scientific- Numbers represented using scientific notation

MaintainerBas van Dijk <>
Safe HaskellNone




Data.Scientific provides a space efficient and arbitrary precision scientific number type.

Scientific numbers are represented using scientific notation. It uses an Integer coefficient c and an Int base10Exponent e (do note that since we're using an Int to represent the exponent these numbers aren't truly arbitrary precision). A scientific number corresponds to the Fractional number: fromInteger c * 10 ^^ e.

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
  • 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.

Note that, in order to do fast magnitude computations (10^e), this module computes the first 1100 powers of 10 and stores them in a top-level array. This means that the first magnitude computation (used in realToFrac among others) is slower but subsequent computations should be O(1).

This module is designed to be imported qualified:

import Data.Scientific as Scientific



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


Eq Scientific 
Fractional Scientific

WARNING: recip and / will diverge when their outputs have an infinite decimal expansion. fromRational will diverge when the input Rational has an infinite decimal expansion.

Data Scientific 
Num Scientific 
Ord Scientific 
Read 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.

RealFrac Scientific 
Show Scientific 
Typeable Scientific 
NFData Scientific 
Hashable Scientific 


scientific :: Integer -> Int -> ScientificSource

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


coefficient :: Scientific -> IntegerSource

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 -> IntSource

The base-10 exponent of a scientific number.


fromFloatDigits :: RealFloat a => a -> ScientificSource

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

toRealFloat :: forall a. RealFloat a => Scientific -> aSource

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.

Always prefer toRealFloat over realToFrac when converting from scientific numbers coming from an untrusted source.

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.



Scientific notation (e.g. 2.3e123).


Standard decimal notation.


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

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

Similar to floatToDigits, toDecimalDigits takes a non-negative 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)


  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 :: Scientific -> ScientificSource

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.