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G.2.6 Accuracy Requirements for Complex Arithmetic
1
In the strict mode, the performance of Numerics.Generic_Complex_Types
and Numerics.Generic_Complex_Elementary_Functions shall be as specified
here.
Implementation Requirements
2
When an exception is not raised, the result
of evaluating a real function of an instance
CT of Numerics.Generic_Complex_Types
(i.e., 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).
3
{result interval (for a component
of the result of evaluating a complex function)} 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 (i.e., 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.
{maximum relative
error (for a component of the result of evaluating a complex function)}
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.
{maximum box error (for a component of the result of
evaluating a complex 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.
{Overflow_Check
[partial]} {check, language-defined
(Overflow_Check)} 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,
3.a
Discussion: The maximum
relative error could be specified separately for each component, but
we do not take advantage of that freedom here.
3.b
Discussion: Note that f
· (1.0 + d) defines a small circular region of the complex
plane centered at f, and the result intervals for the real and
imaginary components of the result define a small rectangular box containing
that circle.
3.c
Reason: Box error is used
when the computation of the result risks loss of significance in a component
due to cancellation.
3.d
Ramification: The components
of a complex function that exhibits bounded relative error in each component
have to have the correct sign. In contrast, one of the components of
a complex function that exhibits bounded box error may have the wrong
sign, since the dimensions of the box containing the result are proportional
to the modulus of the mathematical result and not to either component
of the mathematical result individually. Thus, for example, the box containing
the computed result of a complex function whose mathematical result has
a large modulus but lies very close to the imaginary axis might well
straddle that axis, allowing the real component of the computed result
to have the wrong sign. In this case, the distance between the computed
result and the mathematical result is, nevertheless, a small fraction
of the modulus of the mathematical result.
4
- {Constraint_Error (raised by failure of
run-time check)} 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;
5
- if CT.Real'Machine_Overflows is False, the result
is implementation defined.
5.a
Implementation defined: The
result of a complex arithmetic operation or complex elementary function
reference in overflow situations, when the Machine_Overflows attribute
of the corresponding real type is False.
6
The error bounds for particular complex functions
are tabulated below. In the table, the error bound is given as the coefficient
of CT.Real'Model_Epsilon.
7/1
This paragraph was deleted.
8
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 pure-imaginary type) is less than or equal to the angle threshold. For larger
angles, the accuracy is implementation defined.
8.a
Implementation defined: The
accuracy of certain complex arithmetic operations and certain complex
elementary functions for parameters (or components thereof) beyond the
angle threshold.
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
|
9
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:
10
- 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
PI/2.0.
11
- 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 PI/2.0.
12
- 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 -PI/2.0 (resp.,
PI).
12.a
Discussion: It is possible
to give many other prescribed results in which a component of the parameter
is restricted to a similar model interval when the parameter X is appropriately
restricted to an easily testable portion of the domain. We follow the
proposed ISO/IEC standard for Generic_Complex_Elementary_Functions (for
Ada 83) in not doing so, however.
13
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 to the maximum error bounds, effectively
narrowing the result interval allowed by them.
14
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 pure-imaginary parameter, never
exceeds one.
Implementation Advice
15
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.
Wording Changes from Ada 83
15.a
The semantics of Numerics.Generic_Complex_Types
and Numerics.Generic_Complex_Elementary_Functions differs from Generic_Complex_Types
and Generic_Complex_Elementary_Functions as defined in ISO/IEC CDs 13813 and
13814 (for Ada 83) in ways analogous to those identified for the elementary
functions in G.2.4. In addition, we do not generally
specify the signs of zero results (or result components), although those proposed
standards do.
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