

                                   Annex G
                                 (normative)

                                  Numerics


1     The Numerics Annex specifies

2     features for complex arithmetic, including complex I/O;

3     a mode (``strict mode''), in which the predefined arithmetic operations
      of floating point and fixed point types and the functions and operations
      of various predefined packages have to provide guaranteed accuracy or
      conform to other numeric performance requirements, which the Numerics
      Annex also specifies;

4     a mode (``relaxed mode''), in which no accuracy or other numeric
      performance requirements need be satisfied, as for implementations not
      conforming to the Numerics Annex;

5     models of floating point and fixed point arithmetic on which the
      accuracy requirements of strict mode are based; and

6     the definitions of the model-oriented attributes of floating point types
      that apply in the strict mode.


                            Implementation Advice

7     If Fortran (respectively, C) is widely supported in the target
environment, implementations supporting the Numerics Annex should provide the
child package Interfaces.Fortran (respectively, Interfaces.C) specified in
Annex B and should support a convention_identifier of Fortran (respectively,
C) in the interfacing pragmas (see Annex B), thus allowing Ada programs to
interface with programs written in that language.


G.1 Complex Arithmetic


1     Types and arithmetic operations for complex arithmetic are provided in
Generic_Complex_Types, which is defined in G.1.1. Implementation-defined
approximations to the complex analogs of the mathematical functions known as
the ``elementary functions'' are provided by the subprograms in Generic_-
Complex_Elementary_Functions, which is defined in G.1.2. Both of these library
units are generic children of the predefined package Numerics (see A.5).
Nongeneric equivalents of these generic packages for each of the predefined
floating point types are also provided as children of Numerics.


G.1.1 Complex Types



                              Static Semantics

1     The generic library package Numerics.Generic_Complex_Types has the
following declaration:

2/1   generic
         type Real is digits <>;
      package Ada.Numerics.Generic_Complex_Types is
         pragma Pure(Generic_Complex_Types);

3        type Complex is
            record
               Re, Im : Real'Base;
            end record;

4        type Imaginary is private;

5        i : constant Imaginary;
         j : constant Imaginary;

6        function Re (X : Complex)   return Real'Base;
         function Im (X : Complex)   return Real'Base;
         function Im (X : Imaginary) return Real'Base;

7        procedure Set_Re (X  : in out Complex;
                           Re : in     Real'Base);
         procedure Set_Im (X  : in out Complex;
                           Im : in     Real'Base);
         procedure Set_Im (X  :    out Imaginary;
                           Im : in     Real'Base);

8        function Compose_From_Cartesian (Re, Im : Real'Base) return Complex;
         function Compose_From_Cartesian (Re     : Real'Base) return Complex;
         function Compose_From_Cartesian (Im     : Imaginary) return Complex;

9        function Modulus (X     : Complex) return Real'Base;
         function "abs"   (Right : Complex) return Real'Base renames Modulus;

10       function Argument (X     : Complex)   return Real'Base;
         function Argument (X     : Complex;
                            Cycle : Real'Base) return Real'Base;

11       function Compose_From_Polar (Modulus, Argument        : Real'Base)
            return Complex;
         function Compose_From_Polar (Modulus, Argument, Cycle : Real'Base)
            return Complex;

12       function "+"       (Right : Complex) return Complex;
         function "-"       (Right : Complex) return Complex;
         function Conjugate (X     : Complex) return Complex;

13       function "+" (Left, Right : Complex) return Complex;
         function "-" (Left, Right : Complex) return Complex;
         function "*" (Left, Right : Complex) return Complex;
         function "/" (Left, Right : Complex) return Complex;

14       function "**" (Left : Complex; Right : Integer) return Complex;

15       function "+"       (Right : Imaginary) return Imaginary;
         function "-"       (Right : Imaginary) return Imaginary;
         function Conjugate (X     : Imaginary) return Imaginary renames "-";
         function "abs"     (Right : Imaginary) return Real'Base;

16       function "+" (Left, Right : Imaginary) return Imaginary;
         function "-" (Left, Right : Imaginary) return Imaginary;
         function "*" (Left, Right : Imaginary) return Real'Base;
         function "/" (Left, Right : Imaginary) return Real'Base;

17       function "**" (Left : Imaginary; Right : Integer) return Complex;

18       function "<"  (Left, Right : Imaginary) return Boolean;
         function "<=" (Left, Right : Imaginary) return Boolean;
         function ">"  (Left, Right : Imaginary) return Boolean;
         function ">=" (Left, Right : Imaginary) return Boolean;

19       function "+" (Left : Complex;   Right : Real'Base) return Complex;
         function "+" (Left : Real'Base; Right : Complex)   return Complex;
         function "-" (Left : Complex;   Right : Real'Base) return Complex;
         function "-" (Left : Real'Base; Right : Complex)   return Complex;
         function "*" (Left : Complex;   Right : Real'Base) return Complex;
         function "*" (Left : Real'Base; Right : Complex)   return Complex;
         function "/" (Left : Complex;   Right : Real'Base) return Complex;
         function "/" (Left : Real'Base; Right : Complex)   return Complex;

20       function "+" (Left : Complex;   Right : Imaginary) return Complex;
         function "+" (Left : Imaginary; Right : Complex)   return Complex;
         function "-" (Left : Complex;   Right : Imaginary) return Complex;
         function "-" (Left : Imaginary; Right : Complex)   return Complex;
         function "*" (Left : Complex;   Right : Imaginary) return Complex;
         function "*" (Left : Imaginary; Right : Complex)   return Complex;
         function "/" (Left : Complex;   Right : Imaginary) return Complex;
         function "/" (Left : Imaginary; Right : Complex)   return Complex;

21       function "+" (Left : Imaginary; Right : Real'Base) return Complex;
         function "+" (Left : Real'Base; Right : Imaginary) return Complex;
         function "-" (Left : Imaginary; Right : Real'Base) return Complex;
         function "-" (Left : Real'Base; Right : Imaginary) return Complex;
         function "*" (Left : Imaginary; Right : Real'Base) return Imaginary;
         function "*" (Left : Real'Base; Right : Imaginary) return Imaginary;
         function "/" (Left : Imaginary; Right : Real'Base) return Imaginary;
         function "/" (Left : Real'Base; Right : Imaginary) return Imaginary;

22    private

23       type Imaginary is new Real'Base;
         i : constant Imaginary := 1.0;
         j : constant Imaginary := 1.0;

24    end Ada.Numerics.Generic_Complex_Types;

25/1  The library package Numerics.Complex_Types is declared pure and defines
the same types, constants, and subprograms as Numerics.Generic_Complex_Types,
except that the predefined type Float is systematically substituted for
Real'Base throughout. Nongeneric equivalents of Numerics.Generic_Complex_Types
for each of the other predefined floating point types are defined similarly,
with the names Numerics.Short_Complex_Types, Numerics.Long_Complex_Types, etc.

26    Complex is a visible type with cartesian components.

27    Imaginary is a private type; its full type is derived from Real'Base.

28    The arithmetic operations and the Re, Im, Modulus, Argument, and
Conjugate functions have their usual mathematical meanings. When applied to a
parameter of pure-imaginary type, the ``imaginary-part'' function Im yields
the value of its parameter, as the corresponding real value. The remaining
subprograms have the following meanings:

29    The Set_Re and Set_Im procedures replace the designated component of a
      complex parameter with the given real value; applied to a parameter of
      pure-imaginary type, the Set_Im procedure replaces the value of that
      parameter with the imaginary value corresponding to the given real value.

30    The Compose_From_Cartesian function constructs a complex value from the
      given real and imaginary components. If only one component is given, the
      other component is implicitly zero.

31    The Compose_From_Polar function constructs a complex value from the
      given modulus (radius) and argument (angle). When the value of the
      parameter Modulus is positive (resp., negative), the result is the
      complex value represented by the point in the complex plane lying at a
      distance from the origin given by the absolute value of Modulus and
      forming an angle measured counterclockwise from the positive (resp.,
      negative) real axis given by the value of the parameter Argument.

32    When the Cycle parameter is specified, the result of the Argument
function and the parameter Argument of the Compose_From_Polar function are
measured in units such that a full cycle of revolution has the given value;
otherwise, they are measured in radians.

33    The computed results of the mathematically multivalued functions are
rendered single-valued by the following conventions, which are meant to imply
the principal branch:

34    The result of the Modulus function is nonnegative.

35    The result of the Argument function is in the quadrant containing the
      point in the complex plane represented by the parameter X. This may be
      any quadrant (I through IV); thus, the range of the Argument function is
      approximately -PI to PI (-Cycle/2.0 to Cycle/2.0, if the parameter Cycle
      is specified). When the point represented by the parameter X lies on the
      negative real axis, the result approximates

    36    PI (resp., -PI) when the sign of the imaginary component of X is
          positive (resp., negative), if Real'Signed_Zeros is True;

    37    PI, if Real'Signed_Zeros is False.

38    Because 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

39    The exception Numerics.Argument_Error is raised by the Argument and
Compose_From_Polar functions with specified cycle, signaling a parameter value
outside the domain of the corresponding mathematical function, when the value
of the parameter Cycle is zero or negative.

40    The exception Constraint_Error is raised by the division operator when
the value of the right operand is zero, and by the exponentiation operator
when the value of the left operand is zero and the value of the exponent is
negative, provided that Real'Machine_Overflows is True; when
Real'Machine_Overflows is False, the result is unspecified. Constraint_Error
can also be raised when a finite result overflows (see G.2.6).


                         Implementation Requirements

41    In the implementation of Numerics.Generic_Complex_Types, the range of
intermediate values allowed during the calculation of a final result shall not
be affected by any range constraint of the subtype Real.

42    In the following cases, evaluation of a complex arithmetic operation
shall yield the prescribed result, provided that the preceding rules do not
call for an exception to be raised:

43    The results of the Re, Im, and Compose_From_Cartesian functions are
      exact.

44    The real (resp., imaginary) component of the result of a binary addition
      operator that yields a result of complex type is exact when either of
      its operands is of pure-imaginary (resp., real) type.

45    The real (resp., imaginary) component of the result of a binary
      subtraction operator that yields a result of complex type is exact when
      its right operand is of pure-imaginary (resp., real) type.

46    The real component of the result of the Conjugate function for the
      complex type is exact.

47    When the point in the complex plane represented by the parameter X lies
      on the nonnegative real axis, the Argument function yields a result of
      zero.

48    When the value of the parameter Modulus is zero, the Compose_From_Polar
      function yields a result of zero.

49    When the value of the parameter Argument is equal to a multiple of the
      quarter cycle, the result of the Compose_From_Polar function with
      specified cycle lies on one of the axes. In this case, one of its
      components is zero, and the other has the magnitude of the parameter
      Modulus.

50    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, provided that the exponent is nonzero. When the
      left operand is of pure-imaginary type, one component of the result of
      the exponentiation operator is zero.

51    When the result, or a result component, of any operator of
Numerics.Generic_Complex_Types has a mathematical definition in terms of a
single arithmetic or relational operation, that result or result component
exhibits the accuracy of the corresponding operation of the type Real.

52    Other accuracy requirements for the Modulus, Argument, and
Compose_From_Polar functions, and accuracy requirements for the multiplication
of a pair of complex operands or for division by a complex operand, all of
which apply only in the strict mode, are given in G.2.6.

53    The sign of a zero result or zero result component yielded by a complex
arithmetic operation or function is implementation defined when
Real'Signed_Zeros is True.


                         Implementation Permissions

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

55/1  Implementations may obtain the result of exponentiation of a complex or
pure-imaginary operand by repeated complex multiplication, with arbitrary
association of the factors and with a possible final complex reciprocation
(when the exponent is negative). Implementations are also permitted to obtain
the result of exponentiation of a complex operand, but not of a pure-imaginary
operand, by converting the left operand to a polar representation;
exponentiating the modulus by the given exponent; multiplying the argument by
the given exponent; and reconverting to a cartesian representation. Because of
this implementation freedom, no accuracy requirement is imposed on complex
exponentiation (except for the prescribed results given above, which apply
regardless of the implementation method chosen).


                            Implementation Advice

56    Because the usual mathematical meaning of multiplication of a complex
operand and a real operand is that of the scaling of both components of the
former by the latter, an implementation should not perform this operation by
first promoting the real operand to complex type and then performing a full
complex multiplication. In systems that, in the future, support an Ada binding
to IEC 559:1989, the latter technique will not generate the required result
when one of the components of the complex operand is infinite. (Explicit
multiplication of the infinite component by the zero component obtained during
promotion yields a NaN that propagates into the final result.) Analogous
advice applies in the case of multiplication of a complex operand and a
pure-imaginary operand, and in the case of division of a complex operand by a
real or pure-imaginary operand.

57    Likewise, because the usual mathematical meaning of addition of a
complex operand and a real operand is that the imaginary operand remains
unchanged, an implementation should not perform this operation by first
promoting the real operand to complex type and then performing a full complex
addition. In implementations in which the Signed_Zeros attribute of the
component type is True (and which therefore conform to IEC 559:1989 in regard
to the handling of the sign of zero in predefined arithmetic operations), the
latter technique will not generate the required result when the imaginary
component of the complex operand is a negatively signed zero. (Explicit
addition of the negative zero to the zero obtained during promotion yields a
positive zero.) Analogous advice applies in the case of addition of a complex
operand and a pure-imaginary operand, and in the case of subtraction of a
complex operand and a real or pure-imaginary operand.

58    Implementations in which Real'Signed_Zeros is True should attempt to
provide a rational treatment of the signs of zero results and result
components. As one example, the result of the Argument function should have
the sign of the imaginary component of the parameter X when the point
represented by that parameter lies on the positive real axis; as another, the
sign of the imaginary component of the Compose_From_Polar function should be
the same as (resp., the opposite of) that of the Argument parameter when that
parameter has a value of zero and the Modulus parameter has a nonnegative
(resp., negative) value.


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   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.

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    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. When Complex_Types.Real'Machine_Overflows is
False, the result at poles is unspecified.


                         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.

35    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.

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.


                         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    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). 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.


G.1.3 Complex Input-Output


1     The generic package Text_IO.Complex_IO defines procedures for the
formatted input and output of complex values. The generic actual parameter in
an instantiation of Text_IO.Complex_IO is an instance of
Numerics.Generic_Complex_Types for some floating point subtype. Exceptional
conditions are reported by raising the appropriate exception defined in
Text_IO.


                              Static Semantics

2     The generic library package Text_IO.Complex_IO has the following
declaration:

3     with Ada.Numerics.Generic_Complex_Types;
      generic
         with package Complex_Types is
               new Ada.Numerics.Generic_Complex_Types (<>);
      package Ada.Text_IO.Complex_IO is

4        use Complex_Types;

5        Default_Fore : Field := 2;
         Default_Aft  : Field := Real'Digits - 1;
         Default_Exp  : Field := 3;

6        procedure Get (File  : in  File_Type;
                        Item  : out Complex;
                        Width : in  Field := 0);
         procedure Get (Item  : out Complex;
                        Width : in  Field := 0);

7        procedure Put (File : in File_Type;
                        Item : in Complex;
                        Fore : in Field := Default_Fore;
                        Aft  : in Field := Default_Aft;
                        Exp  : in Field := Default_Exp);
         procedure Put (Item : in Complex;
                        Fore : in Field := Default_Fore;
                        Aft  : in Field := Default_Aft;
                        Exp  : in Field := Default_Exp);

8        procedure Get (From : in  String;
                        Item : out Complex;
                        Last : out Positive);
         procedure Put (To   : out String;
                        Item : in  Complex;
                        Aft  : in  Field := Default_Aft;
                        Exp  : in  Field := Default_Exp);

9     end Ada.Text_IO.Complex_IO;

10    The semantics of the Get and Put procedures are as follows:

11    procedure Get (File  : in  File_Type;
                     Item  : out Complex;
                     Width : in  Field := 0);
      procedure Get (Item  : out Complex;
                     Width : in  Field := 0);

    12/1  The input sequence is a pair of optionally signed real literals
          representing the real and imaginary components of a complex value
          These components have the format defined for the corresponding Get
          procedure of an instance of Text_IO.Float_IO (see A.10.9) for the
          base subtype of Complex_Types.Real. The pair of components may be
          separated by a comma or surrounded by a pair of parentheses or both.
          Blanks are freely allowed before each of the components and before
          the parentheses and comma, if either is used. If the value of the
          parameter Width is zero, then

        13    line and page terminators are also allowed in these places;

        14    the components shall be separated by at least one blank or line
              terminator if the comma is omitted; and

        15    reading stops when the right parenthesis has been read, if the
              input sequence includes a left parenthesis, or when the
              imaginary component has been read, otherwise.

    15.1  If a nonzero value of Width is supplied, then

        16    the components shall be separated by at least one blank if the
              comma is omitted; and

        17    exactly Width characters are read, or the characters (possibly
              none) up to a line terminator, whichever comes first (blanks are
              included in the count).

    18    Returns, in the parameter Item, the value of type Complex that
          corresponds to the input sequence.

    19    The exception Text_IO.Data_Error is raised if the input sequence
          does not have the required syntax or if the components of the
          complex value obtained are not of the base subtype of
          Complex_Types.Real.

20    procedure Put (File : in File_Type;
                     Item : in Complex;
                     Fore : in Field := Default_Fore;
                     Aft  : in Field := Default_Aft;
                     Exp  : in Field := Default_Exp);
      procedure Put (Item : in Complex;
                     Fore : in Field := Default_Fore;
                     Aft  : in Field := Default_Aft;
                     Exp  : in Field := Default_Exp);

    21    Outputs the value of the parameter Item as a pair of decimal
          literals representing the real and imaginary components of the
          complex value, using the syntax of an aggregate. More specifically,

        22    outputs a left parenthesis;

        23    outputs the value of the real component of the parameter Item
              with the format defined by the corresponding Put procedure of an
              instance of Text_IO.Float_IO for the base subtype of
              Complex_Types.Real, using the given values of Fore, Aft, and Exp;

        24    outputs a comma;

        25    outputs the value of the imaginary component of the parameter
              Item with the format defined by the corresponding Put procedure
              of an instance of Text_IO.Float_IO for the base subtype of
              Complex_Types.Real, using the given values of Fore, Aft, and Exp;

        26    outputs a right parenthesis.

27    procedure Get (From : in  String;
                     Item : out Complex;
                     Last : out Positive);

    28    Reads a complex value from the beginning of the given string,
          following the same rule as the Get procedure that reads a complex
          value from a file, but treating the end of the string as a line
          terminator. Returns, in the parameter Item, the value of type
          Complex that corresponds to the input sequence. Returns in Last the
          index value such that From(Last) is the last character read.

    29    The exception Text_IO.Data_Error is raised if the input sequence
          does not have the required syntax or if the components of the
          complex value obtained are not of the base subtype of
          Complex_Types.Real.

30    procedure Put (To   : out String;
                     Item : in  Complex;
                     Aft  : in  Field := Default_Aft;
                     Exp  : in  Field := Default_Exp);

    31    Outputs the value of the parameter Item to the given string as a
          pair of decimal literals representing the real and imaginary
          components of the complex value, using the syntax of an aggregate.
          More specifically,

        32    a left parenthesis, the real component, and a comma are left
              justified in the given string, with the real component having
              the format defined by the Put procedure (for output to a file)
              of an instance of Text_IO.Float_IO for the base subtype of
              Complex_Types.Real, using a value of zero for Fore and the given
              values of Aft and Exp;

        33    the imaginary component and a right parenthesis are right
              justified in the given string, with the imaginary component
              having the format defined by the Put procedure (for output to a
              file) of an instance of Text_IO.Float_IO for the base subtype of
              Complex_Types.Real, using a value for Fore that completely fills
              the remainder of the string, together with the given values of
              Aft and Exp.

    34    The exception Text_IO.Layout_Error is raised if the given string is
          too short to hold the formatted output.


                         Implementation Permissions

35    Other exceptions declared (by renaming) in Text_IO may be raised by the
preceding procedures in the appropriate circumstances, as for the
corresponding procedures of Text_IO.Float_IO.


G.1.4 The Package Wide_Text_IO.Complex_IO



                              Static Semantics

1     Implementations shall also provide the generic library package
Wide_Text_IO.Complex_IO. Its declaration is obtained from that of
Text_IO.Complex_IO by systematically replacing Text_IO by Wide_Text_IO and
String by Wide_String; the description of its behavior is obtained by
additionally replacing references to particular characters (commas,
parentheses, etc.) by those for the corresponding wide characters.


G.2 Numeric Performance Requirements



                         Implementation Requirements

1     Implementations shall provide a user-selectable mode in which the
accuracy and other numeric performance requirements detailed in the following
subclauses are observed. This mode, referred to as the strict mode, may or may
not be the default mode; it directly affects the results of the predefined
arithmetic operations of real types and the results of the subprograms in
children of the Numerics package, and indirectly affects the operations in
other language defined packages. Implementations shall also provide the
opposing mode, which is known as the relaxed mode.


                         Implementation Permissions

2     Either mode may be the default mode.

3     The two modes need not actually be different.


G.2.1 Model of Floating Point Arithmetic


1     In the strict mode, the predefined operations of a floating point type
shall satisfy the accuracy requirements specified here and shall avoid or
signal overflow in the situations described. This behavior is presented in
terms of a model of floating point arithmetic that builds on the concept of
the canonical form (see A.5.3).


                              Static Semantics

2     Associated with each floating point type is an infinite set of model
numbers. The model numbers of a type are used to define the accuracy
requirements that have to be satisfied by certain predefined operations of the
type; through certain attributes of the model numbers, they are also used to
explain the meaning of a user-declared floating point type declaration. The
model numbers of a derived type are those of the parent type; the model
numbers of a subtype are those of its type.

3     The model numbers of a floating point type T are zero and all the values
expressible in the canonical form (for the type T), in which mantissa has
T'Model_Mantissa digits and exponent has a value greater than or equal to
T'Model_Emin. (These attributes are defined in G.2.2.)

4     A model interval of a floating point type is any interval whose bounds
are model numbers of the type. The model interval of a type T associated with
a value v is the smallest model interval of T that includes v. (The model
interval associated with a model number of a type consists of that number
only.)


                         Implementation Requirements

5     The accuracy requirements for the evaluation of certain predefined
operations of floating point types are as follows.

6     An operand interval is the model interval, of the type specified for the
operand of an operation, associated with the value of the operand.

7     For any predefined arithmetic operation that yields a result of a
floating point type T, the required bounds on the result are given by a model
interval of T (called the result interval) defined in terms of the operand
values as follows:

8     The result interval is the smallest model interval of T that includes
      the minimum and the maximum of all the values obtained by applying the
      (exact) mathematical operation to values arbitrarily selected from the
      respective operand intervals.

9     The result interval of an exponentiation is obtained by applying the
above rule to the sequence of multiplications defined by the exponent,
assuming arbitrary association of the factors, and to the final division in
the case of a negative exponent.

10    The result interval of a conversion of a numeric value to a floating
point type T is the model interval of T associated with the operand value,
except when the source expression is of a fixed point type with a small that
is not a power of T'Machine_Radix or is a fixed point multiplication or
division either of whose operands has a small that is not a power of
T'Machine_Radix; in these cases, the result interval is implementation
defined.

11    For any of the foregoing operations, the implementation shall deliver a
value that belongs to the result interval when both bounds of the result
interval are in the safe range of the result type T, as determined by the
values of T'Safe_First and T'Safe_Last; otherwise,

12    if T'Machine_Overflows is True, the implementation shall either deliver
      a value that belongs to the result interval or raise Constraint_Error;

13    if T'Machine_Overflows is False, the result is implementation defined.

14    For any predefined relation on operands of a floating point type T, the
implementation may deliver any value (i.e., either True or False) obtained by
applying the (exact) mathematical comparison to values arbitrarily chosen from
the respective operand intervals.

15    The result of a membership test is defined in terms of comparisons of
the operand value with the lower and upper bounds of the given range or type
mark (the usual rules apply to these comparisons).


                         Implementation Permissions

16    If the underlying floating point hardware implements division as
multiplication by a reciprocal, the result interval for division (and
exponentiation by a negative exponent) is implementation defined.


G.2.2 Model-Oriented Attributes of Floating Point Types


1     In implementations that support the Numerics Annex, the model-oriented
attributes of floating point types shall yield the values defined here, in
both the strict and the relaxed modes. These definitions add conditions to
those in A.5.3.


                              Static Semantics

2     For every subtype S of a floating point type T:

3     S'Model_Mantissa
              Yields the number of digits in the mantissa of the canonical
              form of the model numbers of T (see A.5.3). The value of this
              attribute shall be greater than or equal to Ceiling(d  log(10)
              / log(T'Machine_Radix)) + 1, where d is the requested decimal
              precision of T. In addition, it shall be less than or equal to
              the value of T'Machine_Mantissa. This attribute yields a value
              of the type universal_integer.

4     S'Model_Emin
              Yields the minimum exponent of the canonical form of the model
              numbers of T (see A.5.3). The value of this attribute shall be
              greater than or equal to the value of T'Machine_Emin. This
              attribute yields a value of the type universal_integer.

5     S'Safe_First
              Yields the lower bound of the safe range of T. The value of this
              attribute shall be a model number of T and greater than or equal
              to the lower bound of the base range of T. In addition, if T is
              declared by a floating_point_definition or is derived from such
              a type, and the floating_point_definition includes a
              real_range_specification specifying a lower bound of lb, then
              the value of this attribute shall be less than or equal to lb;
              otherwise, it shall be less than or equal to -10.0 (4  d),
              where d is the requested decimal precision of T. This attribute
              yields a value of the type universal_real.

6     S'Safe_Last
              Yields the upper bound of the safe range of T. The value of this
              attribute shall be a model number of T and less than or equal to
              the upper bound of the base range of T. In addition, if T is
              declared by a floating_point_definition or is derived from such
              a type, and the floating_point_definition includes a
              real_range_specification specifying an upper bound of ub, then
              the value of this attribute shall be greater than or equal to
              ub; otherwise, it shall be greater than or equal to 10.0 (4 
              d), where d is the requested decimal precision of T. This
              attribute yields a value of the type universal_real.

7     S'Model Denotes a function (of a parameter X) whose specification is
              given in A.5.3. If X is a model number of T, the function yields
              X; otherwise, it yields the value obtained by rounding or
              truncating X to either one of the adjacent model numbers of T.
              Constraint_Error is raised if the resulting model number is
              outside the safe range of S. A zero result has the sign of X
              when S'Signed_Zeros is True.

8     Subject to the constraints given above, the values of S'Model_Mantissa
and S'Safe_Last are to be maximized, and the values of S'Model_Emin and
S'Safe_First minimized, by the implementation as follows:

9     First, S'Model_Mantissa is set to the largest value for which values of
      S'Model_Emin, S'Safe_First, and S'Safe_Last can be chosen so that the
      implementation satisfies the strict-mode requirements of G.2.1 in terms
      of the model numbers and safe range induced by these attributes.

10    Next, S'Model_Emin is set to the smallest value for which values of
      S'Safe_First and S'Safe_Last can be chosen so that the implementation
      satisfies the strict-mode requirements of G.2.1 in terms of the model
      numbers and safe range induced by these attributes and the previously
      determined value of S'Model_Mantissa.

11    Finally, S'Safe_First and S'Safe_last are set (in either order) to the
      smallest and largest values, respectively, for which the implementation
      satisfies the strict-mode requirements of G.2.1 in terms of the model
      numbers and safe range induced by these attributes and the previously
      determined values of S'Model_Mantissa and S'Model_Emin.


G.2.3 Model of Fixed Point Arithmetic


1     In the strict mode, the predefined arithmetic operations of a fixed
point type shall satisfy the accuracy requirements specified here and shall
avoid or signal overflow in the situations described.


                         Implementation Requirements

2     The accuracy requirements for the predefined fixed point arithmetic
operations and conversions, and the results of relations on fixed point
operands, are given below.

3     The operands of the fixed point adding operators, absolute value, and
comparisons have the same type. These operations are required to yield exact
results, unless they overflow.

4     Multiplications and divisions are allowed between operands of any two
fixed point types; the result has to be (implicitly or explicitly) converted
to some other numeric type. For purposes of defining the accuracy rules, the
multiplication or division and the conversion are treated as a single
operation whose accuracy depends on three types (those of the operands and the
result). For decimal fixed point types, the attribute T'Round may be used to
imply explicit conversion with rounding (see 3.5.10).

5     When the result type is a floating point type, the accuracy is as given
in G.2.1. For some combinations of the operand and result types in the
remaining cases, the result is required to belong to a small set of values
called the perfect result set; for other combinations, it is required merely
to belong to a generally larger and implementation-defined set of values
called the close result set. When the result type is a decimal fixed point
type, the perfect result set contains a single value; thus, operations on
decimal types are always fully specified.

6     When one operand of a fixed-fixed multiplication or division is of type
universal_real, that operand is not implicitly converted in the usual sense,
since the context does not determine a unique target type, but the accuracy of
the result of the multiplication or division (i.e., whether the result has to
belong to the perfect result set or merely the close result set) depends on
the value of the operand of type universal_real and on the types of the other
operand and of the result.

7     For a fixed point multiplication or division whose (exact) mathematical
result is v, and for the conversion of a value v to a fixed point type, the
perfect result set and close result set are defined as follows:

8     If the result type is an ordinary fixed point type with a small of s,

    9     if v is an integer multiple of s, then the perfect result set
          contains only the value v;

    10    otherwise, it contains the integer multiple of s just below v and
          the integer multiple of s just above v.

11    The close result set is an implementation-defined set of consecutive
      integer multiples of s containing the perfect result set as a subset.

12    If the result type is a decimal type with a small of s,

    13    if v is an integer multiple of s, then the perfect result set
          contains only the value v;

    14    otherwise, if truncation applies then it contains only the integer
          multiple of s in the direction toward zero, whereas if rounding
          applies then it contains only the nearest integer multiple of s
          (with ties broken by rounding away from zero).

15    The close result set is an implementation-defined set of consecutive
      integer multiples of s containing the perfect result set as a subset.

16    If the result type is an integer type,

    17    if v is an integer, then the perfect result set contains only the
          value v;

    18    otherwise, it contains the integer nearest to the value v (if v lies
          equally distant from two consecutive integers, the perfect result
          set contains the one that is further from zero).

19    The close result set is an implementation-defined set of consecutive
      integers containing the perfect result set as a subset.

20    The result of a fixed point multiplication or division shall belong
either to the perfect result set or to the close result set, as described
below, if overflow does not occur. In the following cases, if the result type
is a fixed point type, let s be its small; otherwise, i.e. when the result
type is an integer type, let s be 1.0.

21    For a multiplication or division neither of whose operands is of type
      universal_real, let l and r be the smalls of the left and right
      operands. For a multiplication, if (l  r) / s is an integer or the
      reciprocal of an integer (the smalls are said to be ``compatible'' in
      this case), the result shall belong to the perfect result set;
      otherwise, it belongs to the close result set. For a division, if l / (r
       s) is an integer or the reciprocal of an integer (i.e., the smalls are
      compatible), the result shall belong to the perfect result set;
      otherwise, it belongs to the close result set.

22    For a multiplication or division having one universal_real operand with
      a value of v, note that it is always possible to factor v as an integer
      multiple of a ``compatible'' small, but the integer multiple may be `
      `too big.'' If there exists a factorization in which that multiple is
      less than some implementation-defined limit, the result shall belong to
      the perfect result set; otherwise, it belongs to the close result set.

23    A multiplication P * Q of an operand of a fixed point type F by an
operand of an integer type I, or vice-versa, and a division P / Q of an
operand of a fixed point type F by an operand of an integer type I, are also
allowed. In these cases, the result has a type of F; explicit conversion of
the result is never required. The accuracy required in these cases is the same
as that required for a multiplication F(P * Q) or a division F(P / Q) obtained
by interpreting the operand of the integer type to have a fixed point type
with a small of 1.0.

24    The accuracy of the result of a conversion from an integer or fixed
point type to a fixed point type, or from a fixed point type to an integer
type, is the same as that of a fixed point multiplication of the source value
by a fixed point operand having a small of 1.0 and a value of 1.0, as given by
the foregoing rules. The result of a conversion from a floating point type to
a fixed point type shall belong to the close result set. The result of a
conversion of a universal_real operand to a fixed point type shall belong to
the perfect result set.

25    The possibility of overflow in the result of a predefined arithmetic
operation or conversion yielding a result of a fixed point type T is analogous
to that for floating point types, except for being related to the base range
instead of the safe range. If all of the permitted results belong to the base
range of T, then the implementation shall deliver one of the permitted
results; otherwise,

26    if T'Machine_Overflows is True, the implementation shall either deliver
      one of the permitted results or raise Constraint_Error;

27    if T'Machine_Overflows is False, the result is implementation defined.


G.2.4 Accuracy Requirements for the Elementary Functions


1     In the strict mode, the performance of
Numerics.Generic_Elementary_Functions shall be as specified here.


                         Implementation Requirements

2     When an exception is not raised, the result of evaluating a function in
an instance EF of Numerics.Generic_Elementary_Functions belongs to a result
interval, defined as the smallest model interval of EF.Float_Type that
contains all the values of the form f  (1.0 + d), where f is the exact value
of the corresponding mathematical function at the given parameter values, d is
a real number, and |d| is less than or equal to the function's maximum
relative error. The function delivers a value that belongs to the result
interval when both of its bounds belong to the safe range of EF.Float_Type;
otherwise,

3     if EF.Float_Type'Machine_Overflows is True, the function either delivers
      a value that belongs to the result interval or raises Constraint_Error,
      signaling overflow;

4     if EF.Float_Type'Machine_Overflows is False, the result is
      implementation defined.

5     The maximum relative error exhibited by each function is as follows:

6     2.0  EF.Float_Type'Model_Epsilon, in the case of the Sqrt, Sin, and Cos
      functions;

7     4.0  EF.Float_Type'Model_Epsilon, in the case of the Log, Exp, Tan,
      Cot, and inverse trigonometric functions; and

8     8.0  EF.Float_Type'Model_Epsilon, in the case of the forward and
      inverse hyperbolic functions.

9     The maximum relative error exhibited by the exponentiation operator,
which depends on the values of the operands, is (4.0   +   |Right  log(
Left)|   /   32.0)  EF.Float_Type'Model_Epsilon.

10    The maximum relative error given above applies throughout the domain of
the forward trigonometric functions when the Cycle parameter is specified.
When the Cycle parameter is omitted, the maximum relative error given above
applies only when the absolute value of the angle parameter X is less than or
equal to some implementation-defined angle threshold, which shall be at least
EF.Float_Type'Machine_Radix (Floor(EF.Float_Type'Machine_Mantissa/2)). Beyond
the angle threshold, the accuracy of the forward trigonometric functions is
implementation defined.

11    The prescribed results specified in A.5.1 for certain functions at
particular parameter values take precedence over the maximum relative error
bounds; effectively, they narrow to a single value the result interval allowed
by the maximum relative error bounds. Additional rules with a similar effect
are given by the table below for the inverse trigonometric functions, at
particular parameter values for which the mathematical result is possibly not
a model number of EF.Float_Type (or is, indeed, even transcendental). In each
table entry, the values of the parameters are such that the result lies on the
axis between two quadrants; the corresponding accuracy rule, which takes
precedence over the maximum relative error bounds, is that the result interval
is the model interval of EF.Float_Type associated with the exact mathematical
result given in the table.

12/1  This paragraph was deleted.

13    The last line of the table is meant to apply when
EF.Float_Type'Signed_Zeros is False; the two lines just above it, when
EF.Float_Type'Signed_Zeros is True and the parameter Y has a zero value with
the indicated sign.

14    The amount by which the result of an inverse trigonometric function is
allowed to spill over into a quadrant adjacent to the one corresponding to the
principal branch, as given in A.5.1, is limited. The rule is that the result
belongs to the smallest model interval of EF.Float_Type that contains both
boundaries of the quadrant corresponding to the principal branch. This rule
also takes precedence over the maximum relative error bounds, effectively
narrowing the result interval allowed by them.

          Tightly Approximated Elementary Function Results

          Function 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          Value of X 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          Value of Y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          Exact Result
          when Cycle
          Specified 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          Exact Result
          when Cycle
          Omitted

          Arcsin 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          1.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          n.a. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          Cycle/4.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          PI/2.0
          Arcsin 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          -1.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          n.a. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          -Cycle/4.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          -PI/2.0
          Arccos 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          0.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          n.a. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          Cycle/4.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          PI/2.0
          Arccos 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          -1.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          n.a. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          Cycle/2.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          PI
          Arctan and Arccot 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          0.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          positive 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          Cycle/4.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          PI/2.0
          Arctan and Arccot 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          0.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          negative 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          -Cycle/4.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          -PI/2.0
          Arctan and Arccot 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          negative 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          +0.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          Cycle/2.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          PI
          Arctan and Arccot 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          negative 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          -0.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          -Cycle/2.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          -PI
          Arctan and Arccot 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          negative 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          0.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          Cycle/2.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

          PI
15    Finally, the following specifications also take precedence over the
maximum relative error bounds:

16    The absolute value of the result of the Sin, Cos, and Tanh functions
      never exceeds one.

17    The absolute value of the result of the Coth function is never less than
      one.

18    The result of the Cosh function is never less than one.


                            Implementation Advice

19    The versions of the forward trigonometric functions 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. For the same reason, the version of
Log without a Base parameter should not be implemented by calling the
corresponding version with a Base parameter of Numerics.e.


G.2.5 Performance Requirements for Random Number Generation


1     In the strict mode, the performance of Numerics.Float_Random and
Numerics.Discrete_Random shall be as specified here.


                         Implementation Requirements

2     Two different calls to the time-dependent Reset procedure shall reset
the generator to different states, provided that the calls are separated in
time by at least one second and not more than fifty years.

3     The implementation's representations of generator states and its
algorithms for generating random numbers shall yield a period of at least
2(31)-2; much longer periods are desirable but not required.

4     The implementations of Numerics.Float_Random.Random and
Numerics.Discrete_Random.Random shall pass at least 85% of the individual
trials in a suite of statistical tests. For Numerics.Float_Random, the tests
are applied directly to the floating point values generated (i.e., they are
not converted to integers first), while for Numerics.Discrete_Random they are
applied to the generated values of various discrete types. Each test suite
performs 6 different tests, with each test repeated 10 times, yielding a total
of 60 individual trials. An individual trial is deemed to pass if the
chi-square value (or other statistic) calculated for the observed counts or
distribution falls within the range of values corresponding to the 2.5 and
97.5 percentage points for the relevant degrees of freedom (i.e., it shall be
neither too high nor too low). For the purpose of determining the degrees of
freedom, measurement categories are combined whenever the expected counts are
fewer than 5.


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     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. 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,

4     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.

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.

           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).

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.

