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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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<title>Special Functions (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="Special Functions (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-Interface.html#MPFR-Interface" rel="up" title="MPFR Interface">
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<link href="Input-and-Output-Functions.html#Input-and-Output-Functions" rel="next" title="Input and Output Functions">
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<body lang="en">
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<a name="Special-Functions"></a>
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<div class="header">
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<p>
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Next: <a href="Input-and-Output-Functions.html#Input-and-Output-Functions" accesskey="n" rel="next">Input and Output Functions</a>, Previous: <a href="Comparison-Functions.html#Comparison-Functions" accesskey="p" rel="prev">Comparison Functions</a>, Up: <a href="MPFR-Interface.html#MPFR-Interface" accesskey="u" rel="up">MPFR Interface</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="index-Special-functions"></a>
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<a name="Special-Functions-1"></a>
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<h3 class="section">5.7 Special Functions</h3>
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<p>All those functions, except explicitly stated (for example
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<code>mpfr_sin_cos</code>), return a <a href="Rounding-Modes.html#ternary-value">ternary value</a>, i.e., zero for an
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exact return value, a positive value for a return value larger than the
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exact result, and a negative value otherwise.
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</p>
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<p>Important note: in some domains, computing special functions (either with
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correct or incorrect rounding) is expensive, even for small precision,
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for example the trigonometric and Bessel functions for large argument.
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</p>
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<dl>
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<dt><a name="index-mpfr_005flog"></a>Function: <em>int</em> <strong>mpfr_log</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005flog2"></a>Function: <em>int</em> <strong>mpfr_log2</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005flog10"></a>Function: <em>int</em> <strong>mpfr_log10</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the natural logarithm of <var>op</var>,
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<em>log2(<var>op</var>)</em> or
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<em>log10(<var>op</var>)</em>, respectively,
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rounded in the direction <var>rnd</var>.
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Set <var>rop</var> to +0 if <var>op</var> is 1 (in all rounding modes),
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for consistency with the ISO C99 and IEEE 754-2008 standards.
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Set <var>rop</var> to −Inf if <var>op</var> is ±0
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(i.e., the sign of the zero has no influence on the result).
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fexp"></a>Function: <em>int</em> <strong>mpfr_exp</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fexp2"></a>Function: <em>int</em> <strong>mpfr_exp2</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fexp10"></a>Function: <em>int</em> <strong>mpfr_exp10</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the exponential of <var>op</var>,
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to <em>2 power of <var>op</var></em>
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or to <em>10 power of <var>op</var></em>, respectively,
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rounded in the direction <var>rnd</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fcos"></a>Function: <em>int</em> <strong>mpfr_cos</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fsin"></a>Function: <em>int</em> <strong>mpfr_sin</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005ftan"></a>Function: <em>int</em> <strong>mpfr_tan</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the cosine of <var>op</var>, sine of <var>op</var>,
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tangent of <var>op</var>, rounded in the direction <var>rnd</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fsin_005fcos"></a>Function: <em>int</em> <strong>mpfr_sin_cos</strong> <em>(mpfr_t <var>sop</var>, mpfr_t <var>cop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set simultaneously <var>sop</var> to the sine of <var>op</var> and <var>cop</var> to the
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cosine of <var>op</var>, rounded in the direction <var>rnd</var> with the corresponding
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precisions of <var>sop</var> and <var>cop</var>, which must be different variables.
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Return 0 iff both results are exact, more precisely it returns <em>s+4c</em>
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where <em>s=0</em> if <var>sop</var> is exact, <em>s=1</em> if <var>sop</var> is larger
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than the sine of <var>op</var>, <em>s=2</em> if <var>sop</var> is smaller than the sine
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of <var>op</var>, and similarly for <em>c</em> and the cosine of <var>op</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fsec"></a>Function: <em>int</em> <strong>mpfr_sec</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fcsc"></a>Function: <em>int</em> <strong>mpfr_csc</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fcot"></a>Function: <em>int</em> <strong>mpfr_cot</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the secant of <var>op</var>, cosecant of <var>op</var>,
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cotangent of <var>op</var>, rounded in the direction <var>rnd</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005facos"></a>Function: <em>int</em> <strong>mpfr_acos</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fasin"></a>Function: <em>int</em> <strong>mpfr_asin</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fatan"></a>Function: <em>int</em> <strong>mpfr_atan</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the arc-cosine, arc-sine or arc-tangent of <var>op</var>,
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rounded in the direction <var>rnd</var>.
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Note that since <code>acos(-1)</code> returns the floating-point number closest to
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<em>Pi</em> according to the given rounding mode, this number might not be
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in the output range <em>0 <= <var>rop</var> < \pi</em>
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of the arc-cosine function;
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still, the result lies in the image of the output range
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by the rounding function.
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The same holds for <code>asin(-1)</code>, <code>asin(1)</code>, <code>atan(-Inf)</code>,
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<code>atan(+Inf)</code> or for <code>atan(op)</code> with large <var>op</var> and
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small precision of <var>rop</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fatan2"></a>Function: <em>int</em> <strong>mpfr_atan2</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>y</var>, mpfr_t <var>x</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the arc-tangent2 of <var>y</var> and <var>x</var>,
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rounded in the direction <var>rnd</var>:
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if <code>x > 0</code>, <code>atan2(y, x) = atan (y/x)</code>;
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if <code>x < 0</code>, <code>atan2(y, x) = sign(y)*(Pi - atan (abs(y/x)))</code>,
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thus a number from <em>-Pi</em> to <em>Pi</em>.
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As for <code>atan</code>, in case the exact mathematical result is <em>+Pi</em> or
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<em>-Pi</em>,
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its rounded result might be outside the function output range.
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</p>
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<p><code>atan2(y, 0)</code> does not raise any floating-point exception.
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Special values are handled as described in the ISO C99 and IEEE 754-2008
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standards for the <code>atan2</code> function:
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</p><ul>
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<li> <code>atan2(+0, -0)</code> returns <em>+Pi</em>.
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</li><li> <code>atan2(-0, -0)</code> returns <em>-Pi</em>.
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</li><li> <code>atan2(+0, +0)</code> returns +0.
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</li><li> <code>atan2(-0, +0)</code> returns −0.
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</li><li> <code>atan2(+0, x)</code> returns <em>+Pi</em> for <em>x < 0</em>.
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</li><li> <code>atan2(-0, x)</code> returns <em>-Pi</em> for <em>x < 0</em>.
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</li><li> <code>atan2(+0, x)</code> returns +0 for <em>x > 0</em>.
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</li><li> <code>atan2(-0, x)</code> returns −0 for <em>x > 0</em>.
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</li><li> <code>atan2(y, 0)</code> returns <em>-Pi/2</em> for <em>y < 0</em>.
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</li><li> <code>atan2(y, 0)</code> returns <em>+Pi/2</em> for <em>y > 0</em>.
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</li><li> <code>atan2(+Inf, -Inf)</code> returns <em>+3*Pi/4</em>.
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</li><li> <code>atan2(-Inf, -Inf)</code> returns <em>-3*Pi/4</em>.
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</li><li> <code>atan2(+Inf, +Inf)</code> returns <em>+Pi/4</em>.
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</li><li> <code>atan2(-Inf, +Inf)</code> returns <em>-Pi/4</em>.
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</li><li> <code>atan2(+Inf, x)</code> returns <em>+Pi/2</em> for finite <em>x</em>.
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</li><li> <code>atan2(-Inf, x)</code> returns <em>-Pi/2</em> for finite <em>x</em>.
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</li><li> <code>atan2(y, -Inf)</code> returns <em>+Pi</em> for finite <em>y > 0</em>.
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</li><li> <code>atan2(y, -Inf)</code> returns <em>-Pi</em> for finite <em>y < 0</em>.
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</li><li> <code>atan2(y, +Inf)</code> returns +0 for finite <em>y > 0</em>.
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</li><li> <code>atan2(y, +Inf)</code> returns −0 for finite <em>y < 0</em>.
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</li></ul>
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</dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fcosh"></a>Function: <em>int</em> <strong>mpfr_cosh</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fsinh"></a>Function: <em>int</em> <strong>mpfr_sinh</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005ftanh"></a>Function: <em>int</em> <strong>mpfr_tanh</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the hyperbolic cosine, sine or tangent of <var>op</var>,
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rounded in the direction <var>rnd</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fsinh_005fcosh"></a>Function: <em>int</em> <strong>mpfr_sinh_cosh</strong> <em>(mpfr_t <var>sop</var>, mpfr_t <var>cop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set simultaneously <var>sop</var> to the hyperbolic sine of <var>op</var> and
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<var>cop</var> to the hyperbolic cosine of <var>op</var>,
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rounded in the direction <var>rnd</var> with the corresponding precision of
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<var>sop</var> and <var>cop</var>, which must be different variables.
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Return 0 iff both results are exact (see <code>mpfr_sin_cos</code> for a more
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detailed description of the return value).
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fsech"></a>Function: <em>int</em> <strong>mpfr_sech</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fcsch"></a>Function: <em>int</em> <strong>mpfr_csch</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fcoth"></a>Function: <em>int</em> <strong>mpfr_coth</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the hyperbolic secant of <var>op</var>, cosecant of <var>op</var>,
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cotangent of <var>op</var>, rounded in the direction <var>rnd</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005facosh"></a>Function: <em>int</em> <strong>mpfr_acosh</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fasinh"></a>Function: <em>int</em> <strong>mpfr_asinh</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fatanh"></a>Function: <em>int</em> <strong>mpfr_atanh</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the inverse hyperbolic cosine, sine or tangent of <var>op</var>,
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rounded in the direction <var>rnd</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005ffac_005fui"></a>Function: <em>int</em> <strong>mpfr_fac_ui</strong> <em>(mpfr_t <var>rop</var>, unsigned long int <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the factorial of <var>op</var>, rounded in the direction <var>rnd</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005flog1p"></a>Function: <em>int</em> <strong>mpfr_log1p</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the logarithm of one plus <var>op</var>,
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rounded in the direction <var>rnd</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fexpm1"></a>Function: <em>int</em> <strong>mpfr_expm1</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to <em>the exponential of <var>op</var> followed by a
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subtraction by one</em>, rounded in the direction <var>rnd</var>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005feint"></a>Function: <em>int</em> <strong>mpfr_eint</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the exponential integral of <var>op</var>,
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rounded in the direction <var>rnd</var>.
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For positive <var>op</var>,
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the exponential integral is the sum of Euler’s constant, of the logarithm
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of <var>op</var>, and of the sum for k from 1 to infinity of
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<var>op</var> to the power k, divided by k and factorial(k).
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For negative <var>op</var>, <var>rop</var> is set to NaN
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(this definition for negative argument follows formula 5.1.2 from the
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|
Handbook of Mathematical Functions from Abramowitz and Stegun, a future
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version might use another definition).
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fli2"></a>Function: <em>int</em> <strong>mpfr_li2</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to real part of the dilogarithm of <var>op</var>, rounded in the
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direction <var>rnd</var>. MPFR defines the dilogarithm function as
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<em>the integral of -log(1-t)/t from 0
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to <var>op</var></em>.
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|
</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fgamma"></a>Function: <em>int</em> <strong>mpfr_gamma</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the value of the Gamma function on <var>op</var>, rounded in the
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direction <var>rnd</var>. When <var>op</var> is a negative integer, <var>rop</var> is set
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to NaN.
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|
</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005flngamma"></a>Function: <em>int</em> <strong>mpfr_lngamma</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the value of the logarithm of the Gamma function on <var>op</var>,
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rounded in the direction <var>rnd</var>.
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When <var>op</var> is 1 or 2, set <var>rop</var> to +0 (in all rounding modes).
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When <var>op</var> is an infinity or a nonpositive integer, set <var>rop</var> to +Inf,
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following the general rules on special values.
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When <em>−2<var>k</var>−1 < <var>op</var> < −2<var>k</var></em>,
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<var>k</var> being a nonnegative integer, set <var>rop</var> to NaN.
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See also <code>mpfr_lgamma</code>.
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|
</p></dd></dl>
|
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<dl>
|
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<dt><a name="index-mpfr_005flgamma"></a>Function: <em>int</em> <strong>mpfr_lgamma</strong> <em>(mpfr_t <var>rop</var>, int *<var>signp</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the value of the logarithm of the absolute value of the
|
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Gamma function on <var>op</var>, rounded in the direction <var>rnd</var>. The sign
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(1 or −1) of Gamma(<var>op</var>) is returned in the object pointed to
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by <var>signp</var>.
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When <var>op</var> is 1 or 2, set <var>rop</var> to +0 (in all rounding modes).
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When <var>op</var> is an infinity or a nonpositive integer, set <var>rop</var> to +Inf.
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When <var>op</var> is NaN, −Inf or a negative integer, *<var>signp</var> is
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undefined, and when <var>op</var> is ±0, *<var>signp</var> is the sign of the zero.
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|
</p></dd></dl>
|
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|
<dl>
|
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<dt><a name="index-mpfr_005fdigamma"></a>Function: <em>int</em> <strong>mpfr_digamma</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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|
<dd><p>Set <var>rop</var> to the value of the Digamma (sometimes also called Psi)
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|
function on <var>op</var>, rounded in the direction <var>rnd</var>.
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|
When <var>op</var> is a negative integer, set <var>rop</var> to NaN.
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|
</p></dd></dl>
|
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|
<dl>
|
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|
<dt><a name="index-mpfr_005fzeta"></a>Function: <em>int</em> <strong>mpfr_zeta</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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|
<dt><a name="index-mpfr_005fzeta_005fui"></a>Function: <em>int</em> <strong>mpfr_zeta_ui</strong> <em>(mpfr_t <var>rop</var>, unsigned long <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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|
<dd><p>Set <var>rop</var> to the value of the Riemann Zeta function on <var>op</var>,
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|
rounded in the direction <var>rnd</var>.
|
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|
|
</p></dd></dl>
|
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|
|
<dl>
|
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|
<dt><a name="index-mpfr_005ferf"></a>Function: <em>int</em> <strong>mpfr_erf</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
|
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|
|
<dt><a name="index-mpfr_005ferfc"></a>Function: <em>int</em> <strong>mpfr_erfc</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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|
|
<dd><p>Set <var>rop</var> to the value of the error function on <var>op</var>
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|
|
(resp. the complementary error function on <var>op</var>)
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|
|
rounded in the direction <var>rnd</var>.
|
|
|
|
|
|
</p></dd></dl>
|
|
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|
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|
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|
|
<dl>
|
|
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|
<dt><a name="index-mpfr_005fj0"></a>Function: <em>int</em> <strong>mpfr_j0</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
|
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|
|
<dt><a name="index-mpfr_005fj1"></a>Function: <em>int</em> <strong>mpfr_j1</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
|
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|
|
<dt><a name="index-mpfr_005fjn"></a>Function: <em>int</em> <strong>mpfr_jn</strong> <em>(mpfr_t <var>rop</var>, long <var>n</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
|
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|
|
<dd><p>Set <var>rop</var> to the value of the first kind Bessel function of order 0,
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|
(resp. 1 and <var>n</var>)
|
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|
|
on <var>op</var>, rounded in the direction <var>rnd</var>. When <var>op</var> is
|
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|
|
|
NaN, <var>rop</var> is always set to NaN. When <var>op</var> is plus or minus Infinity,
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|
|
<var>rop</var> is set to +0. When <var>op</var> is zero, and <var>n</var> is not zero,
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|
|
<var>rop</var> is set to +0 or −0 depending on the parity and sign of <var>n</var>,
|
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|
|
and the sign of <var>op</var>.
|
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|
|
|
|
</p></dd></dl>
|
|
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|
|
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|
|
|
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|
|
<dl>
|
|
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|
<dt><a name="index-mpfr_005fy0"></a>Function: <em>int</em> <strong>mpfr_y0</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
|
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|
|
<dt><a name="index-mpfr_005fy1"></a>Function: <em>int</em> <strong>mpfr_y1</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
|
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|
<dt><a name="index-mpfr_005fyn"></a>Function: <em>int</em> <strong>mpfr_yn</strong> <em>(mpfr_t <var>rop</var>, long <var>n</var>, mpfr_t <var>op</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
|
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|
|
|
|
<dd><p>Set <var>rop</var> to the value of the second kind Bessel function of order 0
|
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|
(resp. 1 and <var>n</var>)
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|
|
on <var>op</var>, rounded in the direction <var>rnd</var>. When <var>op</var> is
|
|
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|
|
NaN or negative, <var>rop</var> is always set to NaN. When <var>op</var> is +Inf,
|
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|
|
<var>rop</var> is set to +0. When <var>op</var> is zero, <var>rop</var> is set to +Inf
|
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|
|
or −Inf depending on the parity and sign of <var>n</var>.
|
|
|
|
|
|
</p></dd></dl>
|
|
|
|
|
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|
|
|
|
|
|
<dl>
|
|
|
|
|
|
<dt><a name="index-mpfr_005ffma"></a>Function: <em>int</em> <strong>mpfr_fma</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op1</var>, mpfr_t <var>op2</var>, mpfr_t <var>op3</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
|
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|
|
<dt><a name="index-mpfr_005ffms"></a>Function: <em>int</em> <strong>mpfr_fms</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op1</var>, mpfr_t <var>op2</var>, mpfr_t <var>op3</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
|
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|
|
<dd><p>Set <var>rop</var> to <em>(<var>op1</var> times <var>op2</var>) + <var>op3</var></em>
|
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|
|
(resp. <em>(<var>op1</var> times <var>op2</var>) - <var>op3</var></em>)
|
|
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|
|
rounded in the direction <var>rnd</var>. Concerning special values (signed zeros,
|
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|
|
|
|
infinities, NaN), these functions behave like a multiplication followed by a
|
|
|
|
|
|
separate addition or subtraction. That is, the fused operation matters only
|
|
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|
|
for rounding.
|
|
|
|
|
|
</p></dd></dl>
|
|
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|
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|
|
|
|
|
|
<dl>
|
|
|
|
|
|
<dt><a name="index-mpfr_005fagm"></a>Function: <em>int</em> <strong>mpfr_agm</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>op1</var>, mpfr_t <var>op2</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
|
|
|
|
|
|
<dd><p>Set <var>rop</var> to the arithmetic-geometric mean of <var>op1</var> and <var>op2</var>,
|
|
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|
|
rounded in the direction <var>rnd</var>.
|
|
|
|
|
|
The arithmetic-geometric mean is the common limit of the sequences
|
|
|
|
|
|
<em><var>u</var>_<var>n</var></em> and <em><var>v</var>_<var>n</var></em>,
|
|
|
|
|
|
where <em><var>u</var>_<var>0</var></em>=<var>op1</var>, <em><var>v</var>_<var>0</var></em>=<var>op2</var>,
|
|
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|
|
<em><var>u</var>_(<var>n</var>+1)</em> is the
|
|
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|
|
arithmetic mean of <em><var>u</var>_<var>n</var></em> and <em><var>v</var>_<var>n</var></em>,
|
|
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|
|
|
and <em><var>v</var>_(<var>n</var>+1)</em> is the geometric mean of
|
|
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|
|
<em><var>u</var>_<var>n</var></em> and <em><var>v</var>_<var>n</var></em>.
|
|
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|
|
If any operand is negative, set <var>rop</var> to NaN.
|
|
|
|
|
|
</p></dd></dl>
|
|
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|
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<dl>
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<dt><a name="index-mpfr_005fhypot"></a>Function: <em>int</em> <strong>mpfr_hypot</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>x</var>, mpfr_t <var>y</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the Euclidean norm of <var>x</var> and <var>y</var>,
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i.e., the square root of the sum of the squares of <var>x</var> and <var>y</var>,
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rounded in the direction <var>rnd</var>.
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Special values are handled as described in the ISO C99 (Section F.9.4.3)
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and IEEE 754-2008 (Section 9.2.1) standards:
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If <var>x</var> or <var>y</var> is an infinity, then +Inf is returned in <var>rop</var>,
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even if the other number is NaN.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fai"></a>Function: <em>int</em> <strong>mpfr_ai</strong> <em>(mpfr_t <var>rop</var>, mpfr_t <var>x</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the value of the Airy function Ai
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on <var>x</var>, rounded in the direction <var>rnd</var>.
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When <var>x</var> is
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NaN,
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<var>rop</var> is always set to NaN. When <var>x</var> is +Inf or −Inf,
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<var>rop</var> is +0.
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The current implementation is not intended to be used with large arguments.
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It works with abs(<var>x</var>) typically smaller than 500. For larger arguments,
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other methods should be used and will be implemented in a future version.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fconst_005flog2"></a>Function: <em>int</em> <strong>mpfr_const_log2</strong> <em>(mpfr_t <var>rop</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fconst_005fpi"></a>Function: <em>int</em> <strong>mpfr_const_pi</strong> <em>(mpfr_t <var>rop</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fconst_005feuler"></a>Function: <em>int</em> <strong>mpfr_const_euler</strong> <em>(mpfr_t <var>rop</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dt><a name="index-mpfr_005fconst_005fcatalan"></a>Function: <em>int</em> <strong>mpfr_const_catalan</strong> <em>(mpfr_t <var>rop</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the logarithm of 2, the value of <em>Pi</em>,
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of Euler’s constant 0.577…, of Catalan’s constant 0.915…,
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respectively, rounded in the direction
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<var>rnd</var>. These functions cache the computed values to avoid other
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calculations if a lower or equal precision is requested. To free these caches,
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use <code>mpfr_free_cache</code>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005ffree_005fcache"></a>Function: <em>void</em> <strong>mpfr_free_cache</strong> <em>(void)</em></dt>
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<dd><p>Free various caches used by MPFR internally, in particular the
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caches used by the functions computing constants (<code>mpfr_const_log2</code>,
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<code>mpfr_const_pi</code>,
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<code>mpfr_const_euler</code> and <code>mpfr_const_catalan</code>).
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You should call this function before terminating a thread, even if you did
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not call these functions directly (they could have been called internally).
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpfr_005fsum"></a>Function: <em>int</em> <strong>mpfr_sum</strong> <em>(mpfr_t <var>rop</var>, mpfr_ptr const <var>tab</var>[], unsigned long int <var>n</var>, mpfr_rnd_t <var>rnd</var>)</em></dt>
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<dd><p>Set <var>rop</var> to the sum of all elements of <var>tab</var>, whose size is <var>n</var>,
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rounded in the direction <var>rnd</var>. Warning: for efficiency reasons,
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<var>tab</var> is an array of pointers
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to <code>mpfr_t</code>, not an array of <code>mpfr_t</code>.
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If the returned <code>int</code> value is zero, <var>rop</var> is guaranteed to be the
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exact sum; otherwise <var>rop</var> might be smaller than, equal to, or larger than
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the exact sum (in accordance to the rounding mode).
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However, <code>mpfr_sum</code> does guarantee the result is correctly rounded.
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</p></dd></dl>
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<hr>
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<div class="header">
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<p>
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Next: <a href="Input-and-Output-Functions.html#Input-and-Output-Functions" accesskey="n" rel="next">Input and Output Functions</a>, Previous: <a href="Comparison-Functions.html#Comparison-Functions" accesskey="p" rel="prev">Comparison Functions</a>, Up: <a href="MPFR-Interface.html#MPFR-Interface" accesskey="u" rel="up">MPFR Interface</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</body>
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