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Next: <a href="Exceptions.html#Exceptions" accesskey="n" rel="next">Exceptions</a>, Previous: <a href="Rounding-Modes.html#Rounding-Modes" accesskey="p" rel="prev">Rounding Modes</a>, Up: <a href="MPFR-Basics.html#MPFR-Basics" accesskey="u" rel="up">MPFR Basics</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Floating_002dPoint-Values-on-Special-Numbers-1"></a>
<h3 class="section">4.5 Floating-Point Values on Special Numbers</h3>
<p>This section specifies the floating-point values (of type <code>mpfr_t</code>)
returned by MPFR functions (where by “returned” we mean here the modified
value of the destination object, which should not be mixed with the ternary
return value of type <code>int</code> of those functions).
For functions returning several values (like
<code>mpfr_sin_cos</code>), the rules apply to each result separately.
</p>
<p>Functions can have one or several input arguments. An input point is
a mapping from these input arguments to the set of the MPFR numbers.
When none of its components are NaN, an input point can also be seen
as a tuple in the extended real numbers (the set of the real numbers
with both infinities).
</p>
<p>When the input point is in the domain of the mathematical function, the
result is rounded as described in Section “Rounding Modes” (but see
below for the specification of the sign of an exact zero). Otherwise
the general rules from this section apply unless stated otherwise in
the description of the MPFR function (<a href="MPFR-Interface.html#MPFR-Interface">MPFR Interface</a>).
</p>
<p>When the input point is not in the domain of the mathematical function
but is in its closure in the extended real numbers and the function can
be extended by continuity, the result is the obtained limit.
Examples: <code>mpfr_hypot</code> on (+Inf,0) gives +Inf. But <code>mpfr_pow</code>
cannot be defined on (1,+Inf) using this rule, as one can find
sequences (<em><var>x</var>_<var>n</var></em>,<em><var>y</var>_<var>n</var></em>) such that
<em><var>x</var>_<var>n</var></em> goes to 1, <em><var>y</var>_<var>n</var></em> goes to +Inf
and <em><var>x</var>_<var>n</var> to the <var>y</var>_<var>n</var></em> goes to any
positive value when <var>n</var> goes to the infinity.
</p>
<p>When the input point is in the closure of the domain of the mathematical
function and an input argument is +0 (resp. −0), one considers
the limit when the corresponding argument approaches 0 from above
(resp. below), if possible. If the limit is not defined (e.g.,
<code>mpfr_sqrt</code> and <code>mpfr_log</code> on −0), the behavior is
specified in the description of the MPFR function, but must be consistent
with the rule from the above paragraph (e.g., <code>mpfr_log</code> on ±0
gives −Inf).
</p>
<p>When the result is equal to 0, its sign is determined by considering the
limit as if the input point were not in the domain: If one approaches 0
from above (resp. below), the result is +0 (resp. −0);
for example, <code>mpfr_sin</code> on −0 gives −0 and
<code>mpfr_acos</code> on 1 gives +0 (in all rounding modes).
In the other cases, the sign is specified in the description of the MPFR
function; for example <code>mpfr_max</code> on −0 and +0 gives +0.
</p>
<p>When the input point is not in the closure of the domain of the function,
the result is NaN. Example: <code>mpfr_sqrt</code> on −17 gives NaN.
</p>
<p>When an input argument is NaN, the result is NaN, possibly except when
a partial function is constant on the finite floating-point numbers;
such a case is always explicitly specified in <a href="MPFR-Interface.html#MPFR-Interface">MPFR Interface</a>.
Example: <code>mpfr_hypot</code> on (NaN,0) gives NaN, but <code>mpfr_hypot</code>
on (NaN,+Inf) gives +Inf (as specified in <a href="Special-Functions.html#Special-Functions">Special Functions</a>),
since for any finite or infinite input <var>x</var>, <code>mpfr_hypot</code> on
(<var>x</var>,+Inf) gives +Inf.
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