Module malachite_base::num::arithmetic::traits

Various traits for performing arithmetic operations on numbers.

Traits

• Abs
  Takes the absolute value of a number. Assumes that the number has a representable absolute number.
• AbsAssign
  Replaces a number with its absolute value. Assumes that the number has a representable absolute number.
• AbsDiff
  Subtracts two numbers and takes the absolute value of the difference.
• AbsDiffAssign
  Replaces a number with the absolute value of its difference with another number.
• AddMul
  Adds a number and the product of two other numbers.
• AddMulAssign
  Adds a number and the product of two other numbers, in place.
• ArithmeticCheckedShl
  Left-shifts a number (multiplies it by a power of 2), returning None if the result is not representable.
• ArithmeticCheckedShr
  Right-shifts a number (divides it by a power of 2), returning None if the result is not representable.
• BinomialCoefficient
• Ceiling
  Takes the ceiling of a number.
• CeilingAssign
  Replaces a number with its ceiling.
• CeilingDivAssignMod
  Divides a number by another number in place, taking the quotient and returning the remainder. The quotient is rounded towards positive infinity and the remainder has the opposite sign of the divisor (second input).
• CeilingDivAssignNegMod
  Divides a number by another number in place, taking the ceiling of the quotient and returning the remainder of the negative of the first number divided by the second.
• CeilingDivMod
  Divides a number by another number, returning the quotient and remainder. The quotient is rounded towards positive infinity and the remainder has the opposite sign as the divisor (second input).
• CeilingDivNegMod
  Divides a number by another number, returning the ceiling of the quotient and the remainder of the negative of the first number divided by the second.
• CeilingLogBase
  Calculates the ceiling of the base-$b$ logarithm of a number.
• CeilingLogBase2
  Calculates the ceiling of the base-2 logarithm of a number.
• CeilingLogBasePowerOf2
  Calculates the ceiling of the base-$2^k$ logarithm of a number.
• CeilingMod
  Divides a number by another number, returning just the remainder. The remainder has the opposite sign as the divisor (second number).
• CeilingModAssign
  Divides a number by another number, replacing the first number by the remainder. The remainder has the same sign as the divisor (second number).
• CeilingModPowerOf2
  Divides a number by $2^k$, returning just the remainder. The remainder is non-positive.
• CeilingModPowerOf2Assign
  Divides a number by $2^k$, replacing the number by the remainder. The remainder is non-positive.
• CeilingRoot
  Finds the ceiling of the $n$th root of a number.
• CeilingRootAssign
  Replaces a number with the ceiling of its $n$th root.
• CeilingSqrt
  Finds the ceiling of the square root of a number.
• CeilingSqrtAssign
  Replaces a number with the ceiling of its square root.
• CheckedAbs
  Takes the absolute valie of a number, returning None if the result is not representable.
• CheckedAdd
  Adds two numbers, returning None if the result is not representable.
• CheckedAddMul
  Adds a number and the product of two other numbers, returning None if the result is not representable.
• CheckedBinomialCoefficient
• CheckedDiv
  Divides two numbers, returning None if the result is not representable.
• CheckedDoubleFactorial
  Computes the double factorial of a u64, returning None if the result is too large to be represented. The double factorial of a non-negative integer is the product of all the positive integers that are less than or equal to it and have the same parity as it.
• CheckedFactorial
  Computes the factorial of a u64, returning None if the result is too large to be represented.
• CheckedLcm
  Calculates the LCM (least common multiple) of two numbers, returning None if the result is not representable.
• CheckedLogBase
  Calculates the base-$b$ logarithm of a number, or returns None if the number is not a perfect power of $b$.
• CheckedLogBase2
  Calculates the base-2 logarithm of a number, or returns None if the number is not a perfect power of 2.
• CheckedLogBasePowerOf2
  Calculates the base-$2^k$ logarithm of a number, or returns None if the number is not a perfect power of $2^k$.
• CheckedMul
  Multiplies two numbers, returning None if the result is not representable.
• CheckedMultifactorial
  Computes the $m$-multifactorial of a u64, returning None if the result is too large to be represented. The $m$-multifactorial of a non-negative integer $n$ is the product of all integers $k$ such that $0<k\leq n$ and $k\equiv n \pmod m$.
• CheckedNeg
  Negates a number, returning None if the result is not representable.
• CheckedNextPowerOf2
  Finds the smallest integer power of 2 greater than or equal to a number, returning None if the result is not representable.
• CheckedPow
  Raises a number to a power, returning None if the result is not representable.
• CheckedPrimorial
• CheckedRoot
  Finds the $n$th root of a number, returning None if it is not a perfect $n$th power.
• CheckedSqrt
  Finds the square root of a number, returning None if it is not a perfect square.
• CheckedSquare
  Squares a number, returning None if the result is not representable.
• CheckedSub
  Subtracts two numbers, returning None if the result is not representable.
• CheckedSubMul
  Subtracts a number by the product of two other numbers, returning None if the result is not representable.
• CheckedSubfactorial
  Computes the subfactorial of a u64, returning None if the result is too large to be represented. The subfactorial of a non-negative integer $n$ counts the number of derangements of $n$ elements, which are the permutations in which no element is fixed.
• CoprimeWith
  Determines whether two numbers are coprime.
• DivAssignMod
  Divides a number by another number in place, returning the remainder. The quotient is rounded towards negative infinity, and the remainder has the same sign as the divisor (second input).
• DivAssignRem
  Divides a number by another number in place, returning the remainder. The quotient is rounded towards zero, and the remainder has the same sign as the dividend (first input).
• DivExact
  Divides two numbers, assuming the first exactly divides the second.
• DivExactAssign
  Divides a number by another number in place, assuming the first exactly divides the second.
• DivMod
  Divides two numbers, returning the quotient and remainder. The quotient is rounded towards negative infinity, and the remainder has the same sign as the divisor (second input).
• DivRem
  Divides two numbers, returning the quotient and remainder. The quotient is rounded towards zero, and the remainder has the same sign as the dividend (first input).
• DivRound
  Divides a number by another number and rounds according to a specified rounding mode. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the exact value.
• DivRoundAssign
  Divides a number by another number in place and rounds according to a specified rounding mode. An Ordering is returned, indicating whether the assigned value is less than, equal to, or greater than the exact value.
• DivisibleBy
  Determines whether a number is divisible by another number.
• DivisibleByPowerOf2
  Determines whether a number is divisible by $2^k$.
• DoubleFactorial
  Computes the double factorial of a u64. The double factorial of a non-negative integer is the product of all the positive integers that are less than or equal to it and have the same parity as it.
• EqMod
  Determines whether a number is equivalent to another number modulo $m$.
• EqModPowerOf2
  Determines whether a number is equivalent to another number modulo $2^k$.
• ExtendedGcd
  Computes the GCD (greatest common divisor) of two numbers $a$ and $b$, and also the coefficients $x$ and $y$ in Bézout’s identity $ax+by=\gcd(a,b)$.
• Factorial
  Computes the factorial of a u64.
• Floor
  Takes the floor of a number.
• FloorAssign
  Replaces a number with its floor.
• FloorLogBase
  Calculates the floor of the base-$b$ logarithm of a number.
• FloorLogBase2
  Calculates the floor of the base-2 logarithm of a number.
• FloorLogBasePowerOf2
  Calculates the floor of the base-$2^k$ logarithm of a number.
• FloorRoot
  Finds the floor of the $n$th root of a number.
• FloorRootAssign
  Replaces a number with the floor of its $n$th root.
• FloorSqrt
  Finds the floor of the square root of a number.
• FloorSqrtAssign
  Replaces a number with the floor of its square root.
• Gcd
  Calculates the GCD (greatest common divisor) of two numbers.
• GcdAssign
  Replaces a number with the GCD (greatest common divisor) of it and another number.
• IsPowerOf2
  Determines whether a number is an integer power of 2.
• JacobiSymbol
  Calculates the Jacobi symbol of two numbers.
• KroneckerSymbol
  Calculates the Kronecker symbol of two numbers.
• Lcm
  Calculates the LCM (least common multiple) of two numbers.
• LcmAssign
  Replaces a number with the LCM (least common multiple) of it and another number.
• LegendreSymbol
  Calculates the Legendre symbol of two numbers. Typically the implementations will be identical to those of JacobiSymbol.
• Ln
  Takes the natural logarithm of a number.
• Mod
  Divides a number by another number, returning just the remainder. The remainder has the same sign as the divisor (second number).
• ModAdd
  Adds two numbers modulo a third number $m$. The inputs must be already reduced modulo $m$.
• ModAddAssign
  Adds two numbers modulo a third number $m$, in place. The inputs must be already reduced modulo $m$.
• ModAssign
  Divides a number by another number, replacing the first number by the remainder. The remainder has the same sign as the divisor (second number).
• ModInverse
  Finds the multiplicative inverse of a number modulo another number $m$. The input must be already reduced modulo $m$.
• ModIsReduced
  Checks whether a number is reduced modulo another number $m$.
• ModMul
  Multiplies two numbers modulo a third number $m$. The inputs must be already reduced modulo $m$.
• ModMulAssign
  Multiplies two numbers modulo a third number $m$, in place. The inputs must be already reduced modulo $m$.
• ModMulPrecomputed
  Multiplies two numbers modulo a third number $m$. The inputs must be already reduced modulo $m$.
• ModMulPrecomputedAssign
  Multiplies two numbers modulo a third number $m$, in place.The inputs must be already reduced modulo $m$.
• ModNeg
  Negates a number modulo another number $m$. The input must be already reduced modulo $m$.
• ModNegAssign
  Negates a number modulo another number $m$, in place. The input must be already reduced modulo $m$.
• ModPow
  Raises a number to a power modulo another number $m$. The base must be already reduced modulo $m$.
• ModPowAssign
  Raises a number to a power modulo another number $m$, in place. The base must be already reduced modulo $m$.
• ModPowPrecomputed
  Raises a number to a power modulo another number $m$. The base must be already reduced modulo $m$.
• ModPowPrecomputedAssign
  Raises a number to a power modulo another number $m$, in place. The base must be already reduced modulo $m$.
• ModPowerOf2
  Divides a number by $2^k$, returning just the remainder. The remainder is non-negative.
• ModPowerOf2Add
  Adds two numbers modulo $2^k$. The inputs must be already reduced modulo $2^k$.
• ModPowerOf2AddAssign
  Adds two numbers modulo $2^k$, in place. The inputs must be already reduced modulo $2^k$.
• ModPowerOf2Assign
  Divides a number by $2^k$, replacing the number by the remainder. The remainder is non-negative.
• ModPowerOf2Inverse
  Finds the multiplicative inverse of a number modulo $2^k$. The input must be already reduced modulo $2^k$.
• ModPowerOf2IsReduced
  Checks whether a number is reduced modulo $2^k$.
• ModPowerOf2Mul
  Multiplies two numbers modulo $2^k$. The inputs must be already reduced modulo $2^k$.
• ModPowerOf2MulAssign
  Multiplies two numbers modulo $2^k$, in place. The inputs must be already reduced modulo $2^k$.
• ModPowerOf2Neg
  Negates a number modulo $2^k$. The input must be already reduced modulo $2^k$.
• ModPowerOf2NegAssign
  Negates a number modulo $2^k$ in place. The input must be already reduced modulo $2^k$.
• ModPowerOf2Pow
  Raises a number to a power modulo $2^k$. The base must be already reduced modulo $2^k$.
• ModPowerOf2PowAssign
  Raises a number to a power modulo $2^k$, in place. The base must be already reduced modulo $2^k$.
• ModPowerOf2Shl
  Left-shifts a number (multiplies it by a power of 2) modulo $2^k$. The number must be already reduced modulo $2^k$.
• ModPowerOf2ShlAssign
  Left-shifts a number (multiplies it by a power of 2) modulo $2^k$, in place. The number must be already reduced modulo $2^k$.
• ModPowerOf2Shr
  Right-shifts a number (divides it by a power of 2) modulo $2^k$. The number must be already reduced modulo $2^k$.
• ModPowerOf2ShrAssign
  Right-shifts a number (divides it by a power of 2) modulo $2^k$, in place. The number must be already reduced modulo $2^k$.
• ModPowerOf2Square
  Squares a number modulo $2^k$. The input must be already reduced modulo $2^k$.
• ModPowerOf2SquareAssign
  Squares a number modulo $2^k$ in place. The input must be already reduced modulo $2^k$.
• ModPowerOf2Sub
  Subtracts two numbers modulo $2^k$. The inputs must be already reduced modulo $2^k$.
• ModPowerOf2SubAssign
  Subtracts two numbers modulo $2^k$, in place. The inputs must be already reduced modulo $2^k$.
• ModShl
  Left-shifts a number (multiplies it by a power of 2) modulo another number $m$. The number must be already reduced modulo $m$.
• ModShlAssign
  Left-shifts a number (multiplies it by a power of 2) modulo another number $m$, in place. The number must be already reduced modulo $m$.
• ModShr
  Left-shifts a number (divides it by a power of 2) modulo another number $m$. The number must be already reduced modulo $m$.
• ModShrAssign
  Left-shifts a number (divides it by a power of 2) modulo another number $m$, in place. The number must be already reduced modulo $m$.
• ModSquare
  Squares a number modulo another number $m$. The input must be already reduced modulo $m$.
• ModSquareAssign
  Squares a number modulo another number $m$, in place. The input must be already reduced modulo $m$.
• ModSquarePrecomputed
  Squares a number modulo another number $m$. The input must be already reduced modulo $m$.
• ModSquarePrecomputedAssign
  Squares a number modulo another number $m$, in place. The input must be already reduced modulo $m$.
• ModSub
  Adds two numbers modulo a third number $m$. The inputs must be already reduced modulo $m$.
• ModSubAssign
  Adds two numbers modulo a third number $m$, in place. The inputs must be already reduced modulo $m$.
• Multifactorial
  Computes the $m$-multifactorial of a u64. The $m$-multifactorial of a non-negative integer $n$ is the product of all integers $k$ such that $0<k\leq n$ and $k\equiv n \pmod m$.
• NegAssign
  Replaces a number with its negative. Assumes the result is representable.
• NegMod
  Divides the negative of a number by another number, returning the remainder.
• NegModAssign
  Divides the negative of a number by another number, replacing the first number by the remainder.
• NegModPowerOf2
  Divides the negative of a number by $2^k$, returning the remainder.
• NegModPowerOf2Assign
  Divides the negative of a number by $2^k$, replacing the number by the remainder.
• NextPowerOf2
  Returns the smallest power of 2 greater than or equal to a number. Assumes the result is representable.
• NextPowerOf2Assign
  Replaces a number with the smallest power of 2 greater than or equal it. Assumes the result is representable.
• OverflowingAbs
  Takes the absolute value of a number.
• OverflowingAbsAssign
  Replaces a number with its absolute value.
• OverflowingAdd
  Adds two numbers.
• OverflowingAddAssign
  Adds a number to another number in place.
• OverflowingAddMul
  Adds a number and the product of two other numbers.
• OverflowingAddMulAssign
  Adds a number and the product of two other numbers, in place.
• OverflowingDiv
  Divides two numbers.
• OverflowingDivAssign
  Divides a number by another number in place.
• OverflowingMul
  Multiplies two numbers.
• OverflowingMulAssign
  Multiplies a number by another number in place.
• OverflowingNeg
  Negates a number.
• OverflowingNegAssign
  Negates a number in place.
• OverflowingPow
  Raises a number to a power.
• OverflowingPowAssign
  Raises a number to a power in place.
• OverflowingSquare
  Squares a number.
• OverflowingSquareAssign
  Squares a number in place.
• OverflowingSub
  Subtracts two numbers.
• OverflowingSubAssign
  Subtracts a number by another number in place.
• OverflowingSubMul
  Subtracts a number by the product of two other numbers.
• OverflowingSubMulAssign
  Subtracts a number by the product of two other numbers, in place.
• Parity
  Determines whether a number is even or odd.
• Pow
  Raises a number to a power. Assumes the result is representable.
• PowAssign
  Raises a number to a power in place. Assumes the result is representable.
• PowerOf2
  Raises 2 to a power.
• Primorial
• Reciprocal
  Finds the reciprocal (multiplicative inverse) of a number.
• ReciprocalAssign
  Replaces a number with its reciprocal (multiplicative inverse).
• RemPowerOf2
  Divides a number by $2^k$, returning just the remainder. The remainder has the same sign as the number.
• RemPowerOf2Assign
  Divides a number by $2^k$, replacing the number by the remainder. The remainder has the same sign as the number.
• RootAssignRem
  Replaces a number with the floor of its $n$th root, returning the remainder.
• RootRem
  Finds the floor of the $n$th root of a number, returning both the root and the remainder.
• RotateLeft
  Rotates a number left, inserting the leftmost bits into the right end.
• RotateLeftAssign
  Rotates a number left, inserting the leftmost bits into the right end, in place.
• RotateRight
  Rotates a number right, inserting the leftmost bits into the left end.
• RotateRightAssign
  Rotates a number right, inserting the leftmost bits into the left end, in place.
• RoundToMultiple
  Rounds a number to a multiple of another number, according to a specified rounding mode. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.
• RoundToMultipleAssign
  Rounds a number to a multiple of another number in place, according to a specified rounding mode. Ordering is returned, indicating whether the returned value is less than, equal to, or greater than the original value.
• RoundToMultipleOfPowerOf2
  Rounds a number to a multiple of $2^k$, according to a specified rounding mode. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the original value.
• RoundToMultipleOfPowerOf2Assign
  Rounds a number to a multiple of $2^k$ in place, according to a specified rounding mode. An Ordering is returned, indicating whether the returned value is less than, equal to, or greater than the original value.
• SaturatingAbs
  Takes the absolute value of a number, saturating at the numeric bounds instead of overflowing.
• SaturatingAbsAssign
  Replaces a number with its absolute value, saturating at the numeric bounds instead of overflowing.
• SaturatingAdd
  Adds two numbers, saturating at the numeric bounds instead of overflowing.
• SaturatingAddAssign
  Add a number to another number in place, saturating at the numeric bounds instead of overflowing.
• SaturatingAddMul
  Adds a number and the product of two other numbers, saturating at the numeric bounds instead of overflowing.
• SaturatingAddMulAssign
  Adds a number and the product of two other numbers in place, saturating at the numeric bounds instead of overflowing.
• SaturatingMul
  Multiplies two numbers, saturating at the numeric bounds instead of overflowing.
• SaturatingMulAssign
  Multiplies a number by another number in place, saturating at the numeric bounds instead of overflowing.
• SaturatingNeg
  Negates a number, saturating at the numeric bounds instead of overflowing.
• SaturatingNegAssign
  Negates a number in place, saturating at the numeric bounds instead of overflowing.
• SaturatingPow
  Raises a number to a power, saturating at the numeric bounds instead of overflowing.
• SaturatingPowAssign
  Raises a number to a power in place, saturating at the numeric bounds instead of overflowing.
• SaturatingSquare
  Squares a number, saturating at the numeric bounds instead of overflowing.
• SaturatingSquareAssign
  Squares a number in place, saturating at the numeric bounds instead of overflowing.
• SaturatingSub
  Subtracts two numbers, saturating at the numeric bounds instead of overflowing.
• SaturatingSubAssign
  Subtracts a number by another number in place, saturating at the numeric bounds instead of overflowing.
• SaturatingSubMul
  Subtracts a number by the product of two other numbers, saturating at the numeric bounds instead of overflowing.
• SaturatingSubMulAssign
  Subtracts a number by the product of two other numbers in place, saturating at the numeric bounds instead of overflowing.
• ShlRound
  Left-shifts a number (multiplies it by a power of 2), rounding the result according to a specified rounding mode. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the exact value.
• ShlRoundAssign
  Left-shifts a number (multiplies it by a power of 2) in place, rounding the result according to a specified rounding mode. An Ordering is also returned, indicating whether the assigned value is less than, equal to, or greater than the exact value.
• ShrRound
  Right-shifts a number (divides it by a power of 2), rounding the result according to a specified rounding mode. An Ordering is also returned, indicating whether the returned value is less than, equal to, or greater than the exact value.
• ShrRoundAssign
  Right-shifts a number (divides it by a power of 2) in place, rounding the result according to a specified rounding mode. An Ordering is also returned, indicating whether the assigned value is less than, equal to, or greater than the exact value.
• Sign
  Returns Greater, Equal, or Less, depending on whether a number is positive, zero, or negative, respectively.
• Sqrt
  Takes the square root of a number.
• SqrtAssign
  Replaces a number with its square root.
• SqrtAssignRem
  Replaces a number with the floor of its square root, returning the remainder.
• SqrtRem
  Finds the floor of the square root of a number, returning both the root and the remainder.
• Square
  Squares a number.
• SquareAssign
  Replaces a number with its square.
• SubMul
  Subtracts a number by the product of two other numbers.
• SubMulAssign
  Subtracts a number by the product of two other numbers, in place.
• Subfactorial
  Computes the subfactorial of a u64. The subfactorial of a non-negative integer $n$ counts the number of derangements of $n$ elements, which are the permutations in which no element is fixed.
• UnsignedAbs
  Takes the absolute value of a number and converts to the unsigned equivalent.
• WrappingAbs
  Takes the absolute value of a number, wrapping around at the boundary of the type.
• WrappingAbsAssign
  Replaces a number with its absolute value, wrapping around at the boundary of the type.
• WrappingAdd
  Adds two numbers, wrapping around at the boundary of the type.
• WrappingAddAssign
  Adds a number to another number in place, wrapping around at the boundary of the type.
• WrappingAddMul
  Adds a number and the product of two other numbers, wrapping around at the boundary of the type.
• WrappingAddMulAssign
  Adds a number and the product of two other numbers, in place, wrapping around at the boundary of the type.
• WrappingDiv
  Divides a number by another number, wrapping around at the boundary of the type.
• WrappingDivAssign
  Divides a number by another number in place, wrapping around at the boundary of the type.
• WrappingMul
  Multiplies two numbers, wrapping around at the boundary of the type.
• WrappingMulAssign
  Multiplies a number by another number in place, wrapping around at the boundary of the type.
• WrappingNeg
  Negates a number, wrapping around at the boundary of the type.
• WrappingNegAssign
  Negates a number in place, wrapping around at the boundary of the type.
• WrappingPow
  Raises a number to a power, wrapping around at the boundary of the type.
• WrappingPowAssign
  Raises a number to a power in place, wrapping around at the boundary of the type.
• WrappingSquare
  Squares a number, wrapping around at the boundary of the type.
• WrappingSquareAssign
  Squares a number in place, wrapping around at the boundary of the type.
• WrappingSub
  Subtracts two numbers, wrapping around at the boundary of the type.
• WrappingSubAssign
  Subtracts a number by another number in place, wrapping around at the boundary of the type.
• WrappingSubMul
  Subtracts a number by the product of two other numbers, wrapping around at the boundary of the type.
• WrappingSubMulAssign
  Subtracts a number by the product of two other numbers, in place, wrapping around at the boundary of the type.
• XMulYToZZ
  Multiplies two numbers, returning the product as a pair of Self values.
• XXAddYYToZZ
  Adds two numbers, each composed of two Self values, returning the sum as a pair of Self values.
• XXDivModYToQR
  Computes the quotient and remainder of two numbers. The first is composed of two Self values, and the second of a single one.
• XXSubYYToZZ
  Subtracts two numbers, each composed of two Self values, returing the difference as a pair of Self values.
• XXXAddYYYToZZZ
  Adds two numbers, each composed of three Self values, returning the sum as a triple of Self values.
• XXXSubYYYToZZZ
  Subtracts two numbers, each composed of three Self values, returing the difference as a triple of Self values.
• XXXXAddYYYYToZZZZ
  Adds two numbers, each composed of four Self values, returning the sum as a quadruple of Self values.