G.2.6 Accuracy Requirements for Complex Arithmetic
In the strict mode, the performance of Numerics.Generic_Complex_Types
and Numerics.Generic_Complex_Elementary_Functions shall be as specified
here.
Implementation Requirements
When an exception is not raised, the result of evaluating
a real function of an instance
CT of Numerics.Generic_Complex_Types
(that is, a function that yields a value of subtype
CT.Real'Base
or
CT.Imaginary) belongs to a result interval defined as for a
real elementary function (see
G.2.4).
When an exception is not raised,
each component of the result of evaluating a complex function of such
an instance, or of an instance of Numerics.Generic_Complex_Elementary_Functions
obtained by instantiating the latter with
CT (that is, a function
that yields a value of subtype
CT.Complex), also belongs to a
result interval. The result intervals for the components of the
result are either defined by a
maximum relative error bound or
by a
maximum box error bound.
When the result
interval for the real (resp., imaginary) component is defined by maximum
relative error, it is defined as for that of a real function, relative
to the exact value of the real (resp., imaginary) part of the result
of the corresponding mathematical function.
When
defined by maximum box error, the result interval for a component of
the result is the smallest model interval of
CT.Real that contains
all the values of the corresponding part of
f
· (1.0 +
d), where
f
is the exact complex value of the corresponding mathematical function
at the given parameter values,
d is
complex, and 
d is less than or equal
to the given maximum box error.
The
function delivers a value that belongs to the result interval (or a value
both of whose components belong to their respective result intervals)
when both bounds of the result interval(s) belong to the safe range of
CT.Real; otherwise,
if
CT.Real'Machine_Overflows
is True, the function either delivers a value that belongs to the result
interval (or a value both of whose components belong to their respective
result intervals) or raises Constraint_Error, signaling overflow;
if CT.Real'Machine_Overflows is False, the
result is implementation defined.
The error bounds for particular complex functions
are tabulated in Table G.2. In the table, the error bound is given as
the coefficient of CT.Real'Model_Epsilon.
This paragraph was
deleted.
Table G.2: Error Bounds for Particular Complex Functions
Function or Operator  Nature
of
Result  Nature of
Bound  Error Bound


Modulus  real  max.
rel. error  3.0

Argument  real  max.
rel. error  4.0

Compose_From_Polar  complex  max.
rel. error  3.0

"*" (both operands complex)  complex  max.
box error  5.0

"/" (right operand complex)  complex  max.
box error  13.0

Sqrt  complex  max.
rel. error  6.0

Log  complex  max.
box error  13.0

Exp (complex parameter)  complex  max.
rel. error  7.0

Exp (imaginary parameter)  complex  max.
rel. error  2.0

Sin, Cos, Sinh, and Cosh  complex  max.
rel. error  11.0

Tan, Cot, Tanh, and Coth  complex  max.
rel. error  35.0

inverse trigonometric  complex  max.
rel. error  14.0

inverse hyperbolic  complex  max.
rel. error  14.0

The maximum relative error given above applies throughout
the domain of the Compose_From_Polar function when the Cycle parameter
is specified. When the Cycle parameter is omitted, the maximum relative
error applies only when the absolute value of the parameter Argument
is less than or equal to the angle threshold (see
G.2.4).
For the Exp function, and for the forward hyperbolic (resp., trigonometric)
functions, the maximum relative error given above likewise applies only
when the absolute value of the imaginary (resp., real) component of the
parameter X (or the absolute value of the parameter itself, in the case
of the Exp function with a parameter of pureimaginary type) is less
than or equal to the angle threshold. For larger angles, the accuracy
is implementation defined.
The prescribed results
specified in
G.1.2 for certain functions
at particular parameter values take precedence over the error bounds;
effectively, they narrow to a single value the result interval allowed
by the error bounds for a component of the result. Additional rules with
a similar effect are given below for certain inverse trigonometric and
inverse hyperbolic functions, at particular parameter values for which
a component of the mathematical result is transcendental. In each case,
the accuracy rule, which takes precedence over the error bounds, is that
the result interval for the stated result component is the model interval
of
CT.Real associated with the component's exact mathematical
value. The cases in question are as follows:
When the parameter X has the value zero, the real
(resp., imaginary) component of the result of the Arccot (resp., Arccoth)
function is in the model interval of CT.Real associated with the
value π/2.0.
When the parameter X has the value one, the real
component of the result of the Arcsin function is in the model interval
of CT.Real associated with the value π/2.0.
When the parameter X has the value –1.0,
the real component of the result of the Arcsin (resp., Arccos) function
is in the model interval of CT.Real associated with the value
–π/2.0 (resp., π).
The amount by which a component of the result of
an inverse trigonometric or inverse hyperbolic function is allowed to
spill over into a quadrant adjacent to the one corresponding to the principal
branch, as given in
G.1.2, is limited. The
rule is that the result belongs to the smallest model interval of
CT.Real
that contains both boundaries of the quadrant corresponding to the principal
branch. This rule also takes precedence over the maximum error bounds,
effectively narrowing the result interval allowed by them.
Finally, the results allowed by the error bounds
are narrowed by one further rule: The absolute value of each component
of the result of the Exp function, for a pureimaginary parameter, never
exceeds one.
Implementation Advice
The version of the Compose_From_Polar function without
a Cycle parameter should not be implemented by calling the corresponding
version with a Cycle parameter of 2.0*Numerics.Pi, since this will not
provide the required accuracy in some portions of the domain.
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