Copyright  Bas van Dijk 2013 

License  BSD3 
Maintainer  Bas van Dijk <v.dijk.bas@gmail.com> 
Safe Haskell  Trustworthy 
Language  Haskell2010 
This module provides the number type Scientific
. Scientific numbers are
arbitrary precision and space efficient. They are represented using
scientific notation. The
implementation uses an Integer
coefficient
c
and an Int
base10Exponent
e
. A scientific number corresponds to the Fractional
number:
.fromInteger
c * 10 ^^
e
Note that since we're using an Int
to represent the exponent these numbers
aren't truly arbitrary precision. I intend to change the type of the exponent
to Integer
in a future release.
WARNING: Although Scientific
has instances for all numeric classes the
methods should be used with caution when applied to scientific numbers coming
from untrusted sources. See the warnings of the instances belonging to
Scientific
.
The main application of Scientific
is to be used as the target of parsing
arbitrary precision numbers coming from an untrusted source. The advantages
over using Rational
for this are that:
 A
Scientific
is more efficient to construct. Rational numbers need to be constructed using%
which has to compute thegcd
of thenumerator
anddenominator
. Scientific
is safe against numbers with huge exponents. For example:1e1000000000 ::
will fill up all space and crash your program. Scientific works as expected:Rational
> read "1e1000000000" :: Scientific 1.0e1000000000
 Also, the space usage of converting scientific numbers with huge exponents
to
(like:Integral
sInt
) or
(like:RealFloat
sDouble
orFloat
) will always be bounded by the target type.
This module is designed to be imported qualified:
import qualified Data.Scientific as Scientific
Synopsis
 data Scientific
 scientific :: Integer > Int > Scientific
 coefficient :: Scientific > Integer
 base10Exponent :: Scientific > Int
 isFloating :: Scientific > Bool
 isInteger :: Scientific > Bool
 unsafeFromRational :: Rational > Scientific
 fromRationalRepetend :: Maybe Int > Rational > Either (Scientific, Rational) (Scientific, Maybe Int)
 fromRationalRepetendLimited :: Int > Rational > Either (Scientific, Rational) (Scientific, Maybe Int)
 fromRationalRepetendUnlimited :: Rational > (Scientific, Maybe Int)
 toRationalRepetend :: Scientific > Int > Rational
 floatingOrInteger :: (RealFloat r, Integral i) => Scientific > Either r i
 toRealFloat :: RealFloat a => Scientific > a
 toBoundedRealFloat :: forall a. RealFloat a => Scientific > Either a a
 toBoundedInteger :: forall i. (Integral i, Bounded i) => Scientific > Maybe i
 fromFloatDigits :: RealFloat a => a > Scientific
 scientificP :: ReadP Scientific
 formatScientific :: FPFormat > Maybe Int > Scientific > String
 data FPFormat
 toDecimalDigits :: Scientific > ([Int], Int)
 normalize :: Scientific > Scientific
Documentation
data Scientific Source #
An arbitraryprecision number represented using scientific notation.
This type describes the set of all
which have a finite
decimal expansion.Real
s
A scientific number with coefficient
c
and base10Exponent
e
corresponds to the Fractional
number: fromInteger
c * 10 ^^
e
Instances
Eq Scientific Source #  Scientific numbers can be safely compared for equality. No magnitude 
Defined in Data.Scientific (==) :: Scientific > Scientific > Bool # (/=) :: Scientific > Scientific > Bool #  
Fractional Scientific Source #  WARNING: These methods also compute

Defined in Data.Scientific (/) :: Scientific > Scientific > Scientific # recip :: Scientific > Scientific # fromRational :: Rational > Scientific #  
Data Scientific Source #  
Defined in Data.Scientific gfoldl :: (forall d b. Data d => c (d > b) > d > c b) > (forall g. g > c g) > Scientific > c Scientific # gunfold :: (forall b r. Data b => c (b > r) > c r) > (forall r. r > c r) > Constr > c Scientific # toConstr :: Scientific > Constr # dataTypeOf :: Scientific > DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) > Maybe (c Scientific) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) > Maybe (c Scientific) # gmapT :: (forall b. Data b => b > b) > Scientific > Scientific # gmapQl :: (r > r' > r) > r > (forall d. Data d => d > r') > Scientific > r # gmapQr :: forall r r'. (r' > r > r) > r > (forall d. Data d => d > r') > Scientific > r # gmapQ :: (forall d. Data d => d > u) > Scientific > [u] # gmapQi :: Int > (forall d. Data d => d > u) > Scientific > u # gmapM :: Monad m => (forall d. Data d => d > m d) > Scientific > m Scientific # gmapMp :: MonadPlus m => (forall d. Data d => d > m d) > Scientific > m Scientific # gmapMo :: MonadPlus m => (forall d. Data d => d > m d) > Scientific > m Scientific #  
Num Scientific Source #  WARNING: 
Defined in Data.Scientific (+) :: Scientific > Scientific > Scientific # () :: Scientific > Scientific > Scientific # (*) :: Scientific > Scientific > Scientific # negate :: Scientific > Scientific # abs :: Scientific > Scientific # signum :: Scientific > Scientific # fromInteger :: Integer > Scientific #  
Ord Scientific Source #  Scientific numbers can be safely compared for ordering. No magnitude 
Defined in Data.Scientific compare :: Scientific > Scientific > Ordering # (<) :: Scientific > Scientific > Bool # (<=) :: Scientific > Scientific > Bool # (>) :: Scientific > Scientific > Bool # (>=) :: Scientific > Scientific > Bool # max :: Scientific > Scientific > Scientific # min :: Scientific > Scientific > Scientific #  
Read Scientific Source #  Supports the skipping of parentheses and whitespaces. Example: > read " ( (( 1.0e+3 ) ))" :: Scientific 1000.0 (Note: This 
Defined in Data.Scientific readsPrec :: Int > ReadS Scientific # readList :: ReadS [Scientific] # readPrec :: ReadPrec Scientific # readListPrec :: ReadPrec [Scientific] #  
Real Scientific Source #  WARNING: Avoid applying 
Defined in Data.Scientific toRational :: Scientific > Rational #  
RealFrac Scientific Source #  WARNING: the methods of the 
Defined in Data.Scientific properFraction :: Integral b => Scientific > (b, Scientific) # truncate :: Integral b => Scientific > b # round :: Integral b => Scientific > b # ceiling :: Integral b => Scientific > b # floor :: Integral b => Scientific > b #  
Show Scientific Source #  See 
Defined in Data.Scientific showsPrec :: Int > Scientific > ShowS # show :: Scientific > String # showList :: [Scientific] > ShowS #  
Binary Scientific Source #  Note that in the future I intend to change the type of the 
Defined in Data.Scientific  
NFData Scientific Source #  
Defined in Data.Scientific rnf :: Scientific > () #  
Hashable Scientific Source #  A hash can be safely calculated from a

Defined in Data.Scientific hashWithSalt :: Int > Scientific > Int # hash :: Scientific > Int #  
Lift Scientific Source #  Since: 0.3.7.0 
Defined in Data.Scientific lift :: Scientific > Q Exp # liftTyped :: Scientific > Q (TExp Scientific) # 
Construction
scientific :: Integer > Int > Scientific Source #
scientific c e
constructs a scientific number which corresponds
to the Fractional
number:
.fromInteger
c * 10 ^^
e
Projections
coefficient :: Scientific > Integer Source #
The coefficient of a scientific number.
Note that this number is not necessarily normalized, i.e. it could contain trailing zeros.
Scientific numbers are automatically normalized when pretty printed or
in toDecimalDigits
.
Use normalize
to do manual normalization.
WARNING: coefficient
and base10exponent
violate
substantivity of Eq
.
>>>
let x = scientific 1 2
>>>
let y = scientific 100 0
>>>
x == y
True
but
>>>
(coefficient x == coefficient y, base10Exponent x == base10Exponent y)
(False,False)
base10Exponent :: Scientific > Int Source #
The base10 exponent of a scientific number.
Predicates
isFloating :: Scientific > Bool Source #
Return True
if the scientific is a floating point, False
otherwise.
Also see: floatingOrInteger
.
isInteger :: Scientific > Bool Source #
Return True
if the scientific is an integer, False
otherwise.
Also see: floatingOrInteger
.
Conversions
Rational
unsafeFromRational :: Rational > Scientific Source #
Although fromRational
is unsafe because it will throw errors on
repeating decimals,
unsafeFromRational
is even more unsafe because it will diverge instead (i.e
loop and consume all space). Though it will be more efficient because it
doesn't need to consume space linear in the number of digits in the resulting
scientific to detect the repetition.
Consider using fromRationalRepetend
for these rationals which will detect
the repetition and indicate where it starts.
:: Maybe Int  Optional limit 
> Rational  
> Either (Scientific, Rational) (Scientific, Maybe Int) 
Like fromRational
and unsafeFromRational
, this function converts a
Rational
to a Scientific
but instead of failing or diverging (i.e loop
and consume all space) on
repeating decimals
it detects the repeating part, the repetend, and returns where it starts.
To detect the repetition this function consumes space linear in the number of
digits in the resulting scientific. In order to bound the space usage an
optional limit can be specified. If the number of digits reaches this limit
Left (s, r)
will be returned. Here s
is the Scientific
constructed so
far and r
is the remaining Rational
. toRational s + r
yields the
original Rational
If the limit is not reached or no limit was specified Right (s,
mbRepetendIx)
will be returned. Here s
is the Scientific
without any
repetition and mbRepetendIx
specifies if and where in the fractional part
the repetend begins.
For example:
fromRationalRepetend Nothing (1 % 28) == Right (3.571428e2, Just 2)
This represents the repeating decimal: 0.03571428571428571428...
which is sometimes also unambiguously denoted as 0.03(571428)
.
Here the repetend is enclosed in parentheses and starts at the 3rd digit (index 2)
in the fractional part. Specifying a limit results in the following:
fromRationalRepetend (Just 4) (1 % 28) == Left (3.5e2, 1 % 1400)
You can expect the following property to hold.
forall (mbLimit :: Maybe Int) (r :: Rational). r == (casefromRationalRepetend
mbLimit r of Left (s, r') > toRational s + r' Right (s, mbRepetendIx) > case mbRepetendIx of Nothing > toRational s Just repetendIx >toRationalRepetend
s repetendIx)
fromRationalRepetendLimited Source #
:: Int  limit 
> Rational  
> Either (Scientific, Rational) (Scientific, Maybe Int) 
Like fromRationalRepetend
but always accepts a limit.
fromRationalRepetendUnlimited :: Rational > (Scientific, Maybe Int) Source #
Like fromRationalRepetend
but doesn't accept a limit.
:: Scientific  
> Int  Repetend index 
> Rational 
Converts a Scientific
with a repetend (a repeating part in the fraction),
which starts at the given index, into its corresponding Rational
.
For example to convert the repeating decimal 0.03(571428)
you would use:
toRationalRepetend 0.03571428 2 == 1 % 28
Preconditions for toRationalRepetend s r
:
r >= 0
r < (base10Exponent s)
WARNING: toRationalRepetend
needs to compute the Integer
magnitude:
10^^n
. Where n
is based on the base10Exponent
of the scientific. If
applied to a huge exponent this could fill up all space and crash your
program! So don't apply this function to untrusted input.
The formula to convert the Scientific
s
with a repetend starting at index r
is described in the paper:
turning_repeating_decimals_into_fractions.pdf
and is defined as follows:
(fromInteger nonRepetend + repetend % nines) / fromInteger (10^^r) where c = coefficient s e = base10Exponent s  Size of the fractional part. f = (e)  Size of the repetend. n = f  r m = 10^^n (nonRepetend, repetend) = c `quotRem` m nines = m  1
Also see: fromRationalRepetend
.
Floating & integer
floatingOrInteger :: (RealFloat r, Integral i) => Scientific > Either r i Source #
floatingOrInteger
determines if the scientific is floating point or
integer.
In case it's floatingpoint the scientific is converted to the desired
RealFloat
using toRealFloat
and wrapped in Left
.
In case it's integer to scientific is converted to the desired Integral
and
wrapped in Right
.
WARNING: To convert the scientific to an integral the magnitude 10^e
needs to be computed. If applied to a huge exponent this could take a long
time. Even worse, when the destination type is unbounded (i.e. Integer
) it
could fill up all space and crash your program! So don't apply this function
to untrusted input but use toBoundedInteger
instead.
Also see: isFloating
or isInteger
.
toRealFloat :: RealFloat a => Scientific > a Source #
Safely convert a Scientific
number into a RealFloat
(like a Double
or a
Float
).
Note that this function uses realToFrac
(
)
internally but it guards against computing huge Integer magnitudes (fromRational
. toRational
10^e
)
that could fill up all space and crash your program. If the base10Exponent
of the given Scientific
is too big or too small to be represented in the
target type, Infinity or 0 will be returned respectively. Use
toBoundedRealFloat
which explicitly handles this case by returning Left
.
Always prefer toRealFloat
over realToFrac
when converting from scientific
numbers coming from an untrusted source.
toBoundedRealFloat :: forall a. RealFloat a => Scientific > Either a a Source #
Preciser version of toRealFloat
. If the base10Exponent
of the given
Scientific
is too big or too small to be represented in the target type,
Infinity or 0 will be returned as Left
.
toBoundedInteger :: forall i. (Integral i, Bounded i) => Scientific > Maybe i Source #
Convert a Scientific
to a bounded integer.
If the given Scientific
doesn't fit in the target representation, it will
return Nothing
.
This function also guards against computing huge Integer magnitudes (10^e
)
that could fill up all space and crash your program.
fromFloatDigits :: RealFloat a => a > Scientific Source #
Convert a RealFloat
(like a Double
or Float
) into a Scientific
number.
Note that this function uses floatToDigits
to compute the digits
and exponent of the RealFloat
number. Be aware that the algorithm used in
floatToDigits
doesn't work as expected for some numbers, e.g. as
the Double
1e23
is converted to 9.9999999999999991611392e22
, and that
value is shown as 9.999999999999999e22
rather than the shorter 1e23
; the
algorithm doesn't take the rounding direction for values exactly halfway
between two adjacent representable values into account, so if you have a
value with a short decimal representation exactly halfway between two
adjacent representable values, like 5^23*2^e
for e
close to 23, the
algorithm doesn't know in which direction the short decimal representation
would be rounded and computes more digits
Parsing
scientificP :: ReadP Scientific Source #
A parser for parsing a floatingpoint
number into a Scientific
value. Example:
> import Text.ParserCombinators.ReadP (readP_to_S) > readP_to_S scientificP "3" [(3.0,"")] > readP_to_S scientificP "3.0e2" [(3.0,"e2"),(300.0,"")] > readP_to_S scientificP "+3.0e+2" [(3.0,"e+2"),(300.0,"")] > readP_to_S scientificP "3.0e2" [(3.0,"e2"),(3.0e2,"")]
Note: This parser only parses the number itself; it does not parse any surrounding parentheses or whitespaces.
Pretty printing
:: FPFormat  
> Maybe Int  Number of decimal places to render. 
> Scientific  
> String 
Like show
but provides rendering options.
Control the rendering of floating point numbers.
Exponent  Scientific notation (e.g. 
Fixed  Standard decimal notation. 
Generic  Use decimal notation for values between 
Instances
Enum FPFormat  
Defined in Data.Text.Lazy.Builder.RealFloat  
Read FPFormat  
Show FPFormat  
toDecimalDigits :: Scientific > ([Int], Int) Source #
Similar to floatToDigits
, toDecimalDigits
takes a
positive Scientific
number, and returns a list of digits and
a base10 exponent. In particular, if x>=0
, and
toDecimalDigits x = ([d1,d2,...,dn], e)
then
n >= 1
x = 0.d1d2...dn * (10^^e)
0 <= di <= 9
null $ takeWhile (==0) $ reverse [d1,d2,...,dn]
The last property means that the coefficient will be normalized, i.e. doesn't contain trailing zeros.
Normalization
normalize :: Scientific > Scientific Source #
Normalize a scientific number by dividing out powers of 10 from the
coefficient
and incrementing the base10Exponent
each time.
You should rarely have a need for this function since scientific numbers are
automatically normalized when prettyprinted and in toDecimalDigits
.