MathNot
,
MathAnd
,
MathOr
,
BitAnd, BitOr, BitXor
,
Equals
,
GreaterThan, LessThan
,
Math...
,
Fast...
,
ShiftLeft, ShiftRight
,
IsPromptShown
,
MathLibrary
,
GetTime
.
Built-in (core) functions
Yacas comes with a small core of built-in functions and a large library of
user-defined functions. Some of these core functions are documented in this
chapter.
It is important for a developer to know which functions are built-in and cannot
be redefined or Retract-ed. Also, core functions may be somewhat faster to
execute than functions defined in the script library. All core functions are
listed in the file corefunctions.h in the src/ subdirectory of the Yacas
source tree. The declarations typically look like this:
SetCommand(LispSubtract, "MathSubtract");
|
Here LispSubtract is the Yacas internal name for the function and MathSubtract is the name visible to the Yacas language.
Built-in bodied functions and infix operators are declared in the same file.
MathNot -- built-in logical "not"
Internal function
Calling format:
Description:
Returns "False" if "expression" evaluates
to "True", and vice versa.
MathAnd -- built-in logical "and"
Calling format:
Description:
Lazy logical And: returns True if all args evaluate to
True, and does this by looking at first, and then at the
second argument, until one is False.
If one of the arguments is False, And immediately returns False without
evaluating the rest. This is faster, but also means that none of the
arguments should cause side effects when they are evaluated.
MathOr -- built-in logical "or"
Internal function
Calling format:
MathOr is the basic logical "or" function. Similarly to And, it is
lazy-evaluated. And(...) and Or(...) do also exist, defined in the script
library. You can redefine them as infix operators yourself, so you have the
choice of precedence. In the standard scripts they are in fact declared as
infix operators, so you can write expr1 And expr.
BitAnd, BitOr, BitXor -- bitwise arithmetic
Internal function
Calling format:
BitAnd(n,m)
BitOr(n,m)
BitXor(n,m)
|
Description:
These functions return bitwise "and", "or" and "xor"
of two numbers.
Equals -- check equality
Internal function
Calling format:
Description:
Compares evaluated a and b recursively
(stepping into expressions). So "Equals(a,b)" returns
"True" if the expressions would be printed exactly
the same, and "False" otherwise.
GreaterThan, LessThan -- comparison predicates
Internal function
Calling format:
GreaterThan(a,b)
LessThan(a,b)
|
Parameters:
a, b -- numbers or strings
Description:
Comparing numbers or strings (lexicographically).
Example:
In> LessThan(1,1)
Out> False;
In> LessThan("a","b")
Out> True;
|
Math... -- arbitrary-precision math functions
Internal function
Calling format:
MathGcd(n,m) (Greatest Common Divisor)
MathAdd(x,y) (add two numbers)
MathSubtract(x,y) (subtract two numbers)
MathMultiply(x,y) (multiply two numbers)
MathDivide(x,y) (divide two numbers)
MathSqrt(x) (square root, must be x>=0)
MathFloor(x) (largest integer not larger than x)
MathCeil(x) (smallest integer not smaller than x)
MathAbs(x) (absolute value of x, or |x| )
MathExp(x) (exponential, base 2.718...)
MathLog(x) (natural logarithm, for x>0)
MathPower(x,y) (power, x ^ y)
MathSin(x) (sine)
MathCos(x) (cosine)
MathTan(x) (tangent)
MathSinh(x) (hyperbolic sine)
MathCosh(x) (hyperbolic cosine)
MathTanh(x) (hyperbolic tangent)
MathArcSin(x) (inverse sine)
MathArcCos(x) (inverse cosine)
MathArcTan(x) (inverse tangent)
MathArcSinh(x) (inverse hyperbolic sine)
MathArcCosh(x) (inverse hyperbolic cosine)
MathArcTanh(x) (inverse hyperbolic tangent)
MathDiv(x,y) (integer division, result is an integer)
MathMod(x,y) (remainder of division, or x mod y)
|
Description:
These commands perform the calculation of elementary mathematical functions.
The arguments must be numbers.
The reason for the prefix Math is that
the library needs to define equivalent
non-numerical functions for symbolic computations, such as Exp, Sin and so on.
Note that all functions, such as the MathPower, MathSqrt, MathAdd etc., accept integers as well as floating-point numbers.
The resulting values may be integers or floats.
If the mathematical result is an exact integer, then the integer is returned.
For example, MathSqrt(25) returns the integer 5, and MathPower(2,3) returns the integer 8.
In such cases, the integer result is returned even if the calculation requires more digits than set by Precision.
However, when the result is mathematically not an integer, the functions return a floating-point result which is correct only to the current precision.
Example:
In> Precision(10)
Out> True
In> Sqrt(10)
Out> Sqrt(10)
In> MathSqrt(10)
Out> 3.16227766
In> MathSqrt(490000*2^150)
Out> 26445252304070013196697600
In> MathSqrt(490000*2^150+1)
Out> 0.264452523e26
In> MathPower(2,3)
Out> 8
In> MathPower(2,-3)
Out> 0.125
|
Fast... -- double-precision math functions
Internal function
Calling format:
FastExp(x), FastLog(x) (natural logarithm),
FastPower(x,y),
FastSin(x), FastCos(x), FastTan(x),
FastArcSin(x), FastArcCos(x), FastArcTan(x)
Description:
Versions of these functions using the C++ library. These
should then at least be faster than the arbitrary precision versions.
ShiftLeft, ShiftRight -- built-in bit shifts
Internal function
Calling format:
ShiftLeft(expr,bits)
ShiftRight(expr,bits)
|
Description:
Shift bits to the left or to the right.
IsPromptShown -- test for the Yacas prompt option
Internal function
Calling format:
Description:
Returns False if Yacas has been started with the option to suppress the prompt, and True otherwise.
MathLibrary -- obtain current math library name
Internal function
Calling format:
Description:
Returns a string that describes the currently used arbitrary-precision arithmetic library.
Possible names supported at the moment are "Internal" and "Gmp", indicating the internal math library libyacasnumbers and the GNU Multiple Precision library libgmp.
Example:
In> MathLibrary()
Out> "Internal";
|
GetTime -- measure the time taken by an evaluation
Internal function
Calling format:
Parameters:
expr -- any expression
Description:
The function GetTime(expr) evaluates the expression expr and returns the time needed for the evaluation.
The result is returned as a floating-point number of seconds.
The value of the expression expr is lost.
The result is the "user time" as reported by the OS, not the real ("wall clock") time.
Therefore, any CPU-intensive processes running alongside Yacas will not significantly affect the result of GetTime.
Example:
In> GetTime(Simplify((a*b)/(b*a)))
Out> 0.09;
|
See also:
Time
.
Full listing of core functions
The following Yacas functions are currently declared in corefunctions.h
as core functions.
The list indicates whether a function is a real function (evaluating its arguments) or a macro (not evaluating its arguments).
Also a function can either take a fixed number of arguments or a variable number (say, 2 or more).
Some kernel functions are additionally declared as operators (bodied, prefix, infix, postfix) with the given precedence.
- = -- function, 2 arguments
- @ -- prefix operator, prec. 0
- And -- macro, 1 or more arguments
- ApplyPure -- function, 2 arguments
- ArrayCreate -- function, 2 arguments
- ArrayGet -- function, 2 arguments
- ArraySet -- function, 3 arguments
- ArraySize -- function, 1 argument
- Atom -- function, 1 argument
- BackQuote -- bodied operator, prec. KMaxPrecedence
- BitAnd -- function, 2 arguments
- BitOr -- function, 2 arguments
- BitXor -- function, 2 arguments
- BitsToDigits -- function, 2 arguments
- Bodied -- function, 2 arguments
- CTokenizer -- function, 0 arguments
- CharString -- function, 1 argument
- Check -- macro, 2 arguments
- Clear -- macro, 1 or more arguments
- CommonLispTokenizer -- function, 0 arguments
- ConcatStrings -- function, 1 or more arguments
- Concat -- function, 1 or more arguments
- CurrentFile -- function, 0 arguments
- CurrentLine -- function, 0 arguments
- CustomEval'Expression -- function, 0 arguments
- CustomEval'Locals -- function, 0 arguments
- CustomEval'Result -- function, 0 arguments
- CustomEval'Stop -- function, 0 arguments
- CustomEval -- macro, 4 arguments
- DebugFile -- function, 1 argument
- DebugLine -- function, 1 argument
- DefLoadFunction -- function, 1 argument
- DefLoad -- function, 1 argument
- DefMacroRuleBaseListed -- macro, 2 arguments
- DefMacroRuleBase -- macro, 2 arguments
- DefaultDirectory -- function, 1 argument
- DefaultTokenizer -- function, 0 arguments
- Delete -- function, 2 arguments
- DestructiveDelete -- function, 2 arguments
- DestructiveInsert -- function, 3 arguments
- DestructiveReplace -- function, 3 arguments
- DestructiveReverse -- function, 1 argument
- DigitsToBits -- function, 2 arguments
- DllDirectory -- function, 1 argument
- DllEnumerate -- function, 0 arguments
- DllLoad -- function, 1 argument
- DllUnload -- function, 1 argument
- Equals -- function, 2 arguments
- Eval -- function, 1 argument
- Exit -- function, 0 arguments
- FastAbs -- function, 1 argument
- FastArcCos -- function, 1 argument
- FastArcSin -- function, 1 argument
- FastArcTan -- function, 1 argument
- FastAssoc -- function, 2 arguments
- FastCeil -- function, 1 argument
- FastCos -- function, 1 argument
- FastExp -- function, 1 argument
- FastFloor -- function, 1 argument
- FastIsPrime -- function, 1 argument
- FastLog -- function, 1 argument
- FastMod -- function, 2 arguments
- FastPi -- function, 0 arguments
- FastPower -- function, 2 arguments
- FastSin -- function, 1 argument
- FastSqrt -- function, 1 argument
- FastTan -- function, 1 argument
- FindFile -- function, 1 argument
- FindFunction -- function, 1 argument
- FlatCopy -- function, 1 argument
- FromBase -- function, 2 arguments
- FromFile -- bodied operator, prec. KMaxPrecedence
- FromFile -- macro, 2 arguments
- FromString -- bodied operator, prec. KMaxPrecedence
- FromString -- macro, 2 arguments
- FullForm -- function, 1 argument
- GarbageCollect -- function, 0 arguments
- GenericTypeName -- function, 1 argument
- GetCoreError -- function, 0 arguments
- GetExtraInfo -- function, 1 argument
- GetPrecision -- function, 0 arguments
- GetPrettyPrinter -- function, 0 arguments
- GetTime -- macro, 1 argument
- GreaterThan -- function, 2 arguments
- Head -- function, 1 argument
- HistorySize -- function, 1 argument
- HoldArg -- macro, 2 arguments
- Hold -- macro, 1 argument
- If -- macro, 2 or more arguments
- InDebugMode -- function, 0 arguments
- Infix -- function, 2 arguments
- Insert -- function, 3 arguments
- IsAtom -- function, 1 argument
- IsBodied -- function, 1 argument
- IsBound -- macro, 1 argument
- IsExitRequested -- function, 0 arguments
- IsFunction -- function, 1 argument
- IsGeneric -- function, 1 argument
- IsInfix -- function, 1 argument
- IsInteger -- function, 1 argument
- IsList -- function, 1 argument
- IsNumber -- function, 1 argument
- IsPostfix -- function, 1 argument
- IsPrefix -- function, 1 argument
- IsPromptShown -- function, 0 arguments
- IsString -- function, 1 argument
- LazyGlobal -- macro, 1 argument
- LeftPrecedence -- function, 2 arguments
- Length -- function, 1 argument
- LessThan -- function, 2 arguments
- LispReadListed -- function, 0 arguments
- LispRead -- function, 0 arguments
- Listify -- function, 1 argument
- List -- macro, 1 or more arguments
- Load -- function, 1 argument
- LocalSymbols -- bodied operator, prec. KMaxPrecedence
- LocalSymbols -- macro, 1 or more arguments
- Local -- macro, 1 or more arguments
- MacroClear -- function, 1 or more arguments
- MacroLocal -- function, 1 or more arguments
- MacroRuleBaseListed -- function, 2 arguments
- MacroRuleBase -- function, 2 arguments
- MacroRulePattern -- bodied operator, prec. KMaxPrecedence
- MacroRulePattern -- function, 5 arguments
- MacroRule -- bodied operator, prec. KMaxPrecedence
- MacroRule -- function, 5 arguments
- MacroSet -- macro, 2 arguments
- MathAbs -- function, 1 argument
- MathAdd -- function, 2 arguments
- MathAnd -- macro, 1 or more arguments
- MathArcCos -- function, 1 argument
- MathArcTan -- function, 1 argument
- MathBitCount -- function, 1 argument
- MathCeil -- function, 1 argument
- MathDebugInfo -- function, 1 argument
- MathDivide -- function, 2 arguments
- MathDiv -- function, 2 arguments
- MathFac -- function, 1 argument
- MathFloor -- function, 1 argument
- MathGcd -- function, 2 arguments
- MathGetExactBits -- function, 1 argument
- MathIsSmall -- function, 1 argument
- MathLibrary -- function, 0 arguments
- MathMod -- function, 2 arguments
- MathMultiply -- function, 2 arguments
- MathNegate -- function, 1 argument
- MathNot -- function, 1 argument
- MathNth -- function, 2 arguments
- MathOr -- macro, 1 or more arguments
- MathPi -- function, 0 arguments
- MathSetExactBits -- function, 2 arguments
- MathSign -- function, 1 argument
- MathSubtract -- function, 2 arguments
- MaxEvalDepth -- function, 1 argument
- Not -- function, 1 argument
- OSVersion -- function, 0 arguments
- OpLeftPrecedence -- function, 1 argument
- OpPrecedence -- function, 1 argument
- OpRightPrecedence -- function, 1 argument
- Or -- macro, 1 or more arguments
- PatchLoad -- function, 1 argument
- PatchString -- function, 1 argument
- PatternCreate -- function, 2 arguments
- PatternMatches -- function, 2 arguments
- Postfix -- function, 2 arguments
- Precision -- function, 1 argument
- Prefix -- function, 2 arguments
- PrettyPrinter -- function, 1 or more arguments
- Prog -- macro, 1 or more arguments
- ReadCmdLineString -- function, 1 argument
- ReadToken -- function, 0 arguments
- Read -- function, 0 arguments
- Replace -- function, 3 arguments
- Retract -- function, 2 arguments
- RightAssociative -- function, 1 argument
- RightPrecedence -- function, 2 arguments
- RuleBaseArgList -- function, 2 arguments
- RuleBaseDefined -- function, 2 arguments
- RuleBaseListed -- macro, 2 arguments
- RuleBase -- macro, 2 arguments
- RulePattern -- bodied operator, prec. KMaxPrecedence
- RulePattern -- macro, 5 arguments
- Rule -- bodied operator, prec. KMaxPrecedence
- Rule -- macro, 5 arguments
- Secure -- macro, 1 argument
- SetExtraInfo -- function, 2 arguments
- SetStringMid -- function, 3 arguments
- Set -- macro, 2 arguments
- ShiftLeft -- function, 2 arguments
- ShiftRight -- function, 2 arguments
- StaSiz -- function, 0 arguments
- StringMid -- function, 3 arguments
- String -- function, 1 argument
- Subst -- bodied operator, prec. KMaxPrecedence
- Subst -- function, 3 arguments
- SystemCall -- function, 1 argument
- Tail -- function, 1 argument
- ToBase -- function, 2 arguments
- ToFile -- bodied operator, prec. KMaxPrecedence
- ToFile -- macro, 2 arguments
- ToStdout -- bodied operator, prec. KMaxPrecedence
- ToStdout -- macro, 1 argument
- ToString -- bodied operator, prec. KMaxPrecedence
- ToString -- macro, 1 argument
- TraceRule -- bodied operator, prec. KMaxPrecedence
- TraceRule -- macro, 2 arguments
- TraceStack -- macro, 1 argument
- TrapError -- macro, 2 arguments
- Type -- function, 1 argument
- UnFence -- function, 2 arguments
- UnList -- function, 1 argument
- Use -- function, 1 argument
- Version -- function, 0 arguments
- While -- bodied operator, prec. KMaxPrecedence
- While -- macro, 2 arguments
- WriteString -- function, 1 argument
- Write -- function, 1 or more arguments
- XmlExplodeTag -- function, 1 argument
- XmlTokenizer -- function, 0 arguments
- _ -- infix operator, prec. 0
- _ -- prefix operator, prec. 0
- ` -- macro, 1 argument
- ` -- prefix operator, prec. 0