mathop.h File Reference

Math operations (Mathematical operations) More...

#include "generic.h"
#include <math.h>
#include <float.h>

Macros
#define	VL_E   2.718281828459045
	Euler constant.
#define	VL_LOG_OF_2   0.693147180559945
	Logarithm of 2 (math constant)
#define	VL_PI   3.141592653589793
	Pi (math constant)
#define	VL_EPSILON_F   1.19209290E-07F
	IEEE single precision epsilon (math constant) More...
#define	VL_EPSILON_D   2.220446049250313e-16
	IEEE double precision epsilon (math constant) More...
#define	VL_NAN_F   (vl_nan_f.value)
	IEEE single precision NaN (not signaling)
#define	VL_INFINITY_F   (vl_infinity_f.value)
	IEEE single precision positive infinity (not signaling)
#define	VL_NAN_D   (vl_nan_d.value)
	IEEE double precision NaN (not signaling)
#define	VL_INFINITY_D   (vl_infinity_d.value)
	IEEE double precision positive infinity (not signaling)

Typedefs
typedef float(*	VlFloatVectorComparisonFunction )(vl_size dimension, float const *X, float const *Y)
	Pointer to a function to compare vectors of floats.
typedef double(*	VlDoubleVectorComparisonFunction )(vl_size dimension, double const *X, double const *Y)
	Pointer to a function to compare vectors of doubles.
typedef float(*	VlFloatVector3ComparisonFunction )(vl_size dimension, float const *X, float const *Y, float const *Z)
	Pointer to a function to compare 3 vectors of doubles.
typedef double(*	VlDoubleVector3ComparisonFunction )(vl_size dimension, double const *X, double const *Y, double const *Z)
	Pointer to a function to compare 3 vectors of doubles.
typedef enum _VlVectorComparisonType	VlVectorComparisonType
	Vector comparison types.

Enumerations
enum	_VlVectorComparisonType {   VlDistanceL1, VlDistanceL2, VlDistanceChi2, VlDistanceHellinger,   VlDistanceJS, VlDistanceMahalanobis, VlKernelL1, VlKernelL2,   VlKernelChi2, VlKernelHellinger, VlKernelJS }
	Vector comparison types. More...

Functions
float	vl_mod_2pi_f (float x)
	Fast mod(x, 2 * VL_PI) More...
double	vl_mod_2pi_d (double x)
	Fast mod(x, 2 * VL_PI) More...
long int	vl_floor_f (float x)
	Floor and convert to integer. More...
long int	vl_floor_d (double x)
	Floor and convert to integer. More...
long int	vl_ceil_f (float x)
	Ceil and convert to integer. More...
long int	vl_ceil_d (double x)
	Ceil and convert to integer. More...
long int	vl_round_f (float x)
	Round. More...
long int	vl_round_d (double x)
	Round. More...
float	vl_abs_f (float x)
	Fast abs(x) More...
double	vl_abs_d (double x)
	Fast abs(x) More...
double	vl_log2_d (double x)
	Base-2 logaritghm. More...
float	vl_log2_f (float x)
	Base-2 logaritghm. More...
double	vl_sqrt_d (double x)
	Square root. More...
float	vl_sqrt_f (float x)
	Square root. More...
vl_bool	vl_is_nan_f (float x)
	Check whether a floating point value is NaN. More...
vl_bool	vl_is_nan_d (double x)
	Check whether a floating point value is NaN. More...
vl_bool	vl_is_inf_f (float x)
	Check whether a floating point value is infinity. More...
vl_bool	vl_is_inf_d (double x)
	Check whether a floating point value is infinity. More...
float	vl_fast_atan2_f (float y, float x)
	Fast atan2 approximation. More...
double	vl_fast_atan2_d (double y, double x)
	Fast atan2 approximation. More...
float	vl_fast_resqrt_f (float x)
	Fast resqrt approximation. More...
double	vl_fast_resqrt_d (double x)
	Fast resqrt approximation. More...
float	vl_fast_sqrt_f (float x)
	Fast sqrt approximation. More...
double	vl_fast_sqrt_d (float x)
	Fast sqrt approximation. More...
vl_uint64	vl_fast_sqrt_ui64 (vl_uint64 x)
	Fast integer sqrt approximation. More...
vl_uint32	vl_fast_sqrt_ui32 (vl_uint32 x)
	Fast sqrt approximation. More...
vl_uint16	vl_fast_sqrt_ui16 (vl_uint16 x)
	Fast sqrt approximation. More...
vl_uint8	vl_fast_sqrt_ui8 (vl_uint8 x)
	Fast sqrt approximation. More...
char const *	vl_get_vector_comparison_type_name (int type)
	Get the symbolic name of a vector comparison type. More...
VlFloatVectorComparisonFunction	vl_get_vector_comparison_function_f (VlVectorComparisonType type)
	Get vector comparison function from comparison type. More...
VlDoubleVectorComparisonFunction	vl_get_vector_comparison_function_d (VlVectorComparisonType type)
	Get vector comparison function from comparison type. More...
void	vl_eval_vector_comparison_on_all_pairs_f (float *result, vl_size dimension, float const *X, vl_size numDataX, float const *Y, vl_size numDataY, VlFloatVectorComparisonFunction function)
	Evaluate vector comparison function on all vector pairs. More...
void	vl_eval_vector_comparison_on_all_pairs_d (double *result, vl_size dimension, double const *X, vl_size numDataX, double const *Y, vl_size numDataY, VlDoubleVectorComparisonFunction function)
	Evaluate vector comparison function on all vector pairs. More...

Variables
union {
}	vl_nan_f
	IEEE single precision quiet NaN constant. More...
union {
}	vl_infinity_f
	IEEE single precision infinity constant. More...
union {
}	vl_nan_d
	IEEE double precision quiet NaN constant. More...
union {
}	vl_infinity_d
	IEEE double precision infinity constant. More...

Detailed Description

Author

Andrea Vedaldi, David Novotny

Macro Definition Documentation

#define VL_EPSILON_D   2.220446049250313e-16

1.0 + VL_EPSILON_D is the smallest representable double precision number greater than 1.0. Numerically, VL_EPSILON_D is equal to \( 2^{-52} \).

#define VL_EPSILON_F   1.19209290E-07F

1.0F + VL_EPSILON_F is the smallest representable single precision number greater than 1.0F. Numerically, VL_EPSILON_F is equal to \( 2^{-23} \).

Enumeration Type Documentation

enum _VlVectorComparisonType

Enumerator
VlDistanceL1	l1 distance (squared intersection metric)
VlDistanceL2	squared l2 distance
VlDistanceChi2	squared Chi2 distance
VlDistanceHellinger	squared Hellinger's distance
VlDistanceJS	squared Jensen-Shannon distance
VlDistanceMahalanobis	squared mahalanobis distance
VlKernelL1	intersection kernel
VlKernelL2	l2 kernel
VlKernelChi2	Chi2 kernel
VlKernelHellinger	Hellinger's kernel
VlKernelJS	Jensen-Shannon kernel

Function Documentation

double vl_abs_d ( double  x )

inline

See Also

vl_abs_f

float vl_abs_f ( float  x )

inline

Parameters

x

argument.

Returns

abs(x)

long int vl_ceil_d ( double  x )

inline

See Also

vl_ceil_f

long int vl_ceil_f ( float  x )

inline

Parameters

x

argument.

Returns

lceilf(x)

vl_eval_vector_comparison_on_all_pairs_d	(	double *	result,
		vl_size	dimension,
		double const *	X,
		vl_size	numDataX,
		double const *	Y,
		vl_size	numDataY,
		VlDoubleVectorComparisonFunction	function
	)

See Also

vl_eval_vector_comparison_on_all_pairs_f

vl_eval_vector_comparison_on_all_pairs_f	(	float *	result,
		vl_size	dimension,
		float const *	X,
		vl_size	numDataX,
		float const *	Y,
		vl_size	numDataY,
		VlFloatVectorComparisonFunction	function
	)

Parameters

result	comparison matrix (output).
dimension	number of vector components (rows of X and Y).
X	data matrix X.
Y	data matrix Y.
numDataX	number of vectors in X (columns of X)
numDataY	number of vectros in Y (columns of Y)
function	vector comparison function.

The function evaluates function on all pairs of columns from matrices X and Y, filling a numDataX by numDataY matrix.

If Y is a null pointer the function compares all columns from X with themselves.

double vl_fast_atan2_d ( double  y, double  x  )

inline

See Also

vl_fast_atan2_f

float vl_fast_atan2_f ( float  y, float  x  )

inline

Parameters

y	argument.
x	argument.

The function computes a relatively rough but fast approximation of atan2(y,x).

Algorithm

The algorithm approximates the function \( f(r)=atan((1-r)/(1+r)) \), \( r \in [-1,1] \) with a third order polynomial \( f(r)=c_0 + c_1 r + c_2 r^2 + c_3 r^3 \). To fit the polynomial we impose the constraints

\begin{eqnarray*} f(+1) &=& c_0 + c_1 + c_2 + c_3 = atan(0) = 0,\\ f(-1) &=& c_0 - c_1 + c_2 - c_3 = atan(\infty) = \pi/2,\\ f(0) &=& c_0 = atan(1) = \pi/4. \end{eqnarray*}

The last degree of freedom is fixed by minimizing the \( l^{\infty} \) error, which yields

\[ c_0=\pi/4, \quad c_1=-0.9675, \quad c_2=0, \quad c_3=0.1821, \]

with maximum error of 0.0061 radians at 0.35 degrees.

Returns

Approximation of atan2(y,x).

double vl_fast_resqrt_d ( double  x )

inline

See Also

vl_fast_resqrt_d

float vl_fast_resqrt_f ( float  x )

inline

Parameters

x

argument.

Returns

approximation of resqrt(x).

The function quickly computes an approximation of \( x^{-1/2} \).

Algorithm

The goal is to compute \( y = x^{-1/2} \), which we do by finding the solution of \( 0 = f(y) = y^{-2} - x \) by two Newton steps. Each Newton iteration is given by

\[ y \leftarrow y - \frac{f(y)}{\frac{df(y)}{dy}} = y + \frac{1}{2} (y-xy^3) = \frac{y}{2} \left( 3 - xy^2 \right) \]

which yields a simple polynomial update rule.

The clever bit (attributed to either J. Carmack or G. Tarolli) is the way an initial guess \( y \approx x^{-1/2} \) is chosen.

See Also

Inverse Sqare Root.

double vl_fast_sqrt_d ( float  x )

inline

Fast sqrt approximation.

Parameters

x

argument.

Returns

approximation of sqrt(x).

The function uses vl_fast_resqrt_f (or vl_fast_resqrt_d) to compute x * vl_fast_resqrt_f(x).

float vl_fast_sqrt_f ( float  x )

inline

Parameters

x

argument.

Returns

approximation of sqrt(x).

The function uses vl_fast_resqrt_f (or vl_fast_resqrt_d) to compute x * vl_fast_resqrt_f(x).

vl_uint16 vl_fast_sqrt_ui16 ( vl_uint16  x )

inline

Fast integer sqrt approximation.

Parameters

x	non-negative integer.

Returns

largest integer \(y\) such that \(y^2 \leq x\).

See Also

Algorithm

vl_uint32 vl_fast_sqrt_ui32 ( vl_uint32  x )

inline

Fast integer sqrt approximation.

Parameters

x	non-negative integer.

Returns

largest integer \(y\) such that \(y^2 \leq x\).

See Also

Algorithm

vl_uint64 vl_fast_sqrt_ui64 ( vl_uint64  x )

inline

Parameters

x	non-negative integer.

Returns

largest integer \(y\) such that \(y^2 \leq x\).

See Also

Algorithm

vl_uint8 vl_fast_sqrt_ui8 ( vl_uint8  x )

inline

Fast integer sqrt approximation.

Parameters

x	non-negative integer.

Returns

largest integer \(y\) such that \(y^2 \leq x\).

See Also

Algorithm

long int vl_floor_d ( double  x )

inline

See Also

vl_floor_f

long int vl_floor_f ( float  x )

inline

Parameters

x

argument.

Returns

Similar to (int) floor(x)

vl_get_vector_comparison_function_d

(

VlVectorComparisonType

type

)

See Also

vl_get_vector_comparison_function_f

vl_get_vector_comparison_function_f

(

VlVectorComparisonType

type

)

Parameters

type	vector comparison type.

Returns

comparison function.

char const* vl_get_vector_comparison_type_name ( int  type )

inline

Parameters

type	vector comparison type.

Returns

data symbolic name.

vl_bool vl_is_inf_d ( double  x )

inline

Parameters

x

argument.

Returns

true if x is infinity.

vl_bool vl_is_inf_f ( float  x )

inline

Parameters

x

argument.

Returns

true if x is infinity.

vl_bool vl_is_nan_d ( double  x )

inline

Parameters

x

argument.

Returns

true if x is NaN.

vl_bool vl_is_nan_f ( float  x )

inline

Parameters

x

argument.

Returns

true if x is NaN.

double vl_log2_d ( double  x )

inline

Parameters

x

argument.

Returns

log(x).

float vl_log2_f ( float  x )

inline

Parameters

x

argument.

Returns

log(x).

double vl_mod_2pi_d ( double  x )

inline

See Also

vl_mod_2pi_f

float vl_mod_2pi_f ( float  x )

inline

Parameters

x	input value.

Returns

mod(x, 2 * VL_PI)

The function is optimized for small absolute values of x.

The result is guaranteed to be not smaller than 0. However, due to finite numerical precision and rounding errors, the result can be equal to 2 * VL_PI (for instance, if x is a very small negative number).

long int vl_round_d ( double  x )

inline

Parameters

x

argument.

Returns

lround(x) This function is either the same or similar to C99 lround().

long int vl_round_f ( float  x )

inline

Parameters

x

argument.

Returns

lroundf(x) This function is either the same or similar to C99 lroundf().

double vl_sqrt_d ( double  x )

inline

Parameters

x

argument.

Returns

sqrt(x).

float vl_sqrt_f ( float  x )

inline

Parameters

x

argument.

Returns

sqrt(x).

Variable Documentation

union { ... } vl_infinity_d

Initial value:

=

    { 0x7FF0000000000000ui64 }

union { ... } vl_infinity_f

Initial value:

=
    { 0x7F800000UL }

union { ... } vl_nan_d

Initial value:

=

    { 0x7FF8000000000000ui64 }

union { ... } vl_nan_f

Initial value:

=
    { 0x7FC00000UL }