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<!-- This manual documents how to install and use the Multiple Precision
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Floating-Point Reliable Library, version 3.1.4.
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Copyright 1991, 1993-2016 Free Software Foundation, Inc.
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Permission is granted to copy, distribute and/or modify this document under
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the terms of the GNU Free Documentation License, Version 1.2 or any later
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with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the
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license is included in GNU Free Documentation License. -->
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<head>
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<title>Floating-Point Values on Special Numbers (GNU MPFR 3.1.4)</title>
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<meta name="description" content="How to install and use GNU MPFR, a library for reliable multiple precision
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floating-point arithmetic, version 3.1.4.">
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<meta name="keywords" content="Floating-Point Values on Special Numbers (GNU MPFR 3.1.4)">
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<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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<link href="MPFR-Basics.html#MPFR-Basics" rel="up" title="MPFR Basics">
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<link href="Exceptions.html#Exceptions" rel="next" title="Exceptions">
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<body lang="en">
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<a name="Floating_002dPoint-Values-on-Special-Numbers"></a>
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<div class="header">
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<p>
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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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</div>
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<hr>
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<a name="Floating_002dPoint-Values-on-Special-Numbers-1"></a>
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<h3 class="section">4.5 Floating-Point Values on Special Numbers</h3>
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<p>This section specifies the floating-point values (of type <code>mpfr_t</code>)
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returned by MPFR functions (where by “returned” we mean here the modified
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value of the destination object, which should not be mixed with the ternary
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return value of type <code>int</code> of those functions).
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For functions returning several values (like
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<code>mpfr_sin_cos</code>), the rules apply to each result separately.
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</p>
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<p>Functions can have one or several input arguments. An input point is
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a mapping from these input arguments to the set of the MPFR numbers.
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When none of its components are NaN, an input point can also be seen
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as a tuple in the extended real numbers (the set of the real numbers
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with both infinities).
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</p>
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<p>When the input point is in the domain of the mathematical function, the
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result is rounded as described in Section “Rounding Modes” (but see
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below for the specification of the sign of an exact zero). Otherwise
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the general rules from this section apply unless stated otherwise in
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the description of the MPFR function (<a href="MPFR-Interface.html#MPFR-Interface">MPFR Interface</a>).
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</p>
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<p>When the input point is not in the domain of the mathematical function
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but is in its closure in the extended real numbers and the function can
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be extended by continuity, the result is the obtained limit.
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Examples: <code>mpfr_hypot</code> on (+Inf,0) gives +Inf. But <code>mpfr_pow</code>
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cannot be defined on (1,+Inf) using this rule, as one can find
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sequences (<em><var>x</var>_<var>n</var></em>,<em><var>y</var>_<var>n</var></em>) such that
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<em><var>x</var>_<var>n</var></em> goes to 1, <em><var>y</var>_<var>n</var></em> goes to +Inf
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and <em><var>x</var>_<var>n</var> to the <var>y</var>_<var>n</var></em> goes to any
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positive value when <var>n</var> goes to the infinity.
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</p>
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<p>When the input point is in the closure of the domain of the mathematical
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function and an input argument is +0 (resp. −0), one considers
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the limit when the corresponding argument approaches 0 from above
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(resp. below), if possible. If the limit is not defined (e.g.,
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<code>mpfr_sqrt</code> and <code>mpfr_log</code> on −0), the behavior is
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specified in the description of the MPFR function, but must be consistent
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with the rule from the above paragraph (e.g., <code>mpfr_log</code> on ±0
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gives −Inf).
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</p>
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<p>When the result is equal to 0, its sign is determined by considering the
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limit as if the input point were not in the domain: If one approaches 0
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from above (resp. below), the result is +0 (resp. −0);
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for example, <code>mpfr_sin</code> on −0 gives −0 and
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<code>mpfr_acos</code> on 1 gives +0 (in all rounding modes).
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In the other cases, the sign is specified in the description of the MPFR
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function; for example <code>mpfr_max</code> on −0 and +0 gives +0.
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</p>
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<p>When the input point is not in the closure of the domain of the function,
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the result is NaN. Example: <code>mpfr_sqrt</code> on −17 gives NaN.
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</p>
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<p>When an input argument is NaN, the result is NaN, possibly except when
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a partial function is constant on the finite floating-point numbers;
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such a case is always explicitly specified in <a href="MPFR-Interface.html#MPFR-Interface">MPFR Interface</a>.
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Example: <code>mpfr_hypot</code> on (NaN,0) gives NaN, but <code>mpfr_hypot</code>
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on (NaN,+Inf) gives +Inf (as specified in <a href="Special-Functions.html#Special-Functions">Special Functions</a>),
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since for any finite or infinite input <var>x</var>, <code>mpfr_hypot</code> on
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(<var>x</var>,+Inf) gives +Inf.
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</p>
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<hr>
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<div class="header">
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<p>
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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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</div>
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