namespace Eigen {

/** \eigenManualPage CoeffwiseMathFunctions Catalog of coefficient-wise math functions


<!-- <span style="font-size:300%; color:red; font-weight: 900;">!WORK IN PROGRESS!</span> -->

This table presents a catalog of the coefficient-wise math functions supported by %Eigen.
In this table, \c a, \c b, refer to Array objects or expressions, and \c m refers to a linear algebra Matrix/Vector object. Standard scalar types are abbreviated as follows:
  - \c int: \c i32
  - \c float: \c f
  - \c double: \c d
  - \c std::complex<float>: \c cf
  - \c std::complex<double>: \c cd

For each row, the first column list the equivalent calls for arrays, and matrices when supported. Of course, all functions are available for matrices by first casting it as an array: \c m.array().

The third column gives some hints in the underlying scalar implementation. In most cases, %Eigen does not implement itself the math function but relies on the STL for standard scalar types, or user-provided functions for custom scalar types.
For instance, some simply calls the respective function of the STL while preserving <a href="http://en.cppreference.com/w/cpp/language/adl">argument-dependent lookup</a> for custom types.
The following:
\code
using std::foo;
foo(a[i]);
\endcode
means that the STL's function \c std::foo will be potentially called if it is compatible with the underlying scalar type. If not, then the user must ensure that an overload of the function foo is available for the given scalar type (usually defined in the same namespace as the given scalar type).

<table class="manual-hl">
<tr>
<th>API</th><th>Description</th><th>Default scalar implementation</th><th>SIMD</th>
</tr>
<tr><td colspan="4"></td></tr>
<tr><th colspan="4">Basic operations</th></tr>
<tr>
  <td class="code">
  \anchor cwisetable_abs
  a.\link ArrayBase::abs abs\endlink(); \n
  \link Eigen::abs abs\endlink(a); \n
  m.\link MatrixBase::cwiseAbs cwiseAbs\endlink();
  </td>
  <td>absolute value (\f$ |a_i| \f$) </td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/fabs">std::abs</a>; \n
  abs(a[i]);
  </td>
  <td>SSE2, AVX (i32,f,d)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_inverse
  a.\link ArrayBase::inverse inverse\endlink(); \n
  \link Eigen::inverse inverse\endlink(a); \n
  m.\link MatrixBase::cwiseInverse cwiseInverse\endlink();
  </td>
  <td>inverse value (\f$ 1/a_i \f$) </td>
  <td class="code">
  1/a[i];
  </td>
  <td>All engines (f,d,fc,fd)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_conj
  a.\link ArrayBase::conjugate conjugate\endlink(); \n
  \link Eigen::conj conj\endlink(a); \n
  m.\link MatrixBase::conjugate conjugate\endlink();
  </td>
  <td><a href="https://en.wikipedia.org/wiki/Complex_conjugate">complex conjugate</a> (\f$ \bar{a_i} \f$),\n
  no-op for real </td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/complex/conj">std::conj</a>; \n
  conj(a[i]);
  </td>
  <td>All engines (fc,fd)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_arg
  a.\link ArrayBase::arg arg\endlink(); \n
  \link Eigen::arg arg\endlink(a); \n
  m.\link MatrixBase::cwiseArg cwiseArg\endlink();
  </td>
  <td>phase angle of complex number</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/complex/arg">std::arg</a>; \n
  arg(a[i]);
  </td>
  <td>All engines (fc,fd)</td>
</tr>
<tr>
<th colspan="4">Exponential functions</th>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_exp
  a.\link ArrayBase::exp exp\endlink(); \n
  \link Eigen::exp exp\endlink(a);
  </td>
  <td>\f$ e \f$ raised to the given power (\f$ e^{a_i} \f$) </td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/exp">std::exp</a>; \n
  exp(a[i]);
  </td>
  <td>SSE2, AVX (f,d)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_log
  a.\link ArrayBase::log log\endlink(); \n
  \link Eigen::log log\endlink(a);
  </td>
  <td>natural (base \f$ e \f$) logarithm (\f$ \ln({a_i}) \f$)</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/log">std::log</a>; \n
  log(a[i]);
  </td>
  <td>SSE2, AVX (f)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_log1p
  a.\link ArrayBase::log1p log1p\endlink(); \n
  \link Eigen::log1p log1p\endlink(a);
  </td>
  <td>natural (base \f$ e \f$) logarithm of 1 plus \n the given number (\f$ \ln({1+a_i}) \f$)</td>
  <td>built-in generic implementation based on \c log,\n
  plus \c using <a href="http://en.cppreference.com/w/cpp/numeric/math/log1p">\c std::log1p </a></td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_log10
  a.\link ArrayBase::log10 log10\endlink(); \n
  \link Eigen::log10 log10\endlink(a);
  </td>
  <td>base 10 logarithm (\f$ \log_{10}({a_i}) \f$)</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/log10">std::log10</a>; \n
  log10(a[i]);
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_ldexp
  a.\link ArrayBase::ldexp ldexp\endlink(e); \n
  \link Eigen::ldexp ldexp\endlink(a,e);
  </td>
  <td>multiplies each coefficient by \f$ 2^{e} \f$ for an integer \c e (\f$ a_i \cdot 2^{e} \f$); \n
  exact — the floating-point exponent is adjusted directly, so the result \n
  is correct (including denormals) even when \f$ 2^{e} \f$ itself is not representable</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/ldexp">std::ldexp</a>; \n
  ldexp(a[i],e);
  </td>
  <td>SSE2, AVX (f,d)</td>
</tr>
<tr>
<th colspan="4">Power functions</th>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_pow
  a.\link ArrayBase::pow pow\endlink(b); \n
  \link ArrayBase::pow(const Eigen::ArrayBase< Derived > &x, const Eigen::ArrayBase< ExponentDerived > &exponents) pow\endlink(a,b);
  </td>
  <!-- For some reason Doxygen thinks that pow is in ArrayBase namespace -->
  <td>raises a number to the given power (\f$ a_i ^ {b_i} \f$) \n \c a and \c b can be either an array or scalar.</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/pow">std::pow</a>; \n
  pow(a[i],b[i]);\n
  (plus builtin for integer types)</td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_sqrt
  a.\link ArrayBase::sqrt sqrt\endlink(); \n
  \link Eigen::sqrt sqrt\endlink(a);\n
  m.\link MatrixBase::cwiseSqrt cwiseSqrt\endlink();
  </td>
  <td>computes square root (\f$ \sqrt a_i \f$)</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/sqrt">std::sqrt</a>; \n
  sqrt(a[i]);</td>
  <td>SSE2, AVX (f,d)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_cbrt
  a.\link ArrayBase::cbrt cbrt\endlink(); \n
  \link Eigen::cbrt cbrt\endlink(a);\n
  m.\link MatrixBase::cwiseCbrt cwiseCbrt\endlink();
  </td>
  <td>computes cube root (\f$ \sqrt[3]{ a_i }\f$)</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/cbrt">std::cbrt</a>; \n
  cbrt(a[i]);</td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_rsqrt
  a.\link ArrayBase::rsqrt rsqrt\endlink(); \n
  \link Eigen::rsqrt rsqrt\endlink(a);
  </td>
  <td><a href="https://en.wikipedia.org/wiki/Fast_inverse_square_root">reciprocal square root</a> (\f$ 1/{\sqrt a_i} \f$)</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/sqrt">std::sqrt</a>; \n
  1/sqrt(a[i]); \n
  </td>
  <td>SSE2, AVX, AltiVec, ZVector (f,d)\n
  (approx + 1 Newton iteration)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_square
  a.\link ArrayBase::square square\endlink(); \n
  \link Eigen::square square\endlink(a);
  </td>
  <td>computes square power (\f$ a_i^2 \f$)</td>
  <td class="code">
  a[i]*a[i]</td>
  <td>All (i32,f,d,cf,cd)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_cube
  a.\link ArrayBase::cube cube\endlink(); \n
  \link Eigen::cube cube\endlink(a);
  </td>
  <td>computes cubic power (\f$ a_i^3 \f$)</td>
  <td class="code">
  a[i]*a[i]*a[i]</td>
  <td>All (i32,f,d,cf,cd)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_abs2
  a.\link ArrayBase::abs2 abs2\endlink(); \n
  \link Eigen::abs2 abs2\endlink(a);\n
  m.\link MatrixBase::cwiseAbs2 cwiseAbs2\endlink();
  </td>
  <td>computes the squared absolute value (\f$ |a_i|^2 \f$)</td>
  <td class="code">
  real:    a[i]*a[i] \n
  complex:  real(a[i])*real(a[i]) \n
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; + imag(a[i])*imag(a[i])</td>
  <td>All (i32,f,d)</td>
</tr>
<tr>
<th colspan="4">Trigonometric functions</th>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_sin
  a.\link ArrayBase::sin sin\endlink(); \n
  \link Eigen::sin sin\endlink(a);
  </td>
  <td>computes sine</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/sin">std::sin</a>; \n
  sin(a[i]);</td>
  <td>SSE2, AVX (f)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_cos
  a.\link ArrayBase::cos cos\endlink(); \n
  \link Eigen::cos cos\endlink(a);
  </td>
  <td>computes cosine</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/cos">std::cos</a>; \n
  cos(a[i]);</td>
  <td>SSE2, AVX (f)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_tan
  a.\link ArrayBase::tan tan\endlink(); \n
  \link Eigen::tan tan\endlink(a);
  </td>
  <td>computes tangent</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/tan">std::tan</a>; \n
  tan(a[i]);</td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_asin
  a.\link ArrayBase::asin asin\endlink(); \n
  \link Eigen::asin asin\endlink(a);
  </td>
  <td>computes arc sine (\f$ \sin^{-1} a_i \f$)</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/asin">std::asin</a>; \n
  asin(a[i]);</td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_acos
  a.\link ArrayBase::acos acos\endlink(); \n
  \link Eigen::acos acos\endlink(a);
  </td>
  <td>computes arc cosine  (\f$ \cos^{-1} a_i \f$)</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/acos">std::acos</a>; \n
  acos(a[i]);</td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_atan
  a.\link ArrayBase::atan atan\endlink(); \n
  \link Eigen::atan atan\endlink(a);
  </td>
  <td>computes arc tangent (\f$ \tan^{-1} a_i \f$)</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/atan">std::atan</a>; \n
  atan(a[i]);</td>
  <td></td>
</tr>
<tr>
<th colspan="4">Hyperbolic functions</th>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_sinh
  a.\link ArrayBase::sinh sinh\endlink(); \n
  \link Eigen::sinh sinh\endlink(a);
  </td>
  <td>computes hyperbolic sine</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/sinh">std::sinh</a>; \n
  sinh(a[i]);</td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_cosh
  a.\link ArrayBase::cosh cohs\endlink(); \n
  \link Eigen::cosh cosh\endlink(a);
  </td>
  <td>computes hyperbolic cosine</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/cosh">std::cosh</a>; \n
  cosh(a[i]);</td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_tanh
  a.\link ArrayBase::tanh tanh\endlink(); \n
  \link Eigen::tanh tanh\endlink(a);
  </td>
  <td>computes hyperbolic tangent</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/tanh">std::tanh</a>; \n
  tanh(a[i]);</td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_asinh
  a.\link ArrayBase::asinh asinh\endlink(); \n
  \link Eigen::asinh asinh\endlink(a);
  </td>
  <td>computes inverse hyperbolic sine</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/asinh">std::asinh</a>; \n
  asinh(a[i]);</td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_acosh
  a.\link ArrayBase::acosh cohs\endlink(); \n
  \link Eigen::acosh acosh\endlink(a);
  </td>
  <td>computes hyperbolic cosine</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/acosh">std::acosh</a>; \n
  acosh(a[i]);</td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_atanh
  a.\link ArrayBase::atanh atanh\endlink(); \n
  \link Eigen::atanh atanh\endlink(a);
  </td>
  <td>computes hyperbolic tangent</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/atanh">std::atanh</a>; \n
  atanh(a[i]);</td>
  <td></td>
</tr>
<tr>
<th colspan="4">Nearest integer floating point operations</th>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_ceil
  a.\link ArrayBase::ceil ceil\endlink(); \n
  \link Eigen::ceil ceil\endlink(a);
  </td>
  <td>nearest integer not less than the given value</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/ceil">std::ceil</a>; \n
  ceil(a[i]);</td>
  <td>SSE4,AVX,ZVector (f,d)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_floor
  a.\link ArrayBase::floor floor\endlink(); \n
  \link Eigen::floor floor\endlink(a);
  </td>
  <td>nearest integer not greater than the given value</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/floor">std::floor</a>; \n
  floor(a[i]);</td>
  <td>SSE4,AVX,ZVector (f,d)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_round
  a.\link ArrayBase::round round\endlink(); \n
  \link Eigen::round round\endlink(a);
  </td>
  <td>nearest integer, \n rounding away from zero in halfway cases</td>
  <td>built-in generic implementation \n based on \c floor and \c ceil,\n
  plus \c using <a href="http://en.cppreference.com/w/cpp/numeric/math/round">\c std::round </a></td>
  <td>SSE4,AVX,ZVector (f,d)</td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_rint
  a.\link ArrayBase::rint rint\endlink(); \n
  \link Eigen::rint rint\endlink(a);
  </td>
  <td>nearest integer, \n rounding to nearest even in halfway cases</td>
  <td>built-in generic implementation using <a href="http://en.cppreference.com/w/cpp/numeric/math/rint">\c std::rint </a>
  or <a href="http://en.cppreference.com/w/c/numeric/math/rint">\c rintf </a>; </td>
  <td>SSE4,AVX (f,d)</td>
</tr>
<tr>
<th colspan="4">Floating point manipulation functions</th>
</tr>
<tr>
<th colspan="4">Classification and comparison</th>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_isfinite
  a.\link ArrayBase::isFinite isFinite\endlink(); \n
  \link Eigen::isfinite isfinite\endlink(a);
  </td>
  <td>checks if the given number has finite value</td>
  <td>built-in generic implementation,\n
  plus \c using <a href="http://en.cppreference.com/w/cpp/numeric/math/isfinite">\c std::isfinite </a></td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_isinf
  a.\link ArrayBase::isInf isInf\endlink(); \n
  \link Eigen::isinf isinf\endlink(a);
  </td>
  <td>checks if the given number is infinite</td>
  <td>built-in generic implementation,\n
  plus \c using <a href="http://en.cppreference.com/w/cpp/numeric/math/isinf">\c std::isinf </a></td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_isnan
  a.\link ArrayBase::isNaN isNaN\endlink(); \n
  \link Eigen::isnan isnan\endlink(a);
  </td>
  <td>checks if the given number is not a number</td>
  <td>built-in generic implementation,\n
  plus \c using <a href="http://en.cppreference.com/w/cpp/numeric/math/isnan">\c std::isnan </a></td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_less
  a.\c operator<(b); \n
  a.\c operator<(s);
  </td>
  <td>coefficient-wise less than comparison (\f$ a_i \lt b_i \f$). \n
  \c a and \c b can be either an array or scalar.</td>
  <td class="code">
  a[i] < b[i];
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_less_equal
  a.\c operator<=(b); \n
  a.\c operator<=(s);
  </td>
  <td>coefficient-wise less than or equal comparison (\f$ a_i \le b_i \f$). \n
  \c a and \c b can be either an array or scalar.</td>
  <td class="code">
  a[i] <= b[i];
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_greater
  a.\c operator>(b); \n
  a.\c operator>(s);
  </td>
  <td>coefficient-wise greater than comparison (\f$ a_i \gt b_i \f$). \n
  \c a and \c b can be either an array or scalar.</td>
  <td class="code">
  a[i] > b[i];
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_greater_equal
  a.\c operator>=(b); \n
  a.\c operator>=(s);
  </td>
  <td>coefficient-wise greater than or equal comparison (\f$ a_i \ge b_i \f$). \n
  \c a and \c b can be either an array or scalar.</td>
  <td class="code">
  a[i] >= b[i];
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_equal
  a.\c operator==(b); \n
  a.\c operator==(s);
  </td>
  <td>coefficient-wise equality comparison (\f$ a_i = b_i \f$). \n
  \c a and \c b can be either an array or scalar. \n
  \warning Performs exact comparison; prefer \c isApprox() for floating-point types.</td>
  <td class="code">
  a[i] == b[i];
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_not_equal
  a.\c operator!=(b); \n
  a.\c operator!=(s);
  </td>
  <td>coefficient-wise not-equal comparison (\f$ a_i \ne b_i \f$). \n
  \c a and \c b can be either an array or scalar. \n
  \warning Performs exact comparison; prefer \c isApprox() for floating-point types.</td>
  <td class="code">
  a[i] != b[i];
  </td>
  <td></td>
</tr>
<tr>
<th colspan="4">Error and gamma functions</th>
</tr>
<tr> <td colspan="4">  Require \c \#include \c <contrib/Eigen/SpecialFunctions> </td></tr>
<tr>
  <td class="code">
  \anchor cwisetable_erf
  a.\link ArrayBase::erf erf\endlink(); \n
  \link Eigen::erf erf\endlink(a);
  </td>
  <td>error function</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/erf">std::erf</a> \n
  erf(a[i]);
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_erfc
  a.\link ArrayBase::erfc erfc\endlink(); \n
  \link Eigen::erfc erfc\endlink(a);
  </td>
  <td>complementary error function</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/erfc">std::erfc</a> \n
  erfc(a[i]);
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_lgamma
  a.\link ArrayBase::lgamma lgamma\endlink(); \n
  \link Eigen::lgamma lgamma\endlink(a);
  </td>
  <td>natural logarithm of the gamma function</td>
  <td class="code">
  using <a href="http://en.cppreference.com/w/cpp/numeric/math/lgamma">std::lgamma</a> \n
  lgamma(a[i]);
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_digamma
  a.\link ArrayBase::digamma digamma\endlink(); \n
  digamma(a);
  </td>
  <td><a href="https://en.wikipedia.org/wiki/Digamma_function">logarithmic derivative of the gamma function</a></td>
  <td>
  built-in for float and double
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_igamma
  igamma(a,x);
  </td>
  <td><a href="https://en.wikipedia.org/wiki/Incomplete_gamma_function">lower incomplete gamma integral</a>
  \n \f$ \gamma(a_i,x_i)= \frac{1}{|a_i|} \int_{0}^{x_i}e^{\text{-}t} t^{a_i-1} \mathrm{d} t \f$</td>
  <td>
  built-in for float and double,\n built-in generic fallback
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_igammac
  igammac(a,x);
  </td>
  <td><a href="https://en.wikipedia.org/wiki/Incomplete_gamma_function">upper incomplete gamma integral</a>
  \n \f$ \Gamma(a_i,x_i) = \frac{1}{|a_i|} \int_{x_i}^{\infty}e^{\text{-}t} t^{a_i-1} \mathrm{d} t \f$</td>
  <td>
  built-in for float and double,\n built-in generic fallback
  </td>
  <td></td>
</tr>
<tr>
<th colspan="4">Special functions</th>
</tr>
<tr> <td colspan="4">  Require \c \#include \c <contrib/Eigen/SpecialFunctions> </td></tr>
<tr>
  <td class="code">
  \anchor cwisetable_polygamma
  polygamma(n,x);
  </td>
  <td><a href="https://en.wikipedia.org/wiki/Polygamma_function">n-th derivative of digamma at x</a></td>
  <td>
  built-in generic based on\n <a href="#cwisetable_lgamma">\c lgamma </a>,
  <a href="#cwisetable_digamma"> \c digamma </a>
  and <a href="#cwisetable_zeta">\c zeta </a>.
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_betainc
  betainc(a,b,x);
  </td>
  <td><a href="https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function">regularized incomplete beta function</a></td>
  <td>
  built-in for float and double,\n built-in generic fallback
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_zeta
  zeta(a,b); \n
  a.\link ArrayBase::zeta zeta\endlink(b);
  </td>
  <td><a href="https://en.wikipedia.org/wiki/Hurwitz_zeta_function">Hurwitz zeta function</a>
  \n \f$ \zeta(a_i,b_i)=\sum_{k=0}^{\infty}(b_i+k)^{\text{-}a_i} \f$</td>
  <td>
  built-in for float and double
  </td>
  <td></td>
</tr>
<tr>
  <td class="code">
  \anchor cwisetable_ndtri
  a.\link ArrayBase::ndtri ndtri\endlink(); \n
  \link Eigen::ndtri ndtri\endlink(a);
  </td>
  <td>Inverse of the CDF of the Normal distribution function</td>
  <td>
  built-in for float and double
  </td>
  <td></td>
</tr>
<tr><td colspan="4"></td></tr>
</table>

\n

\section CoeffwiseMathFunctionsAccuracy Accuracy of vectorized math functions

The following tables summarize the accuracy of %Eigen's vectorized implementations measured
in units of ULP (Unit in the Last Place) on an x86-64 system (Intel Xeon, GCC) with SSE2 SIMD
target. The reference values were computed using
<a href="https://www.mpfr.org/">MPFR</a> at 128-bit precision.
Float results are exhaustive over all ~4.28 billion finite representable values.
Double results sample ~2.88 billion values using a geometric stepping factor of 10<sup>-6</sup>.

These numbers may differ for other SIMD targets (AVX, AVX512, NEON, SVE, etc.)
since each has its own packet math implementations. Functions marked "delegates to std"
do not have a custom vectorized implementation for the tested SIMD target &mdash; they call
the standard library function element-by-element.

The full histograms for each function can be generated with the \c ulp_accuracy tool
in <tt>test/ulp_accuracy/</tt>.

\subsection CoeffwiseMathFunctionsAccuracy_float Float precision

<table class="manual-hl">
<tr><th>Function</th><th>Max |ULP|</th><th>Mean |ULP|</th><th>% Exact</th><th>Notes</th></tr>
<tr><th colspan="5">Trigonometric</th></tr>
<tr><td>sin</td>  <td>2</td>  <td>0.087</td> <td>91.3%</td> <td>\ref CoeffwiseMathFunctionsAccuracy_note1 "1"</td></tr>
<tr><td>cos</td>  <td>2</td>  <td>0.088</td> <td>91.2%</td> <td>\ref CoeffwiseMathFunctionsAccuracy_note1 "1"</td></tr>
<tr><td>tan</td>  <td>5</td>  <td>0.238</td> <td>77.3%</td> <td>\ref CoeffwiseMathFunctionsAccuracy_note1 "1"</td></tr>
<tr><td>asin</td> <td>4</td>  <td>0.726</td> <td>51.3%</td> <td></td></tr>
<tr><td>acos</td> <td>4</td>  <td>0.057</td> <td>95.0%</td> <td></td></tr>
<tr><td>atan</td> <td>4</td>  <td>0.061</td> <td>94.0%</td> <td></td></tr>
<tr><th colspan="5">Hyperbolic</th></tr>
<tr><td>sinh</td>  <td>2</td>  <td>0.017</td> <td>98.3%</td> <td></td></tr>
<tr><td>cosh</td>  <td>2</td>  <td>0.004</td> <td>99.6%</td> <td></td></tr>
<tr><td>tanh</td>  <td>6</td>  <td>0.030</td> <td>97.2%</td> <td></td></tr>
<tr><td>asinh</td> <td>2</td>  <td>0.145</td> <td>85.5%</td> <td></td></tr>
<tr><td>acosh</td> <td>2</td>  <td>0.057</td> <td>94.3%</td> <td></td></tr>
<tr><td>atanh</td> <td>2</td>  <td>0.004</td> <td>99.6%</td> <td></td></tr>
<tr><th colspan="5">Exponential / Logarithmic</th></tr>
<tr><td>exp</td>   <td>1</td>  <td>0.018</td> <td>98.2%</td> <td></td></tr>
<tr><td>exp2</td>  <td>6</td>  <td>0.034</td> <td>97.3%</td> <td></td></tr>
<tr><td>expm1</td> <td>5</td>  <td>0.060</td> <td>94.6%</td> <td></td></tr>
<tr><td>log</td>   <td>3</td>  <td>0.120</td> <td>88.0%</td> <td></td></tr>
<tr><td>log1p</td> <td>5</td>  <td>0.134</td> <td>87.5%</td> <td></td></tr>
<tr><td>log10</td> <td>2</td>  <td>0.007</td> <td>99.3%</td> <td></td></tr>
<tr><td>log2</td>  <td>5</td>  <td>0.005</td> <td>99.5%</td> <td></td></tr>
<tr><th colspan="5">Error / Special</th></tr>
<tr><td>erf</td>      <td>7</td>     <td>0.332</td> <td>67.5%</td> <td></td></tr>
<tr><td>erfc</td>     <td>8</td>     <td>0.010</td> <td>99.2%</td> <td></td></tr>
<tr><td>lgamma</td>   <td colspan="3"><em>delegates to std</em></td> <td>\ref CoeffwiseMathFunctionsAccuracy_note3 "3"</td></tr>
<tr><th colspan="5">Other</th></tr>
<tr><td>logistic</td> <td>7</td>     <td>0.040</td> <td>97.0%</td> <td></td></tr>
<tr><td>sqrt</td>     <td>0</td>     <td>0.000</td> <td>100%</td>  <td>Uses hardware sqrt</td></tr>
<tr><td>cbrt</td>     <td>2</td>     <td>0.552</td> <td>49.1%</td> <td></td></tr>
<tr><td>rsqrt</td>    <td>&infin;</td> <td>0.114</td> <td>88.2%</td> <td>\ref CoeffwiseMathFunctionsAccuracy_note4 "4"</td></tr>
</table>

\subsection CoeffwiseMathFunctionsAccuracy_double Double precision

<table class="manual-hl">
<tr><th>Function</th><th>Max |ULP|</th><th>Mean |ULP|</th><th>% Exact</th><th>Notes</th></tr>
<tr><th colspan="5">Trigonometric</th></tr>
<tr><td>sin</td>  <td>13,879,755</td> <td>0.093</td> <td>93.2%</td> <td>\ref CoeffwiseMathFunctionsAccuracy_note1 "1"</td></tr>
<tr><td>cos</td>  <td>2,024,130</td>  <td>0.043</td> <td>98.2%</td> <td>\ref CoeffwiseMathFunctionsAccuracy_note1 "1"</td></tr>
<tr><td>tan</td>  <td>13,879,755</td> <td>0.128</td> <td>92.7%</td> <td>\ref CoeffwiseMathFunctionsAccuracy_note1 "1"</td></tr>
<tr><td>asin</td> <td>1</td>          <td>&lt;0.001</td> <td>&gt;99.9%</td> <td></td></tr>
<tr><td>acos</td> <td>1</td>          <td>&lt;0.001</td> <td>100%</td> <td></td></tr>
<tr><td>atan</td> <td>5</td>          <td>0.013</td> <td>98.8%</td> <td></td></tr>
<tr><th colspan="5">Hyperbolic</th></tr>
<tr><td>sinh</td>  <td>2</td>  <td>0.004</td> <td>99.6%</td> <td></td></tr>
<tr><td>cosh</td>  <td>2</td>  <td>0.001</td> <td>99.9%</td> <td></td></tr>
<tr><td>tanh</td>  <td>8</td>  <td>0.008</td> <td>99.3%</td> <td></td></tr>
<tr><td>asinh</td> <td>2</td>  <td>0.098</td> <td>90.2%</td> <td></td></tr>
<tr><td>acosh</td> <td>2</td>  <td>0.047</td> <td>95.3%</td> <td></td></tr>
<tr><td>atanh</td> <td>2</td>  <td>&lt;0.001</td> <td>&gt;99.9%</td> <td></td></tr>
<tr><th colspan="5">Exponential / Logarithmic</th></tr>
<tr><td>exp</td>   <td>2</td>   <td>0.001</td> <td>99.9%</td> <td></td></tr>
<tr><td>exp2</td>  <td>214</td> <td>0.107</td> <td>99.6%</td> <td>\ref CoeffwiseMathFunctionsAccuracy_note2 "2"</td></tr>
<tr><td>expm1</td> <td>3</td>   <td>0.010</td> <td>99.1%</td> <td></td></tr>
<tr><td>log</td>   <td>2</td>   <td>0.147</td> <td>85.3%</td> <td></td></tr>
<tr><td>log1p</td> <td>3</td>   <td>0.097</td> <td>90.6%</td> <td></td></tr>
<tr><td>log10</td> <td>2</td>   <td>0.001</td> <td>99.9%</td> <td></td></tr>
<tr><td>log2</td>  <td>2</td>   <td>&lt;0.001</td> <td>99.9%</td> <td></td></tr>
<tr><th colspan="5">Error / Special</th></tr>
<tr><td>erf</td>      <td>&infin;</td> <td>0.050</td> <td>70.5%</td> <td>\ref CoeffwiseMathFunctionsAccuracy_note5 "5"</td></tr>
<tr><td>erfc</td>     <td>11</td>      <td>0.002</td> <td>99.9%</td> <td></td></tr>
<tr><td>lgamma</td>   <td colspan="3"><em>delegates to std</em></td> <td>\ref CoeffwiseMathFunctionsAccuracy_note3 "3"</td></tr>
<tr><th colspan="5">Other</th></tr>
<tr><td>logistic</td> <td>3</td>       <td>0.008</td> <td>99.2%</td> <td></td></tr>
<tr><td>sqrt</td>     <td>0</td>       <td>0.000</td> <td>100%</td>  <td>Uses hardware sqrt</td></tr>
<tr><td>cbrt</td>     <td>2</td>       <td>0.119</td> <td>88.1%</td> <td></td></tr>
<tr><td>rsqrt</td>    <td>&infin;</td> <td>0.135</td> <td>86.5%</td> <td>\ref CoeffwiseMathFunctionsAccuracy_note4 "4"</td></tr>
</table>

\subsection CoeffwiseMathFunctionsAccuracy_notes Notes

\anchor CoeffwiseMathFunctionsAccuracy_note1
<b>1. sin/cos/tan argument reduction:</b>
%Eigen's vectorized sin, cos, and tan use a Cody-Waite argument reduction scheme that
subtracts multiples of &pi;/2 from the input. For very large arguments (|x| &gt; ~10<sup>4</sup>
in float, |x| &gt; ~10 in double), this reduction loses precision, producing
occasional large ULP errors. The mean error remains low because most representable
values are small. Applications that need high accuracy for large arguments should
perform argument reduction in user code before calling these functions.

\anchor CoeffwiseMathFunctionsAccuracy_note2
<b>2. exp2 double precision:</b>
The exp2 implementation for double shows a max error of 214 ULP near the overflow
boundary (x &asymp; 1022). The mean error is still low (0.107 ULP, 99.6% exact), so
the large max error affects only inputs very close to overflow.

\anchor CoeffwiseMathFunctionsAccuracy_note3
<b>3. lgamma:</b>
The vectorized lgamma delegates to the standard library function \c std::lgamma for
this SIMD target (SSE2) and therefore has the same accuracy as the platform's C math library.

\anchor CoeffwiseMathFunctionsAccuracy_note4
<b>4. rsqrt max=&infin;:</b>
The infinite max ULP is due to a sign disagreement at a single subnormal input:
rsqrt(-0) returns -&infin; in %Eigen but the MPFR reference produces NaN (rsqrt of a negative
value). Ignoring this edge case, the implementation is accurate (&lt; 2 ULP) everywhere else.
For float, 16.8 million subnormal negative inputs (0.4%) also produce &plusmn;&infin; vs NaN.
The mean error excluding these outliers is well below 1 ULP.

\anchor CoeffwiseMathFunctionsAccuracy_note5
<b>5. erf double &infin;:</b>
The vectorized erf for double returns NaN for &plusmn;&infin; instead of the correct &plusmn;1.
This produces an infinite max ULP error at a single input value. Excluding &plusmn;&infin;,
the max error is 3 ULP and the mean is 0.050 ULP.

*/

}