StdReal¶
This module can be imported directly or as a part of the StdEnv module.
It provides definition for many critical operations and functions on real numbers.
Visit StdReal on Cloogle for source code of this module.
Units¶
The real number units follow their definitions in mathematics.
Zero¶
Implementation
// CLEAN
zero :: Real
zero :== 0.0
Definition
Represents the additive identity of real numbers.
One¶
Implementation
// CLEAN
one :: Real
one :== 1.0
Definition
Represents the multiplicative identity of real numbers.
Infinity¶
Implementation
// CLEAN
Infinity :: Real
Infinity :== 1E9999
Definition
Represents numerical values which are too large for CLEAN represent.
Not-a-number¶
Implementation
// CLEAN
NaN :: Real
NaN :== 1E9999 + (-1E9999)
Definition
Represents numerical values which cannot be represented by CLEAN.
See NaN for additional information.
Math operations¶
The definition and behavior of these operators follow their mathematic counterpart.
It is worth noting that, natively, these operations only work with real number operands, but the programmer can define additional behaviors and functionalities.
Addition¶
Signature
// CLEAN
(+) infixl 6 :: Real Real -> Real
(+) a b = ...
Behavior
Adds a and b.
Usage
// CLEAN
2.5 + 1.5 // 4.0
2.5 + (-1.5) // 1.5
(-2.5) + 1.5 // -1.0
(-2.5) + (-1.5) // -4.0
Subtraction¶
Signature
// CLEAN
(-) infixl 6 :: Real Real -> Real
(-) a b = ...
Behavior
Subtracts b from a.
Usage
// CLEAN
2.5 - 1.5 // 1.0
2.5 - (-1.5) // 4.0
(-2.5) - 1.5 // -1.5
(-2.5) - (-1.5) // -1.0
Multiplication¶
Signature
// CLEAN
(*) infixl 7 :: Real Real -> Real
(*) a b = ...
Behavior
Multiplies a with b.
Usage
// CLEAN
2.5 * 1.5 // 3.75
2.5 * (-1.5) // -3.75
(-2.5) * 1.5 // -3.75
(-2.5) * (-1.5) // 3.75
Division¶
Signature
// CLEAN
(/) infixl 7 :: Real Real -> Real
(/) a b = ...
Behavior
Divides a with b.
Usage
// CLEAN
2.5 / 1.5 // 1.66666666666667
2.5 / (-1.5) // -1.66666666666667
(-2.5) / 1.5 // -1.66666666666667
(-2.5) / (-1.5) // 1.66666666666667
1.0 / 0.0 // #INF
(-1.0) / 0.0 // -#INF
0.0 / 0.0 // #NaN
Exponentiation¶
Signature
// CLEAN
(^) infixr 8 :: Real Real -> Real
(^) a b = ...
Behavior
Raises a to the power of b.
Usage
// CLEAN
2.5 ^ 1.5 // 3.95284707521047
2.5 ^ (-1.5) // 0.25298221281347
(-2.5) ^ 1.5 // #NAN
(-2.5) ^ (-1.5) // #NAN
0.0 ^ 0.0 // 1.0
0.0 ^ 1.0 // 0.0
0.0 ^ (-1.0) // #INF
Math functions¶
This function is a unary operator, but there is no such thing as a unary operator in CLEAN, so to keep things tidy, I have decided to place it here.
Negation¶
Signature
// CLEAN
~ :: Real -> Real
~ a = ...
Behavior
Inverts the sign of a.
Usage
// CLEAN
~ 1.0 // -1.0
~ 0.0 // 0.0
~(-1.0) // 1.0
sign¶
Signature
// CLEAN
sign :: Real -> Int
sign a = ...
Behavior
Returns the sign of a.
Usage
// CLEAN
sign 1.0 // 1
sign 0.0 // 0
sign (-1.0) // -1
abs¶
Signature
// CLEAN
abs :: Real -> Real
abs a = ...
Behavior
Returns the absolute value of a.
Usage
// CLEAN
abs 1.0 // 1.0
abs 0.0 // 0.0
abs (-1.0) // 1.0
ln¶
Signature
// CLEAN
ln :: Real -> Real
ln a = ...
Behavior
Returns the natural logarithm of a.
Usage
// CLEAN
ln 2.5 // 0.916290731874155
ln 1.5 // -1.405465108108164
ln 0.0 // -#INF
ln (-1.5) // #NAN
ln (-2.5) // #NAN
log10¶
Signature
// CLEAN
log10 :: Real -> Real
log10 a = ...
Behavior
Returns the logarithm base ten of a.
Usage
// CLEAN
log10 2.5 // 0.397940008672038
log10 1.5 // -0.176091259055681
log10 0.0 // -#INF
log10 (-1.5) // #NAN
log10 (-2.5) // #NAN
exp¶
Signature
// CLEAN
exp :: Real -> Real
exp a = ...
Behavior
Raises \(e\) to the power of a.
See Euler's number.
Usage
// CLEAN
exp 2.5 // 12.1824939607035
exp 1.5 // 4.48168907033806
exp 0.0 // 1.0
exp (-1.5) // 0.22313016014843
exp (-2.5) // 0.0820849986238988
sqrt¶
Signature
// CLEAN
sqrt :: Real -> Real
sqrt a = ...
Behavior
Returns the square root of a.
Usage
// CLEAN
sqrt 2.5 // 1.58113883008419
sqrt 1.5 // 1.22474487139159
sqrt 0.0 // 0.0
sqrt (-1.5) // #NAN
sqrt (-2.5) // #NAN
Relational operations¶
Equality comparison between two real numbers seems to be unaffected by rounding errors.
Equal to¶
Signature
// CLEAN
(==) infix 4 :: Real Real -> Bool
(==) a b = ...
Behavior
Returns true if a is equal to b.
Otherwise, returns false.
Usage
// CLEAN
5.0 == 2.0 // False
(-5.0) == 2.0 // False
5.0 == (-2.0) // False
(-5.0) == (-2.0) // False
(-2.0) == (-2.0) // True
Not equal to¶
Signature
// CLEAN
(<>) infix 4 :: Real Real -> Bool
(<>) a b = ...
Behavior
Returns true if a is not equal to b.
Otherwise, returns false.
Usage
// CLEAN
5.0 <> 2.0 // True
(-5.0) <> 2.0 // True
5.0 <> (-2.0) // True
(-5.0) <> (-2.0) // True
(-2.0) <> (-2.0) // False
Less than¶
Signature
// CLEAN
(<) infix 4 :: Real Real -> Bool
(<) a b = ...
Behavior
Returns true if a is strictly less than b.
Otherwise, returns false.
Usage
// CLEAN
5.0 < 2.0 // False
(-5.0) < 2.0 // True
5.0 < (-2.0) // False
(-5.0) < (-2.0) // True
(-2.0) < (-2.0) // False
Less than or equal to¶
Signature
// CLEAN
(<=) infix 4 :: Real Real -> Bool
(<=) a b = ...
Behavior
Returns true if a is less than or equal to b.
Otherwise, returns false.
Usage
// CLEAN
5.0 <= 2.0 // False
(-5.0) <= 2.0 // True
5.0 <= (-2.0) // False
(-5.0) <= (-2.0) // True
(-2.0) <= (-2.0) // True
Greater than¶
Signature
// CLEAN
(>) infix 4 :: Real Real -> Bool
(>) a b = ...
Behavior
Returns true if a is strictly greater than b.
Otherwise, returns false.
Usage
// CLEAN
5.0 > 2.0 // True
(-5.0) > 2.0 // False
5.0 > (-2.0) // True
(-5.0) > (-2.0) // False
(-2.0) > (-2.0) // False
Greater than or equal to¶
Signature
// CLEAN
(>=) infix 4 :: Real Real -> Bool
(>=) a b = ...
Behavior
Returns true if a is greater than or equal to b.
Otherwise, returns false.
Usage
// CLEAN
5.0 >= 2.0 // True
(-5.0) >= 2.0 // False
5.0 >= (-2.0) // True
(-5.0) >= (-2.0) // False
(-2.0) >= (-2.0) // True
Trigonometric Functions¶
sin¶
Signature
// CLEAN
sin :: Real -> Real
sin a = ...
Behavior
Returns the sine of a.
Usage
// CLEAN
sin 2.5 // 0.598472144103956
sin 1.5 // 0.997494986604054
sin 0.0 // 0.0
sin (-1.5) // -0.997494986604054
sin (-2.5) // -0.598472144103956
cos¶
Signature
// CLEAN
cos :: Real -> Real
cos a = ...
Behavior
Returns the cosine of a.
Usage
// CLEAN
cos 2.5 // -0.801143615546934
cos 1.5 // 0.0707372016677029
cos 0.0 // 1.0
cos (-1.5) // 0.0707372016677029
cos (-2.5) // -0.801143615546934
tan¶
Signature
// CLEAN
tan :: Real -> Real
tan a = ...
Behavior
Returns the tangent of a.
Usage
// CLEAN
tan 2.5 // - 0.74702229723866
tan 1.5 // 14.1014199471717
tan 0.0 // 0.0
tan (-1.5) // -14.1014199471717
tan (-2.5) // 0.74702229723866
asin¶
Signature
// CLEAN
asin :: Real -> Real
asin a = ...
Behavior
Returns the inverse sine of a.
Usage
// CLEAN
asin 2.5 // #NAN
asin 1.5 // #NAN
asin 0.5 // 0.523598775598299
asin 0.0 // 0.0
asin (-0.5) // -0.523598775598299
asin (-1.5) // #NAN
asin (-2.5) // #NAN
acos¶
Signature
// CLEAN
acos :: Real -> Real
acos a = ...
Behavior
Returns the inverse cosine of a.
Usage
// CLEAN
acos 2.5 // #NAN
acos 1.5 // #NAN
acos 0.5 // 1.0471975511966
acos 0.0 // 1.5707963267949
acos (-0.5) // 2.0943951023932
acos (-1.5) // #NAN
acos (-2.5) // #NAN
atan¶
Signature
// CLEAN
atan :: Real -> Real
atan a = ...
Behavior
Returns the inverse tangent of a.
Usage
// CLEAN
atan 2.5 // 1.19028994968253
atan 1.5 // 0.982793723247329
atan 0.5 // 0.463647609000806
atan 0.0 // 0.0
atan (-0.5) // -0.463647609000806
atan (-1.5) // -0.982793723247329
atan (-2.5) // -1.19028994968253
sinh¶
Signature
// CLEAN
sinh :: Real -> Real
sinh a = ...
Behavior
Returns the hyperbolic sine of a.
Usage
// CLEAN
sinh 2.5 // 6.05020448103979
sinh 1.5 // 2.12927945509482
sinh 0.5 // 0.521095305493747
sinh 0.0 // 0.0
sinh (-0.5) // -0.521095305493747
sinh (-1.5) // -2.12927945509482
sinh (-2.5) // -6.05020448103979
cosh¶
Signature
// CLEAN
cosh :: Real -> Real
cosh a = ...
Behavior
Returns the hyperbolic cosine of a.
Usage
// CLEAN
cosh 2.5 // 6.13228947966369
cosh 1.5 // 2.35240961524325
cosh 0.5 // 1.12762596520638
cosh 0.0 // 1.0
cosh (-0.5) // -1.12762596520638
cosh (-1.5) // -2.35240961524325
cosh (-2.5) // -6.13228947966369
tanh¶
Signature
// CLEAN
tanh :: Real -> Real
tanh a = ...
Behavior
Returns the hyperbolic tangent of a.
Usage
// CLEAN
tanh 2.5 // 0.98661429815143
tanh 1.5 // 0.905148253644866
tanh 0.5 // 0.46211715726001
tanh 0.0 // 0.0
tanh (-0.5) // -0.46211715726001
tanh (-1.5) // -0.905148253644866
tanh (-2.5) // -0.98661429815143
asinh¶
Signature
// CLEAN
asinh :: Real -> Real
asinh a = ...
Behavior
Returns the inverse hyperbolic sine of a.
Usage
// CLEAN
asinh 2.5 // 1.6472311463711
asinh 1.5 // 1.19476321728711
asinh 0.5 // 0.481211825059604
asinh 0.0 // 0.0
asinh (-0.5) // -0.481211825059604
asinh (-1.5) // -1.19476321728711
asinh (-2.5) // -1.6472311463711
acosh¶
Signature
// CLEAN
acosh :: Real -> Real
acosh a = ...
Behavior
Returns the inverse hyperbolic cosine of a.
Usage
// CLEAN
acosh 2.5 // 1.56679923697241
acosh 1.5 // 0.962423650119207
acosh 0.5 // #NAN
acosh 0.0 // #NAN
acosh (-0.5) // #NAN
acosh (-1.5) // #NAN
acosh (-2.5) // #NAN
atanh¶
Signature
// CLEAN
atanh :: Real -> Real
atanh a = ...
Behavior
Returns the inverse hyperbolic tangent of a.
Usage
// CLEAN
atanh 2.5 // #NAN
atanh 1.5 // #NAN
atanh 0.5 // 0.549306144334055
atanh 0.0 // 0.0
atanh (-0.5) // -0.549306144334055
atanh (-1.5) // #NAN
atanh (-2.5) // #NAN
Validator functions¶
isNaN¶
Signature
// CLEAN
isNaN :: Real -> Bool
isNaN a = ...
Behavior
Returns true if a is a NaN value.
Otherwise, returns false.
Usage
// CLEAN
isNaN 2.0 // False
isNaN 0.0 // False
isNaN (sqrt (-1.0)) // True
isNaN (ln (-1.0)) // True
isNaN ((-1.0) / 0.0) // False
isNaN ( 0.0 ^ (-1.0)) // False
isInfinity¶
Signature
// CLEAN
isInfinity :: Real -> Bool
isInfinity a = ...
Behavior
Returns true if a is an Infinity value.
Otherwise, returns false.
Usage
// CLEAN
isInfinity 2.0 // False
isInfinity 0.0 // False
isInfinity (sqrt (-1.0)) // False
isInfinity (ln (-1.0)) // False
isInfinity ((-1.0) / 0.0) // True
isInfinity ( 0.0 ^ (-1.0)) // True
isFinite¶
Signature
// CLEAN
isFinite :: Real -> Bool
isFinite a = ...
Behavior
Returns true if a is representable.
Otherwise, returns false.
Usage
// CLEAN
isFinite 2.0 // True
isFinite 0.0 // True
isFinite (sqrt (-1.0)) // False
isFinite (ln (-1.0)) // False
isFinite ((-1.0) / 0.0) // False
isFinite ( 0.0 ^ (-1.0)) // False
Conversions to real number¶
Natively, two types can convert to real numbers, namely, integers and strings.
From integer¶
Signature
// CLEAN
toReal :: Int -> Real
toReal a = ...
Behavior
Sets the decimal places of a to 0.
Usage
// CLEAN
toReal 2 // 2.0
toReal 1 // 1.0
toReal 0 // 0.0
toReal (-1) // -1.0
toReal (-2) // -2.0
From string¶
Signature
// CLEAN
toReal :: {#Char} -> Real
toReal a = ...
Behavior
It parses a as a real number.
Usage
// CLEAN
toReal "1.0" // 1.0
toReal "1" // 1.0
toReal "0" // 0.0
toReal "-1" // -1.0
toReal "-1.0" // 1.0
Conversions from real number¶
The function responsible for type conversion is overloaded, it has the same parameter but different return type, so to use it, the desired type must be unambiguous.
To integer¶
Signature
// CLEAN
fromReal :: Real -> Int
fromReal a = ...
Behavior
Rounds a to its nearest integer.
Usage
// CLEAN
expr :: Int
expr = fromReal 1.9 // 2
expr = fromReal 1.4 // 1
expr = fromReal (-1.4) // -1
expr = fromReal (-1.9) // -2
To string¶
Signature
// CLEAN
fromReal :: Real -> {#Char}
fromReal a = ...
Behavior
Returns a string representation of a.
Usage
// CLEAN
expr :: {#Char}
expr = fromReal 1.9 // "1.9"
expr = fromReal 1.4 // "1.4"
expr = fromReal (-1.4) // "-1.4"
expr = fromReal (-1.9) // "-1.9"
entier¶
Signature
// CLEAN
entier :: Real -> Int
entier a = ...
Behavior
Rounds a down to its nearest integer.
Usage
// CLEAN
entier 1.5 // 1
entier 0.0 // 0
entier (-1.5) // -2