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

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,

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;

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

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

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

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

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

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.

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 implementationdefined 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.

The prescribed results specified in See section A.5.1 Elementary Functions, 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.

++
 
 Tightly Approximated Elementary Function Results 
 
+++++ +
     
    Exact  Exact 
    Result  Result 
  Value of  Value of  when Cycle  when 
 Function  X  Y  Specified  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 & Arccot  0.0  positive  Cycle/4.0  Pi/2.0 
     
 Arctan & Arccot  0.0  negative  Cycle/4.0  Pi/2.0 
     
 Arctan & Arccot  negative  +0.0  Cycle/2.0  Pi 
     
 Arctan & Arccot  negative  0.0  Cycle/2.0  Pi 
     
 Arctan & Arccot  negative  0.0  Cycle/2.0  Pi 
     
++++++

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.

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 See section A.5.1 Elementary Functions, 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.

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

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

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

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

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.
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