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

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

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

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

Alias for mulBoundedL

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

Alias for mulBoundedL

Alias for flip mulBoundedL

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

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

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

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

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

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

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

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

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

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

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 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

Apply negate to every other element, starting with the second

>>> alternateSign [1..5]
[1,-2,3,-4,5]

Division rounding to the nearest integer and rounding half integers to the nearest even integer.

log2 x returns the base 2 logarithm of x rounded towards zero.

The input must be positive

log10 x returns the base 10 logarithm of x rounded towards zero.

The input must be positive

isqrt x returns the square root of x rounded towards zero.

The input must not be negative

Return a string representing a decimal number within 2^-p of the value represented by the given CReal p.

How many decimal digits are required to represent a number to within 2^-p

rationalToDecimal p x returns a string representing x at p decimal places.