{-# LANGUAGE Trustworthy #-}{-# LANGUAGE ScopedTypeVariables #-}{-# LANGUAGE PolyKinds #-}{-# LANGUAGE FlexibleInstances #-}------------------------------------------------------------------------------- |-- Module : Data.Fixed-- Copyright : (c) Ashley Yakeley 2005, 2006, 2009-- License : BSD-style (see the file libraries/base/LICENSE)---- Maintainer : Ashley Yakeley <ashley@semantic.org>-- Stability : experimental-- Portability : portable---- This module defines a \"Fixed\" type for fixed-precision arithmetic.-- The parameter to 'Fixed' is any type that's an instance of 'HasResolution'.-- 'HasResolution' has a single method that gives the resolution of the 'Fixed'-- type.---- This module also contains generalisations of 'div', 'mod', and 'divMod' to-- work with any 'Real' instance.-------------------------------------------------------------------------------moduleData.Fixed(div' ,mod' ,divMod' ,Fixed (..),HasResolution (..),showFixed ,E0 ,Uni ,E1 ,Deci ,E2 ,Centi ,E3 ,Milli ,E6 ,Micro ,E9 ,Nano ,E12 ,Pico )whereimportData.Data importGHC.TypeLits (KnownNat ,natVal )importGHC.Read importText.ParserCombinators.ReadPrec importText.Read.Lex default()-- avoid any defaulting shenanigans-- | Generalisation of 'div' to any instance of 'Real'div' ::(Real a ,Integral b )=>a ->a ->b div' :: a -> a -> b
div' a
n a
d =Rational -> b
forall a b. (RealFrac a, Integral b) => a -> b
floor ((a -> Rational
forall a. Real a => a -> Rational
toRational a
n )Rational -> Rational -> Rational
forall a. Fractional a => a -> a -> a
/ (a -> Rational
forall a. Real a => a -> Rational
toRational a
d ))-- | Generalisation of 'divMod' to any instance of 'Real'divMod' ::(Real a ,Integral b )=>a ->a ->(b ,a )divMod' :: a -> a -> (b, a)
divMod' a
n a
d =(b
f ,a
n a -> a -> a
forall a. Num a => a -> a -> a
- (b -> a
forall a b. (Integral a, Num b) => a -> b
fromIntegral b
f )a -> a -> a
forall a. Num a => a -> a -> a
* a
d )wheref :: b
f =a -> a -> b
forall a b. (Real a, Integral b) => a -> a -> b
div' a
n a
d -- | Generalisation of 'mod' to any instance of 'Real'mod' ::(Real a )=>a ->a ->a mod' :: a -> a -> a
mod' a
n a
d =a
n a -> a -> a
forall a. Num a => a -> a -> a
- (Integer -> a
forall a. Num a => Integer -> a
fromInteger Integer
f )a -> a -> a
forall a. Num a => a -> a -> a
* a
d wheref :: Integer
f =a -> a -> Integer
forall a b. (Real a, Integral b) => a -> a -> b
div' a
n a
d -- | The type parameter should be an instance of 'HasResolution'.newtypeFixed (a ::k )=MkFixed Integerderiving(Eq-- ^ @since 2.01,Ord-- ^ @since 2.01)-- We do this because the automatically derived Data instance requires (Data a) context.-- Our manual instance has the more general (Typeable a) context.tyFixed ::DataType tyFixed :: DataType
tyFixed =String -> [Constr] -> DataType
mkDataType String
"Data.Fixed.Fixed"[Constr
conMkFixed ]conMkFixed ::Constr conMkFixed :: Constr
conMkFixed =DataType -> String -> [String] -> Fixity -> Constr
mkConstr DataType
tyFixed String
"MkFixed"[]Fixity
Prefix -- | @since 4.1.0.0instance(Typeable k ,Typeable a )=>Data (Fixed (a ::k ))wheregfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b)
-> (forall g. g -> c g) -> Fixed a -> c (Fixed a)
gfoldl forall d b. Data d => c (d -> b) -> d -> c b
k forall g. g -> c g
z (MkFixed Integer
a )=c (Integer -> Fixed a) -> Integer -> c (Fixed a)
forall d b. Data d => c (d -> b) -> d -> c b
k ((Integer -> Fixed a) -> c (Integer -> Fixed a)
forall g. g -> c g
z Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed )Integer
a gunfold :: (forall b r. Data b => c (b -> r) -> c r)
-> (forall r. r -> c r) -> Constr -> c (Fixed a)
gunfold forall b r. Data b => c (b -> r) -> c r
k forall r. r -> c r
z Constr
_=c (Integer -> Fixed a) -> c (Fixed a)
forall b r. Data b => c (b -> r) -> c r
k ((Integer -> Fixed a) -> c (Integer -> Fixed a)
forall r. r -> c r
z Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed )dataTypeOf :: Fixed a -> DataType
dataTypeOf Fixed a
_=DataType
tyFixed toConstr :: Fixed a -> Constr
toConstr Fixed a
_=Constr
conMkFixed classHasResolution (a ::k )whereresolution ::p a ->Integer-- | For example, @Fixed 1000@ will give you a 'Fixed' with a resolution of 1000.instanceKnownNat n =>HasResolution n whereresolution :: p n -> Integer
resolution p n
_=Proxy n -> Integer
forall (n :: Nat) (proxy :: Nat -> *).
KnownNat n =>
proxy n -> Integer
natVal (Proxy n
forall k (t :: k). Proxy t
Proxy ::Proxy n )withType ::(Proxy a ->f a )->f a withType :: (Proxy a -> f a) -> f a
withType Proxy a -> f a
foo =Proxy a -> f a
foo Proxy a
forall k (t :: k). Proxy t
Proxy withResolution ::(HasResolution a )=>(Integer->f a )->f a withResolution :: (Integer -> f a) -> f a
withResolution Integer -> f a
foo =(Proxy a -> f a) -> f a
forall k (a :: k) (f :: k -> *). (Proxy a -> f a) -> f a
withType (Integer -> f a
foo (Integer -> f a) -> (Proxy a -> Integer) -> Proxy a -> f a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Proxy a -> Integer
forall k (a :: k) (p :: k -> *). HasResolution a => p a -> Integer
resolution )-- | @since 2.01instanceEnum (Fixed a )wheresucc :: Fixed a -> Fixed a
succ (MkFixed Integer
a )=Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer
forall a. Enum a => a -> a
succ Integer
a )pred :: Fixed a -> Fixed a
pred (MkFixed Integer
a )=Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer
forall a. Enum a => a -> a
pred Integer
a )toEnum :: Int -> Fixed a
toEnum =Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Fixed a) -> (Int -> Integer) -> Int -> Fixed a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Int -> Integer
forall a. Enum a => Int -> a
toEnum fromEnum :: Fixed a -> Int
fromEnum (MkFixed Integer
a )=Integer -> Int
forall a. Enum a => a -> Int
fromEnum Integer
a enumFrom :: Fixed a -> [Fixed a]
enumFrom (MkFixed Integer
a )=(Integer -> Fixed a) -> [Integer] -> [Fixed a]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> [Integer]
forall a. Enum a => a -> [a]
enumFrom Integer
a )enumFromThen :: Fixed a -> Fixed a -> [Fixed a]
enumFromThen (MkFixed Integer
a )(MkFixed Integer
b )=(Integer -> Fixed a) -> [Integer] -> [Fixed a]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer -> [Integer]
forall a. Enum a => a -> a -> [a]
enumFromThen Integer
a Integer
b )enumFromTo :: Fixed a -> Fixed a -> [Fixed a]
enumFromTo (MkFixed Integer
a )(MkFixed Integer
b )=(Integer -> Fixed a) -> [Integer] -> [Fixed a]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer -> [Integer]
forall a. Enum a => a -> a -> [a]
enumFromTo Integer
a Integer
b )enumFromThenTo :: Fixed a -> Fixed a -> Fixed a -> [Fixed a]
enumFromThenTo (MkFixed Integer
a )(MkFixed Integer
b )(MkFixed Integer
c )=(Integer -> Fixed a) -> [Integer] -> [Fixed a]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer -> Integer -> [Integer]
forall a. Enum a => a -> a -> a -> [a]
enumFromThenTo Integer
a Integer
b Integer
c )-- | @since 2.01instance(HasResolution a )=>Num (Fixed a )where(MkFixed Integer
a )+ :: Fixed a -> Fixed a -> Fixed a
+ (MkFixed Integer
b )=Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
+ Integer
b )(MkFixed Integer
a )- :: Fixed a -> Fixed a -> Fixed a
- (MkFixed Integer
b )=Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Integer
b )fa :: Fixed a
fa @(MkFixed Integer
a )* :: Fixed a -> Fixed a -> Fixed a
* (MkFixed Integer
b )=Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
div (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
b )(Fixed a -> Integer
forall k (a :: k) (p :: k -> *). HasResolution a => p a -> Integer
resolution Fixed a
fa ))negate :: Fixed a -> Fixed a
negate (MkFixed Integer
a )=Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer
forall a. Num a => a -> a
negate Integer
a )abs :: Fixed a -> Fixed a
abs (MkFixed Integer
a )=Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer
forall a. Num a => a -> a
abs Integer
a )signum :: Fixed a -> Fixed a
signum (MkFixed Integer
a )=Integer -> Fixed a
forall a. Num a => Integer -> a
fromInteger (Integer -> Integer
forall a. Num a => a -> a
signum Integer
a )fromInteger :: Integer -> Fixed a
fromInteger Integer
i =(Integer -> Fixed a) -> Fixed a
forall k (a :: k) (f :: k -> *).
HasResolution a =>
(Integer -> f a) -> f a
withResolution (\Integer
res ->Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer
i Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
res ))-- | @since 2.01instance(HasResolution a )=>Real (Fixed a )wheretoRational :: Fixed a -> Rational
toRational fa :: Fixed a
fa @(MkFixed Integer
a )=(Integer -> Rational
forall a. Real a => a -> Rational
toRational Integer
a )Rational -> Rational -> Rational
forall a. Fractional a => a -> a -> a
/ (Integer -> Rational
forall a. Real a => a -> Rational
toRational (Fixed a -> Integer
forall k (a :: k) (p :: k -> *). HasResolution a => p a -> Integer
resolution Fixed a
fa ))-- | @since 2.01instance(HasResolution a )=>Fractional (Fixed a )wherefa :: Fixed a
fa @(MkFixed Integer
a )/ :: Fixed a -> Fixed a -> Fixed a
/ (MkFixed Integer
b )=Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
div (Integer
a Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* (Fixed a -> Integer
forall k (a :: k) (p :: k -> *). HasResolution a => p a -> Integer
resolution Fixed a
fa ))Integer
b )recip :: Fixed a -> Fixed a
recip fa :: Fixed a
fa @(MkFixed Integer
a )=Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
div (Integer
res Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
res )Integer
a )whereres :: Integer
res =Fixed a -> Integer
forall k (a :: k) (p :: k -> *). HasResolution a => p a -> Integer
resolution Fixed a
fa fromRational :: Rational -> Fixed a
fromRational Rational
r =(Integer -> Fixed a) -> Fixed a
forall k (a :: k) (f :: k -> *).
HasResolution a =>
(Integer -> f a) -> f a
withResolution (\Integer
res ->Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Rational -> Integer
forall a b. (RealFrac a, Integral b) => a -> b
floor (Rational
r Rational -> Rational -> Rational
forall a. Num a => a -> a -> a
* (Integer -> Rational
forall a. Real a => a -> Rational
toRational Integer
res ))))-- | @since 2.01instance(HasResolution a )=>RealFrac (Fixed a )whereproperFraction :: Fixed a -> (b, Fixed a)
properFraction Fixed a
a =(b
i ,Fixed a
a Fixed a -> Fixed a -> Fixed a
forall a. Num a => a -> a -> a
- (b -> Fixed a
forall a b. (Integral a, Num b) => a -> b
fromIntegral b
i ))wherei :: b
i =Fixed a -> b
forall a b. (RealFrac a, Integral b) => a -> b
truncate Fixed a
a truncate :: Fixed a -> b
truncate Fixed a
f =Rational -> b
forall a b. (RealFrac a, Integral b) => a -> b
truncate (Fixed a -> Rational
forall a. Real a => a -> Rational
toRational Fixed a
f )round :: Fixed a -> b
round Fixed a
f =Rational -> b
forall a b. (RealFrac a, Integral b) => a -> b
round (Fixed a -> Rational
forall a. Real a => a -> Rational
toRational Fixed a
f )ceiling :: Fixed a -> b
ceiling Fixed a
f =Rational -> b
forall a b. (RealFrac a, Integral b) => a -> b
ceiling (Fixed a -> Rational
forall a. Real a => a -> Rational
toRational Fixed a
f )floor :: Fixed a -> b
floor Fixed a
f =Rational -> b
forall a b. (RealFrac a, Integral b) => a -> b
floor (Fixed a -> Rational
forall a. Real a => a -> Rational
toRational Fixed a
f )chopZeros ::Integer->String chopZeros :: Integer -> String
chopZeros Integer
0=String
""chopZeros Integer
a |Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
mod Integer
a Integer
10Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
==Integer
0=Integer -> String
chopZeros (Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
div Integer
a Integer
10)chopZeros Integer
a =Integer -> String
forall a. Show a => a -> String
show Integer
a -- only works for positive ashowIntegerZeros ::Bool->Int->Integer->String showIntegerZeros :: Bool -> Int -> Integer -> String
showIntegerZeros Bool
TrueInt
_Integer
0=String
""showIntegerZeros Bool
chopTrailingZeros Int
digits Integer
a =Int -> Char -> String
forall a. Int -> a -> [a]
replicate (Int
digits Int -> Int -> Int
forall a. Num a => a -> a -> a
- String -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length String
s )Char
'0'String -> String -> String
forall a. [a] -> [a] -> [a]
++ String
s' wheres :: String
s =Integer -> String
forall a. Show a => a -> String
show Integer
a s' :: String
s' =ifBool
chopTrailingZeros thenInteger -> String
chopZeros Integer
a elseString
s withDot ::String ->String withDot :: String -> String
withDot String
""=String
""withDot String
s =Char
'.'Char -> String -> String
forall a. a -> [a] -> [a]
:String
s -- | First arg is whether to chop off trailing zerosshowFixed ::(HasResolution a )=>Bool->Fixed a ->String showFixed :: Bool -> Fixed a -> String
showFixed Bool
chopTrailingZeros fa :: Fixed a
fa @(MkFixed Integer
a )|Integer
a Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
<Integer
0=String
"-"String -> String -> String
forall a. [a] -> [a] -> [a]
++ (Bool -> Fixed a -> String
forall k (a :: k). HasResolution a => Bool -> Fixed a -> String
showFixed Bool
chopTrailingZeros (Fixed a -> Fixed a -> Fixed a
forall a. a -> a -> a
asTypeOf (Integer -> Fixed a
forall k (a :: k). Integer -> Fixed a
MkFixed (Integer -> Integer
forall a. Num a => a -> a
negate Integer
a ))Fixed a
fa ))showFixed Bool
chopTrailingZeros fa :: Fixed a
fa @(MkFixed Integer
a )=(Integer -> String
forall a. Show a => a -> String
show Integer
i )String -> String -> String
forall a. [a] -> [a] -> [a]
++ (String -> String
withDot (Bool -> Int -> Integer -> String
showIntegerZeros Bool
chopTrailingZeros Int
digits Integer
fracNum ))whereres :: Integer
res =Fixed a -> Integer
forall k (a :: k) (p :: k -> *). HasResolution a => p a -> Integer
resolution Fixed a
fa (Integer
i ,Integer
d )=Integer -> Integer -> (Integer, Integer)
forall a. Integral a => a -> a -> (a, a)
divMod Integer
a Integer
res -- enough digits to be unambiguousdigits :: Int
digits =Double -> Int
forall a b. (RealFrac a, Integral b) => a -> b
ceiling (Double -> Double -> Double
forall a. Floating a => a -> a -> a
logBase Double
10(Integer -> Double
forall a. Num a => Integer -> a
fromInteger Integer
res )::Double)maxnum :: Integer
maxnum =Integer
10Integer -> Int -> Integer
forall a b. (Num a, Integral b) => a -> b -> a
^ Int
digits -- read floors, so show must ceil for `read . show = id` to hold. See #9240fracNum :: Integer
fracNum =Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
divCeil (Integer
d Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
* Integer
maxnum )Integer
res divCeil :: a -> a -> a
divCeil a
x a
y =(a
x a -> a -> a
forall a. Num a => a -> a -> a
+ a
y a -> a -> a
forall a. Num a => a -> a -> a
- a
1)a -> a -> a
forall a. Integral a => a -> a -> a
`div` a
y -- | @since 2.01instance(HasResolution a )=>Show (Fixed a )whereshowsPrec :: Int -> Fixed a -> String -> String
showsPrec Int
p Fixed a
n =Bool -> (String -> String) -> String -> String
showParen (Int
p Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
>Int
6Bool -> Bool -> Bool
&&Fixed a
n Fixed a -> Fixed a -> Bool
forall a. Ord a => a -> a -> Bool
<Fixed a
0)((String -> String) -> String -> String)
-> (String -> String) -> String -> String
forall a b. (a -> b) -> a -> b
$ String -> String -> String
showString (String -> String -> String) -> String -> String -> String
forall a b. (a -> b) -> a -> b
$ Bool -> Fixed a -> String
forall k (a :: k). HasResolution a => Bool -> Fixed a -> String
showFixed Bool
FalseFixed a
n -- | @since 4.3.0.0instance(HasResolution a )=>Read (Fixed a )wherereadPrec :: ReadPrec (Fixed a)
readPrec =(Lexeme -> ReadPrec (Fixed a)) -> ReadPrec (Fixed a)
forall a. Num a => (Lexeme -> ReadPrec a) -> ReadPrec a
readNumber Lexeme -> ReadPrec (Fixed a)
forall k (a :: k). HasResolution a => Lexeme -> ReadPrec (Fixed a)
convertFixed readListPrec :: ReadPrec [Fixed a]
readListPrec =ReadPrec [Fixed a]
forall a. Read a => ReadPrec [a]
readListPrecDefault readList :: ReadS [Fixed a]
readList =ReadS [Fixed a]
forall a. Read a => ReadS [a]
readListDefault convertFixed ::foralla .HasResolution a =>Lexeme ->ReadPrec (Fixed a )convertFixed :: Lexeme -> ReadPrec (Fixed a)
convertFixed (Number Number
n )|Just (Integer
i ,Integer
f )<-Integer -> Number -> Maybe (Integer, Integer)
numberToFixed Integer
e Number
n =Fixed a -> ReadPrec (Fixed a)
forall (m :: * -> *) a. Monad m => a -> m a
return (Integer -> Fixed a
forall a. Num a => Integer -> a
fromInteger Integer
i Fixed a -> Fixed a -> Fixed a
forall a. Num a => a -> a -> a
+ (Integer -> Fixed a
forall a. Num a => Integer -> a
fromInteger Integer
f Fixed a -> Fixed a -> Fixed a
forall a. Fractional a => a -> a -> a
/ (Fixed a
10Fixed a -> Integer -> Fixed a
forall a b. (Num a, Integral b) => a -> b -> a
^ Integer
e )))wherer :: Integer
r =Proxy a -> Integer
forall k (a :: k) (p :: k -> *). HasResolution a => p a -> Integer
resolution (Proxy a
forall k (t :: k). Proxy t
Proxy ::Proxy a )-- round 'e' up to help make the 'read . show == id' property-- possible also for cases where 'resolution' is not a-- power-of-10, such as e.g. when 'resolution = 128'e :: Integer
e =Double -> Integer
forall a b. (RealFrac a, Integral b) => a -> b
ceiling (Double -> Double -> Double
forall a. Floating a => a -> a -> a
logBase Double
10(Integer -> Double
forall a. Num a => Integer -> a
fromInteger Integer
r )::Double)convertFixed Lexeme
_=ReadPrec (Fixed a)
forall a. ReadPrec a
pfail dataE0 -- | @since 4.1.0.0instanceHasResolution E0 whereresolution :: p E0 -> Integer
resolution p E0
_=Integer
1-- | resolution of 1, this works the same as IntegertypeUni =Fixed E0 dataE1 -- | @since 4.1.0.0instanceHasResolution E1 whereresolution :: p E1 -> Integer
resolution p E1
_=Integer
10-- | resolution of 10^-1 = .1typeDeci =Fixed E1 dataE2 -- | @since 4.1.0.0instanceHasResolution E2 whereresolution :: p E2 -> Integer
resolution p E2
_=Integer
100-- | resolution of 10^-2 = .01, useful for many monetary currenciestypeCenti =Fixed E2 dataE3 -- | @since 4.1.0.0instanceHasResolution E3 whereresolution :: p E3 -> Integer
resolution p E3
_=Integer
1000-- | resolution of 10^-3 = .001typeMilli =Fixed E3 dataE6 -- | @since 2.01instanceHasResolution E6 whereresolution :: p E6 -> Integer
resolution p E6
_=Integer
1000000-- | resolution of 10^-6 = .000001typeMicro =Fixed E6 dataE9 -- | @since 4.1.0.0instanceHasResolution E9 whereresolution :: p E9 -> Integer
resolution p E9
_=Integer
1000000000-- | resolution of 10^-9 = .000000001typeNano =Fixed E9 dataE12 -- | @since 2.01instanceHasResolution E12 whereresolution :: p E12 -> Integer
resolution p E12
_=Integer
1000000000000-- | resolution of 10^-12 = .000000000001typePico =Fixed E12