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

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

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)

The base-10 exponent of a scientific number.

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.

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.

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 (3.571428e-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)

Like fromRationalRepetend but always accepts a limit.

Like fromRationalRepetend but doesn't accept a limit.

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.

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 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.

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.

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.

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.

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

A parser for parsing a floating-point 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 "-3.0e-2"
[(-3.0,"e-2"),(-3.0e-2,"")]

Note: This parser only parses the number itself; it does not parse any surrounding parentheses or whitespaces.

Like show but provides rendering options.

Control the rendering of floating point numbers.

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.

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.