Safe Haskell	None
Language	Haskell2010

Data.CReal.Internal

Contents

The CReal type
- Simple utilities
More efficient variants of common functions
Utilities for operating inside CReals
- Integer operations
Utilities for converting CReals to Strings

Description

This module exports a bunch of utilities for working inside the CReal datatype. One should be careful to maintain the CReal invariant when using these functions

Synopsis

The CReal type

newtype CReal n Source

The type CReal represents a fast binary Cauchy sequence. This is a Cauchy sequence with the invariant that the pth element will be within 2^-p of the true value. Internally this sequence is represented as a function from Ints to Integers.

Constructors

CR (Int -> Integer)

Instances

KnownNat n => Eq (CReal n) Source	Values of type `CReal p` are compared for equality at precision `p`. This may cause values which differ by less than 2^-p to compare as equal. `>>>` `0 == (0.1 :: CReal 1)`True
Methods (==) :: CReal n -> CReal n -> Bool (/=) :: CReal n -> CReal n -> Bool
Floating (CReal n) Source
Methods pi :: CReal n exp :: CReal n -> CReal n log :: CReal n -> CReal n sqrt :: CReal n -> CReal n (**) :: CReal n -> CReal n -> CReal n logBase :: CReal n -> CReal n -> CReal n sin :: CReal n -> CReal n cos :: CReal n -> CReal n tan :: CReal n -> CReal n asin :: CReal n -> CReal n acos :: CReal n -> CReal n atan :: CReal n -> CReal n sinh :: CReal n -> CReal n cosh :: CReal n -> CReal n tanh :: CReal n -> CReal n asinh :: CReal n -> CReal n acosh :: CReal n -> CReal n atanh :: CReal n -> CReal n
Fractional (CReal n) Source	Taking the reciprocal of zero will not terminate
Methods (/) :: CReal n -> CReal n -> CReal n recip :: CReal n -> CReal n fromRational :: Rational -> CReal n
Num (CReal n) Source	`signum (x :: CReal p)` returns the sign of `x` at precision `p`. It's important to remember that this may not represent the actual sign of `x` if the distance between `x` and zero is less than 2^-`p`. This is a little bit of a fudge, but it's probably better than failing to terminate when trying to find the sign of zero. The class still respects the abs-signum law though. `>>>` `signum (0.1 :: CReal 2)`0.0 `>>>` `signum (0.1 :: CReal 3)`1.0
Methods (+) :: CReal n -> CReal n -> CReal n (-) :: CReal n -> CReal n -> CReal n (*) :: CReal n -> CReal n -> CReal n negate :: CReal n -> CReal n abs :: CReal n -> CReal n signum :: CReal n -> CReal n fromInteger :: Integer -> CReal n
KnownNat n => Ord (CReal n) Source	Like equality values of type `CReal p` are compared at precision `p`.
Methods compare :: CReal n -> CReal n -> Ordering (<) :: CReal n -> CReal n -> Bool (<=) :: CReal n -> CReal n -> Bool (>) :: CReal n -> CReal n -> Bool (>=) :: CReal n -> CReal n -> Bool max :: CReal n -> CReal n -> CReal n min :: CReal n -> CReal n -> CReal n
KnownNat n => Read (CReal n) Source	The instance of Read will read an optionally signed number expressed in decimal scientific notation
Methods readsPrec :: Int -> ReadS (CReal n) readList :: ReadS [CReal n] readPrec :: ReadPrec (CReal n) readListPrec :: ReadPrec [CReal n]
KnownNat n => Real (CReal n) Source	`toRational` returns the CReal n evaluated at a precision of 2^-n
Methods toRational :: CReal n -> Rational
KnownNat n => RealFloat (CReal n) Source	Several of the functions in this class (`floatDigits`, `floatRange`, `exponent`, `significand`) only make sense for floats represented by a mantissa and exponent. These are bound to error. `atan2 y x atPrecision p` performs the comparison to determine the quadrant at precision p. This can cause atan2 to be slightly slower than atan
Methods floatRadix :: CReal n -> Integer floatDigits :: CReal n -> Int floatRange :: CReal n -> (Int, Int) decodeFloat :: CReal n -> (Integer, Int) encodeFloat :: Integer -> Int -> CReal n exponent :: CReal n -> Int significand :: CReal n -> CReal n scaleFloat :: Int -> CReal n -> CReal n isNaN :: CReal n -> Bool isInfinite :: CReal n -> Bool isDenormalized :: CReal n -> Bool isNegativeZero :: CReal n -> Bool isIEEE :: CReal n -> Bool atan2 :: CReal n -> CReal n -> CReal n
KnownNat n => RealFrac (CReal n) Source
Methods properFraction :: Integral b => CReal n -> (b, CReal n) truncate :: Integral b => CReal n -> b round :: Integral b => CReal n -> b ceiling :: Integral b => CReal n -> b floor :: Integral b => CReal n -> b
KnownNat n => Show (CReal n) Source	A CReal with precision p is shown as a decimal number d such that d is within 2^-p of the true value. `>>>` `show (47176870 :: CReal 0)`"47176870" `>>>` `show (pi :: CReal 230)`"3.1415926535897932384626433832795028841971693993751058209749445923078164"
Methods showsPrec :: Int -> CReal n -> ShowS show :: CReal n -> String showList :: [CReal n] -> ShowS

Simple utilities

atPrecision :: CReal n -> Int -> Integer Source

x `atPrecision` p returns the numerator of the pth element in the Cauchy sequence represented by x. The denominator is 2^p.

>>> 10 `atPrecision` 10
10240

crealPrecision :: KnownNat n => CReal n -> Int Source

crealPrecision x returns the type level parameter representing x's default precision.

>>> crealPrecision (1 :: CReal 10)
10

More efficient variants of common functions

Multiplicative

mulBounded :: CReal n -> CReal n -> CReal n infixl 7 Source

A more efficient multiply with the restriction that both values must be in the closed range [-1..1]

(.*.) :: CReal n -> CReal n -> CReal n infixl 7 Source

Alias for mulBoundedL

mulBoundedL :: CReal n -> CReal n -> CReal n infixl 7 Source

A more efficient multiply with the restriction that the first argument must be in the closed range [-1..1]

(.*) :: CReal n -> CReal n -> CReal n infixl 7 Source

Alias for mulBoundedL

(*.) :: CReal n -> CReal n -> CReal n infixl 7 Source

Alias for flip mulBoundedL

recipBounded :: CReal n -> CReal n Source

A more efficient recip with the restriction that the input must have absolute value greater than or equal to 1

shiftL :: CReal n -> Int -> CReal n infixl 8 Source

x `shiftL` n is equal to x multiplied by 2^n

n can be negative or zero

This can be faster than doing the multiplication

shiftR :: CReal n -> Int -> CReal n infixl 8 Source

x `shiftR` n is equal to x divided by 2^n

n can be negative or zero

This can be faster than doing the division

square :: CReal n -> CReal n Source

Return the square of the input, more efficient than (*)

Exponential

expBounded :: CReal n -> CReal n Source

A more efficient exp with the restriction that the input must be in the closed range [-1..1]

expPosNeg :: CReal n -> (CReal n, CReal n) Source

expPosNeg x returns @(exp x, exp (-x))#

logBounded :: CReal n -> CReal n Source

A more efficient log with the restriction that the input must be in the closed range [2/3..2]

Trigonometric

atanBounded :: CReal n -> CReal n Source

A more efficient atan with the restriction that the input must be in the closed range [-1..1]

sinBounded :: CReal n -> CReal n Source

A more efficient sin with the restriction that the input must be in the closed range [-1..1]

cosBounded :: CReal n -> CReal n Source

A more efficient cos with the restriction that the input must be in the closed range [-1..1]

Utilities for operating inside CReals

crMemoize :: (Int -> Integer) -> CReal n Source

crMemoize takes a fast binary Cauchy sequence and returns a CReal represented by that sequence which will memoize the values at each precision. This is essential for getting good performance.

powerSeries :: [Rational] -> (Int -> Int) -> CReal n -> CReal n Source

powerSeries q f x atPrecision p will evaluate the power series with coefficients q up to the coefficient at index f p at value x

f should be a function such that the CReal invariant is maintained. This means that if the power series y = a[0] + a[1] + a[2] + ... is evaluated at precision p then the sum of every a[n] for n > f p must be less than 2^-p.

This is used by all the bounded transcendental functions.

>>> let (!) x = product [2..x]
>>> powerSeries [1 % (n!) | n <- [0..]] (max 5) 1 :: CReal 218
2.718281828459045235360287471352662497757247093699959574966967627724

alternateSign :: Num a => [a] -> [a] Source

Apply negate to every other element, starting with the second