GeographicLib: Math.cpp Source File

GeographicLib 2.7

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Math.cpp
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1/**
2 * \file Math.cpp
3 * \brief Implementation for GeographicLib::Math class
4 *
5 * Copyright (c) Charles Karney (2015-2024) <karney@alum.mit.edu> and licensed
6 * under the MIT/X11 License. For more information, see
7 * https://geographiclib.sourceforge.io/
8 **********************************************************************/
9
10#include <GeographicLib/Math.hpp>
11
12namespace GeographicLib {
13
14 using namespace std;
15
16 void Math::dummy() {
17 static_assert(GEOGRAPHICLIB_PRECISION >= 1, "Bad value of precision");
18 }
19
20 int Math::digits() {
21#if GEOGRAPHICLIB_PRECISION == 5
22 return numeric_limits<real>::digits();
23#else
24 return numeric_limits<real>::digits;
25#endif
26 }
27
28 int Math::set_digits(int ndigits) {
29#if GEOGRAPHICLIB_PRECISION >= 5
30# if GEOGRAPHICLIB_PRECISION > 5
31 // This sets ndigits = GEOGRAPHICLIB_PRECISION
32 ndigits = numeric_limits<real>::digits;
33# endif
34 mpfr::mpreal::set_default_prec(ndigits >= 2 ? ndigits : 2);
35#else
36 (void) ndigits;
37#endif
38 return digits();
39 }
40
41 int Math::digits10() {
42#if GEOGRAPHICLIB_PRECISION == 5
43 return numeric_limits<real>::digits10();
44#else
45 return numeric_limits<real>::digits10;
46#endif
47 }
48
49 int Math::extra_digits() {
50 return
51 digits10() > numeric_limits<double>::digits10 ?
52 digits10() - numeric_limits<double>::digits10 : 0;
53 }
54
55 template<typename T> T Math::sum(T u, T v, T& t) {
56 GEOGRAPHICLIB_VOLATILE T s = u + v;
57 GEOGRAPHICLIB_VOLATILE T up = s - v;
58 GEOGRAPHICLIB_VOLATILE T vpp = s - up;
59 up -= u;
60 vpp -= v;
61 // if s = 0, then t = 0 and give t the same sign as s
62 // mpreal needs T(0) here
63 t = s != 0 ? T(0) - (up + vpp) : s;
64 // u + v = s + t
65 // = round(u + v) + t
66 return s;
67 }
68
69 template<typename T> T Math::AngNormalize(T x) {
70 T y = remainder(x, T(td));
71#if GEOGRAPHICLIB_PRECISION == 4
72 // boost-quadmath doesn't set the sign of 0 correctly, see
73 // https://github.com/boostorg/multiprecision/issues/426
74 // Fixed by https://github.com/boostorg/multiprecision/pull/428
75 if (y == 0) y = copysign(y, x);
76#endif
77 return fabs(y) == T(hd) ? copysign(T(hd), x) : y;
78 }
79
80 template<typename T> T Math::AngDiff(T x, T y, T& e) {
81 // Use remainder instead of AngNormalize, since we treat boundary cases
82 // later taking account of the error
83 T d = sum(remainder(-x, T(td)), remainder( y, T(td)), e);
84 // This second sum can only change d if abs(d) < 128, so don't need to
85 // apply remainder yet again.
86 d = sum(remainder(d, T(td)), e, e);
87 // Fix the sign if d = -180, 0, 180.
88 if (d == 0 || fabs(d) == hd)
89 // If e == 0, take sign from y - x
90 // else (e != 0, implies d = +/-180), d and e must have opposite signs
91 d = copysign(d, e == 0 ? y - x : -e);
92 return d;
93 }
94
95 template<typename T> T Math::AngRound(T x) {
96 static const T z = T(1)/T(16);
97 GEOGRAPHICLIB_VOLATILE T y = fabs(x);
98 GEOGRAPHICLIB_VOLATILE T w = z - y;
99 // The compiler mustn't "simplify" z - (z - y) to y
100 y = w > 0 ? z - w : y;
101 return copysign(y, x);
102 }
103
104 template<typename T> void Math::sincosd(T x, T& sinx, T& cosx) {
105 // In order to minimize round-off errors, this function exactly reduces
106 // the argument to the range [-45, 45] before converting it to radians.
107 int q = 0;
108 T d = remquo(x, T(qd), &q), // now abs(r) <= 45
109 r = d * degree<T>();
110 // g++ -O turns these two function calls into a call to sincos
111 T s = sin(r), c = cos(r);
112 if (2 * fabs(d) == qd) {
113 c = sqrt(1/T(2));
114 s = copysign(c, r);
115 } else if (3 * fabs(d) == qd) {
116 c = sqrt(T(3))/2;
117 s = copysign(1/T(2), r);
118 }
119 switch (unsigned(q) & 3U) {
120 case 0U: sinx = s; cosx = c; break;
121 case 1U: sinx = c; cosx = -s; break;
122 case 2U: sinx = -s; cosx = -c; break;
123 default: sinx = -c; cosx = s; break; // case 3U
124 }
125 // http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf
126 // mpreal needs T(0) here
127 cosx += T(0); // special values from F.10.1.12
128 if (sinx == 0) sinx = copysign(sinx, x); // special values from F.10.1.13
129 }
130
131 template<typename T> void Math::sincosde(T x, T t, T& sinx, T& cosx) {
132 // In order to minimize round-off errors, this function exactly reduces
133 // the argument to the range [-45, 45] before converting it to radians.
134 // This implementation allows x outside [-180, 180], but implementations in
135 // other languages may not.
136 int q = 0;
137 T d = AngRound(remquo(x, T(qd), &q) + t), // now abs(r) <= 45
138 r = d * degree<T>();
139 // g++ -O turns these two function calls into a call to sincos
140 T s = sin(r), c = cos(r);
141 if (2 * fabs(d) == qd) {
142 c = sqrt(1/T(2));
143 s = copysign(c, r);
144 } else if (3 * fabs(d) == qd) {
145 c = sqrt(T(3))/2;
146 s = copysign(1/T(2), r);
147 }
148 switch (unsigned(q) & 3U) {
149 case 0U: sinx = s; cosx = c; break;
150 case 1U: sinx = c; cosx = -s; break;
151 case 2U: sinx = -s; cosx = -c; break;
152 default: sinx = -c; cosx = s; break; // case 3U
153 }
154 // http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf
155 // mpreal needs T(0) here
156 cosx += T(0); // special values from F.10.1.12
157 // Should we copy the sign from x or x+t? Given that t is small, there's
158 // only a distinction if x+t == +/- 0. Here are the cases
159 // x t x-sign (x+t)-sign
160 // <0 >0 -0 +0 different
161 // >0 <0 +0 +0
162 // -0 -0 -0 -0
163 // -0 +0 -0 +0 different
164 // +0 -0 +0 +0
165 // +0 +0 +0 +0
166 // On balance, taking the sign from x is better, particularly for the case
167 // x = -0, t = +0. This choice also avoids the bias towards +0 that x+t
168 // gives.
169 if (sinx == 0) sinx = copysign(sinx, x); // special values from F.10.1.13
170 }
171
172 template<typename T> T Math::sind(T x) {
173 // See sincosd
174 int q = 0;
175 T d = remquo(x, T(qd), &q), // now abs(r) <= 45
176 r = d * degree<T>();
177 unsigned p = unsigned(q);
178 // r = p & 1U ? cos(r) : sin(r); replaced by ...
179 r = p & 1U ? (2 * fabs(d) == qd ? sqrt(1/T(2)) :
180 (3 * fabs(d) == qd ? sqrt(T(3))/2 : cos(r))) :
181 copysign(2 * fabs(d) == qd ? sqrt(1/T(2)) :
182 (3 * fabs(d) == qd ? 1/T(2) : sin(r)), r);
183 if (p & 2U) r = -r;
184 if (r == 0) r = copysign(r, x);
185 return r;
186 }
187
188 template<typename T> T Math::cosd(T x) {
189 // See sincosd
190 int q = 0;
191 T d = remquo(x, T(qd), &q), // now abs(r) <= 45
192 r = d * degree<T>();
193 unsigned p = unsigned(q + 1);
194 r = p & 1U ? (2 * fabs(d) == qd ? sqrt(1/T(2)) :
195 (3 * fabs(d) == qd ? sqrt(T(3))/2 : cos(r))) :
196 copysign(2 * fabs(d) == qd ? sqrt(1/T(2)) :
197 (3 * fabs(d) == qd ? 1/T(2) : sin(r)), r);
198 if (p & 2U) r = -r;
199 // mpreal needs T(0) here
200 return T(0) + r;
201 }
202
203 template<typename T> T Math::tand(T x) {
204 static const T overflow = 1 / sq(numeric_limits<T>::epsilon());
205 T s, c;
206 sincosd(x, s, c);
207 // http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf
208 T r = s / c; // special values from F.10.1.14
209 return clamp(r, -overflow, overflow);
210 }
211
212 template<typename T> T Math::atan2d(T y, T x) {
213 // In order to minimize round-off errors, this function rearranges the
214 // arguments so that result of atan2 is in the range [-pi/4, pi/4] before
215 // converting it to degrees and mapping the result to the correct quadrant.
216 // With mpreal we could use T(mpfr::atan2u(y, x, td)); but we're not ready
217 // for this yet.
218 int q = 0;
219 if (fabs(y) > fabs(x)) { swap(x, y); q = 2; }
220 if (signbit(x)) { x = -x; ++q; }
221 // here x >= 0 and x >= abs(y), so angle is in [-pi/4, pi/4]
222 // Replace atan2(y, x) / degree<T>() by this to ensure that special values
223 // (45, 90, etc.) are returned.
224 T ang = (atan2(y, x) / pi<T>()) * T(hd);
225 switch (q) {
226 case 1: ang = copysign(T(hd), y) - ang; break;
227 case 2: ang = qd - ang; break;
228 case 3: ang = -qd + ang; break;
229 default: break;
230 }
231 return ang;
232 }
233
234 template<typename T> T Math::atand(T x)
235 { return atan2d(x, T(1)); }
236
237 template<typename T> T Math::eatanhe(T x, T es) {
238 return es > 0 ? es * atanh(es * x) : -es * atan(es * x);
239 }
240
241 template<typename T> T Math::taupf(T tau, T es) {
242 // Need this test, otherwise tau = +/-inf gives taup = nan.
243 if (isfinite(tau)) {
244 T tau1 = hypot(T(1), tau),
245 sig = sinh( eatanhe(tau / tau1, es ) );
246 return hypot(T(1), sig) * tau - sig * tau1;
247 } else
248 return tau;
249 }
250
251 template<typename T> T Math::tauf(T taup, T es) {
252 static const int numit = 5;
253 // min iterations = 1, max iterations = 2; mean = 1.95
254 static const T tol = sqrt(numeric_limits<T>::epsilon()) / 10;
255 static const T taumax = 2 / sqrt(numeric_limits<T>::epsilon());
256 T e2m = 1 - sq(es),
257 // To lowest order in e^2, taup = (1 - e^2) * tau = _e2m * tau; so use
258 // tau = taup/e2m as a starting guess. Only 1 iteration is needed for
259 // |lat| < 3.35 deg, otherwise 2 iterations are needed. If, instead, tau
260 // = taup is used the mean number of iterations increases to 1.999 (2
261 // iterations are needed except near tau = 0).
262 //
263 // For large tau, taup = exp(-es*atanh(es)) * tau. Use this as for the
264 // initial guess for |taup| > 70 (approx |phi| > 89deg). Then for
265 // sufficiently large tau (such that sqrt(1+tau^2) = |tau|), we can exit
266 // with the intial guess and avoid overflow problems. This also reduces
267 // the mean number of iterations slightly from 1.963 to 1.954.
268 tau = fabs(taup) > 70 ? taup * exp(eatanhe(T(1), es)) : taup/e2m,
269 stol = tol * fmax(T(1), fabs(taup));
270 if (!(fabs(tau) < taumax)) return tau; // handles +/-inf and nan
271 for (int i = 0;
272 i < numit ||
273 GEOGRAPHICLIB_PANIC("Convergence failure in Math::tauf");
274 ++i) {
275 T taupa = taupf(tau, es),
276 dtau = (taup - taupa) * (1 + e2m * sq(tau)) /
277 ( e2m * hypot(T(1), tau) * hypot(T(1), taupa) );
278 tau += dtau;
279 if (!(fabs(dtau) >= stol))
280 break;
281 }
282 return tau;
283 }
284
285 template<typename T> T Math::hypot3(T x, T y, T z) {
286#if GEOGRAPHICLIB_PRECISION == 4
287 // Boost implementation is given by
288 // https://github.com/boostorg/math/pull/1318
289 // might make its way into 1.90 or later
290 return hypot(hypot(x, y), z);
291#else
292 return hypot(x, y, z);
293#endif
294 }
295
296 template<typename T> T Math::clamp(T x, T a, T b) {
297 // Use max/min here (instead of fmax/fmin) to preserve NaN
298 return min(max(x, a), b);
299 }
300
301 template<typename T> T Math::NaN() {
302 if constexpr (numeric_limits<T>::has_quiet_NaN)
303 return numeric_limits<T>::quiet_NaN();
304 else
305 return (numeric_limits<T>::max)();
306 }
307
308 template<typename T> T Math::infinity() {
309 if constexpr (numeric_limits<T>::has_infinity)
310 return numeric_limits<T>::infinity();
311 else
312 return (numeric_limits<T>::max)();
313 }
314
315 /// \cond SKIP
316 // Instantiate
317#define GEOGRAPHICLIB_MATH_INSTANTIATE(T) \
318 template T GEOGRAPHICLIB_EXPORT Math::sum <T>(T, T, T&); \
319 template T GEOGRAPHICLIB_EXPORT Math::AngNormalize <T>(T); \
320 template T GEOGRAPHICLIB_EXPORT Math::AngDiff <T>(T, T, T&); \
321 template T GEOGRAPHICLIB_EXPORT Math::AngRound <T>(T); \
322 template void GEOGRAPHICLIB_EXPORT Math::sincosd <T>(T, T&, T&); \
323 template void GEOGRAPHICLIB_EXPORT Math::sincosde <T>(T, T, T&, T&); \
324 template T GEOGRAPHICLIB_EXPORT Math::sind <T>(T); \
325 template T GEOGRAPHICLIB_EXPORT Math::cosd <T>(T); \
326 template T GEOGRAPHICLIB_EXPORT Math::tand <T>(T); \
327 template T GEOGRAPHICLIB_EXPORT Math::atan2d <T>(T, T); \
328 template T GEOGRAPHICLIB_EXPORT Math::atand <T>(T); \
329 template T GEOGRAPHICLIB_EXPORT Math::eatanhe <T>(T, T); \
330 template T GEOGRAPHICLIB_EXPORT Math::taupf <T>(T, T); \
331 template T GEOGRAPHICLIB_EXPORT Math::tauf <T>(T, T); \
332 template T GEOGRAPHICLIB_EXPORT Math::hypot3 <T>(T, T, T); \
333 template T GEOGRAPHICLIB_EXPORT Math::clamp <T>(T, T, T); \
334 template T GEOGRAPHICLIB_EXPORT Math::NaN <T>(); \
335 template T GEOGRAPHICLIB_EXPORT Math::infinity <T>();
336
337 // Instantiate with the standard floating type
338 GEOGRAPHICLIB_MATH_INSTANTIATE(float)
339 GEOGRAPHICLIB_MATH_INSTANTIATE(double)
340#if GEOGRAPHICLIB_HAVE_LONG_DOUBLE
341 // Instantiate if long double is distinct from double
342 GEOGRAPHICLIB_MATH_INSTANTIATE(long double)
343#endif
344#if GEOGRAPHICLIB_PRECISION > 3
345 // Instantiate with the high precision type
346 GEOGRAPHICLIB_MATH_INSTANTIATE(Math::real)
347#endif
348
349#undef GEOGRAPHICLIB_MATH_INSTANTIATE
350
351 // Also we need int versions for Utility::nummatch
352#define GEOGRAPHICLIB_MATH_INSTANTIATE2(T) \
353 template T GEOGRAPHICLIB_EXPORT Math::NaN <T>(); \
354 template T GEOGRAPHICLIB_EXPORT Math::infinity<T>();
355
356 GEOGRAPHICLIB_MATH_INSTANTIATE2(int)
357 GEOGRAPHICLIB_MATH_INSTANTIATE2(unsigned long long)
358
359#undef GEOGRAPHICLIB_MATH_INSTANTIATE2
360 /// \endcond
361
362} // 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