Copyright  Bas van Dijk 2013 

License  BSD3 
Maintainer  Bas van Dijk <v.dijk.bas@gmail.com> 
Safe Haskell  None 
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.
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.
WARNING: Although Scientific
is an instance of Fractional
, the methods
are only partially defined! Specifically recip
and /
will diverge
(i.e. loop and consume all space) when their outputs have an infinite decimal
expansion. fromRational
will diverge when the input Rational
has an
infinite decimal expansion. Consider using fromRationalRepetend
for these
rationals which will detect the repetition and indicate where it starts.
This module is designed to be imported qualified:
import Data.Scientific as Scientific
 data Scientific
 scientific :: Integer > Int > Scientific
 coefficient :: Scientific > Integer
 base10Exponent :: Scientific > Int
 isFloating :: Scientific > Bool
 isInteger :: Scientific > Bool
 fromRationalRepetend :: Maybe Int > Rational > Either (Scientific, 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
 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
Eq Scientific Source  
Fractional Scientific Source  WARNING:

Data Scientific Source  
Num Scientific Source  
Ord Scientific Source  
Read Scientific Source  
Real Scientific Source  WARNING: Avoid applying 
RealFrac Scientific Source  
Show Scientific Source  
Binary Scientific Source  
NFData Scientific Source  
Hashable Scientific Source 
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.
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
:: Maybe Int  Optional limit 
> Rational  
> Either (Scientific, Rational) (Scientific, Maybe Int) 
Like fromRational
, this function converts a Rational
to a Scientific
but instead of 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)
:: 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)
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
.
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
.
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
Pretty printing
:: FPFormat  
> Maybe Int  Number of decimal places to render. 
> Scientific  
> String 
Like show
but provides rendering options.
data FPFormat :: *
Control the rendering of floating point numbers.
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
.