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openvins_linux/global_fusion/Thirdparty/GeographicLib/include/Math.hpp
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/**
* \file Math.hpp
* \brief Header for GeographicLib::Math class
*
* Copyright (c) Charles Karney (2008-2017) <charles@karney.com> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
// Constants.hpp includes Math.hpp. Place this include outside Math.hpp's
// include guard to enforce this ordering.
#include "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. With OSX the
// version is GNUC version 4.2 and __cplusplus is set to 201103, so remove the
// version check on GNUC.
# if defined(__GNUC__) && __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/version.hpp>
#if BOOST_VERSION >= 105600
#include <boost/cstdfloat.hpp>
#endif
#include <boost/multiprecision/float128.hpp>
#include <boost/math/special_functions.hpp>
__float128 fmaq(__float128, __float128, __float128);
#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 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. See also
* Utility::set_digits for caveats about when this routine should be
* called.
**********************************************************************/
static 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 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 int extra_digits() {
return
digits10() > std::numeric_limits<double>::digits10 ?
digits10() - std::numeric_limits<double>::digits10 : 0;
}
/**
* true if the machine is big-endian.
**********************************************************************/
static const bool bigendian = GEOGRAPHICLIB_WORDS_BIGENDIAN;
/**
* @tparam T the type of the returned value.
* @return &pi;.
**********************************************************************/
template<typename T> static T pi() {
using std::atan2;
static const T pi = atan2(T(0), T(-1));
return pi;
}
/**
* A synonym for pi<real>().
**********************************************************************/
static 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 T degree() {
static const T degree = pi<T>() / 180;
return degree;
}
/**
* A synonym for degree<real>().
**********************************************************************/
static 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 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 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) https://doi.org/10.1147/rd.276.0577
// and A. A. Dubrulle (1983) https://doi.org/10.1147/rd.276.0582
#endif
}
/**
* exp(\e x) &minus; 1 accurate near \e x = 0.
*
* @tparam T the type of the argument and the returned value.
* @param[in] x
* @return exp(\e x) &minus; 1.
**********************************************************************/
template<typename T> static T expm1(T x) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::expm1; return expm1(x);
#else
using std::exp; using std::abs; using std::log;
GEOGRAPHICLIB_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 T log1p(T x) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::log1p; return log1p(x);
#else
using std::log;
GEOGRAPHICLIB_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 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 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 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
}
/**
* Fused multiply and add.
*
* @tparam T the type of the arguments and the returned value.
* @param[in] x
* @param[in] y
* @param[in] z
* @return <i>xy</i> + <i>z</i>, correctly rounded (on those platforms with
* support for the <code>fma</code> instruction).
*
* On platforms without the <code>fma</code> instruction, no attempt is
* made to improve on the result of a rounded multiplication followed by a
* rounded addition.
**********************************************************************/
template<typename T> static T fma(T x, T y, T z) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::fma; return fma(x, y, z);
#else
return x * y + z;
#endif
}
/**
* Normalize a two-vector.
*
* @tparam T the type of the argument and the returned value.
* @param[in,out] x on output set to <i>x</i>/hypot(<i>x</i>, <i>y</i>).
* @param[in,out] y on output set to <i>y</i>/hypot(<i>x</i>, <i>y</i>).
**********************************************************************/
template<typename T> static void norm(T& x, T& y)
{ T h = hypot(x, y); x /= h; y /= h; }
/**
* 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 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;
}
/**
* Evaluate a polynomial.
*
* @tparam T the type of the arguments and returned value.
* @param[in] N the order of the polynomial.
* @param[in] p the coefficient array (of size \e N + 1).
* @param[in] x the variable.
* @return the value of the polynomial.
*
* Evaluate <i>y</i> = &sum;<sub><i>n</i>=0..<i>N</i></sub>
* <i>p</i><sub><i>n</i></sub> <i>x</i><sup><i>N</i>&minus;<i>n</i></sup>.
* Return 0 if \e N &lt; 0. Return <i>p</i><sub>0</sub>, if \e N = 0 (even
* if \e x is infinite or a nan). The evaluation uses Horner's method.
**********************************************************************/
template<typename T> static T polyval(int N, const T p[], T x)
// This used to employ Math::fma; but that's too slow and it seemed not to
// improve the accuracy noticeably. This might change when there's direct
// hardware support for fma.
{ T y = N < 0 ? 0 : *p++; while (--N >= 0) y = y * x + *p++; return y; }
/**
* Normalize an 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([&minus;180&deg;, 180&deg;].
*
* The range of \e x is unrestricted.
**********************************************************************/
template<typename T> static T AngNormalize(T x) {
#if GEOGRAPHICLIB_CXX11_MATH && GEOGRAPHICLIB_PRECISION != 4
using std::remainder;
x = remainder(x, T(360)); return x != -180 ? x : 180;
#else
using std::fmod;
T y = fmod(x, T(360));
#if defined(_MSC_VER) && _MSC_VER < 1900
// Before version 14 (2015), Visual Studio had problems dealing
// with -0.0. Specifically
// VC 10,11,12 and 32-bit compile: fmod(-0.0, 360.0) -> +0.0
// sincosd has a similar fix.
// python 2.7 on Windows 32-bit machines has the same problem.
if (x == 0) y = x;
#endif
return y <= -180 ? y + 360 : (y <= 180 ? y : y - 360);
#endif
}
/**
* Normalize a latitude.
*
* @tparam T the type of the argument and returned value.
* @param[in] x the angle in degrees.
* @return x if it is in the range [&minus;90&deg;, 90&deg;], otherwise
* return NaN.
**********************************************************************/
template<typename T> static T LatFix(T x)
{ using std::abs; return abs(x) > 90 ? NaN<T>() : x; }
/**
* The exact difference of two angles reduced to
* (&minus;180&deg;, 180&deg;].
*
* @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.
* @param[out] e the error term in degrees.
* @return \e d, the truncated value of \e y &minus; \e x.
*
* This computes \e z = \e y &minus; \e x exactly, reduced to
* (&minus;180&deg;, 180&deg;]; and then sets \e z = \e d + \e e where \e d
* is the nearest representable number to \e z and \e e is the truncation
* error. If \e d = &minus;180, then \e e &gt; 0; If \e d = 180, then \e e
* &le; 0.
**********************************************************************/
template<typename T> static T AngDiff(T x, T y, T& e) {
#if GEOGRAPHICLIB_CXX11_MATH && GEOGRAPHICLIB_PRECISION != 4
using std::remainder;
T t, d = AngNormalize(sum(remainder(-x, T(360)),
remainder( y, T(360)), t));
#else
T t, d = AngNormalize(sum(AngNormalize(-x), AngNormalize(y), t));
#endif
// Here y - x = d + t (mod 360), exactly, where d is in (-180,180] and
// abs(t) <= eps (eps = 2^-45 for doubles). The only case where the
// addition of t takes the result outside the range (-180,180] is d = 180
// and t > 0. The case, d = -180 + eps, t = -eps, can't happen, since
// sum would have returned the exact result in such a case (i.e., given t
// = 0).
return sum(d == 180 && t > 0 ? -180 : d, t, e);
}
/**
* Difference of two angles reduced to [&minus;180&deg;, 180&deg;]
*
* @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 &minus; \e x, reduced to the range [&minus;180&deg;,
* 180&deg;].
*
* The result is equivalent to computing the difference exactly, reducing
* it to (&minus;180&deg;, 180&deg;] and rounding the result. Note that
* this prescription allows &minus;180&deg; to be returned (e.g., if \e x
* is tiny and negative and \e y = 180&deg;).
**********************************************************************/
template<typename T> static T AngDiff(T x, T y)
{ T e; return AngDiff(x, y, e); }
/**
* Coarsen a value close to zero.
*
* @tparam T the type of the argument and returned value.
* @param[in] x
* @return the coarsened value.
*
* The makes the smallest gap in \e x = 1/16 - nextafter(1/16, 0) =
* 1/2<sup>57</sup> for reals = 0.7 pm on the earth if \e x is an angle in
* degrees. (This is about 1000 times more resolution than we get with
* angles around 90&deg;.) We use this to avoid having to deal with near
* singular cases when \e x is non-zero but tiny (e.g.,
* 10<sup>&minus;200</sup>). This converts -0 to +0; however tiny negative
* numbers get converted to -0.
**********************************************************************/
template<typename T> static T AngRound(T x) {
using std::abs;
static const T z = 1/T(16);
if (x == 0) return 0;
GEOGRAPHICLIB_VOLATILE T y = abs(x);
// The compiler mustn't "simplify" z - (z - y) to y
y = y < z ? z - (z - y) : y;
return x < 0 ? -y : y;
}
/**
* Evaluate the sine and cosine function with the argument in degrees
*
* @tparam T the type of the arguments.
* @param[in] x in degrees.
* @param[out] sinx sin(<i>x</i>).
* @param[out] cosx cos(<i>x</i>).
*
* The results obey exactly the elementary properties of the trigonometric
* functions, e.g., sin 9&deg; = cos 81&deg; = &minus; sin 123456789&deg;.
* If x = &minus;0, then \e sinx = &minus;0; this is the only case where
* &minus;0 is returned.
**********************************************************************/
template<typename T> static void sincosd(T x, T& sinx, T& cosx) {
// In order to minimize round-off errors, this function exactly reduces
// the argument to the range [-45, 45] before converting it to radians.
using std::sin; using std::cos;
T r; int q;
#if GEOGRAPHICLIB_CXX11_MATH && GEOGRAPHICLIB_PRECISION <= 3 && \
!defined(__GNUC__)
// Disable for gcc because of bug in glibc version < 2.22, see
// https://sourceware.org/bugzilla/show_bug.cgi?id=17569
// Once this fix is widely deployed, should insert a runtime test for the
// glibc version number. For example
// #include <gnu/libc-version.h>
// std::string version(gnu_get_libc_version()); => "2.22"
using std::remquo;
r = remquo(x, T(90), &q);
#else
using std::fmod; using std::floor;
r = fmod(x, T(360));
q = int(floor(r / 90 + T(0.5)));
r -= 90 * q;
#endif
// now abs(r) <= 45
r *= degree();
// Possibly could call the gnu extension sincos
T s = sin(r), c = cos(r);
#if defined(_MSC_VER) && _MSC_VER < 1900
// Before version 14 (2015), Visual Studio had problems dealing
// with -0.0. Specifically
// VC 10,11,12 and 32-bit compile: fmod(-0.0, 360.0) -> +0.0
// VC 12 and 64-bit compile: sin(-0.0) -> +0.0
// AngNormalize has a similar fix.
// python 2.7 on Windows 32-bit machines has the same problem.
if (x == 0) s = x;
#endif
switch (unsigned(q) & 3U) {
case 0U: sinx = s; cosx = c; break;
case 1U: sinx = c; cosx = -s; break;
case 2U: sinx = -s; cosx = -c; break;
default: sinx = -c; cosx = s; break; // case 3U
}
// Set sign of 0 results. -0 only produced for sin(-0)
if (x != 0) { sinx += T(0); cosx += T(0); }
}
/**
* Evaluate the sine function with the argument in degrees
*
* @tparam T the type of the argument and the returned value.
* @param[in] x in degrees.
* @return sin(<i>x</i>).
**********************************************************************/
template<typename T> static T sind(T x) {
// See sincosd
using std::sin; using std::cos;
T r; int q;
#if GEOGRAPHICLIB_CXX11_MATH && GEOGRAPHICLIB_PRECISION <= 3 && \
!defined(__GNUC__)
using std::remquo;
r = remquo(x, T(90), &q);
#else
using std::fmod; using std::floor;
r = fmod(x, T(360));
q = int(floor(r / 90 + T(0.5)));
r -= 90 * q;
#endif
// now abs(r) <= 45
r *= degree();
unsigned p = unsigned(q);
r = p & 1U ? cos(r) : sin(r);
if (p & 2U) r = -r;
if (x != 0) r += T(0);
return r;
}
/**
* Evaluate the cosine function with the argument in degrees
*
* @tparam T the type of the argument and the returned value.
* @param[in] x in degrees.
* @return cos(<i>x</i>).
**********************************************************************/
template<typename T> static T cosd(T x) {
// See sincosd
using std::sin; using std::cos;
T r; int q;
#if GEOGRAPHICLIB_CXX11_MATH && GEOGRAPHICLIB_PRECISION <= 3 && \
!defined(__GNUC__)
using std::remquo;
r = remquo(x, T(90), &q);
#else
using std::fmod; using std::floor;
r = fmod(x, T(360));
q = int(floor(r / 90 + T(0.5)));
r -= 90 * q;
#endif
// now abs(r) <= 45
r *= degree();
unsigned p = unsigned(q + 1);
r = p & 1U ? cos(r) : sin(r);
if (p & 2U) r = -r;
return T(0) + r;
}
/**
* Evaluate the tangent function with the argument in degrees
*
* @tparam T the type of the argument and the returned value.
* @param[in] x in degrees.
* @return tan(<i>x</i>).
*
* If \e x = &plusmn;90&deg;, then a suitably large (but finite) value is
* returned.
**********************************************************************/
template<typename T> static T tand(T x) {
static const T overflow = 1 / sq(std::numeric_limits<T>::epsilon());
T s, c;
sincosd(x, s, c);
return c != 0 ? s / c : (s < 0 ? -overflow : overflow);
}
/**
* Evaluate the atan2 function with the result in degrees
*
* @tparam T the type of the arguments and the returned value.
* @param[in] y
* @param[in] x
* @return atan2(<i>y</i>, <i>x</i>) in degrees.
*
* The result is in the range (&minus;180&deg; 180&deg;]. N.B.,
* atan2d(&plusmn;0, &minus;1) = +180&deg;; atan2d(&minus;&epsilon;,
* &minus;1) = &minus;180&deg;, for &epsilon; positive and tiny;
* atan2d(&plusmn;0, +1) = &plusmn;0&deg;.
**********************************************************************/
template<typename T> static T atan2d(T y, T x) {
// In order to minimize round-off errors, this function rearranges the
// arguments so that result of atan2 is in the range [-pi/4, pi/4] before
// converting it to degrees and mapping the result to the correct
// quadrant.
using std::atan2; using std::abs;
int q = 0;
if (abs(y) > abs(x)) { std::swap(x, y); q = 2; }
if (x < 0) { x = -x; ++q; }
// here x >= 0 and x >= abs(y), so angle is in [-pi/4, pi/4]
T ang = atan2(y, x) / degree();
switch (q) {
// Note that atan2d(-0.0, 1.0) will return -0. However, we expect that
// atan2d will not be called with y = -0. If need be, include
//
// case 0: ang = 0 + ang; break;
//
// and handle mpfr as in AngRound.
case 1: ang = (y >= 0 ? 180 : -180) - ang; break;
case 2: ang = 90 - ang; break;
case 3: ang = -90 + ang; break;
}
return ang;
}
/**
* Evaluate the atan function with the result in degrees
*
* @tparam T the type of the argument and the returned value.
* @param[in] x
* @return atan(<i>x</i>) in degrees.
**********************************************************************/
template<typename T> static T atand(T x)
{ return atan2d(x, T(1)); }
/**
* Evaluate <i>e</i> atanh(<i>e x</i>)
*
* @tparam T the type of the argument and the returned value.
* @param[in] x
* @param[in] es the signed eccentricity = sign(<i>e</i><sup>2</sup>)
* sqrt(|<i>e</i><sup>2</sup>|)
* @return <i>e</i> atanh(<i>e x</i>)
*
* If <i>e</i><sup>2</sup> is negative (<i>e</i> is imaginary), the
* expression is evaluated in terms of atan.
**********************************************************************/
template<typename T> static T eatanhe(T x, T es);
/**
* Copy the sign.
*
* @tparam T the type of the argument.
* @param[in] x gives the magitude of the result.
* @param[in] y gives the sign of the result.
* @return value with the magnitude of \e x and with the sign of \e y.
*
* This routine correctly handles the case \e y = &minus;0, returning
* &minus|<i>x</i>|.
**********************************************************************/
template<typename T> static T copysign(T x, T y) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::copysign; return copysign(x, y);
#else
using std::abs;
// NaN counts as positive
return abs(x) * (y < 0 || (y == 0 && 1/y < 0) ? -1 : 1);
#endif
}
/**
* tan&chi; in terms of tan&phi;
*
* @tparam T the type of the argument and the returned value.
* @param[in] tau &tau; = tan&phi;
* @param[in] es the signed eccentricity = sign(<i>e</i><sup>2</sup>)
* sqrt(|<i>e</i><sup>2</sup>|)
* @return &tau;&prime; = tan&chi;
*
* See Eqs. (7--9) of
* C. F. F. Karney,
* <a href="https://doi.org/10.1007/s00190-011-0445-3">
* Transverse Mercator with an accuracy of a few nanometers,</a>
* J. Geodesy 85(8), 475--485 (Aug. 2011)
* (preprint
* <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a>).
**********************************************************************/
template<typename T> static T taupf(T tau, T es);
/**
* tan&phi; in terms of tan&chi;
*
* @tparam T the type of the argument and the returned value.
* @param[in] taup &tau;&prime; = tan&chi;
* @param[in] es the signed eccentricity = sign(<i>e</i><sup>2</sup>)
* sqrt(|<i>e</i><sup>2</sup>|)
* @return &tau; = tan&phi;
*
* See Eqs. (19--21) of
* C. F. F. Karney,
* <a href="https://doi.org/10.1007/s00190-011-0445-3">
* Transverse Mercator with an accuracy of a few nanometers,</a>
* J. Geodesy 85(8), 475--485 (Aug. 2011)
* (preprint
* <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a>).
**********************************************************************/
template<typename T> static T tauf(T taup, T es);
/**
* 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 bool isfinite(T x) {
#if GEOGRAPHICLIB_CXX11_MATH
using std::isfinite; return isfinite(x);
#else
using std::abs;
#if defined(_MSC_VER)
return abs(x) <= (std::numeric_limits<T>::max)();
#else
// There's a problem using MPFR C++ 3.6.3 and g++ -std=c++14 (reported on
// 2015-05-04) with the parens around std::numeric_limits<T>::max. Of
// course, these parens are only needed to deal with Windows stupidly
// defining max as a macro. So don't insert the parens on non-Windows
// platforms.
return abs(x) <= std::numeric_limits<T>::max();
#endif
#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 T NaN() {
#if defined(_MSC_VER)
return std::numeric_limits<T>::has_quiet_NaN ?
std::numeric_limits<T>::quiet_NaN() :
(std::numeric_limits<T>::max)();
#else
return std::numeric_limits<T>::has_quiet_NaN ?
std::numeric_limits<T>::quiet_NaN() :
std::numeric_limits<T>::max();
#endif
}
/**
* A synonym for NaN<real>().
**********************************************************************/
static 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 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 T infinity() {
#if defined(_MSC_VER)
return std::numeric_limits<T>::has_infinity ?
std::numeric_limits<T>::infinity() :
(std::numeric_limits<T>::max)();
#else
return std::numeric_limits<T>::has_infinity ?
std::numeric_limits<T>::infinity() :
std::numeric_limits<T>::max();
#endif
}
/**
* A synonym for infinity<real>().
**********************************************************************/
static 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 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 real hypot(real x, real y)
{ return boost::math::hypot(x, y, boost_special_functions_policy()); }
static real expm1(real x)
{ return boost::math::expm1(x, boost_special_functions_policy()); }
static real log1p(real x)
{ return boost::math::log1p(x, boost_special_functions_policy()); }
static real asinh(real x)
{ return boost::math::asinh(x, boost_special_functions_policy()); }
static real atanh(real x)
{ return boost::math::atanh(x, boost_special_functions_policy()); }
static real cbrt(real x)
{ return boost::math::cbrt(x, boost_special_functions_policy()); }
static real fma(real x, real y, real z)
{ return fmaq(__float128(x), __float128(y), __float128(z)); }
static real copysign(real x, real y)
{ return boost::math::copysign(x, y); }
static bool isnan(real x) { return boost::math::isnan(x); }
static bool isfinite(real x) { return boost::math::isfinite(x); }
#endif
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_MATH_HPP
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