GHC.Num.Integer

Compute absolute value of an Integer

Add two Integers

Bitwise AND operation

Fake 2's complement for negative values (might be slow)

Integer for which only n-th bit is set

Positive Integer for which only n-th bit is set

Check Integer invariants

Check Integer invariants

Compare two Integer

Binary complement of the

Decode a Double# into (# Integer mantissa, Int# exponent #)

Simultaneous integerDiv and integerMod.

Divisor must be non-zero otherwise the GHC runtime will terminate with a division-by-zero fault.

Simultaneous integerDiv and integerMod.

Divisor must be non-zero otherwise the GHC runtime will terminate with a division-by-zero fault.

Encode (Integer mantissa, Int exponent) into a Double

Encode (# Integer mantissa, Int# exponent #) into a Double#

Encode (# Integer mantissa, Int# exponent #) into a Float#

TODO: Not sure if it's worth to write Float optimized versions here

Equal predicate.

Equal predicate.

Read an Integer (without sign) in base-256 representation from an Addr#.

The size is given in bytes.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Null higher limbs are automatically trimed.

Read an Integer (without sign) in base-256 representation from an Addr#.

The size is given in bytes.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Null higher limbs are automatically trimed.

Create a positive Integer from a BigNat

Create a negative Integer from a BigNat

Create an Integer from a sign-bit and a BigNat

Read an Integer (without sign) in base-256 representation from a ByteArray#.

The size is given in bytes.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Null higher limbs are automatically trimed.

Read an Integer (without sign) in base-256 representation from a ByteArray#.

The size is given in bytes.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Null higher limbs are automatically trimed.

Create an Integer from an Int

Create an Integer from an Int#

Convert an Int64# into an Integer

Convert a Natural into an Integer

Convert a Word into an Integer

Convert a Word# into an Integer

Convert a Word64# into an Integer

Convert a list of Word into an Integer

Create a negative Integer with the given Word magnitude

Create an Integer from a sign and a Word magnitude

Compute greatest common divisor.

Get the extended GCD of two integers.

`integerGcde a b` returns (g,x,y) where * ax + by = g = |gcd a b|

Get the extended GCD of two integers.

`integerGcde# a b` returns (# g,x,y #) where * ax + by = g = |gcd a b|

Greater-or-equal predicate.

Greater-or-equal predicate.

Greater predicate.

Greater predicate.

Negative predicate

Negative predicate

One predicate

Indicate if the value is a power of two and which one

Zero predicate

Compute least common multiple.

Lower-or-equal predicate.

Lower-or-equal predicate.

Base 2 logarithm (floor)

For numbers <= 0, return 0

Base 2 logarithm (floor)

For numbers <= 0, return 0

Logarithm (floor) for an arbitrary base

For numbers <= 0, return 0

Logarithm (floor) for an arbitrary base

For numbers <= 0, return 0

Logarithm (floor) for an arbitrary base

For numbers <= 0, return 0

Logarithm (floor) for an arbitrary base

For numbers <= 0, return 0

Lower predicate.

Lower predicate.

Multiply two Integers

Not-equal predicate.

Not-equal predicate.

Negate Integer.

One edge-case issue to take into account is that Int's range is not symmetric around 0. I.e. minBound+maxBound = -1

IP is used iff n > maxBound::Int IN is used iff n < minBound::Int

Integer One

Bitwise OR operation

Fake 2's complement for negative values (might be slow)

Count number of set bits. For negative arguments returns the negated population count of the absolute value.

Computes the modular exponentiation.

integerPowMod# b e m behaves as follows:

• If m > 1 and e >= 0, it returns an integer y with 0 <= y < m and y congruent to b^e modulo m.

• If m > 1 and e < 0, it uses integerRecipMod# to try to find a modular multiplicative inverse b' (which only exists if gcd b m = 1) and then caculates (b')^(-e) modulo m (note that -e > 0); if the inverse does not exist then it fails.

• If m = 1, it returns 0 for all b and e.

• If m = 0, it fails.

NB. Successful evaluation returns a value of the form (# n | #); failure is indicated by returning (# | () #).

Simultaneous integerQuot and integerRem.

Divisor must be non-zero otherwise the GHC runtime will terminate with a division-by-zero fault.

Simultaneous integerQuot and integerRem.

Divisor must be non-zero otherwise the GHC runtime will terminate with a division-by-zero fault.

Computes the modular inverse.

integerRecipMod# x m behaves as follows:

• If m > 1 and gcd x m = 1, it returns an integer y with 0 < y < m such that x*y is congruent to 1 modulo m.

• If m > 1 and gcd x m > 1, it fails.

• If m = 1, it returns 0 for all x. The computation effectively takes place in the zero ring, which has a single element 0 with 0+0 = 0*0 = 0: the element 0 is the multiplicative identity element and is its own multiplicative inverse.

• If m = 0, it fails.

NB. Successful evaluation returns a value of the form (# n | #); failure is indicated by returning (# | () #).

Shift-left operation

Remember that bits are stored in sign-magnitude form, hence the behavior of negative Integers is different from negative Int's behavior.

Shift-left operation

Shift-right operation

Fake 2's complement for negative values (might be slow)

Shift-right operation

Fake 2's complement for negative values (might be slow)

Return -1, 0, and 1 depending on whether argument is negative, zero, or positive, respectively

Return -1#, 0#, and 1# depending on whether argument is negative, zero, or positive, respectively

Compute the number of digits of the Integer (without the sign) in the given base.

base must be > 1

Square a Integer

Subtract one Integer from another.

Test if n-th bit is set. For negative Integers it tests the n-th bit of the negated argument.

Fake 2's complement for negative values (might be slow)

Test if n-th bit is set.

Fake 2's complement for negative values (might be slow)

Write an Integer (without sign) to addr in base-256 representation and return the number of bytes written.

The endianness is selected with the Bool# parameter: write most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Write an Integer (without sign) to addr in base-256 representation and return the number of bytes written.

The endianness is selected with the Bool# parameter: write most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Convert an Integer into a BigNat.

Return 0 for negative Integers.

Convert an Integer into a sign-bit and a BigNat

Truncates Integer to least-significant Int#

Truncates Integer to least-significant Int#

Convert an Integer into an Int64#

Write an Integer (without sign) in base-256 representation and return the number of bytes written.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Write an Integer (without sign) in base-256 representation and return the number of bytes written.

The endianness is selected with the Bool# parameter: most significant byte first (big-endian) if 1# or least significant byte first (little-endian) if 0#.

Convert an Integer into a Natural

Return absolute value

Convert an Integer into a Natural

Return 0 for negative Integers.

Convert an Integer into a Natural

Throw an Underflow exception if input is negative.

Truncate an Integer into a Word

Truncate an Integer into a Word

Convert an Integer into a Word64#

Bitwise XOR operation

Fake 2's complement for negative values (might be slow)

Integer Zero

Arbitrary precision integers. In contrast with fixed-size integral types such as Int, the Integer type represents the entire infinite range of integers.

Integers are stored in a kind of sign-magnitude form, hence do not expect two's complement form when using bit operations.

If the value is small (i.e., fits into an Int), the IS constructor is used. Otherwise IP and IN constructors are used to store a BigNat representing the positive or the negative value magnitude, respectively.

Invariant: IP and IN are used iff the value does not fit in IS.