivative
This free function will attempt to perform a numerical differentiation
of a single variable compiled expression at a given point for a given
epsilon, using a variant of Newton's difference quotient called the
five-point stencil method. The derivative function has two overloads,
where the variable of differentiation can either be passed as a
reference or as a name in string form. Example usage of the derivative
function is as follows:
typedef exprtk::parser<T> parser_t;
typedef exprtk::expression<T> expression_t;
typedef exprtk::symbol_table<T> symbol_table_t;
std::string expression_string = "sqrt(1 - (x^2))";
T x = T(0);
symbol_table_t symbol_table;
symbol_table.add_variable("x",x);
expression_t expression;
expression.register_symbol_table(symbol_table);
parser_t parser;
parser.compile(expression_string,expression);
....
// Differentiate expression at value of x = 12.3 using a reference
// to the x variable
x = T(12.3);
T derivative1 = exprtk::derivative(expression, x);
// Differentiate expression where value x = 45.6 using name
// of the x variable
x = T(45.6);
T derivative2 = exprtk::derivative(expression, "x");
(f) second_derivative
This free function will attempt to perform a numerical second
derivative of a single variable compiled expression at a given point
for a given epsilon, using a variant of Newton's difference quotient
method. The second_derivative function has two overloads, where the
variable of differentiation can either be passed as a reference or as
a name in string form. Example usage of the second_derivative function
is as follows:
typedef exprtk::parser<T> parser_t;
typedef exprtk::expression<T> expression_t;
typedef exprtk::symbol_table<T> symbol_table_t;
std::string expression_string = "sqrt(1 - (x^2))";
T x = T(0);
symbol_table_t symbol_table;
symbol_table.add_variable("x",x);
expression_t expression;
expression.register_symbol_table(symbol_table);
parser_t parser;
parser.compile(expression_string,expression);
....
// Second derivative of expression where value of x = 12.3 using a
// reference to x variable
x = T(12.3);
T derivative1 = exprtk::second_derivative(expression,x);
// Second derivative of expression where value of x = 45.6 using
// name of x variable
x = T(45.6);
T derivative2 = exprtk::second_derivative(expression, "x");
(g) third_derivative
This free function will attempt to perform a numerical third
derivative of a single variable compiled expression at a given point
for a given epsilon, using a variant of Newton's difference quotient
method. The third_derivative function has two overloads, where the
variable of differentiation can either be passed as a reference or as
a name in string form. Example usage of the third_derivative function
is as follows:
typedef exprtk::parser<T> parser_t;
typedef exprtk::expression<T> expression_t;
typedef exprtk::symbol_table<T> symbol_table_t;
std::string expression_string = "sqrt(1 - (x^2))";
T x = T(0);
symbol_table_t symbol_table;
symbol_table.add_variable("x",x);
expression_t expression;
expression.register_symbol_table(symbol_table);
parser_t parser;
parser.compile(expression_string,expression);
....
// Third derivative of expression where value of x = 12.3 using a
// reference to the x variable
x = T(12.3);
T derivative1 = exprtk::third_derivative(expression, x);
// Third derivative of expression where value of x = 45.6 using
// name of the x variable
x = T(45.6);
T derivative2 = exprtk::third_derivative(expression, "x");
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 24 - BENCHMARKING]
As part of the ExprTk package there is an expression benchmark utility
named 'exprtk_benchmark'. The utility attempts to determine expression
evaluation speed (or rate of evaluations - evals per second), by
evaluating each expression numerous times and mutating the underlying
variables of the expression between each evaluation. The utility
assumes any valid ExprTk expression (containing conditionals, loops
etc), however it will only make use of a predefined set of scalar
variables, namely: a, b, c, x, y, z and w. That being said expressions
themselves can contain any number of local variables, vectors or
strings. There are two modes of operation:
(1) Default
(2) User Specified Expressions
(1) Default
The default mode is enabled simply by executing the exprtk_benchmark
binary with no command line parameters. In this mode a predefined set
of expressions will be evaluated in three phases:
(a) ExprTk evaluation
(b) Native evaluation
(c) ExprTk parse
In the first two phases (a and b) a list of predefined (hard-coded)
expressions will be evaluated using both ExprTk and native mode
implementations. This is done so as to compare evaluation times
between ExprTk and native implementations. The set of expressions used
are as follows:
(01) (y + x)
(02) 2 * (y + x)
(03) (2 * y + 2 * x)
(04) ((1.23 * x^2) / y) - 123.123
(05) (y + x / y) * (x - y / x)
(06) x / ((x + y) + (x - y)) / y
(07) 1 - ((x * y) + (y / x)) - 3
(08) (5.5 + x) + (2 * x - 2 / 3 * y) * (x / 3 + y / 4) + (y + 7.7)
(09) 1.1x^1 + 2.2y^2 - 3.3x^3 + 4.4y^15 - 5.5x^23 + 6.6y^55
(10) sin(2 * x) + cos(pi / y)
(11) 1 - sin(2 * x) + cos(pi / y)
(12) sqrt(111.111 - sin(2 * x) + cos(pi / y) / 333.333)
(13) (x^2 / sin(2 * pi / y)) - x / 2
(14) x + (cos(y - sin(2 / x * pi)) - sin(x - cos(2 * y / pi))) - y
(15) clamp(-1.0, sin(2 * pi * x) + cos(y / 2 * pi), +1.0)
(16) max(3.33, min(sqrt(1 - sin(2 * x) + cos(pi / y) / 3), 1.11))
(17) if((y + (x * 2.2)) <= (x + y + 1.1), x - y, x*y) + 2 * pi / x
The third and final phase (c), is used to determine average
compilation rates (compiles per second) for expressions of varying
complexity. Each expression is compiled 100K times and the average for
each expression is output.
(2) User Specified Expressions
In this mode two parameters are passed to the utility via the command
line:
(a) A name of a text file containing one expression per line
(b) An integer representing the number of evaluations per expression
An example execution of the benchmark utility in this mode is as
follows:
./exprtk_benchmark my_expressions.txt 1000000
The above invocation will load the expressions from the file
'my_expressions.txt' and will then proceed to evaluate each expression
one million times, varying the above mentioned variables (x, y, z
etc.) between each evaluation, and at the end of each expression round
a print out of running times, result of a single evaluation and total
sum of results is provided as demonstrated below:
Expression 1 of 7 4.770 ns 47700 ns ( 9370368.0) '((((x+y)+z)))'
Expression 2 of 7 4.750 ns 47500 ns ( 1123455.9) '((((x+y)-z)))'
Expression 3 of 7 4.766 ns 47659 ns (21635410.7) '((((x+y)*z)))'
Expression 4 of 7 5.662 ns 56619 ns ( 1272454.9) '((((x+y)/z)))'
Expression 5 of 7 4.950 ns 49500 ns ( 4123455.9) '((((x-y)+z)))'
Expression 6 of 7 7.581 ns 75810 ns (-4123455.9) '((((x-y)-z)))'
Expression 7 of 7 4.801 ns 48010 ns ( 0.0) '((((x-y)*z)))'
The benchmark utility can be very useful when investigating evaluation
efficiency issues with ExprTk or simply during the prototyping of
expressions. As an example, lets take the following expression:
1 / sqrt(2x) * e^(3y)
Lets say we would like to determine which sub-part of the expression
takes the most time to evaluate and perhaps attempt to rework the
expression based on the results. In order to do this we will create a
text file called 'test.txt' and then proceed to make some educated
guesses about how to break the expression up into its more
'interesting' sub-parts which we will then add as one expression per
line to the file. An example breakdown may be as follows:
1 / sqrt(2x) * e^(3y)
1 / sqrt(2x)
e^(3y)
The benchmark with the given file, where each expression will be
evaluated 100K times can be executed as follows:
./exprtk_benchmark test.txt 100000
Expr 1 of 3 90.340 ns 9034000 ns (296417859.3) '1/sqrt(2x)*e^(3y)'
Expr 2 of 3 11.100 ns 1109999 ns ( 44267.3) '1/sqrt(2x)'
Expr 3 of 3 77.830 ns 7783000 ns (615985286.6) 'e^(3y)'
[*] Number Of Evals: 300000
[*] Total Time: 0.018sec
[*] Total Single Eval Time: 0.000ms
From the results above we conclude that the third expression (e^(3y))
consumes the largest amount of time. The variable 'e', as used in both
the benchmark and in the expression, is an approximation of the
transcendental mathematical constant e (2.71828182845904...) hence the
sub-expression should perhaps be modified to use the generally more
efficient built-in 'exp' function.
./exprtk_benchmark test.txt 1000000
Expr 1 of 5 86.563 ns 8656300ns (296417859.6) '1/sqrt(2x)*e^(3y)'
Expr 2 of 5 40.506 ns 4050600ns (296417859.6) '1/sqrt(2x)*exp(3y)'
Expr 3 of 5 14.248 ns 1424799ns ( 44267.2) '1/sqrt(2x)'
Expr 4 of 5 88.840 ns 8884000ns (615985286.9) 'e^(3y)'
Expr 5 of 5 29.267 ns 2926699ns (615985286.9) 'exp(3y)'
[*] Number Of Evals: 5000000
[*] Total Time: 0.260sec
[*] Total Single Eval Time: 0.000ms
The above output demonstrates the results from making the previously
mentioned modification to the expression. As can be seen the new form
of the expression using the 'exp' function reduces the evaluation time
by over 50%, in other words increases the evaluation rate by two fold.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 25 - EXPRTK NOTES]
The following is a list of facts and suggestions one may want to take
into account when using ExprTk:
(00) Precision and performance of expression evaluations are the
dominant principles of the ExprTk library.
(01) ExprTk uses a rudimentary imperative programming model with
syntax based on languages such as Pascal and C. Furthermore
ExprTk is an LL(2) type grammar and is processed using a
recursive descent parsing algorithm.
(02) Supported types are float, double, long double and MPFR/GMP.
(03) Standard mathematical operator precedence is applied (BEDMAS).
(04) 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).
(05) Supported user defined types are numeric and string
variables, numeric vectors and functions.
(06) All reserved words, keywords, variable, vector, string and
function names are case-insensitive.
(07) Variable, vector, string variable and function names must begin
with a letter (A-Z or a-z), then can be comprised of any
combination of letters, digits, underscores and dots. (eg: x,
var1 or power_func99, person.age, item.size.0)
(08) Expression lengths and sub-expression lists are limited only by
storage capacity.
(09) 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 utilise objects, such as variables,
of that symbol-table, otherwise the result will be undefined
behavior.
(10) Equal and Nequal are normalised-epsilon equality routines,
which use epsilons of 0.0000000001 and 0.000001 for double and
float types respectively.
(11) All trigonometric functions assume radian input unless stated
otherwise.
(12) Expressions may contain white-space characters such as space,
tabs, new-lines, control-feed et al.
('\n', '\r', '\t', '\b', '\v', '\f')
(13) Strings may be comprised of any combination of letters, digits
special characters including (~!@#$%^&*()[]|=+ ,./?<>;:"`~_) or
hexadecimal escaped sequences (eg: \0x30) and must be enclosed
with single-quotes.
eg: 'Frankly my dear, \0x49 do n0t give a damn!'
(14) User defined normal functions can have up to 20 parameters,
where as user defined generic-functions and vararg-functions
can have an unlimited number of parameters.
(15) The inbuilt polynomial functions can be at most of degree 12.
(16) Where appropriate constant folding optimisations may be applied.
(eg: The expression '2 + (3 - (x / y))' becomes '5 - (x / y)')
(17) If the strength reduction compilation option has been enabled,
then where applicable strength reduction optimisations may be
applied.
(18) String processing capabilities are available by default. To
turn them off, the following needs to be defined at compile
time: exprtk_disable_string_capabilities
(19) Composited functions can call themselves or any other functions
that have been defined prior to their own definition.
(20) Recursive calls made from within composited functions will have
a stack size bound by the stack of the executing architecture.
(21) User defined functions by default are assumed to have side
effects. As such an "all constant parameter" invocation of such
functions wont result in constant folding. If the function has
no side effects then that can be noted during the constructor
of the ifunction allowing it to be constant folded where
appropriate.
(22) The entity relationship between symbol_table and an expression
is many-to-many. However the intended 'typical' use-case where
possible, is to have a single symbol table manage the variable
and function requirements of multiple expressions.
(23) 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 and suboptimal approach would
be to recompile an expression from its string form every time
it requires evaluating.
(24) It is strongly recommended that the return value of method
invocations from the parser and symbol_table types be taken
into account. Specifically the 'compile' method of the parser
and the 'add_xxx' set of methods of the symbol_table as they
denote either the success or failure state of the invoked call.
Continued processing from a failed state without having first
rectified the underlying issue will in turn result in further
failures and undefined behaviours.
(25) The following are examples of compliant floating point value
representations:
(1) 12345 (5) -123.456
(2) +123.456e+12 (6) 123.456E-12
(3) +012.045e+07 (7) .1234
(4) 123.456f (8) -321.654E+3L
(26) Expressions may contain any of the following comment styles:
(1) // .... \n
(2) # .... \n
(3) /* .... */
(27) The 'null' value type is a special non-zero type that
incorporates specific semantics when undergoing operations with
the standard numeric type. The following is a list of type and
boolean results associated with the use of 'null':
(1) null +,-,*,/,% x --> x
(2) x +,-,*,/,% null --> x
(3) null +,-,*,/,% null --> null
(4) null == null --> true
(5) null == x --> true
(6) x == null --> true
(7) x != null --> false
(8) null != null --> false
(9) null != x --> false
(28) The following is a list of reserved words and symbols used by
ExprTk. Attempting to add a variable or custom function to a
symbol table using any of the reserved words will result in a
failure.
abs, acos, acosh, and, asin, asinh, atan, atan2, atanh, avg,
break, case, ceil, clamp, continue, cosh, cos, cot, csc,
default, deg2grad, deg2rad, else, equal, erfc, erf, exp,
expm1, false, floor, for, frac, grad2deg, hypot, iclamp, if,
ilike, in, inrange, in, like, log, log10, log1p, log2, logn,
mand, max, min, mod, mor, mul, nand, ncdf, nor, not,
not_equal, not, null, or, pow, rad2deg, repeat, return,
root, roundn, round, sec, sgn, shl, shr, sinc, sinh, sin,
sqrt, sum, swap, switch, tanh, tan, true, trunc, until, var,
while, xnor, xor, xor
(29) Every valid ExprTk statement is a "value returning" expression.
Unlike some languages that limit the types of expressions that
can be performed in certain situations, in ExprTk any valid
expression can be used in any "value consuming" context. eg:
var y := 3;
for (var x := switch
{
case 1 : 7;
case 2 : -1 + ~{var x{};};
default : y > 2 ? 3 : 4;
};
x != while (y > 0) { y -= 1; };
x -= {
if (min(x,y) < 2 * max(x,y))
x + 2;
else
x + y - 3;
}
)
{
(x + y) / (x - y);
}
(30) It is recommended when prototyping expressions that the ExprTk
REPL be utilised, as it supports all the features available in
the library, including complete error analysis, benchmarking
and dependency dumps etc which allows for rapid
coding/prototyping and debug cycles without the hassle of
having to recompile test programs with expressions that have
been hard-coded. It's also a good source of truth for how the
library's various features can be applied.
(31) For performance considerations, one should assume the actions
of expression, symbol table and parser instance instantiation
and destruction, and the expression compilation process itself
to be of high latency. Hence none of them should be part of any
performance critical code paths, and should instead occur
entirely either before or after such code paths.
(32) Deep copying an expression instance for the purposes of
persisting to disk or otherwise transmitting elsewhere with the
intent to 'resurrect' the expression instance later on is not
possible due to the reasons described in the final note of
Section 10. The recommendation is to instead simply persist the
string form of the expression and compile the expression at
run-time on the target.
(33) Before jumping in and using ExprTk, do take the time to peruse
the documentation and all of the examples, both in the main and
the extras distributions. Having an informed general view of
what can and can't be done, and how something should be done
with ExprTk, will likely result in a far more productive and
enjoyable programming experience.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 26 - SIMPLE EXPRTK EXAMPLE]
The following is a simple yet complete example demonstrating typical
usage of the ExprTk Library. The example instantiates a symbol table
object, adding to it three variables named x, y and z, and a custom
user defined function, that accepts only two parameters, named myfunc.
The example then proceeds to instantiate an expression object and
register to it the symbol table instance.
A parser is then instantiated, and the string representation of the
expression and the expression object are passed to the parser's
compile method for compilation. If an error occurred during
compilation, the compile method will return false, leading to a series
of error diagnostics being printed to stdout. Otherwise the newly
compiled expression is evaluated by invoking the expression object's
value method, and subsequently printing the result of the computation
to stdout.
--- 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 occurred. Attempt to
// print all errors to 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 ---
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 27 - BUILD OPTIONS]
When building ExprTk there are a number of defines that will enable or
disable certain features and capabilities. The defines can either be
part of a compiler command line switch or scoped around the include to
the ExprTk header. The defines are as follows:
(01) exprtk_enable_debugging
(02) exprtk_disable_comments
(03) exprtk_disable_break_continue
(04) exprtk_disable_sc_andor
(05) exprtk_disable_return_statement
(06) exprtk_disable_enhanced_features
(07) exprtk_disable_string_capabilities
(08) exprtk_disable_superscalar_unroll
(09) exprtk_disable_rtl_io_file
(10) exprtk_disable_rtl_vecops
(11) exprtk_disable_caseinsensitivity
(01) exprtk_enable_debugging
This define will enable printing of debug information to stdout during
the compilation process.
(02) exprtk_disable_comments
This define will disable the ability for expressions to have comments.
Expressions that have comments when parsed with a build that has this
option, will result in a compilation failure.
(03) exprtk_disable_break_continue
This define will disable the loop-wise 'break' and 'continue'
capabilities. Any expression that contains those keywords will result
in a compilation failure.
(04) exprtk_disable_sc_andor
This define will disable the short-circuit '&' (and) and '|' (or)
operators
(05) exprtk_disable_return_statement
This define will disable use of return statements within expressions.
(06) exprtk_disable_enhanced_features
This define will disable all enhanced features such as strength
reduction and special function optimisations and expression specific
type instantiations. This feature will reduce compilation times and
binary sizes but will also result in massive performance degradation
of expression evaluations.
(07) exprtk_disable_string_capabilities
This define will disable all string processing capabilities. Any
expression that contains a string or string related syntax will result
in a compilation failure.
(08) exprtk_disable_superscalar_unroll
This define will set the loop unroll batch size to 4 operations per
loop instead of the default 8 operations. This define is used in
operations that involve vectors and aggregations over vectors. When
targeting non-superscalar architectures, it may be recommended to
build using this particular option if efficiency of evaluations is of
concern.
(09) exprtk_disable_rtl_io_file
This define will disable the file I/O RTL package features. When
present, any attempts to register the file I/O package with a given
symbol table will fail causing a compilation error.
(10) exprtk_disable_rtl_vecops
This define will disable the extended vector operations RTL package
features. When present, any attempts to register the vector operations
package with a given symbol table will fail causing a compilation
error.
(11) exprtk_disable_caseinsensitivity
This define will disable case-insensitivity when matching variables
and functions. Furthermore all reserved and keywords will only be
acknowledged when in all lower-case.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 28 - FILES]
The source distribution of ExprTk is comprised of the following set of
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
(15) exprtk_simple_example_11.cpp
(16) exprtk_simple_example_12.cpp
(17) exprtk_simple_example_13.cpp
(18) exprtk_simple_example_14.cpp
(19) exprtk_simple_example_15.cpp
(20) exprtk_simple_example_16.cpp
(21) exprtk_simple_example_17.cpp
(22) exprtk_simple_example_18.cpp
(23) exprtk_simple_example_19.cpp
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[SECTION 29 - LANGUAGE STRUCTURE]
The following are the various language structures available within
ExprTk and their structural representations.
(00) If Statement
(01) Else Statement
(02) Ternary Statement
(03) While Loop
(04) Repeat Until Loop
(05) For Loop
(06) Switch Statement
(07) Multi Subexpression Statement
(08) Multi Case-Consequent Statement
(09) Variable Definition Statement
(10) Vector Definition Statement
(11) String Definition Statement
(12) Range Statement
(13) Return Statement
(00) - If Statement
+-------------------------------------------------------------+
| |
| [if] ---> [(] ---> [condition] -+-> [,] -+ |
| | | |
| +---------------<---------------+ | |
| | | |
| | +------------------<------------------+ |
| | | |
| | +--> [consequent] ---> [,] ---> [alternative] ---> [)] |
| | |
| +--> [)] --+-> [{] ---> [expression*] ---> [}] --+ |
| | | |
| | +---------<----------+ |
| +----<-----+ | |
| | v |
| +--> [consequent] --> [;] -{*}-> [else-statement] |
| |
+-------------------------------------------------------------+
(01) - Else Statement
+-------------------------------------------------------------+
| |
| [else] -+-> [alternative] ---> [;] |
| | |
| +--> [{] ---> [expression*] ---> [}] |
| | |
| +--> [if-statement] |
| |
+-------------------------------------------------------------+
(02) - Ternary Statement
+-------------------------------------------------------------+
| |
| [condition] ---> [?] ---> [consequent] ---> [:] --+ |
| | |
| +------------------------<------------------------+ |
| | |
| +--> [alternative] --> [;] |
| |
+-------------------------------------------------------------+
(03) - While Loop
+-------------------------------------------------------------+
| |
| [while] ---> [(] ---> [condition] ---> [)] ---+ |
| | |
| +----------------------<----------------------+ |
| | |
| +--> [{] ---> [expression*] ---> [}] |
| |
+-------------------------------------------------------------+
(04) - Repeat Until Loop
+-------------------------------------------------------------+
| |
| [repeat] ---> [expression*] ---+ |
| | |
| +--------------<---------------+ |
| | |
| +--> [until] ---> [(] ---> [condition] --->[)] |
| |
+-------------------------------------------------------------+
(05) - For Loop
+-------------------------------------------------------------+
| |
| [for] ---> [(] -+-> [initialise expression] --+--+ |
| | | | |
| +------------->---------------+ v |
| | |
| +-----------------------<------------------------+ |
| | |
| +--> [;] -+-> [condition] -+-> [;] ---+ |
| | | | |
| +------->--------+ v |
| | |
| +------------------<---------+--------+ |
| | | |
| +--> [increment expression] -+-> [)] --+ |
| | |
| +------------------<-------------------+ |
| | |
| +--> [{] ---> [expression*] ---> [}] |
| |
+-------------------------------------------------------------+
(06) - Switch Statement
+-------------------------------------------------------------+
| |
| [switch] ---> [{] ---+ |
| | |
| +---------<----------+-----------<-----------+ |
| | | |
| +--> [case] ---> [condition] ---> [:] ---+ | |
| | | |
| +-------------------<--------------------+ | |
| | | |
| +--> [consequent] ---> [;] --------->--------+ |
| | | |
| | | |
| +--> [default] ---> [consequent] ---> [;] ---+ |
| | | |
| +---------------------<----------------------+ |
| | |
| +--> [}] |
| |
+-------------------------------------------------------------+
(07) - Multi Subexpression Statement
+-------------------------------------------------------------+
| |
| +--------------<---------------+ |
| | | |
| [~] ---> [{\(] -+-> [expression] -+-> [;\,] ---+ |
| | |
| +----------------<----------------+ |
| | |
| +--> [}\)] |
| |
+-------------------------------------------------------------+
(08) - Multi Case-Consequent Statement
+-------------------------------------------------------------+
| |
| [[*]] ---> [{] ---+ |
| | |
| +--------<--------+--------------<----------+ |
| | | |
| +--> [case] ---> [condition] ---> [:] ---+ | |
| | | |
| +-------------------<--------------------+ | |
| | | |
| +--> [consequent] ---> [;] ---+------>------+ |
| | |
| +--> [}] |
| |
+-------------------------------------------------------------+
(09) - Variable Definition Statement
+-------------------------------------------------------------+
| |
| [var] ---> [symbol] -+-> [:=] -+-> [expression] -+-> [;] |
| | | | |
| | +-----> [{}] -->--+ |
| | | |
| +------------->-------------+ |
| |
+-------------------------------------------------------------+
(10) - Vector Definition Statement
+-------------------------------------------------------------+
| |
| [var] ---> [symbol] ---> [[] ---> [constant] ---> []] --+ |
| | |
| +---------------------------<---------------------------+ |
| | |
| | +--------->---------+ |
| | | | |
| +--> [:=] ---> [{] -+-+-> [expression] -+-> [}] ---> [;] |
| | | |
| +--<--- [,] <-----+ |
| |
+-------------------------------------------------------------+
(11) - String Definition Statement
+-------------------------------------------------------------+
| |
| [var] --> [symbol] --> [:=] --> [str-expression] ---> [;] |
| |
+-------------------------------------------------------------+
(12) - Range Statement
+-------------------------------------------------------------+
| |
| +-------->--------+ |
| | | |
| [[] -+-> [expression] -+-> [:] -+-> [expression] -+--> []] |
| | | |
| +-------->--------+ |
| |
+-------------------------------------------------------------+
(13) - Return Statement
+-------------------------------------------------------------+
| |
| [return] ---> [[] -+-> [expression] -+-> []] ---> [;] |
| | | |
| +--<--- [,] <-----+ |
| |
+-------------------------------------------------------------+