Mathematical functions needed by GeographicLib. More...

#include <Math.hpp>

Public Types
typedef double	extended

typedef double	real

Static Public Member Functions
template<typename T >
static T	AngDiff (T x, T y, T &e)

template<typename T >
static T	AngDiff (T x, T y)

template<typename T >
static T	AngNormalize (T x)

template<typename T >
static T	AngRound (T x)

template<typename T >
static T	asinh (T x)

template<typename T >
static T	atan2d (T y, T x)

template<typename T >
static T	atand (T x)

template<typename T >
static T	atanh (T x)

template<typename T >
static T	cbrt (T x)

template<typename T >
static T	copysign (T x, T y)

template<typename T >
static T	cosd (T x)

template<typename T >
static T	degree ()

static real	degree ()

static int	digits ()

static int	digits10 ()

template<typename T >
static T	eatanhe (T x, T es)

template<typename T >
static T	expm1 (T x)

static int	extra_digits ()

template<typename T >
static T	fma (T x, T y, T z)

template<typename T >
static T	hypot (T x, T y)

template<typename T >
static T	infinity ()

static real	infinity ()

template<typename T >
static bool	isfinite (T x)

template<typename T >
static bool	isnan (T x)

template<typename T >
static T	LatFix (T x)

template<typename T >
static T	log1p (T x)

template<typename T >
static T	NaN ()

static real	NaN ()

template<typename T >
static void	norm (T &x, T &y)

template<typename T >
static T	pi ()

static real	pi ()

template<typename T >
static T	polyval (int N, const T p[], T x)

static int	set_digits (int ndigits)

template<typename T >
static void	sincosd (T x, T &sinx, T &cosx)

template<typename T >
static T	sind (T x)

template<typename T >
static T	sq (T x)

template<typename T >
static T	sum (T u, T v, T &t)

template<typename T >
static T	swab (T x)

template<typename T >
static T	tand (T x)

template<typename T >
static T	tauf (T taup, T es)

template<typename T >
static T	taupf (T tau, T es)

Static Public Attributes
static const bool	bigendian = GEOGRAPHICLIB_WORDS_BIGENDIAN

Private Member Functions
void	dummy ()

	Math ()

Detailed Description

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:

// Example of using the GeographicLib::Math class
#include <iostream>
#include <exception>
#include <GeographicLib/Math.hpp>
using namespace std;
using namespace GeographicLib;
int main() {
  try {
    cout << Math::pi() << " " << Math::sq(Math::pi()) << "\n";
  }
  catch (const exception& e) {
    cerr << "Caught exception: " << e.what() << "\n";
    return 1;
  }
}

Definition at line 102 of file Math.hpp.

Member Typedef Documentation

typedef double GeographicLib::Math::extended

Definition at line 119 of file Math.hpp.

typedef double GeographicLib::Math::real

The real type for GeographicLib. Nearly all the testing has been done with real = double. However, the algorithms should also work with float and long double (where available). (CAUTION: reasonable accuracy typically cannot be obtained using floats.)

Definition at line 129 of file Math.hpp.

Constructor & Destructor Documentation

GeographicLib::Math::Math ( )

private

Member Function Documentation

template<typename T >

static T GeographicLib::Math::AngDiff	(	T	x,
		T	y,
		T &	e
	)

inlinestatic

The exact difference of two angles reduced to (−180°, 180°].

Template Parameters

T	the type of the arguments and returned value.

Parameters

[in]	x	the first angle in degrees.
[in]	y	the second angle in degrees.
[out]	e	the error term in degrees.

Returns: d, the truncated value of y − x.

This computes z = y − x exactly, reduced to (−180°, 180°]; and then sets z = d + e where d is the nearest representable number to z and e is the truncation error. If d = −180, then e > 0; If d = 180, then e ≤ 0.

Definition at line 486 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::AngDiff	(	T	x,
		T	y
	)

inlinestatic

Difference of two angles reduced to [−180°, 180°]

Template Parameters

T	the type of the arguments and returned value.

Parameters

[in]	x	the first angle in degrees.
[in]	y	the second angle in degrees.

Returns: y − x, reduced to the range [−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 x is tiny and negative and y = 180°).

Definition at line 517 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::AngNormalize ( T x )

inlinestatic

Normalize an angle.

Template Parameters

T	the type of the argument and returned value.

Parameters

[in] x the angle in degrees.

Returns: the angle reduced to the range([−180°, 180°].

The range of x is unrestricted.

Definition at line 440 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::AngRound ( T x )

inlinestatic

Coarsen a value close to zero.

Template Parameters

T	the type of the argument and returned value.

Parameters

[in] x

Returns: the coarsened value.

The makes the smallest gap in x = 1/16 - nextafter(1/16, 0) = 1/2⁵⁷ for reals = 0.7 pm on the earth if x is an angle in degrees. (This is about 1000 times more resolution than we get with angles around 90°.) We use this to avoid having to deal with near singular cases when x is non-zero but tiny (e.g., 10⁻²⁰⁰). This converts -0 to +0; however tiny negative numbers get converted to -0.

Definition at line 535 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::asinh ( T x )

inlinestatic

The inverse hyperbolic sine function.

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x

Returns: asinh(x).

Definition at line 311 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::atan2d	(	T	y,
		T	x
	)

inlinestatic

Evaluate the atan2 function with the result in degrees

Template Parameters

T	the type of the arguments and the returned value.

Parameters

[in]	y
[in]	x

Returns: atan2(y, x) in degrees.

The result is in the range (−180° 180°]. N.B., atan2d(±0, −1) = +180°; atan2d(−ε, −1) = −180°, for ε positive and tiny; atan2d(±0, +1) = ±0°.

Definition at line 691 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::atand ( T x )

inlinestatic

Evaluate the atan function with the result in degrees

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x

Returns: atan(x) in degrees.

Definition at line 723 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::atanh ( T x )

inlinestatic

The inverse hyperbolic tangent function.

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x

Returns: atanh(x).

Definition at line 328 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::cbrt ( T x )

inlinestatic

The cube root function.

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x

Returns: the real cube root of x.

Definition at line 345 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::copysign	(	T	x,
		T	y
	)

inlinestatic

Copy the sign.

Template Parameters

T	the type of the argument.

Parameters

[in]	x	gives the magitude of the result.
[in]	y	gives the sign of the result.

Returns: value with the magnitude of x and with the sign of y.

This routine correctly handles the case y = −0, returning &minus|x|.

Definition at line 751 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::cosd ( T x )

inlinestatic

Evaluate the cosine function with the argument in degrees

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x in degrees.

Returns: cos(x).

Definition at line 639 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::degree ( )

inlinestatic

Template Parameters

T	the type of the returned value.

Returns: the number of radians in a degree.

Definition at line 216 of file Math.hpp.

static real GeographicLib::Math::degree ( )

inlinestatic

A synonym for degree<real>().

Definition at line 223 of file Math.hpp.

static int GeographicLib::Math::digits ( )

inlinestatic

Returns: the number of bits of precision in a real number.

Definition at line 145 of file Math.hpp.

static int GeographicLib::Math::digits10 ( )

inlinestatic

Returns: the number of decimal digits of precision in a real number.

Definition at line 175 of file Math.hpp.

void GeographicLib::Math::dummy ( )

inlineprivate

Definition at line 104 of file Math.hpp.

template<typename T >

T GeographicLib::Math::eatanhe	(	T	x,
		T	es
	)

static

Evaluate e atanh(e x)

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in]	x
[in]	es	the signed eccentricity = sign(e²) sqrt(\|e²\|)

Returns: e atanh(e x)

If e² is negative (e is imaginary), the expression is evaluated in terms of atan.

Definition at line 21 of file Math.cpp.

template<typename T >

static T GeographicLib::Math::expm1 ( T x )

inlinestatic

exp(x) − 1 accurate near x = 0.

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x

Returns: exp(x) − 1.

Definition at line 265 of file Math.hpp.

static int GeographicLib::Math::extra_digits ( )

inlinestatic

Number of additional decimal digits of precision for real relative to double (0 for float).

Definition at line 187 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::fma	(	T	x,
		T	y,
		T	z
	)

inlinestatic

Fused multiply and add.

Template Parameters

T	the type of the arguments and the returned value.

Parameters

[in]	x
[in]	y
[in]	z

Returns: xy + z, correctly rounded (on those platforms with support for the fma instruction).

On platforms without the fma instruction, no attempt is made to improve on the result of a rounded multiplication followed by a rounded addition.

Definition at line 369 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::hypot	(	T	x,
		T	y
	)

inlinestatic

The hypotenuse function avoiding underflow and overflow.

Template Parameters

T	the type of the arguments and the returned value.

Parameters

[in]	x
[in]	y

Returns: sqrt(x² + y²).

Definition at line 243 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::infinity ( )

inlinestatic

Infinity

Template Parameters

T	the type of the returned value.

Returns: infinity if available, otherwise return the max real.

Definition at line 867 of file Math.hpp.

static real GeographicLib::Math::infinity ( )

inlinestatic

A synonym for infinity<real>().

Definition at line 881 of file Math.hpp.

template<typename T >

static bool GeographicLib::Math::isfinite ( T x )

inlinestatic

Test for finiteness.

Template Parameters

T	the type of the argument.

Parameters

[in] x

Returns: true if number is finite, false if NaN or infinite.

Definition at line 806 of file Math.hpp.

template<typename T >

static bool GeographicLib::Math::isnan ( T x )

inlinestatic

Test for NaN.

Template Parameters

T	the type of the argument.

Parameters

[in] x

Returns: true if argument is a NaN.

Definition at line 853 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::LatFix ( T x )

inlinestatic

Normalize a latitude.

Template Parameters

T	the type of the argument and returned value.

Parameters

[in] x the angle in degrees.

Returns: x if it is in the range [−90°, 90°], otherwise return NaN.

Definition at line 467 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::log1p ( T x )

inlinestatic

log(1 + x) accurate near x = 0.

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x

Returns: log(1 + x).

Definition at line 288 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::NaN ( )

inlinestatic

The NaN (not a number)

Template Parameters

T	the type of the returned value.

Returns: NaN if available, otherwise return the max real of type T.

Definition at line 830 of file Math.hpp.

static real GeographicLib::Math::NaN ( )

inlinestatic

A synonym for NaN<real>().

Definition at line 844 of file Math.hpp.

template<typename T >

static void GeographicLib::Math::norm	(	T &	x,
		T &	y
	)

inlinestatic

Normalize a two-vector.

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in,out]	x	on output set to x/hypot(x, y).
[in,out]	y	on output set to y/hypot(x, y).

Definition at line 384 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::pi ( )

inlinestatic

Template Parameters

T	the type of the returned value.

Returns: π.

Definition at line 202 of file Math.hpp.

static real GeographicLib::Math::pi ( )

inlinestatic

A synonym for pi<real>().

Definition at line 210 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::polyval	(	int	N,
		const T	p[],
		T	x
	)

inlinestatic

Evaluate a polynomial.

Template Parameters

T	the type of the arguments and returned value.

Parameters

[in]	N	the order of the polynomial.
[in]	p	the coefficient array (of size N + 1).
[in]	x	the variable.

Returns: the value of the polynomial.

Evaluate y = ∑_n=0..N p_n x^N−n. Return 0 if N < 0. Return p₀, if N = 0 (even if x is infinite or a nan). The evaluation uses Horner's method.

Definition at line 425 of file Math.hpp.

static int GeographicLib::Math::set_digits ( int ndigits )

inlinestatic

Set the binary precision of a real number.

Parameters

[in] ndigits the number of bits of precision.

Returns: 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.

Definition at line 163 of file Math.hpp.

template<typename T >

static void GeographicLib::Math::sincosd	(	T	x,
		T &	sinx,
		T &	cosx
	)

inlinestatic

Evaluate the sine and cosine function with the argument in degrees

Template Parameters

T	the type of the arguments.

Parameters

[in]	x	in degrees.
[out]	sinx	sin(x).
[out]	cosx	cos(x).

The results obey exactly the elementary properties of the trigonometric functions, e.g., sin 9° = cos 81° = − sin 123456789°. If x = −0, then sinx = −0; this is the only case where −0 is returned.

Definition at line 558 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::sind ( T x )

inlinestatic

Evaluate the sine function with the argument in degrees

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x in degrees.

Returns: sin(x).

Definition at line 609 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::sq ( T x )

inlinestatic

Square a number.

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x

Returns: x².

Definition at line 232 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::sum	(	T	u,
		T	v,
		T &	t
	)

inlinestatic

The error-free sum of two numbers.

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in]	u
[in]	v
[out]	t	the exact error given by (u + v) - s.

Returns: s = round(u + v).

See D. E. Knuth, TAOCP, Vol 2, 4.2.2, Theorem B. (Note that t can be the same as one of the first two arguments.)

Definition at line 399 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::swab ( T x )

inlinestatic

Swap the bytes of a quantity

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x

Returns: x with its bytes swapped.

Definition at line 890 of file Math.hpp.

template<typename T >

static T GeographicLib::Math::tand ( T x )

inlinestatic

Evaluate the tangent function with the argument in degrees

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in] x in degrees.

Returns: tan(x).

If x = ±90°, then a suitably large (but finite) value is returned.

Definition at line 671 of file Math.hpp.

template<typename T >

T GeographicLib::Math::tauf	(	T	taup,
		T	es
	)

static

tanφ in terms of tanχ

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in]	taup	τ′ = tanχ
[in]	es	the signed eccentricity = sign(e²) sqrt(\|e²\|)

Returns: τ = tanφ

See Eqs. (19–21) of C. F. F. Karney, Transverse Mercator with an accuracy of a few nanometers, J. Geodesy 85(8), 475–485 (Aug. 2011) (preprint arXiv:1002.1417).

Definition at line 31 of file Math.cpp.

template<typename T >

T GeographicLib::Math::taupf	(	T	tau,
		T	es
	)

static

tanχ in terms of tanφ

Template Parameters

T	the type of the argument and the returned value.

Parameters

[in]	tau	τ = tanφ
[in]	es	the signed eccentricity = sign(e²) sqrt(\|e²\|)

Returns: τ′ = tanχ

See Eqs. (7–9) of C. F. F. Karney, Transverse Mercator with an accuracy of a few nanometers, J. Geodesy 85(8), 475–485 (Aug. 2011) (preprint arXiv:1002.1417).

Definition at line 25 of file Math.cpp.

Member Data Documentation

const bool GeographicLib::Math::bigendian = GEOGRAPHICLIB_WORDS_BIGENDIAN

static

true if the machine is big-endian.

Definition at line 196 of file Math.hpp.

The documentation for this class was generated from the following files:

Public Types

Static Public Member Functions

Static Public Attributes

Private Member Functions

Detailed Description

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation