Numeric.Decimal.Operation

Eventually most or all of the arithmetic operations described in the General Decimal Arithmetic Specification will be provided here. For now, the operations are mostly limited to those exposed through various class methods.

It is suggested to import this module qualified to avoid Prelude name clashes:

import qualified Numeric.Decimal.Operation as Op

Note that it is not usually necessary to import this module unless you want to use operations unavailable through class methods, or you need precise control over the handling of exceptional conditions.

Arithmetic operations

This section describes the arithmetic operations on, and some other functions of, numbers, including subnormal numbers, negative zeros, and special values (see also IEEE 754 §5 and §6).

abs :: (Precision p, Rounding r) => Decimal a b -> Arith p r (Decimal p r)

abs takes one operand. If the operand is negative, the result is the same as using the minus operation on the operand. Otherwise, the result is the same as using the plus operation on the operand.

Note that the result of this operation is affected by context and may set flags. The copyAbs operation may be used if this is not desired.

add :: (Precision p, Rounding r) => Decimal a b -> Decimal c d -> Arith p r (Decimal p r)

add takes two operands. If either operand is a special value then the general rules apply.

Otherwise, the operands are added.

The result is then rounded to precision digits if necessary, counting from the most significant digit of the result.

subtract :: (Precision p, Rounding r) => Decimal a b -> Decimal c d -> Arith p r (Decimal p r)

subtract takes two operands. If either operand is a special value then the general rules apply.

Otherwise, the operands are added after inverting the sign used for the second operand.

The result is then rounded to precision digits if necessary, counting from the most significant digit of the result.

compare :: (Precision p, Rounding r) => Decimal a b -> Decimal c d -> Arith p r (Decimal p r)

compare takes two operands and compares their values numerically. If either operand is a special value then the general rules apply. No flags are set unless an operand is a signaling NaN.

Otherwise, the operands are compared, returning -1 if the first is less than the second, 0 if they are equal, or 1 if the first is greater than the second.

compareSignal :: (Precision p, Rounding r) => Decimal a b -> Decimal c d -> Arith p r (Decimal p r)

compareSignal takes two operands and compares their values numerically. This operation is identical to compare, except that if neither operand is a signaling NaN then any quiet NaN operand is treated as though it were a signaling NaN. (That is, all NaNs signal, with signaling NaNs taking precedence over quiet NaNs.)

divide :: (FinitePrecision p, Rounding r) => Decimal a b -> Decimal c d -> Arith p r (Decimal p r)

divide takes two operands. If either operand is a special value then the general rules apply.

Otherwise, if the divisor is zero then either the Division undefined condition is raised (if the dividend is zero) and the result is NaN, or the Division by zero condition is raised and the result is an Infinity with a sign which is the exclusive or of the signs of the operands.

Otherwise, a “long division” is effected.

The result is then rounded to precision digits, if necessary, according to the rounding algorithm and taking into account the remainder from the division.

exp :: FinitePrecision p => Decimal a b -> Arith p r (Decimal p RoundHalfEven)

exp takes one operand. If the operand is a NaN then the general rules for special values apply.

Otherwise, the result is e raised to the power of the operand, with the following cases:

• If the operand is -Infinity, the result is 0 and exact.
• If the operand is a zero, the result is 1 and exact.
• If the operand is +Infinity, the result is +Infinity and exact.
• Otherwise the result is inexact and will be rounded using the round-half-even algorithm. The coefficient will have exactly precision digits (unless the result is subnormal). These inexact results should be correctly rounded, but may be up to 1 ulp (unit in last place) in error.

fusedMultiplyAdd :: (Precision p, Rounding r) => Decimal a b -> Decimal c d -> Decimal e f -> Arith p r (Decimal p r)

fusedMultiplyAdd takes three operands; the first two are multiplied together, using multiply, with sufficient precision and exponent range that the result is exact and unrounded. No flags are set by the multiplication unless one of the first two operands is a signaling NaN or one is a zero and the other is an infinity.

Unless the multiplication failed, the third operand is then added to the result of that multiplication, using add, under the current context.

In other words, fusedMultiplyAdd x y z delivers a result which is (x × y) + z with only the one, final, rounding.

ln :: FinitePrecision p => Decimal a b -> Arith p r (Decimal p RoundHalfEven)

ln takes one operand. If the operand is a NaN then the general rules for special values apply.

Otherwise, the operand must be a zero or positive, and the result is the natural (base e) logarithm of the operand, with the following cases:

• If the operand is a zero, the result is -Infinity and exact.
• If the operand is +Infinity, the result is +Infinity and exact.
• If the operand equals one, the result is 0 and exact.
• Otherwise the result is inexact and will be rounded using the round-half-even algorithm. The coefficient will have exactly precision digits (unless the result is subnormal). These inexact results should be correctly rounded, but may be up to 1 ulp (unit in last place) in error.

log10 :: FinitePrecision p => Decimal a b -> Arith p r (Decimal p RoundHalfEven)

log10 takes one operand. If the operand is a NaN then the general rules for special values apply.

Otherwise, the operand must be a zero or positive, and the result is the base 10 logarithm of the operand, with the following cases:

• If the operand is a zero, the result is -Infinity and exact.
• If the operand is +Infinity, the result is +Infinity and exact.
• If the operand equals an integral power of ten (including 10^0 and negative powers) and there is sufficient precision to hold the integral part of the result, the result is an integer (with an exponent of 0) and exact.
• Otherwise the result is inexact and will be rounded using the round-half-even algorithm. The coefficient will have exactly precision digits (unless the result is subnormal). These inexact results should be correctly rounded, but may be up to 1 ulp (unit in last place) in error.

max :: (Precision p, Rounding r) => Decimal a b -> Decimal a b -> Arith p r (Decimal a b)

max takes two operands, compares their values numerically, and returns the maximum. If either operand is a NaN then the general rules apply, unless one is a quiet NaN and the other is numeric, in which case the numeric operand is returned.

maxMagnitude :: (Precision p, Rounding r) => Decimal a b -> Decimal a b -> Arith p r (Decimal a b)

maxMagnitude takes two operands and compares their values numerically with their sign ignored and assumed to be 0.

If, without signs, the first operand is the larger then the original first operand is returned (that is, with the original sign). If, without signs, the second operand is the larger then the original second operand is returned. Otherwise the result is the same as from the max operation.

min :: (Precision p, Rounding r) => Decimal a b -> Decimal a b -> Arith p r (Decimal a b)

min takes two operands, compares their values numerically, and returns the minimum. If either operand is a NaN then the general rules apply, unless one is a quiet NaN and the other is numeric, in which case the numeric operand is returned.

minMagnitude :: (Precision p, Rounding r) => Decimal a b -> Decimal a b -> Arith p r (Decimal a b)

minMagnitude takes two operands and compares their values numerically with their sign ignored and assumed to be 0.

If, without signs, the first operand is the smaller then the original first operand is returned (that is, with the original sign). If, without signs, the second operand is the smaller then the original second operand is returned. Otherwise the result is the same as from the min operation.

minus :: (Precision p, Rounding r) => Decimal a b -> Arith p r (Decimal p r)

minus takes one operand, and corresponds to the prefix minus operator in programming languages.

Note that the result of this operation is affected by context and may set flags. The copyNegate operation may be used instead of minus if this is not desired.

plus :: (Precision p, Rounding r) => Decimal a b -> Arith p r (Decimal p r)

plus takes one operand, and corresponds to the prefix plus operator in programming languages.

Note that the result of this operation is affected by context and may set flags.

multiply :: (Precision p, Rounding r) => Decimal a b -> Decimal c d -> Arith p r (Decimal p r)

multiply takes two operands. If either operand is a special value then the general rules apply. Otherwise, the operands are multiplied together (“long multiplication”), resulting in a number which may be as long as the sum of the lengths of the two operands.

The result is then rounded to precision digits if necessary, counting from the most significant digit of the result.

power :: (FinitePrecision p, Rounding r) => Decimal a b -> Decimal c d -> Arith p r (Decimal p r)

power takes two operands, and raises a number (the left-hand operand) to a power (the right-hand operand). If either operand is a special value then the general rules apply, except in certain cases.

quantize :: (Precision p, Rounding r) => Decimal p r -> Decimal a b -> Arith p r (Decimal p r)

quantize takes two operands. If either operand is a special value then the general rules apply, except that if either operand is infinite and the other is finite an Invalid operation condition is raised and the result is NaN, or if both are infinite then the result is the first operand.

Otherwise (both operands are finite), quantize returns the number which is equal in value (except for any rounding) and sign to the first (left-hand) operand and which has an exponent set to be equal to the exponent of the second (right-hand) operand.

The coefficient of the result is derived from that of the left-hand operand. It may be rounded using the current rounding setting (if the exponent is being increased), multiplied by a positive power of ten (if the exponent is being decreased), or is unchanged (if the exponent is already equal to that of the right-hand operand).

Unlike other operations, if the length of the coefficient after the quantize operation would be greater than precision then an Invalid operation condition is raised. This guarantees that, unless there is an error condition, the exponent of the result of a quantize is always equal to that of the right-hand operand.

Also unlike other operations, quantize will never raise Underflow, even if the result is subnormal and inexact.

reduce :: (Precision p, Rounding r) => Decimal a b -> Arith p r (Decimal p r)

reduce takes one operand. It has the same semantics as the plus operation, except that if the final result is finite it is reduced to its simplest form, with all trailing zeros removed and its sign preserved.

squareRoot :: FinitePrecision p => Decimal a b -> Arith p r (Decimal p RoundHalfEven)

squareRoot takes one operand. If the operand is a special value then the general rules apply.

Otherwise, the ideal exponent of the result is defined to be half the exponent of the operand (rounded to an integer, towards -Infinity, if necessary) and then:

If the operand is less than zero an Invalid operation condition is raised.

If the operand is greater than zero, the result is the square root of the operand. If no rounding is necessary (the exact result requires precision digits or fewer) then the the coefficient and exponent giving the correct value and with the exponent closest to the ideal exponent is used. If the result must be inexact, it is rounded using the round-half-even algorithm and the coefficient will have exactly precision digits (unless the result is subnormal), and the exponent will be set to maintain the correct value.

Otherwise (the operand is equal to zero), the result will be the zero with the same sign as the operand and with the ideal exponent.

Miscellaneous operations

This section describes miscellaneous operations on decimal numbers, including non-numeric comparisons, sign and other manipulations, and logical operations.

Some operations return a boolean value that is described as 0 or 1 in the documentation below. For reasons of efficiency, and as permitted by the General Decimal Arithmetic Specification, these operations return a Bool in this implementation, but can be converted to Decimal via fromBool.

canonical :: Decimal a b -> Arith p r (Decimal a b)

canonical takes one operand. The result has the same value as the operand but always uses a canonical encoding. The definition of canonical is implementation-defined; if more than one internal encoding for a given NaN, Infinity, or finite number is possible then one “preferred” encoding is deemed canonical. This operation then returns the value using that preferred encoding.

If all possible operands have just one internal encoding each, then canonical always returns the operand unchanged (that is, it has the same effect as copy). This operation is unaffected by context and is quiet – no flags are changed in the context.

class_ :: Precision a => Decimal a b -> Arith p r Class

class_ takes one operand. The result is an indication of the class of the operand, where the class is one of ten possibilities, corresponding to one of the strings "sNaN" (signaling NaN), "NaN" (quiet NaN), "-Infinity" (negative infinity), "-Normal" (negative normal finite number), "-Subnormal" (negative subnormal finite number), "-Zero" (negative zero), "+Zero" (non-negative zero), "+Subnormal" (positive subnormal finite number), "+Normal" (positive normal finite number), or "+Infinity" (positive infinity). This operation is quiet; no flags are changed in the context.

Note that unlike the special values in the model, the sign of any NaN is ignored in the classification, as required by IEEE 754.

data Class

Constructors

NumberClass Sign NumberClass

Number (finite or infinite)

NaNClass NaNClass

Not a number (quiet or signaling)

Instances

Eq Class

Methods

(==) :: Class -> Class -> Bool

(/=) :: Class -> Class -> Bool

Show Class

Methods

showsPrec :: Int -> Class -> ShowS

show :: Class -> String

showList :: [Class] -> ShowS

data Sign

Constructors

Pos

Positive or non-negative

Neg

Negative

Instances

Enum Sign

Methods

succ :: Sign -> Sign

pred :: Sign -> Sign

toEnum :: Int -> Sign

fromEnum :: Sign -> Int

enumFrom :: Sign -> [Sign]

enumFromThen :: Sign -> Sign -> [Sign]

enumFromTo :: Sign -> Sign -> [Sign]

enumFromThenTo :: Sign -> Sign -> Sign -> [Sign]

Eq Sign

Methods

(==) :: Sign -> Sign -> Bool

(/=) :: Sign -> Sign -> Bool

data NumberClass

Constructors

ZeroClass

Zero

SubnormalClass

Subnormal finite number

NormalClass

Normal finite number

InfinityClass

Infinity

Instances

Eq NumberClass

Methods

(==) :: NumberClass -> NumberClass -> Bool

(/=) :: NumberClass -> NumberClass -> Bool

data NaNClass

Constructors

QNaNClass

Not a number (quiet)

SNaNClass

Not a number (signaling)

Instances

Eq NaNClass

Methods

(==) :: NaNClass -> NaNClass -> Bool

(/=) :: NaNClass -> NaNClass -> Bool

copy :: Decimal a b -> Arith p r (Decimal a b)

copy takes one operand. The result is a copy of the operand. This operation is unaffected by context and is quiet – no flags are changed in the context.

copyAbs :: Decimal a b -> Arith p r (Decimal a b)

copyAbs takes one operand. The result is a copy of the operand with the sign set to 0. Unlike the abs operation, this operation is unaffected by context and is quiet – no flags are changed in the context.

copyNegate :: Decimal a b -> Arith p r (Decimal a b)

copyNegate takes one operand. The result is a copy of the operand with the sign inverted (a sign of 0 becomes 1 and vice versa). Unlike the minus operation, this operation is unaffected by context and is quiet – no flags are changed in the context.

copySign :: Decimal a b -> Decimal c d -> Arith p r (Decimal a b)

copySign takes two operands. The result is a copy of the first operand with the sign set to be the same as the sign of the second operand. This operation is unaffected by context and is quiet – no flags are changed in the context.

isCanonical :: Decimal a b -> Arith p r Bool

isCanonical takes one operand. The result is 1 if the operand is canonical; otherwise it is 0. The definition of canonical is implementation-defined; if more than one internal encoding for a given NaN, Infinity, or finite number is possible then one “preferred” encoding is deemed canonical. This operation then tests whether the internal encoding is that preferred encoding.

If all possible operands have just one internal encoding each, then isCanonical always returns 1. This operation is unaffected by context and is quiet – no flags are changed in the context.

isFinite :: Decimal a b -> Arith p r Bool

isFinite takes one operand. The result is 1 if the operand is neither infinite nor a NaN (that is, it is a normal number, a subnormal number, or a zero); otherwise it is 0. This operation is unaffected by context and is quiet – no flags are changed in the context.

isInfinite :: Decimal a b -> Arith p r Bool

isInfinite takes one operand. The result is 1 if the operand is an Infinity; otherwise it is 0. This operation is unaffected by context and is quiet – no flags are changed in the context.

isNaN :: Decimal a b -> Arith p r Bool

isNaN takes one operand. The result is 1 if the operand is a NaN (quiet or signaling); otherwise it is 0. This operation is unaffected by context and is quiet – no flags are changed in the context.

isNormal :: Precision a => Decimal a b -> Arith p r Bool

isNormal takes one operand. The result is 1 if the operand is a positive or negative normal number; otherwise it is 0. This operation is quiet; no flags are changed in the context.

isQNaN :: Decimal a b -> Arith p r Bool

isQNaN takes one operand. The result is 1 if the operand is a quiet NaN; otherwise it is 0. This operation is unaffected by context and is quiet – no flags are changed in the context.

isSigned :: Decimal a b -> Arith p r Bool

isSigned takes one operand. The result is 1 if the sign of the operand is 1; otherwise it is 0. This operation is unaffected by context and is quiet – no flags are changed in the context.

isSNaN :: Decimal a b -> Arith p r Bool

isSNaN takes one operand. The result is 1 if the operand is a signaling NaN; otherwise it is 0. This operation is unaffected by context and is quiet – no flags are changed in the context.

isSubnormal :: Precision a => Decimal a b -> Arith p r Bool

isSubnormal takes one operand. The result is 1 if the operand is a positive or negative subnormal number; otherwise it is 0. This operation is quiet; no flags are changed in the context.

isZero :: Decimal a b -> Arith p r Bool

isZero takes one operand. The result is 1 if the operand is a zero; otherwise it is 0. This operation is unaffected by context and is quiet – no flags are changed in the context.

logb :: (Precision p, Rounding r) => Decimal a b -> Arith p r (Decimal p r)

logb takes one operand. If the operand is a NaN then the general arithmetic rules apply. If the operand is infinite then +Infinity is returned. If the operand is a zero, then -Infinity is returned and the Division by zero exceptional condition is raised.

Otherwise, the result is the integer which is the exponent of the magnitude of the most significant digit of the operand (as though the operand were truncated to a single digit while maintaining the value of that digit and without limiting the resulting exponent). All results are exact unless an integer result does not fit in the available precision.

radix :: Precision p => Arith p r (Decimal p r)

radix takes no operands. The result is the radix (base) in which arithmetic is effected; for this specification the result will have the value 10.

sameQuantum :: Decimal a b -> Decimal c d -> Arith p r Bool

sameQuantum takes two operands, and returns 1 if the two operands have the same exponent or 0 otherwise. The result is never affected by either the sign or the coefficient of either operand.

If either operand is a special value, 1 is returned only if both operands are NaNs or both are infinities.

sameQuantum does not change any flags in the context.

shift :: Precision p => Decimal p a -> Decimal b c -> Arith p r (Decimal p a)

shift takes two operands. The second operand must be an integer (with an exponent of 0) in the range -precision through precision. If the first operand is a NaN then the general arithmetic rules apply, and if it is infinite then the result is the Infinity unchanged.

Otherwise (the first operand is finite) the result has the same sign and exponent as the first operand, and a coefficient which is a shifted copy of the digits in the coefficient of the first operand. The number of places to shift is taken from the absolute value of the second operand, with the shift being to the left if the second operand is positive or to the right otherwise. Digits shifted into the coefficient are zeros.

The only flag that might be set is invalid-operation (set if the first operand is an sNaN or the second is not valid).

The rotate operation can be used to rotate rather than shift a coefficient.