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A.5.1 Elementary Functions

1
Implementation-defined approximations to the mathematical functions known as the “elementary functions” are provided by the subprograms in Numerics.Generic_Elementary_Functions. Nongeneric equivalents of this generic package for each of the predefined floating point types are also provided as children of Numerics. 
1.a
Implementation defined: The accuracy actually achieved by the elementary functions.

Static Semantics

2
The generic library package Numerics.Generic_Elementary_Functions has the following declaration:
3
generic
   type Float_Type is digits <>;

package Ada.Numerics.Generic_Elementary_Functions is
   pragma Pure(Generic_Elementary_Functions);
4
   function Sqrt    (X           : Float_Type'Base) return Float_Type'Base;
   function Log     (X           : Float_Type'Base) return Float_Type'Base;
   function Log     (X, Base     : Float_Type'Base) return Float_Type'Base;
   function Exp     (X           : Float_Type'Base) return Float_Type'Base;
   function "**"    (Left, Right : Float_Type'Base) return Float_Type'Base;
5
   function Sin     (X           : Float_Type'Base) return Float_Type'Base;
   function Sin     (X, Cycle    : Float_Type'Base) return Float_Type'Base;
   function Cos     (X           : Float_Type'Base) return Float_Type'Base;
   function Cos     (X, Cycle    : Float_Type'Base) return Float_Type'Base;
   function Tan     (X           : Float_Type'Base) return Float_Type'Base;
   function Tan     (X, Cycle    : Float_Type'Base) return Float_Type'Base;
   function Cot     (X           : Float_Type'Base) return Float_Type'Base;
   function Cot     (X, Cycle    : Float_Type'Base) return Float_Type'Base;
6
   function Arcsin  (X           : Float_Type'Base) return Float_Type'Base;
   function Arcsin  (X, Cycle    : Float_Type'Base) return Float_Type'Base;
   function Arccos  (X           : Float_Type'Base) return Float_Type'Base;
   function Arccos  (X, Cycle    : Float_Type'Base) return Float_Type'Base;
   function Arctan  (Y           : Float_Type'Base;
                     X           : Float_Type'Base := 1.0)
                                                    return Float_Type'Base;
   function Arctan  (Y           : Float_Type'Base;
                     X           : Float_Type'Base := 1.0;
                     Cycle       : Float_Type'Base) return Float_Type'Base;
   function Arccot  (X           : Float_Type'Base;
                     Y           : Float_Type'Base := 1.0)
                                                    return Float_Type'Base;
   function Arccot  (X           : Float_Type'Base;
                     Y           : Float_Type'Base := 1.0;
                     Cycle       : Float_Type'Base) return Float_Type'Base;
7
   function Sinh    (X           : Float_Type'Base) return Float_Type'Base;
   function Cosh    (X           : Float_Type'Base) return Float_Type'Base;
   function Tanh    (X           : Float_Type'Base) return Float_Type'Base;
   function Coth    (X           : Float_Type'Base) return Float_Type'Base;
   function Arcsinh (X           : Float_Type'Base) return Float_Type'Base;
   function Arccosh (X           : Float_Type'Base) return Float_Type'Base;
   function Arctanh (X           : Float_Type'Base) return Float_Type'Base;
   function Arccoth (X           : Float_Type'Base) return Float_Type'Base;
8
end Ada.Numerics.Generic_Elementary_Functions;
9/1
{8652/0020} {AI95-00126-01} The library package Numerics.Elementary_Functions is declared pure and defines the same subprograms as Numerics.Generic_Elementary_Functions, except that the predefined type Float is systematically substituted for Float_Type'Base throughout. Nongeneric equivalents of Numerics.Generic_Elementary_Functions for each of the other predefined floating point types are defined similarly, with the names Numerics.Short_Elementary_Functions, Numerics.Long_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 functions have their usual mathematical meanings. When the Base parameter is specified, the Log function computes the logarithm to the given base; otherwise, it computes the natural logarithm. When the Cycle parameter is specified, the parameter X of the forward trigonometric functions (Sin, Cos, Tan, and Cot) and the results of the inverse trigonometric functions (Arcsin, Arccos, Arctan, and Arccot) are measured in units such that a full cycle of revolution has the given value; otherwise, they are measured in radians.
11
The computed results of the mathematically multivalued functions are rendered single-valued by the following conventions, which are meant to imply the principal branch: 
12
The results of the Sqrt and Arccosh functions and that of the exponentiation operator are nonnegative.
13
The result of the Arcsin function is in the quadrant containing the point (1.0, x), where x is the value of the parameter X. This quadrant is I or IV; thus, the range of the Arcsin function is approximately –π/2.0 to π/2.0 (–Cycle/4.0 to Cycle/4.0, if the parameter Cycle is specified).
14
The result of the Arccos function is in the quadrant containing the point (x, 1.0), where x is the value of the parameter X. This quadrant is I or II; thus, the Arccos function ranges from 0.0 to approximately π (Cycle/2.0, if the parameter Cycle is specified).
15
The results of the Arctan and Arccot functions are in the quadrant containing the point (x, y), where x and y are the values of the parameters X and Y, respectively. This may be any quadrant (I through IV) when the parameter X (resp., Y) of Arctan (resp., Arccot) is specified, but it is restricted to quadrants I and IV (resp., I and II) when that parameter is omitted. Thus, the range when that parameter is specified is approximately –π to π (–Cycle/2.0 to Cycle/2.0, if the parameter Cycle is specified); when omitted, the range of Arctan (resp., Arccot) is that of Arcsin (resp., Arccos), as given above. When the point (x, y) lies on the negative x-axis, the result approximates 
16
π (resp., –π) when the sign of the parameter Y is positive (resp., negative), if Float_Type'Signed_Zeros is True;
17
π, if Float_Type'Signed_Zeros is False.
18
(In the case of the inverse trigonometric functions, in which a result lying on or near one of the axes may not be exactly representable, the approximation inherent in computing the result may place it in an adjacent quadrant, close to but on the wrong side of the axis.) 

Dynamic Semantics

19
The exception Numerics.Argument_Error is raised, signaling a parameter value outside the domain of the corresponding mathematical function, in the following cases: 
20
by any forward or inverse trigonometric function with specified cycle, when the value of the parameter Cycle is zero or negative;
21
by the Log function with specified base, when the value of the parameter Base is zero, one, or negative;
22
by the Sqrt and Log functions, when the value of the parameter X is negative;
23
by the exponentiation operator, when the value of the left operand is negative or when both operands have the value zero;
24
by the Arcsin, Arccos, and Arctanh functions, when the absolute value of the parameter X exceeds one;
25
by the Arctan and Arccot functions, when the parameters X and Y both have the value zero;
26
by the Arccosh function, when the value of the parameter X is less than one; and
27
by the Arccoth function, when the absolute value of the parameter X is less than one. 
28
The exception Constraint_Error is raised, signaling a pole of the mathematical function (analogous to dividing by zero), in the following cases, provided that Float_Type'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 value of the exponent is negative;
31
by the Tan function with specified cycle, when the value of the parameter X is an odd multiple of the quarter cycle;
32
by the Cot function with specified cycle, when the value of the parameter X is zero or a multiple of the half cycle; and
33
by the Arctanh and Arccoth functions, when the absolute value of the parameter X is one. 
34
[Constraint_Error can also be raised when a finite result overflows (see G.2.4); this may occur for parameter values sufficiently near poles, and, in the case of some of the functions, for parameter values with sufficiently large magnitudes.] When Float_Type'Machine_Overflows is False, the result at poles is unspecified. 
34.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. 
34.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 both the predefined arithmetic operations and implementations of the elementary functions to deliver signed infinities (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 elementary functions to deliver signed infinities (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 elementary functions, when sensible, given parameters with infinite values. 
35
When one parameter of a function with multiple parameters represents a pole and another is outside the function's domain, the latter takes precedence (i.e., Numerics.Argument_Error is raised). 

Implementation Requirements

36
In the implementation of Numerics.Generic_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 Float_Type. 
36.a
Implementation Note: Implementations of Numerics.Generic_Elementary_Functions written in Ada should therefore avoid declaring local variables of subtype Float_Type; the subtype Float_Type'Base should be used instead. 
37
In the following cases, evaluation of an elementary function shall yield the prescribed result, provided that the preceding rules do not call for an exception to be raised: 
38
When the parameter X has the value zero, the Sqrt, Sin, Arcsin, Tan, Sinh, Arcsinh, Tanh, and Arctanh functions yield a result of zero, and the Exp, Cos, and Cosh functions yield a result of one.
39
When the parameter X has the value one, the Sqrt function yields a result of one, and the Log, Arccos, and Arccosh functions yield a result of zero.
40
When the parameter Y has the value zero and the parameter X has a positive value, the Arctan and Arccot functions yield a result of zero.
41
The results of the Sin, Cos, Tan, and Cot functions with specified cycle are exact when the mathematical result is zero; those of the first two are also exact when the mathematical result is ± 1.0.
42
Exponentiation by a zero exponent yields the value one. Exponentiation by a unit exponent yields the value of the left operand. Exponentiation of the value one yields the value one. Exponentiation of the value zero yields the value zero. 
43
Other accuracy requirements for the elementary functions, which apply only in implementations conforming to the Numerics Annex, and then only in the “strict” mode defined there (see G.2), are given in G.2.4.
44
When Float_Type'Signed_Zeros is True, the sign of a zero result shall be as follows: 
45
A prescribed zero result delivered at the origin by one of the odd functions (Sin, Arcsin, Sinh, Arcsinh, Tan, Arctan or Arccot as a function of Y when X is fixed and positive, Tanh, and Arctanh) has the sign of the parameter X (Y, in the case of Arctan or Arccot).
46
A prescribed zero result delivered by one of the odd functions away from the origin, or by some other elementary function, has an implementation-defined sign. 
46.a
Implementation defined: The sign of a zero result from some of the operators or functions in Numerics.Generic_Elementary_Functions, when Float_Type'Signed_Zeros is True.
47
[A zero result that is not a prescribed result (i.e., one that results from rounding or underflow) has the correct mathematical sign.] 
47.a
Reason: This is a consequence of the rules specified in IEC 559:1989 as they apply to underflow situations with traps disabled. 

Implementation Permissions

48
The nongeneric equivalent packages may, but need not, be actual instantiations of the generic package for the appropriate predefined type. 

Wording Changes from Ada 83

48.a
The semantics of Numerics.Generic_Elementary_Functions differs from Generic_Elementary_Functions as defined in ISO/IEC DIS 11430 (for Ada 83) in the following ways: 
48.b
The generic package is a child unit of the package defining the Argument_Error exception.
48.c
DIS 11430 specified names for the nongeneric equivalents, if provided. Here, those nongeneric equivalents are required.
48.d
Implementations are not allowed to impose an optional restriction that the generic actual parameter associated with Float_Type be unconstrained. (In view of the ability to declare variables of subtype Float_Type'Base in implementations of Numerics.Generic_Elementary_Functions, this flexibility is no longer needed.)
48.e
The sign of a prescribed zero result at the origin of the odd functions is specified, when Float_Type'Signed_Zeros is True. This conforms with recommendations of Kahan and other numerical analysts.
48.f
The dependence of Arctan and Arccot on the sign of a parameter value of zero is tied to the value of Float_Type'Signed_Zeros.
48.g
Sqrt is prescribed to yield a result of one when its parameter has the value one. This guarantee makes it easier to achieve certain prescribed results of the complex elementary functions (see G.1.2, “Complex Elementary Functions”).
48.h
Conformance to accuracy requirements is conditional.

Wording Changes from Ada 95

48.i/2
{8652/0020} {AI95-00126-01} Corrigendum: Explicitly stated that the nongeneric equivalents of Generic_Elementary_Functions are pure. 

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