/usr/include/GeographicLib/Math.hpp is in libgeographic-dev 1.37-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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* \file Math.hpp
* \brief Header for GeographicLib::Math class
*
* Copyright (c) Charles Karney (2008-2011) <charles@karney.com> and licensed
* under the MIT/X11 License. For more information, see
* http://geographiclib.sourceforge.net/
**********************************************************************/
// Constants.hpp includes Math.hpp. Place this include outside Math.hpp's
// include guard to enforce this ordering.
#include <GeographicLib/Constants.hpp>
#if !defined(GEOGRAPHICLIB_MATH_HPP)
#define GEOGRAPHICLIB_MATH_HPP 1
/**
* Are C++11 math functions available?
**********************************************************************/
#if !defined(GEOGRAPHICLIB_CXX11_MATH)
// Recent versions of g++ -std=c++11 (4.7 and later?) set __cplusplus to 201103
// and support the new C++11 mathematical functions, std::atanh, etc. However
// the Android toolchain, which uses g++ -std=c++11 (4.8 as of 2014-03-11,
// according to Pullan Lu), does not support std::atanh. Android toolchains
// might define __ANDROID__ or ANDROID; so need to check both.
# if defined(__GNUC__) && __GNUC__ == 4 && __GNUC_MINOR__ >= 7 \
&& __cplusplus >= 201103 && \
!(defined(__ANDROID__) || defined(ANDROID) || defined(__CYGWIN__))
# define GEOGRAPHICLIB_CXX11_MATH 1
// Visual C++ 12 supports these functions
# elif defined(_MSC_VER) && _MSC_VER >= 1800
# define GEOGRAPHICLIB_CXX11_MATH 1
# else
# define GEOGRAPHICLIB_CXX11_MATH 0
# endif
#endif
#if !defined(GEOGRAPHICLIB_WORDS_BIGENDIAN)
# define GEOGRAPHICLIB_WORDS_BIGENDIAN 0
#endif
#if !defined(GEOGRAPHICLIB_HAVE_LONG_DOUBLE)
# define GEOGRAPHICLIB_HAVE_LONG_DOUBLE 0
#endif
#if !defined(GEOGRAPHICLIB_PRECISION)
/**
* The precision of floating point numbers used in %GeographicLib. 1 means
* float (single precision); 2 (the default) means double; 3 means long double;
* 4 is reserved for quadruple precision. Nearly all the testing has been
* carried out with doubles and that's the recommended configuration. In order
* for long double to be used, GEOGRAPHICLIB_HAVE_LONG_DOUBLE needs to be
* defined. Note that with Microsoft Visual Studio, long double is the same as
* double.
**********************************************************************/
# define GEOGRAPHICLIB_PRECISION 2
#endif
#include <cmath>
#include <algorithm>
#include <limits>
#if GEOGRAPHICLIB_PRECISION == 4
#include <boost/multiprecision/float128.hpp>
#include <boost/math/special_functions/hypot.hpp>
#include <boost/math/special_functions/expm1.hpp>
#include <boost/math/special_functions/log1p.hpp>
#include <boost/math/special_functions/atanh.hpp>
#include <boost/math/special_functions/asinh.hpp>
#include <boost/math/special_functions/cbrt.hpp>
#elif GEOGRAPHICLIB_PRECISION == 5
#include <mpreal.h>
#endif
#if GEOGRAPHICLIB_PRECISION > 3
// volatile keyword makes no sense for multiprec types
#define GEOGRAPHICLIB_VOLATILE
// Signal a convergence failure with multiprec types by throwing an exception
// at loop exit.
#define GEOGRAPHICLIB_PANIC \
(throw GeographicLib::GeographicErr("Convergence failure"), false)
#else
#define GEOGRAPHICLIB_VOLATILE volatile
// Ignore convergence failures with standard floating points types by allowing
// loop to exit cleanly.
#define GEOGRAPHICLIB_PANIC false
#endif
namespace GeographicLib {
/**
* \brief Mathematical functions needed by %GeographicLib
*
* Define mathematical functions in order to localize system dependencies and
* to provide generic versions of the functions. In addition define a real
* type to be used by %GeographicLib.
*
* Example of use:
* \include example-Math.cpp
**********************************************************************/
class GEOGRAPHICLIB_EXPORT Math {
private:
void dummy() {
GEOGRAPHICLIB_STATIC_ASSERT(GEOGRAPHICLIB_PRECISION >= 1 &&
GEOGRAPHICLIB_PRECISION <= 5,
"Bad value of precision");
}
Math(); // Disable constructor
public:
#if GEOGRAPHICLIB_HAVE_LONG_DOUBLE
/**
* The extended precision type for real numbers, used for some testing.
* This is long double on computers with this type; otherwise it is double.
**********************************************************************/
typedef long double extended;
#else
typedef double extended;
#endif
#if GEOGRAPHICLIB_PRECISION == 2
/**
* The real type for %GeographicLib. Nearly all the testing has been done
* with \e real = double. However, the algorithms should also work with
* float and long double (where available). (<b>CAUTION</b>: reasonable
* accuracy typically cannot be obtained using floats.)
**********************************************************************/
typedef double real;
#elif GEOGRAPHICLIB_PRECISION == 1
typedef float real;
#elif GEOGRAPHICLIB_PRECISION == 3
typedef extended real;
#elif GEOGRAPHICLIB_PRECISION == 4
typedef boost::multiprecision::float128 real;
#elif GEOGRAPHICLIB_PRECISION == 5
typedef mpfr::mpreal real;
#else
typedef double real;
#endif
/**
* @return the number of bits of precision in a real number.
**********************************************************************/
static inline int digits() {
#if GEOGRAPHICLIB_PRECISION != 5
return std::numeric_limits<real>::digits;
#else
return std::numeric_limits<real>::digits();
#endif
}
/**
* Set the binary precision of a real number.
*
* @param[in] ndigits the number of bits of precision.
* @return the resulting number of bits of precision.
*
* This only has an effect when GEOGRAPHICLIB_PRECISION == 5.
**********************************************************************/
static inline int set_digits(int ndigits) {
#if GEOGRAPHICLIB_PRECISION != 5
(void)ndigits;
#else
mpfr::mpreal::set_default_prec(ndigits >= 2 ? ndigits : 2);
#endif
return digits();
}
/**
* @return the number of decimal digits of precision in a real number.
**********************************************************************/
static inline int digits10() {
#if GEOGRAPHICLIB_PRECISION != 5
return std::numeric_limits<real>::digits10;
#else
return std::numeric_limits<real>::digits10();
#endif
}
/**
* Number of additional decimal digits of precision for real relative to
* double (0 for float).
**********************************************************************/
static inline int extra_digits() {
return
digits10() > std::numeric_limits<double>::digits10 ?
digits10() - std::numeric_limits<double>::digits10 : 0;
}
#if GEOGRAPHICLIB_PRECISION <= 3
/**
* Number of additional decimal digits of precision of real relative to
* double (0 for float).
*
* <b>DEPRECATED</b>: use extra_digits() instead
**********************************************************************/
static const int extradigits =
std::numeric_limits<real>::digits10 >
std::numeric_limits<double>::digits10 ?
std::numeric_limits<real>::digits10 -
std::numeric_limits<double>::digits10 : 0;
#endif
/**
* true if the machine is big-endian.
**********************************************************************/
static const bool bigendian = GEOGRAPHICLIB_WORDS_BIGENDIAN;
/**
* @tparam T the type of the returned value.
* @return π.
**********************************************************************/
template<typename T> static inline T pi() {
using std::atan2;
static const T pi = atan2(T(0), T(-1));
return pi;
}
/**
* A synonym for pi<real>().
**********************************************************************/
static inline real pi() { return pi<real>(); }
/**
* @tparam T the type of the returned value.
* @return the number of radians in a degree.
**********************************************************************/
template<typename T> static inline T degree() {
static const T degree = pi<T>() / 180;
return degree;
}
/**
* A synonym for degree<real>().
**********************************************************************/
static inline real degree() { return degree<real>(); }
/**
* Square a number.
*
* @tparam T the type of the argument and the returned value.
* @param[in] x
* @return <i>x</i><sup>2</sup>.
**********************************************************************/
template<typename T> static inline T sq(T x)
{ return x * x; }
/**
* The hypotenuse function avoiding underflow and overflow.
*
* @tparam T the type of the arguments and the returned value.
* @param[in] x
* @param[in] y
* @return sqrt(<i>x</i><sup>2</sup> + <i>y</i><sup>2</sup>).
**********************************************************************/
template<typename T> static inline T hypot(T x, T y) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::hypot; return hypot(x, y);
#else
using std::abs; using std::sqrt;
x = abs(x); y = abs(y);
if (x < y) std::swap(x, y); // Now x >= y >= 0
y /= (x ? x : 1);
return x * sqrt(1 + y * y);
// For an alternative (square-root free) method see
// C. Moler and D. Morrision (1983) http://dx.doi.org/10.1147/rd.276.0577
// and A. A. Dubrulle (1983) http://dx.doi.org/10.1147/rd.276.0582
#endif
}
/**
* exp(\e x) − 1 accurate near \e x = 0.
*
* @tparam T the type of the argument and the returned value.
* @param[in] x
* @return exp(\e x) − 1.
**********************************************************************/
template<typename T> static inline T expm1(T x) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::expm1; return expm1(x);
#else
using std::exp; using std::abs; using std::log;
volatile T
y = exp(x),
z = y - 1;
// The reasoning here is similar to that for log1p. The expression
// mathematically reduces to exp(x) - 1, and the factor z/log(y) = (y -
// 1)/log(y) is a slowly varying quantity near y = 1 and is accurately
// computed.
return abs(x) > 1 ? z : (z == 0 ? x : x * z / log(y));
#endif
}
/**
* log(1 + \e x) accurate near \e x = 0.
*
* @tparam T the type of the argument and the returned value.
* @param[in] x
* @return log(1 + \e x).
**********************************************************************/
template<typename T> static inline T log1p(T x) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::log1p; return log1p(x);
#else
using std::log;
volatile T
y = 1 + x,
z = y - 1;
// Here's the explanation for this magic: y = 1 + z, exactly, and z
// approx x, thus log(y)/z (which is nearly constant near z = 0) returns
// a good approximation to the true log(1 + x)/x. The multiplication x *
// (log(y)/z) introduces little additional error.
return z == 0 ? x : x * log(y) / z;
#endif
}
/**
* The inverse hyperbolic sine function.
*
* @tparam T the type of the argument and the returned value.
* @param[in] x
* @return asinh(\e x).
**********************************************************************/
template<typename T> static inline T asinh(T x) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::asinh; return asinh(x);
#else
using std::abs; T y = abs(x); // Enforce odd parity
y = log1p(y * (1 + y/(hypot(T(1), y) + 1)));
return x < 0 ? -y : y;
#endif
}
/**
* The inverse hyperbolic tangent function.
*
* @tparam T the type of the argument and the returned value.
* @param[in] x
* @return atanh(\e x).
**********************************************************************/
template<typename T> static inline T atanh(T x) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::atanh; return atanh(x);
#else
using std::abs; T y = abs(x); // Enforce odd parity
y = log1p(2 * y/(1 - y))/2;
return x < 0 ? -y : y;
#endif
}
/**
* The cube root function.
*
* @tparam T the type of the argument and the returned value.
* @param[in] x
* @return the real cube root of \e x.
**********************************************************************/
template<typename T> static inline T cbrt(T x) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::cbrt; return cbrt(x);
#else
using std::abs; using std::pow;
T y = pow(abs(x), 1/T(3)); // Return the real cube root
return x < 0 ? -y : y;
#endif
}
/**
* The error-free sum of two numbers.
*
* @tparam T the type of the argument and the returned value.
* @param[in] u
* @param[in] v
* @param[out] t the exact error given by (\e u + \e v) - \e s.
* @return \e s = round(\e u + \e v).
*
* See D. E. Knuth, TAOCP, Vol 2, 4.2.2, Theorem B. (Note that \e t can be
* the same as one of the first two arguments.)
**********************************************************************/
template<typename T> static inline T sum(T u, T v, T& t) {
GEOGRAPHICLIB_VOLATILE T s = u + v;
GEOGRAPHICLIB_VOLATILE T up = s - v;
GEOGRAPHICLIB_VOLATILE T vpp = s - up;
up -= u;
vpp -= v;
t = -(up + vpp);
// u + v = s + t
// = round(u + v) + t
return s;
}
/**
* Normalize an angle (restricted input range).
*
* @tparam T the type of the argument and returned value.
* @param[in] x the angle in degrees.
* @return the angle reduced to the range [−180°, 180°).
*
* \e x must lie in [−540°, 540°).
**********************************************************************/
template<typename T> static inline T AngNormalize(T x)
{ return x >= 180 ? x - 360 : (x < -180 ? x + 360 : x); }
/**
* Normalize an arbitrary angle.
*
* @tparam T the type of the argument and returned value.
* @param[in] x the angle in degrees.
* @return the angle reduced to the range [−180°, 180°).
*
* The range of \e x is unrestricted.
**********************************************************************/
template<typename T> static inline T AngNormalize2(T x)
{ using std::fmod; return AngNormalize<T>(fmod(x, T(360))); }
/**
* Difference of two angles reduced to [−180°, 180°]
*
* @tparam T the type of the arguments and returned value.
* @param[in] x the first angle in degrees.
* @param[in] y the second angle in degrees.
* @return \e y − \e x, reduced to the range [−180°,
* 180°].
*
* \e x and \e y must both lie in [−180°, 180°]. The result
* is equivalent to computing the difference exactly, reducing it to
* (−180°, 180°] and rounding the result. Note that this
* prescription allows −180° to be returned (e.g., if \e x is
* tiny and negative and \e y = 180°).
**********************************************************************/
template<typename T> static inline T AngDiff(T x, T y) {
T t, d = sum(-x, y, t);
if ((d - T(180)) + t > T(0)) // y - x > 180
d -= T(360); // exact
else if ((d + T(180)) + t <= T(0)) // y - x <= -180
d += T(360); // exact
return d + t;
}
/**
* Test for finiteness.
*
* @tparam T the type of the argument.
* @param[in] x
* @return true if number is finite, false if NaN or infinite.
**********************************************************************/
template<typename T> static inline bool isfinite(T x) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::isfinite; return isfinite(x);
#else
using std::abs;
return abs(x) <= (std::numeric_limits<T>::max)();
#endif
}
/**
* The NaN (not a number)
*
* @tparam T the type of the returned value.
* @return NaN if available, otherwise return the max real of type T.
**********************************************************************/
template<typename T> static inline T NaN() {
return std::numeric_limits<T>::has_quiet_NaN ?
std::numeric_limits<T>::quiet_NaN() :
(std::numeric_limits<T>::max)();
}
/**
* A synonym for NaN<real>().
**********************************************************************/
static inline real NaN() { return NaN<real>(); }
/**
* Test for NaN.
*
* @tparam T the type of the argument.
* @param[in] x
* @return true if argument is a NaN.
**********************************************************************/
template<typename T> static inline bool isnan(T x) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::isnan; return isnan(x);
#else
return x != x;
#endif
}
/**
* Infinity
*
* @tparam T the type of the returned value.
* @return infinity if available, otherwise return the max real.
**********************************************************************/
template<typename T> static inline T infinity() {
return std::numeric_limits<T>::has_infinity ?
std::numeric_limits<T>::infinity() :
(std::numeric_limits<T>::max)();
}
/**
* A synonym for infinity<real>().
**********************************************************************/
static inline real infinity() { return infinity<real>(); }
/**
* Swap the bytes of a quantity
*
* @tparam T the type of the argument and the returned value.
* @param[in] x
* @return x with its bytes swapped.
**********************************************************************/
template<typename T> static inline T swab(T x) {
union {
T r;
unsigned char c[sizeof(T)];
} b;
b.r = x;
for (int i = sizeof(T)/2; i--; )
std::swap(b.c[i], b.c[sizeof(T) - 1 - i]);
return b.r;
}
#if GEOGRAPHICLIB_PRECISION == 4
typedef boost::math::policies::policy
< boost::math::policies::domain_error
<boost::math::policies::errno_on_error>,
boost::math::policies::pole_error
<boost::math::policies::errno_on_error>,
boost::math::policies::overflow_error
<boost::math::policies::errno_on_error>,
boost::math::policies::evaluation_error
<boost::math::policies::errno_on_error> >
boost_special_functions_policy;
static inline real hypot(real x, real y)
{ return boost::math::hypot(x, y, boost_special_functions_policy()); }
static inline real expm1(real x)
{ return boost::math::expm1(x, boost_special_functions_policy()); }
static inline real log1p(real x)
{ return boost::math::log1p(x, boost_special_functions_policy()); }
static inline real asinh(real x)
{ return boost::math::asinh(x, boost_special_functions_policy()); }
static inline real atanh(real x)
{ return boost::math::atanh(x, boost_special_functions_policy()); }
static inline real cbrt(real x)
{ return boost::math::cbrt(x, boost_special_functions_policy()); }
static inline bool isnan(real x) { return boost::math::isnan(x); }
static inline bool isfinite(real x) { return boost::math::isfinite(x); }
#endif
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_MATH_HPP
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