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Math.cpp
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/**
* \file Math.cpp
* \brief Implementation for GeographicLib::Math class
*
* Copyright (c) Charles Karney (2015-2024) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#include <GeographicLib/Math.hpp>
namespace GeographicLib {
using namespace std;
void Math::dummy() {
static_assert(GEOGRAPHICLIB_PRECISION >= 1, "Bad value of precision");
}
int Math::digits() {
#if GEOGRAPHICLIB_PRECISION == 5
return numeric_limits<real>::digits();
#else
return numeric_limits<real>::digits;
#endif
}
int Math::set_digits(int ndigits) {
#if GEOGRAPHICLIB_PRECISION >= 5
# if GEOGRAPHICLIB_PRECISION > 5
// This sets ndigits = GEOGRAPHICLIB_PRECISION
ndigits = numeric_limits<real>::digits;
# endif
mpfr::mpreal::set_default_prec(ndigits >= 2 ? ndigits : 2);
#else
(void) ndigits;
#endif
return digits();
}
int Math::digits10() {
#if GEOGRAPHICLIB_PRECISION == 5
return numeric_limits<real>::digits10();
#else
return numeric_limits<real>::digits10;
#endif
}
int Math::extra_digits() {
return
digits10() > numeric_limits<double>::digits10 ?
digits10() - numeric_limits<double>::digits10 : 0;
}
template<typename T> T Math::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;
// if s = 0, then t = 0 and give t the same sign as s
// mpreal needs T(0) here
t = s != 0 ? T(0) - (up + vpp) : s;
// u + v = s + t
// = round(u + v) + t
return s;
}
template<typename T> T Math::AngNormalize(T x) {
T y = remainder(x, T(td));
#if GEOGRAPHICLIB_PRECISION == 4
// boost-quadmath doesn't set the sign of 0 correctly, see
// https://github.com/boostorg/multiprecision/issues/426
// Fixed by https://github.com/boostorg/multiprecision/pull/428
if (y == 0) y = copysign(y, x);
#endif
return fabs(y) == T(hd) ? copysign(T(hd), x) : y;
}
template<typename T> T Math::AngDiff(T x, T y, T& e) {
// Use remainder instead of AngNormalize, since we treat boundary cases
// later taking account of the error
T d = sum(remainder(-x, T(td)), remainder( y, T(td)), e);
// This second sum can only change d if abs(d) < 128, so don't need to
// apply remainder yet again.
d = sum(remainder(d, T(td)), e, e);
// Fix the sign if d = -180, 0, 180.
if (d == 0 || fabs(d) == hd)
// If e == 0, take sign from y - x
// else (e != 0, implies d = +/-180), d and e must have opposite signs
d = copysign(d, e == 0 ? y - x : -e);
return d;
}
template<typename T> T Math::AngRound(T x) {
static const T z = T(1)/T(16);
GEOGRAPHICLIB_VOLATILE T y = fabs(x);
GEOGRAPHICLIB_VOLATILE T w = z - y;
// The compiler mustn't "simplify" z - (z - y) to y
y = w > 0 ? z - w : y;
return copysign(y, x);
}
template<typename T> void Math::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.
int q = 0;
T d = remquo(x, T(qd), &q), // now abs(r) <= 45
r = d * degree<T>();
// g++ -O turns these two function calls into a call to sincos
T s = sin(r), c = cos(r);
if (2 * fabs(d) == qd) {
c = sqrt(1/T(2));
s = copysign(c, r);
} else if (3 * fabs(d) == qd) {
c = sqrt(T(3))/2;
s = copysign(1/T(2), r);
}
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
}
// http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf
// mpreal needs T(0) here
cosx += T(0); // special values from F.10.1.12
if (sinx == 0) sinx = copysign(sinx, x); // special values from F.10.1.13
}
template<typename T> void Math::sincosde(T x, T t, 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.
// This implementation allows x outside [-180, 180], but implementations in
// other languages may not.
int q = 0;
T d = AngRound(remquo(x, T(qd), &q) + t), // now abs(r) <= 45
r = d * degree<T>();
// g++ -O turns these two function calls into a call to sincos
T s = sin(r), c = cos(r);
if (2 * fabs(d) == qd) {
c = sqrt(1/T(2));
s = copysign(c, r);
} else if (3 * fabs(d) == qd) {
c = sqrt(T(3))/2;
s = copysign(1/T(2), r);
}
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
}
// http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf
// mpreal needs T(0) here
cosx += T(0); // special values from F.10.1.12
// Should we copy the sign from x or x+t? Given that t is small, there's
// only a distinction if x+t == +/- 0. Here are the cases
// x t x-sign (x+t)-sign
// <0 >0 -0 +0 different
// >0 <0 +0 +0
// -0 -0 -0 -0
// -0 +0 -0 +0 different
// +0 -0 +0 +0
// +0 +0 +0 +0
// On balance, taking the sign from x is better, particularly for the case
// x = -0, t = +0. This choice also avoids the bias towards +0 that x+t
// gives.
if (sinx == 0) sinx = copysign(sinx, x); // special values from F.10.1.13
}
template<typename T> T Math::sind(T x) {
// See sincosd
int q = 0;
T d = remquo(x, T(qd), &q), // now abs(r) <= 45
r = d * degree<T>();
unsigned p = unsigned(q);
// r = p & 1U ? cos(r) : sin(r); replaced by ...
r = p & 1U ? (2 * fabs(d) == qd ? sqrt(1/T(2)) :
(3 * fabs(d) == qd ? sqrt(T(3))/2 : cos(r))) :
copysign(2 * fabs(d) == qd ? sqrt(1/T(2)) :
(3 * fabs(d) == qd ? 1/T(2) : sin(r)), r);
if (p & 2U) r = -r;
if (r == 0) r = copysign(r, x);
return r;
}
template<typename T> T Math::cosd(T x) {
// See sincosd
int q = 0;
T d = remquo(x, T(qd), &q), // now abs(r) <= 45
r = d * degree<T>();
unsigned p = unsigned(q + 1);
r = p & 1U ? (2 * fabs(d) == qd ? sqrt(1/T(2)) :
(3 * fabs(d) == qd ? sqrt(T(3))/2 : cos(r))) :
copysign(2 * fabs(d) == qd ? sqrt(1/T(2)) :
(3 * fabs(d) == qd ? 1/T(2) : sin(r)), r);
if (p & 2U) r = -r;
// mpreal needs T(0) here
return T(0) + r;
}
template<typename T> T Math::tand(T x) {
static const T overflow = 1 / sq(numeric_limits<T>::epsilon());
T s, c;
sincosd(x, s, c);
// http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf
T r = s / c; // special values from F.10.1.14
return clamp(r, -overflow, overflow);
}
template<typename T> T Math::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.
// With mpreal we could use T(mpfr::atan2u(y, x, td)); but we're not ready
// for this yet.
int q = 0;
if (fabs(y) > fabs(x)) { swap(x, y); q = 2; }
if (signbit(x)) { x = -x; ++q; }
// here x >= 0 and x >= abs(y), so angle is in [-pi/4, pi/4]
// Replace atan2(y, x) / degree<T>() by this to ensure that special values
// (45, 90, etc.) are returned.
T ang = (atan2(y, x) / pi<T>()) * T(hd);
switch (q) {
case 1: ang = copysign(T(hd), y) - ang; break;
case 2: ang = qd - ang; break;
case 3: ang = -qd + ang; break;
default: break;
}
return ang;
}
template<typename T> T Math::atand(T x)
{ return atan2d(x, T(1)); }
template<typename T> T Math::eatanhe(T x, T es) {
return es > 0 ? es * atanh(es * x) : -es * atan(es * x);
}
template<typename T> T Math::taupf(T tau, T es) {
// Need this test, otherwise tau = +/-inf gives taup = nan.
if (isfinite(tau)) {
T tau1 = hypot(T(1), tau),
sig = sinh( eatanhe(tau / tau1, es ) );
return hypot(T(1), sig) * tau - sig * tau1;
} else
return tau;
}
template<typename T> T Math::tauf(T taup, T es) {
static const int numit = 5;
// min iterations = 1, max iterations = 2; mean = 1.95
static const T tol = sqrt(numeric_limits<T>::epsilon()) / 10;
static const T taumax = 2 / sqrt(numeric_limits<T>::epsilon());
T e2m = 1 - sq(es),
// To lowest order in e^2, taup = (1 - e^2) * tau = _e2m * tau; so use
// tau = taup/e2m as a starting guess. Only 1 iteration is needed for
// |lat| < 3.35 deg, otherwise 2 iterations are needed. If, instead, tau
// = taup is used the mean number of iterations increases to 1.999 (2
// iterations are needed except near tau = 0).
//
// For large tau, taup = exp(-es*atanh(es)) * tau. Use this as for the
// initial guess for |taup| > 70 (approx |phi| > 89deg). Then for
// sufficiently large tau (such that sqrt(1+tau^2) = |tau|), we can exit
// with the intial guess and avoid overflow problems. This also reduces
// the mean number of iterations slightly from 1.963 to 1.954.
tau = fabs(taup) > 70 ? taup * exp(eatanhe(T(1), es)) : taup/e2m,
stol = tol * fmax(T(1), fabs(taup));
if (!(fabs(tau) < taumax)) return tau; // handles +/-inf and nan
for (int i = 0;
i < numit ||
GEOGRAPHICLIB_PANIC("Convergence failure in Math::tauf");
++i) {
T taupa = taupf(tau, es),
dtau = (taup - taupa) * (1 + e2m * sq(tau)) /
( e2m * hypot(T(1), tau) * hypot(T(1), taupa) );
tau += dtau;
if (!(fabs(dtau) >= stol))
break;
}
return tau;
}
template<typename T> T Math::hypot3(T x, T y, T z) {
#if GEOGRAPHICLIB_PRECISION == 4
// Boost implementation is given by
// https://github.com/boostorg/math/pull/1318
// might make its way into 1.90 or later
return hypot(hypot(x, y), z);
#else
return hypot(x, y, z);
#endif
}
template<typename T> T Math::clamp(T x, T a, T b) {
// Use max/min here (instead of fmax/fmin) to preserve NaN
return min(max(x, a), b);
}
template<typename T> T Math::NaN() {
if constexpr (numeric_limits<T>::has_quiet_NaN)
return numeric_limits<T>::quiet_NaN();
else
return (numeric_limits<T>::max)();
}
template<typename T> T Math::infinity() {
if constexpr (numeric_limits<T>::has_infinity)
return numeric_limits<T>::infinity();
else
return (numeric_limits<T>::max)();
}
/// \cond SKIP
// Instantiate
#define GEOGRAPHICLIB_MATH_INSTANTIATE(T) \
template T GEOGRAPHICLIB_EXPORT Math::sum <T>(T, T, T&); \
template T GEOGRAPHICLIB_EXPORT Math::AngNormalize <T>(T); \
template T GEOGRAPHICLIB_EXPORT Math::AngDiff <T>(T, T, T&); \
template T GEOGRAPHICLIB_EXPORT Math::AngRound <T>(T); \
template void GEOGRAPHICLIB_EXPORT Math::sincosd <T>(T, T&, T&); \
template void GEOGRAPHICLIB_EXPORT Math::sincosde <T>(T, T, T&, T&); \
template T GEOGRAPHICLIB_EXPORT Math::sind <T>(T); \
template T GEOGRAPHICLIB_EXPORT Math::cosd <T>(T); \
template T GEOGRAPHICLIB_EXPORT Math::tand <T>(T); \
template T GEOGRAPHICLIB_EXPORT Math::atan2d <T>(T, T); \
template T GEOGRAPHICLIB_EXPORT Math::atand <T>(T); \
template T GEOGRAPHICLIB_EXPORT Math::eatanhe <T>(T, T); \
template T GEOGRAPHICLIB_EXPORT Math::taupf <T>(T, T); \
template T GEOGRAPHICLIB_EXPORT Math::tauf <T>(T, T); \
template T GEOGRAPHICLIB_EXPORT Math::hypot3 <T>(T, T, T); \
template T GEOGRAPHICLIB_EXPORT Math::clamp <T>(T, T, T); \
template T GEOGRAPHICLIB_EXPORT Math::NaN <T>(); \
template T GEOGRAPHICLIB_EXPORT Math::infinity <T>();
// Instantiate with the standard floating type
GEOGRAPHICLIB_MATH_INSTANTIATE(float)
GEOGRAPHICLIB_MATH_INSTANTIATE(double)
#if GEOGRAPHICLIB_HAVE_LONG_DOUBLE
// Instantiate if long double is distinct from double
GEOGRAPHICLIB_MATH_INSTANTIATE(long double)
#endif
#if GEOGRAPHICLIB_PRECISION > 3
// Instantiate with the high precision type
GEOGRAPHICLIB_MATH_INSTANTIATE(Math::real)
#endif
#undef GEOGRAPHICLIB_MATH_INSTANTIATE
// Also we need int versions for Utility::nummatch
#define GEOGRAPHICLIB_MATH_INSTANTIATE2(T) \
template T GEOGRAPHICLIB_EXPORT Math::NaN <T>(); \
template T GEOGRAPHICLIB_EXPORT Math::infinity<T>();
GEOGRAPHICLIB_MATH_INSTANTIATE2(int)
GEOGRAPHICLIB_MATH_INSTANTIATE2(unsigned long long)
#undef GEOGRAPHICLIB_MATH_INSTANTIATE2
/// \endcond
} // namespace GeographicLib
ang
GeographicLib::Angle ang
Definition Geod3Solve.cpp:26
Math.hpp
Header for GeographicLib::Math class.
GEOGRAPHICLIB_VOLATILE
#define GEOGRAPHICLIB_VOLATILE
Definition Math.hpp:64
GEOGRAPHICLIB_PANIC
#define GEOGRAPHICLIB_PANIC(msg)
Definition Math.hpp:67
GEOGRAPHICLIB_PRECISION
#define GEOGRAPHICLIB_PRECISION
Definition Math.hpp:35
GeographicLib::Math::degree
static T degree()
Definition Math.hpp:197
GeographicLib::Math::tand
static T tand(T x)
Definition Math.cpp:203
GeographicLib::Math::sincosd
static void sincosd(T x, T &sinx, T &cosx)
Definition Math.cpp:104
GeographicLib::Math::atan2d
static T atan2d(T y, T x)
Definition Math.cpp:212
GeographicLib::Math::AngRound
static T AngRound(T x)
Definition Math.cpp:95
GeographicLib::Math::sq
static T sq(T x)
Definition Math.hpp:209
GeographicLib::Math::sum
static T sum(T u, T v, T &t)
Definition Math.cpp:55
GeographicLib::Math::sind
static T sind(T x)
Definition Math.cpp:172
GeographicLib::Math::qd
static constexpr int qd
degrees per quarter turn
Definition Math.hpp:142
GeographicLib::Math::clamp
static T clamp(T x, T a, T b)
Definition Math.cpp:296
GeographicLib::Math::tauf
static T tauf(T taup, T es)
Definition Math.cpp:251
GeographicLib::Math::hypot3
static T hypot3(T x, T y, T z)
Definition Math.cpp:285
GeographicLib::Math::AngNormalize
static T AngNormalize(T x)
Definition Math.cpp:69
GeographicLib::Math::digits10
static int digits10()
Definition Math.cpp:41
GeographicLib::Math::atand
static T atand(T x)
Definition Math.cpp:234
GeographicLib::Math::digits
static int digits()
Definition Math.cpp:20
GeographicLib::Math::infinity
static T infinity()
Definition Math.cpp:308
GeographicLib::Math::td
static constexpr int td
degrees per turn
Definition Math.hpp:146
GeographicLib::Math::sincosde
static void sincosde(T x, T t, T &sinx, T &cosx)
Definition Math.cpp:131
GeographicLib::Math::taupf
static T taupf(T tau, T es)
Definition Math.cpp:241
GeographicLib::Math::pi
static T pi()
Definition Math.hpp:187
GeographicLib::Math::NaN
static T NaN()
Definition Math.cpp:301
GeographicLib::Math::AngDiff
static T AngDiff(T x, T y, T &e)
Definition Math.cpp:80
GeographicLib::Math::hd
static constexpr int hd
degrees per half turn
Definition Math.hpp:145
GeographicLib::Math::eatanhe
static T eatanhe(T x, T es)
Definition Math.cpp:237
GeographicLib::Math::set_digits
static int set_digits(int ndigits)
Definition Math.cpp:28
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib::Math::cosd
static T cosd(T x)
Definition Math.cpp:188
GeographicLib::Math::extra_digits
static int extra_digits()
Definition Math.cpp:49
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12
std::swap
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)
Definition NearestNeighbor.hpp:819