Documentation>C API
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

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
xargument.
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
xargument.
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 
)
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
resultcomparison matrix (output).
dimensionnumber of vector components (rows of X and Y).
Xdata matrix X.
Ydata matrix Y.
numDataXnumber of vectors in X (columns of X)
numDataYnumber of vectros in Y (columns of Y)
functionvector 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
yargument.
xargument.

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
float vl_fast_resqrt_f ( float  x)
inline
Parameters
xargument.
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
xargument.
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
xargument.
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
xnon-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
xnon-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
xnon-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
xnon-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
xargument.
Returns
Similar to (int) floor(x)
vl_get_vector_comparison_function_d ( VlVectorComparisonType  type)
vl_get_vector_comparison_function_f ( VlVectorComparisonType  type)
Parameters
typevector comparison type.
Returns
comparison function.
char const* vl_get_vector_comparison_type_name ( int  type)
inline
Parameters
typevector comparison type.
Returns
data symbolic name.
vl_bool vl_is_inf_d ( double  x)
inline
Parameters
xargument.
Returns
true if x is infinity.
vl_bool vl_is_inf_f ( float  x)
inline
Parameters
xargument.
Returns
true if x is infinity.
vl_bool vl_is_nan_d ( double  x)
inline
Parameters
xargument.
Returns
true if x is NaN.
vl_bool vl_is_nan_f ( float  x)
inline
Parameters
xargument.
Returns
true if x is NaN.
double vl_log2_d ( double  x)
inline
Parameters
xargument.
Returns
log(x).
float vl_log2_f ( float  x)
inline
Parameters
xargument.
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
xinput 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
xargument.
Returns
lround(x) This function is either the same or similar to C99 lround().
long int vl_round_f ( float  x)
inline
Parameters
xargument.
Returns
lroundf(x) This function is either the same or similar to C99 lroundf().
double vl_sqrt_d ( double  x)
inline
Parameters
xargument.
Returns
sqrt(x).
float vl_sqrt_f ( float  x)
inline
Parameters
xargument.
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 }