Data.Scientific

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 the gcd of the numerator and denominator.
• Scientific is safe against numbers with huge exponents. For example: 1e1000000000 :: Rational will fill up all space and crash your program. Scientific works as expected:

> read "1e1000000000" :: Scientific
1.0e1000000000

• Also, the space usage of converting scientific numbers with huge exponents to Integrals (like: Int) or RealFloats (like: Double or Float) 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

Documentation

data Scientific

An arbitrary-precision number represented using scientific notation.

This type describes the set of all Reals which have a finite decimal expansion.

A scientific number with coefficient c and base10Exponent e corresponds to the Fractional number: fromInteger c * 10 ^^ e

Instances

Eq Scientific

Methods

(==) :: Scientific -> Scientific -> Bool

(/=) :: Scientific -> Scientific -> Bool

Fractional Scientific

WARNING: recip and / will diverge (i.e. loop and consume all space) when their outputs are repeating decimals.

fromRational will diverge when the input Rational is a repeating decimal. Consider using fromRationalRepetend for these rationals which will detect the repetition and indicate where it starts.

Methods

(/) :: Scientific -> Scientific -> Scientific

recip :: Scientific -> Scientific

fromRational :: Rational -> Scientific

Data Scientific

Methods

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 :: (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

Methods

(+) :: Scientific -> Scientific -> Scientific

(-) :: Scientific -> Scientific -> Scientific

(*) :: Scientific -> Scientific -> Scientific

negate :: Scientific -> Scientific

abs :: Scientific -> Scientific

signum :: Scientific -> Scientific

fromInteger :: Integer -> Scientific

Ord Scientific

Methods

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

Methods

readsPrec :: Int -> ReadS Scientific

readList :: ReadS [Scientific]

readPrec :: ReadPrec Scientific

readListPrec :: ReadPrec [Scientific]

Real Scientific

WARNING: toRational needs to compute the Integer magnitude: 10^e. If applied to a huge exponent this could fill up all space and crash your program!

Avoid applying toRational (or realToFrac) to scientific numbers coming from an untrusted source and use toRealFloat instead. The latter guards against excessive space usage.

Methods

toRational :: Scientific -> Rational

RealFrac Scientific

Methods

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

Methods

showsPrec :: Int -> Scientific -> ShowS

show :: Scientific -> String

showList :: [Scientific] -> ShowS

Binary Scientific

Methods

put :: Scientific -> Put

get :: Get Scientific

putList :: [Scientific] -> Put

NFData Scientific

Methods

rnf :: Scientific -> ()

Hashable Scientific

Methods

hashWithSalt :: Int -> Scientific -> Int

hash :: Scientific -> Int

Construction

scientific :: Integer -> Int -> Scientific

scientific c e constructs a scientific number which corresponds to the Fractional number: fromInteger c * 10 ^^ e.

Projections

coefficient :: Scientific -> Integer

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

The base-10 exponent of a scientific number.

Predicates

isFloating :: Scientific -> Bool

Return True if the scientific is a floating point, False otherwise.

Also see: floatingOrInteger.

isInteger :: Scientific -> Bool

Return True if the scientific is an integer, False otherwise.

Also see: floatingOrInteger.

Conversions

fromRationalRepetend

Arguments

:: 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 (3.571428e-2, 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 (3.5e-2, 1 % 1400)

You can expect the following property to hold.

 forall (mbLimit :: Maybe Int) (r :: Rational).
 r == (case fromRationalRepetend mbLimit r of
        Left (s, r') -> toRational s + r'
        Right (s, mbRepetendIx) ->
          case mbRepetendIx of
            Nothing         -> toRational s
            Just repetendIx -> toRationalRepetend s repetendIx)

toRationalRepetend

Arguments

:: 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

floatingOrInteger determines if the scientific is floating point or integer. In case it's floating-point the scientific is converted to the desired RealFloat using toRealFloat.

Also see: isFloating or isInteger.

toRealFloat :: RealFloat a => Scientific -> a

Safely convert a Scientific number into a RealFloat (like a Double or a Float).

Note that this function uses realToFrac (fromRational . toRational) internally but it guards against computing huge Integer magnitudes (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

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

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

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 half-way between two adjacent representable values into account, so if you have a value with a short decimal representation exactly half-way 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

formatScientific

Arguments

:: 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.

Constructors

Exponent

Scientific notation (e.g. 2.3e123).

Fixed

Standard decimal notation.

Generic

Use decimal notation for values between 0.1 and 9,999,999, and scientific notation otherwise.

Instances

Enum FPFormat

Methods

succ :: FPFormat -> FPFormat

pred :: FPFormat -> FPFormat

toEnum :: Int -> FPFormat

fromEnum :: FPFormat -> Int

enumFrom :: FPFormat -> [FPFormat]

enumFromThen :: FPFormat -> FPFormat -> [FPFormat]

enumFromTo :: FPFormat -> FPFormat -> [FPFormat]

enumFromThenTo :: FPFormat -> FPFormat -> FPFormat -> [FPFormat]

Read FPFormat

Methods

readsPrec :: Int -> ReadS FPFormat

readList :: ReadS [FPFormat]

readPrec :: ReadPrec FPFormat

readListPrec :: ReadPrec [FPFormat]

Show FPFormat

Methods

showsPrec :: Int -> FPFormat -> ShowS

show :: FPFormat -> String

showList :: [FPFormat] -> ShowS

toDecimalDigits :: Scientific -> ([Int], Int)

Similar to floatToDigits, toDecimalDigits takes a positive Scientific number, and returns a list of digits and a base-10 exponent. In particular, if x>=0, and

toDecimalDigits x = ([d1,d2,...,dn], e)

then

1. n >= 1

2. x = 0.d1d2...dn * (10^^e)

3. 0 <= di <= 9

4. 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

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 pretty-printed and in toDecimalDigits.