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G.1.2 Complex Elementary Functions
Static Semantics
1
The generic library
package Numerics.Generic_Complex_Elementary_Functions has the following
declaration:
2
with Ada.Numerics.Generic_Complex_Types;
generic
with package Complex_Types is
new Ada.Numerics.Generic_Complex_Types (<>);
use Complex_Types;
package Ada.Numerics.Generic_Complex_Elementary_Functions is
pragma Pure(Generic_Complex_Elementary_Functions);
3
function Sqrt (X : Complex) return Complex;
function Log (X : Complex) return Complex;
function Exp (X : Complex) return Complex;
function Exp (X : Imaginary) return Complex;
function "**" (Left : Complex; Right : Complex) return Complex;
function "**" (Left : Complex; Right : Real'Base) return Complex;
function "**" (Left : Real'Base; Right : Complex) return Complex;
4
function Sin (X : Complex) return Complex;
function Cos (X : Complex) return Complex;
function Tan (X : Complex) return Complex;
function Cot (X : Complex) return Complex;
5
function Arcsin (X : Complex) return Complex;
function Arccos (X : Complex) return Complex;
function Arctan (X : Complex) return Complex;
function Arccot (X : Complex) return Complex;
6
function Sinh (X : Complex) return Complex;
function Cosh (X : Complex) return Complex;
function Tanh (X : Complex) return Complex;
function Coth (X : Complex) return Complex;
7
function Arcsinh (X : Complex) return Complex;
function Arccosh (X : Complex) return Complex;
function Arctanh (X : Complex) return Complex;
function Arccoth (X : Complex) return Complex;
8
end Ada.Numerics.Generic_Complex_Elementary_Functions;
9/1
{
8652/0020}
{Ada.Numerics.Complex_Elementary_Functions} The
library package Numerics.Complex_Elementary_Functions
is declared pure and
defines the same subprograms as Numerics.Generic_Complex_Elementary_Functions,
except that the predefined type Float is systematically substituted for Real'Base,
and the Complex and Imaginary types exported by Numerics.Complex_Types are systematically
substituted for Complex and Imaginary, throughout. Nongeneric equivalents of
Numerics.Generic_Complex_Elementary_Functions corresponding to each of the other
predefined floating point types are defined similarly, with the names Numerics.Short_Complex_Elementary_Functions,
Numerics.Long_Complex_Elementary_Functions, etc.
9.a
Reason: The nongeneric
equivalents are provided to allow the programmer to construct simple
mathematical applications without being required to understand and use
generics.
10
The overloading of the Exp function
for the pure-imaginary type is provided to give the user an alternate way to
compose a complex value from a given modulus and argument. In addition to Compose_From_Polar(Rho,
Theta) (see
G.1.1), the programmer may write Rho
* Exp(i * Theta).
11
The imaginary (resp., real) component of the
parameter X of the forward hyperbolic (resp., trigonometric) functions
and of the Exp function (and the parameter X, itself, in the case of
the overloading of the Exp function for the pure-imaginary type) represents
an angle measured in radians, as does the imaginary (resp., real) component
of the result of the Log and inverse hyperbolic (resp., trigonometric)
functions.
12
The functions have
their usual mathematical meanings. However, the arbitrariness inherent
in the placement of branch cuts, across which some of the complex elementary
functions exhibit discontinuities, is eliminated by the following conventions:
13
- The imaginary component of the result of the Sqrt and Log
functions is discontinuous as the parameter X crosses the negative real
axis.
14
- The result of the exponentiation operator when the left
operand is of complex type is discontinuous as that operand crosses the
negative real axis.
15
- The real (resp., imaginary) component of the result of
the Arcsin and Arccos (resp., Arctanh) functions is discontinuous as
the parameter X crosses the real axis to the left of -1.0 or the right
of 1.0.
16
- The real (resp., imaginary) component of the result of
the Arctan (resp., Arcsinh) function is discontinuous as the parameter
X crosses the imaginary axis below -i or above i.
17
- The real component of the result of the Arccot function
is discontinuous as the parameter X crosses the imaginary axis between
-i and i.
18
- The imaginary component of the Arccosh function is discontinuous
as the parameter X crosses the real axis to the left of 1.0.
19
- The imaginary component of the result of the Arccoth function
is discontinuous as the parameter X crosses the real axis between -1.0
and 1.0.
20
The computed results
of the mathematically multivalued functions are rendered single-valued
by the following conventions, which are meant to imply the principal
branch:
21
- The real component of the result of the Sqrt and Arccosh
functions is nonnegative.
22
- The same convention applies to the imaginary component of the
result of the Log function as applies to the result of the natural-cycle version
of the Argument function of Numerics.Generic_Complex_Types (see G.1.1).
23
- The range of the real (resp., imaginary) component of the
result of the Arcsin and Arctan (resp., Arcsinh and Arctanh) functions
is approximately -PI/2.0 to PI/2.0.
24
- The real (resp., imaginary) component of the result of
the Arccos and Arccot (resp., Arccoth) functions ranges from 0.0 to approximately
PI.
25
- The range of the imaginary component of the result of the
Arccosh function is approximately -PI to PI.
26
In addition, the exponentiation operator inherits
the single-valuedness of the Log function.
Dynamic Semantics
27
The exception Numerics.Argument_Error is raised
by the exponentiation operator, signaling a parameter value outside the
domain of the corresponding mathematical function, when the value of
the left operand is zero and the real component of the exponent (or the
exponent itself, when it is of real type) is zero.
28
{Division_Check
[partial]} {check, language-defined
(Division_Check)} {Constraint_Error
(raised by failure of run-time check)} The
exception Constraint_Error is raised, signaling a pole of the mathematical
function (analogous to dividing by zero), in the following cases, provided
that Complex_Types.Real'Machine_Overflows is True:
29
- by the Log, Cot, and Coth functions, when the value of
the parameter X is zero;
30
- by the exponentiation operator, when the value of the left
operand is zero and the real component of the exponent (or the exponent
itself, when it is of real type) is negative;
31
- by the Arctan and Arccot functions, when the value of the
parameter X is ± i;
32
- by the Arctanh and Arccoth functions, when the value of
the parameter X is ± 1.0.
33
[Constraint_Error can also be raised
when a finite result overflows (see
G.2.6); this
may occur for parameter values sufficiently
near poles, and, in the case
of some of the functions, for parameter values having components of sufficiently
large magnitude.]
{unspecified [partial]} When
Complex_Types.Real'Machine_Overflows is False, the result at poles is unspecified.
33.a
Reason: The purpose of
raising Constraint_Error (rather than Numerics.Argument_Error) at the
poles of a function, when Float_Type'Machine_Overflows is True, is to
provide continuous behavior as the actual parameters of the function
approach the pole and finally reach it.
33.b
Discussion: It is anticipated
that an Ada binding to IEC 559:1989 will be developed in the future.
As part of such a binding, the Machine_Overflows attribute of a conformant
floating point type will be specified to yield False, which will permit
implementations of the complex elementary functions to deliver results
with an infinite component (and set the overflow flag defined by the
binding) instead of raising Constraint_Error in overflow situations,
when traps are disabled. Similarly, it is appropriate for the complex
elementary functions to deliver results with an infinite component (and
set the zero-divide flag defined by the binding) instead of raising Constraint_Error
at poles, when traps are disabled. Finally, such a binding should also
specify the behavior of the complex elementary functions, when sensible,
given parameters with infinite components.
Implementation Requirements
34
In the implementation of Numerics.Generic_Complex_Elementary_Functions,
the range of intermediate values allowed during the calculation of a
final result shall not be affected by any range constraint of the subtype
Complex_Types.Real.
34.a
Implementation Note: Implementations
of Numerics.Generic_Complex_Elementary_Functions written in Ada should
therefore avoid declaring local variables of subtype Complex_Types.Real;
the subtype Complex_Types.Real'Base should be used instead.
35
{prescribed
result (for the evaluation of a complex elementary function)}
In the following cases, evaluation of a complex elementary
function shall yield the
prescribed result (or a result having
the prescribed component), provided that the preceding rules do not call
for an exception to be raised:
36
- When the parameter X has the value zero, the Sqrt, Sin,
Arcsin, Tan, Arctan, Sinh, Arcsinh, Tanh, and Arctanh functions yield
a result of zero; the Exp, Cos, and Cosh functions yield a result of
one; the Arccos and Arccot functions yield a real result; and the Arccoth
function yields an imaginary result.
37
- When the parameter X has the value one, the Sqrt function
yields a result of one; the Log, Arccos, and Arccosh functions yield
a result of zero; and the Arcsin function yields a real result.
38
- When the parameter X has the value -1.0, the Sqrt function
yields the result
39
- i (resp., -i), when the sign of the imaginary
component of X is positive (resp., negative), if Complex_Types.Real'Signed_Zeros
is True;
40
- i, if Complex_Types.Real'Signed_Zeros is False;
41
- the Log function yields an imaginary result; and the Arcsin
and Arccos functions yield a real result.
42
- When the parameter X has the value ± i, the
Log function yields an imaginary result.
43
- Exponentiation by a zero exponent yields the value one.
Exponentiation by a unit exponent yields the value of the left operand
(as a complex value). Exponentiation of the value one yields the value
one. Exponentiation of the value zero yields the value zero.
43.a
Discussion: It is possible
to give many other prescribed results restricting the result to the real
or imaginary axis when the parameter X is appropriately restricted to
easily testable portions of the domain. We follow the proposed ISO/IEC
standard for Generic_Complex_Elementary_Functions (for Ada 83), CD 13813,
in not doing so, however.
44
Other accuracy requirements for the
complex elementary functions, which apply only in the strict mode, are given
in
G.2.6.
45
The sign of a zero result or zero result component
yielded by a complex elementary function is implementation defined when
Complex_Types.Real'Signed_Zeros is True.
45.a
Implementation defined: The
sign of a zero result (or a component thereof) from any operator or function
in Numerics.Generic_Complex_Elementary_Functions, when Complex_Types.Real'Signed_Zeros
is True.
Implementation Permissions
46
The nongeneric equivalent packages may, but need
not, be actual instantiations of the generic package with the appropriate
predefined nongeneric equivalent of Numerics.Generic_Complex_Types; if
they are, then the latter shall have been obtained by actual instantiation
of Numerics.Generic_Complex_Types.
47
The exponentiation operator may be implemented
in terms of the Exp and Log functions. Because this implementation yields
poor accuracy in some parts of the domain, no accuracy requirement is
imposed on complex exponentiation.
48
{unspecified [partial]}
The implementation of the Exp function of a complex
parameter X is allowed to raise the exception Constraint_Error, signaling
overflow, when the real component of X exceeds an unspecified threshold
that is approximately log(Complex_Types.Real'Safe_Last). This permission
recognizes the impracticality of avoiding overflow in the marginal case
that the exponential of the real component of X exceeds the safe range
of Complex_Types.Real but both components of the final result do not.
Similarly, the Sin and Cos (resp., Sinh and Cosh) functions are allowed
to raise the exception Constraint_Error, signaling overflow, when the
absolute value of the imaginary (resp., real) component of the parameter
X exceeds an unspecified threshold that is approximately log(Complex_Types.Real'Safe_Last)
+ log(2.0).
{unspecified [partial]} This
permission recognizes the impracticality of avoiding overflow in the
marginal case that the hyperbolic sine or cosine of the imaginary (resp.,
real) component of X exceeds the safe range of Complex_Types.Real but
both components of the final result do not.
Implementation Advice
49
Implementations in which Complex_Types.Real'Signed_Zeros
is True should attempt to provide a rational treatment of the signs of
zero results and result components. For example, many of the complex
elementary functions have components that are odd functions of one of
the parameter components; in these cases, the result component should
have the sign of the parameter component at the origin. Other complex
elementary functions have zero components whose sign is opposite that
of a parameter component at the origin, or is always positive or always
negative.
Wording Changes from Ada 83
49.a
The
semantics of Numerics.Generic_Complex_Elementary_Functions differs from
Generic_Complex_Elementary_Functions as defined in ISO/IEC CD 13814 (for
Ada 83) in the following ways:
49.b
- The generic package is a child unit of the package defining
the Argument_Error exception.
49.c
- The proposed Generic_Complex_Elementary_Functions standard
(for Ada 83) specified names for the nongeneric equivalents, if provided.
Here, those nongeneric equivalents are required.
49.d
- The generic package imports an instance of Numerics.Generic_Complex_Types
rather than a long list of individual types and operations exported by
such an instance.
49.e
- The dependence of the imaginary component of the Sqrt and
Log functions on the sign of a zero parameter component is tied to the
value of Complex_Types.Real'Signed_Zeros.
49.f
- Conformance to accuracy requirements is conditional.
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