464 lines
22 KiB
Plaintext
464 lines
22 KiB
Plaintext
C++ Mathematical Expression Toolkit Library
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[INTRODUCTION]
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The C++ Mathematical Expression Library (ExprTk) is a simple to use,
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easy to integrate and extremely efficient mathematical expression
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parsing and evaluation engine. The parsing engine supports various
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kinds of functional and logic processing semantics and is very easily
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extendible.
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[CAPABILITIES]
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The ExprTk evaluator supports the following fundamental mathematical
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operations, functions and processes:
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(1) Basic operators: +, -, *, /, %, ^
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(2) Functions: min, max, avg, sum, abs, ceil, floor, round,
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roundn, exp, log, log10, logn, log1p, root,
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sqrt, clamp, inrange, sgn, erf, erfc, frac,
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trunc
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(3) Trigonometry: sin, cos, tan, acos, asin, atan, atan2, cosh,
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cot, csc, sec, sinh, tanh, rad2deg, deg2rad,
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deg2grad, grad2deg, hypot
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(4) Equalities &
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Inequalities: =, ==, <>, !=, <, <=, >, >=
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(5) Boolean logic: and, or, xor, not, nand, nor, shr, shl, true,
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false
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(6) Conditional &
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Loop statement: if-then-else, while
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(7) Assignment: :=
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(8) String
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processing: in, like, ilike
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(9) Calculus: numerical integration and differentiation
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[EXAMPLE EXPRESSIONS]
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The following is a short sample of the types of mathematical
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expressions that can be parsed and evaluated using the ExprTk library.
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(01) sqrt(1 - (x^2))
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(02) clamp(-1,sin(2 * pi * x) + cos(y / 2 * pi),+1)
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(03) sin(2 * x)
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(04) if(((x + 2) == 3) and ((y + 5) <= 9),1 + w, 2 / z)
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(05) inrange(-2,m,+2) == if(({-2 <= m} and [m <= +2]),1,0)
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(06) ({1/1}*[1/2]+(1/3))-{1/4}^[1/5]+(1/6)-({1/7}+[1/8]*(1/9))
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(07) a * exp(2 * t) + c
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(08) z := x + sin(2 * pi / y)
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(09) u := 2 * (pi * z) / (w := x + cos(y / pi))
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(10) 2x + 3y + 4z + 5w == 2 * x + 3 * y + 4 * z + 5 * w
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(11) 3(x + y) / 2 + 1 == 3 * (x + y) / 2 + 1
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(12) (x + y)3 + 1 / 4 == (x + y) * 3 + 1 / 4
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(13) (x + y)z + 1 / 2 == (x + y) * z + 1 / 2
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(14) (sin(x/pi)cos(2y) + 1)==(sin(x / pi) * cos(2 * y) + 1)
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(15) 25x^5 - 35x^4 - 15x^3 + 40x^2 - 15x + 1
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(16) if (avg(x,y) <= x + y, x - y, x * y) + 2 * pi / x
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(17) fib_i := fib_i + (x := y + 0 * (fib_i := x + (y := fib_i)))
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(18) while (x <= 100) { x := x + 1 }
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(19) x <= 'abc123' and (y in 'AString') or ('1x2y3z' != z)
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(20) (x like '*123*') or ('a123b' ilike y)
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[COPYRIGHT NOTICE]
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Free use of the Mathematical Expression Toolkit Library is permitted
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under the guidelines and in accordance with the most current version
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of the Common Public License.
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http://www.opensource.org/licenses/cpl1.0.php
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[DOWNLOADS & UPDATES]
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All updates and the most recent version of the C++ Mathematical
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Expression Library can be found at:
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(1) http://www.partow.net/programming/exprtk/index.html
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(2) svn checkout http://exprtk.googlecode.com/svn/ exprtk
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[INSTALLATION]
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(1) exprtk.hpp should be placed in a project or system include path
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(e.g: /usr/include/).
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[COMPILATION]
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(1) For a complete build: make clean all
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(2) For a PGO build: make clean pgo
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(3) To strip executables: make strip_bin
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[COMPILER COMPATIBILITY]
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(*) GNU Compiler Collection (4.3+)
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(*) Intel<65> C++ Compiler (9.x+)
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(*) Clang/LLVM (1.1+)
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(*) PGI C++ (10.x+)
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(*) Microsoft Visual Studio C++ Compiler (8.1+)
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(*) Comeau C++ Compiler (4.3+)
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[BUILT-IN OPERATIONS & FUNCTIONS]
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(0) Basic Operators
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+-----------+--------------------------------------------------------+
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| OPERATOR | DEFINITION |
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+-----------+--------------------------------------------------------+
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| + | Addition between x and y. (eg: x + y) |
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+-----------+--------------------------------------------------------+
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| - | Subtraction between x and y. (eg: x - y) |
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+-----------+--------------------------------------------------------+
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| * | Multiplication between x and y. (eg: x * y) |
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+-----------+--------------------------------------------------------+
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| / | Division between x and y (eg: x / y) |
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+-----------+--------------------------------------------------------+
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| % | Modulus of x with respect to y. (eg: x % y) |
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+-----------+--------------------------------------------------------+
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| ^ | x to the power of y. (eg: x ^ y) |
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+-----------+--------------------------------------------------------+
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| := | Assign the value of x to y. (eg: y := x) |
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| | where y is a variable type. |
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+-----------+--------------------------------------------------------+
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(1) Equalities & Inequalities
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+-----------+--------------------------------------------------------+
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| OPERATOR | DEFINITION |
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+-----------+--------------------------------------------------------+
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| == or = | True only if x is strictly equal to y. (eg: x == y) |
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+-----------+--------------------------------------------------------+
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| <> or != | True only if x does not equal y (eg: x <> y or x != y) |
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+-----------+--------------------------------------------------------+
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| < | True only if x less than y. (eg: x < y) |
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+-----------+--------------------------------------------------------+
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| <= | True only if x less than or equal to y. (eg: x <= y) |
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+-----------+--------------------------------------------------------+
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| > | True only if x greater than y. (eg: x > y) |
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+-----------+--------------------------------------------------------+
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| >= | True only if x greater than or equal to y (eg: x >= y) |
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+-----------+--------------------------------------------------------+
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(2) Boolean Operations
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+-----------+--------------------------------------------------------+
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| OPERATOR | DEFINITION |
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+-----------+--------------------------------------------------------+
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| true | True state or any value other than zero (typically 1). |
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+-----------+--------------------------------------------------------+
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| false | False state, value of zero. |
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+-----------|--------------------------------------------------------+
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| and | Logical AND, True only if x and y are both true. |
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| | (eg: x and y) |
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+-----------+--------------------------------------------------------+
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| nand | Logical NAND, True only if either x or y is false. |
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| | (eg: x nand y) |
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+-----------+--------------------------------------------------------+
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| nor | Logical NOR, True only if the result of x or y is false|
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| | (eg: x nor y) |
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+-----------+--------------------------------------------------------+
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| not | Logical NOT, Negate the logical sense of the input. |
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| | (eg: not(x and y) == x nand y) |
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+-----------+--------------------------------------------------------+
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| or | Logical OR, True if either x or y is true. (eg: x or y)|
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+-----------+--------------------------------------------------------+
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| xor | Local XOR, True only if the logical states of x and y |
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| | differ. (eg: x xor y) |
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+-----------+--------------------------------------------------------+
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| if | If x is true then return y else return z. |
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| | (eg: if(x, y, z) or if((x + 1) > 2y, z + 1, w / v)) |
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+-----------+--------------------------------------------------------+
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(3) General Purpose Functions
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+-----------+--------------------------------------------------------+
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| FUNCTION | DEFINITION |
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+-----------+--------------------------------------------------------+
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| abs | Absolute value of x. |
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+-----------+--------------------------------------------------------+
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| avg | The average of all the inputs. |
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| | (eg: avg(x,y,z,w) == (x+y+z+w)/4) |
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+-----------+--------------------------------------------------------+
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| ceil | Smallest integer that is greater than or equal to x. |
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+-----------+--------------------------------------------------------+
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| clamp | Clamp x in range between r0 and r1, where r0 < r1. |
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| | (eg: clamp(r0,x,r1) |
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+-----------+--------------------------------------------------------+
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| equal | Equality test between x and y using normalized epsilon |
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+-----------+--------------------------------------------------------+
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| erf | Error function of x |
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+-----------+--------------------------------------------------------+
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| erfc | Complimentary error function of x |
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+-----------+--------------------------------------------------------+
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| exp | e to the power of x |
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+-----------+--------------------------------------------------------+
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| floor | Largest integer that is less than or equal to x. |
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+-----------+--------------------------------------------------------+
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| frac | Fractional portion of x |
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+-----------+--------------------------------------------------------+
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| hypot | Hypotenuse of x and y (eg: hypot(x,y)) |
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+-----------+--------------------------------------------------------+
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| log | Natural logarithm of x |
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+-----------+--------------------------------------------------------+
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| log10 | Base 10 logarithm of x |
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+-----------+--------------------------------------------------------+
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| logn | Base N logarithm of x (eg: logn(1235,8)) |
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| | where n > 0 and is an integer. |
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+-----------+--------------------------------------------------------+
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| log1p | Natural logarithm of 1 + x (eg: log1p(x)) |
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| | where x is very small. |
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+-----------+--------------------------------------------------------+
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| nequal | Not-equal test between x and y using normalized epsilon|
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+-----------+--------------------------------------------------------+
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| root | Nth-Root of x (eg: root(x,3)) |
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| | where n > 0 and is an integer. |
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+-----------+--------------------------------------------------------+
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| round | Round x to the nearest integer. |
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+-----------+--------------------------------------------------------+
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| roundn | Round x to the n decimal places (eg: roundn(x,4)) |
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| | where n > 0 and is an integer. |
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+-----------+--------------------------------------------------------+
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| sgn | Sign of x, -1 where x < 0, +1 where x > 0, else zero. |
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+-----------+--------------------------------------------------------+
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| sqrt | Square root of x, where x > 0 |
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+-----------+--------------------------------------------------------+
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| sum | The sum of all the inputs. |
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| | (eg: sum(x,y,z,w,v) == (x+y+z+w+v)) |
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+-----------+--------------------------------------------------------+
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| trunc | Integer portion of x |
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+-----------+--------------------------------------------------------+
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(4) Trigonometry Functions
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+-----------+--------------------------------------------------------+
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| FUNCTION | DEFINITION |
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+-----------+--------------------------------------------------------+
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| acos | Arc cosine of x expressed in radians. Interval [-1,+1] |
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+-----------+--------------------------------------------------------+
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| asin | Arc sine of x expressed in radians. Interval [-1,+1] |
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+-----------+--------------------------------------------------------+
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| atan | Arc tangent of x expressed in radians. Interval [-1,+1]|
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+-----------+--------------------------------------------------------+
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| atan2 | Arc tangent of x expressed in radians. Interval [-1,+1]|
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+-----------+--------------------------------------------------------+
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| cos | Cosine of x |
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+-----------+--------------------------------------------------------+
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| cosh | Hyperbolic cosine of x |
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+-----------+--------------------------------------------------------+
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| cot | Cotangent of x |
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+-----------+--------------------------------------------------------+
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| csc | Cosecant of x |
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+-----------+--------------------------------------------------------+
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| sec | Secant of x |
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+-----------+--------------------------------------------------------+
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| sin | Sine of x |
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+-----------+--------------------------------------------------------+
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| sinh | Hyperbolic sine of x |
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+-----------+--------------------------------------------------------+
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| tan | Tangent of x |
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+-----------+--------------------------------------------------------+
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| tanh | Hyperbolic tangent of x |
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+-----------+--------------------------------------------------------+
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| rad2deg | Convert x from radians to degrees |
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+-----------+--------------------------------------------------------+
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| deg2rad | Convert x from degrees to radians |
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+-----------+--------------------------------------------------------+
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| deg2grad | Convert x from degrees to gradians |
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+-----------+--------------------------------------------------------+
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| grad2deg | Convert x from gradians to degrees |
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+-----------+--------------------------------------------------------+
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(5) String Processing
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+-----------+--------------------------------------------------------+
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| FUNCTION | DEFINITION |
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+-----------+--------------------------------------------------------+
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| in | True only if x is a substring of y |
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| | (eg: x in y or 'abc' in 'abcdefgh') |
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+-----------+--------------------------------------------------------+
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| like | True only if the string x matches the pattern y. |
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| | Available wildcard characters are '*' and '?' denoting |
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| | zero or more and zero or one matches respectively. |
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| | (eg: x like y or 'abcdefgh' like 'a?d*') |
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+-----------+--------------------------------------------------------+
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| like | True only if the string x matches the pattern y in a |
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| | case insensitive manner. Available wildcard characters |
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| | are '*' and '?' denoting zero or more and zero or one |
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| | matches respectively. |
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| | (eg: x ilike y or 'abcdefgh' like 'a?d*') |
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+-----------+--------------------------------------------------------+
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[SPECIAL FUNCTIONS]
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The purpose of special functions in ExprTk is to provide compiler
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generated equivalents of common mathematical expressions which can be
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invoked by using the 'special function' syntax (eg: $f12(x,y,z) or
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$f24(x,y,z,w)). Where possible, for sub-expressions that are comprised
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of combinations of variables and literal values the ExprTk compiler
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will perform a best effort attempt to replace such expressions with
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special functions.
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Special functions dramatically decrease the total evaluation time of
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expressions which would otherwise have been written using the common
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form by reducing the total number of nodes in the evaluation tree of
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an expression and by also leveraging the compiler's ability to
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correctly optimize such expressions for a given architecture.
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3-Parameter 4-Parameter
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+-------------+-------------+ +--------------+------------------+
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| Prototype | Operation | | Prototype | Operation |
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+-------------+-------------+ +--------------+------------------+
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$f00(x,y,z) | (x + y) / z $f46(x,y,z,w) | x + ((y + z) / w)
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$f01(x,y,z) | (x + y) * z $f47(x,y,z,w) | x + ((y + z) * w)
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$f02(x,y,z) | (x + y) - z $f48(x,y,z,w) | x + ((y - z) / w)
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$f03(x,y,z) | (x + y) + z $f49(x,y,z,w) | x + ((y - z) * w)
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$f04(x,y,z) | (x - y) / z $f50(x,y,z,w) | x + ((y * z) / w)
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$f05(x,y,z) | (x - y) * z $f51(x,y,z,w) | x + ((y * z) * w)
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$f06(x,y,z) | (x * y) + z $f52(x,y,z,w) | x + ((y / z) + w)
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$f07(x,y,z) | (x * y) - z $f53(x,y,z,w) | x + ((y / z) / w)
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$f08(x,y,z) | (x * y) / z $f54(x,y,z,w) | x + ((y / z) * w)
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$f09(x,y,z) | (x * y) * z $f55(x,y,z,w) | x - ((y + z) / w)
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$f10(x,y,z) | (x / y) + z $f56(x,y,z,w) | x - ((y + z) * w)
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$f11(x,y,z) | (x / y) - z $f57(x,y,z,w) | x - ((y - z) / w)
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$f12(x,y,z) | (x / y) / z $f58(x,y,z,w) | x - ((y - z) * w)
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$f13(x,y,z) | (x / y) * z $f59(x,y,z,w) | x - ((y * z) / w)
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$f14(x,y,z) | x / (y + z) $f60(x,y,z,w) | x - ((y * z) * w)
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$f15(x,y,z) | x / (y - z) $f61(x,y,z,w) | x - ((y / z) / w)
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$f16(x,y,z) | x / (y * z) $f62(x,y,z,w) | x - ((y / z) * w)
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$f17(x,y,z) | x / (y / z) $f63(x,y,z,w) | ((x + y) * z) - w
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$f18(x,y,z) | x * (y + z) $f64(x,y,z,w) | ((x - y) * z) - w
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$f19(x,y,z) | x * (y - z) $f65(x,y,z,w) | ((x * y) * z) - w
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$f20(x,y,z) | x * (y * z) $f66(x,y,z,w) | ((x / y) * z) - w
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$f21(x,y,z) | x * (y / z) $f67(x,y,z,w) | ((x + y) / z) - w
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$f22(x,y,z) | x - (y / z) $f68(x,y,z,w) | ((x - y) / z) - w
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$f23(x,y,z) | x - (y / z) $f69(x,y,z,w) | ((x * y) / z) - w
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$f24(x,y,z) | x - (y * z) $f70(x,y,z,w) | ((x / y) / z) - w
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$f25(x,y,z) | x + (y * z) $f71(x,y,z,w) | (x * y) + (z * w)
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$f26(x,y,z) | x + (y / z) $f72(x,y,z,w) | (x * y) - (z * w)
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$f27(x,y,z) | x + (y + z) $f73(x,y,z,w) | (x * y) + (z / w)
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$f28(x,y,z) | x + (y - z) $f74(x,y,z,w) | (x * y) - (z / w)
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$f29(x,y,z) | x * y^2 + z $f75(x,y,z,w) | (x / y) + (z / w)
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$f30(x,y,z) | x * y^3 + z $f76(x,y,z,w) | (x / y) - (z / w)
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$f31(x,y,z) | x * y^4 + z $f77(x,y,z,w) | (x / y) - (z * w)
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$f32(x,y,z) | x * y^5 + z $f78(x,y,z,w) | x / (y + (z * w))
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$f33(x,y,z) | x * y^6 + z $f79(x,y,z,w) | x / (y - (z * w))
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$f34(x,y,z) | x * y^7 + z $f80(x,y,z,w) | x * (y + (z * w))
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$f35(x,y,z) | x * y^8 + z $f81(x,y,z,w) | x * (y - (z * w))
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$f36(x,y,z) | x * y^9 + z $f82(x,y,z,w) | x * y^2 + z * w^2
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$f37(x,y,z) | x * log(y)+z $f83(x,y,z,w) | x * y^3 + z * w^3
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$f38(x,y,z) | x * log(y)-z $f84(x,y,z,w) | x * y^4 + z * w^4
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$f39(x,y,z) | x * log10(y)+z $f85(x,y,z,w) | x * y^5 + z * w^5
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$f40(x,y,z) | x * log10(y)-z $f86(x,y,z,w) | x * y^6 + z * w^6
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$f41(x,y,z) | x * sin(y)+z $f87(x,y,z,w) | x * y^7 + z * w^7
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$f42(x,y,z) | x * sin(y)-z $f88(x,y,z,w) | x * y^8 + z * w^8
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$f43(x,y,z) | x * cos(y)+z $f89(x,y,z,w) | x * y^9 + z * w^9
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$f44(x,y,z) | x * cos(y)-z $f90(x,y,z,w) | (x and y) ? z : w
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$f45(x,y,z) | x ? y : z $f91(x,y,z,w) | (x or y) ? z : w
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$f92(x,y,z,w) | (x < y) ? z : w
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$f93(x,y,z,w) | (x <= y) ? z : w
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$f94(x,y,z,w) | (x > y) ? z : w
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$f95(x,y,z,w) | (x >= y) ? z : w
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$f96(x,y,z,w) | (x == y) ? z : w
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$f97(x,y,z,w) | x*sin(y) + z*cos(w)
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|
||
|
||
|
||
[EXPRTK NOTES]
|
||
(0) Supported types are float, double and long double.
|
||
|
||
(1) Standard mathematical operator precedence is applied (BEDMAS).
|
||
|
||
(2) All variables and functions are case-insensitive
|
||
|
||
(3) Expression lengths are limited only by storage capacity.
|
||
|
||
(4) Equal/Nequal routines use epsilons of 0.00000000001 and 0.000001
|
||
for double and float types respectively.
|
||
|
||
(5) All trigonometric functions assume radian input unless
|
||
stated otherwise.
|
||
|
||
(6) Expressions can contain white-space characters such as
|
||
space, tabs, new-lines, control-feed et al.
|
||
('\n', '\r', '\t', '\b', '\v', '\f')
|
||
|
||
(7) User defined functions can have up to 20 parameters.
|
||
|
||
(8) Polynomial functions can be at most of degree 10.
|
||
|
||
|
||
|
||
[SIMPLE EXPRTK EXAMPLE]
|
||
--- snip ---
|
||
#include <cstdio>
|
||
#include <string>
|
||
|
||
#include "exprtk.hpp"
|
||
|
||
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 [sin(x/pi)^3 + cos(pi/y)^4]";
|
||
|
||
double x = 1.1;
|
||
double y = 2.2;
|
||
double z = 3.3;
|
||
|
||
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);
|
||
|
||
expression_t expression;
|
||
expression.register_symbol_table(symbol_table);
|
||
|
||
parser_t parser;
|
||
|
||
if (!parser.compile(expression_str,expression))
|
||
{
|
||
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)
|
||
{
|
||
error_t error = parser.get_error(i);
|
||
printf("Err: %02d Pos: %02d Type: [%s] Msg: %s Expr: %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;
|
||
}
|
||
|
||
double result = expression.value();
|
||
|
||
printf("Result: %10.5f\n",result);
|
||
|
||
return 0;
|
||
}
|
||
--- snip ---
|
||
|
||
|
||
|
||
[FILES]
|
||
(00) Makefile
|
||
(01) readme.txt
|
||
(02) exprtk.hpp
|
||
(03) exprtk_test.cpp
|
||
(04) exprtk_benchmark.cpp
|