scientific-0.3.4.0: Numbers represented using scientific notation

Copyright Bas van Dijk 2013 BSD3 Bas van Dijk None Haskell2010

Data.Scientific

Description

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`

Synopsis

# Documentation

data Scientific Source

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

 Source Source 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. Source Source Source Source Source 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. Source Source Source Source Source

# Construction

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

# Projections

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.

The base-10 exponent of a scientific number.

# Predicates

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

Also see: `floatingOrInteger`.

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

Also see: `floatingOrInteger`.

# Conversions

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)
```

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 Source

`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 Source

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 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 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

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

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

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 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`.