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basicMath.h
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/*!
* \file basicMath.h
* \author Basilio Noris
* \date 27-11-2006
*/
#ifndef _BASICMATH_H_
#define _BASICMATH_H_
#include <iostream>
#define PIf 3.1415926535897932384626433832795f
//#define Vec2 CvPoint2D32f
#define Vec3 CvPoint3D32f
#define vec2(a,b) cvPoint2D32f(a,b)
#define vec3(a,b,c) cvPoint3D32f(a,b,c)
#define Truncate(a, b) a = (a>=b ? b : a)
#define point2(a) cvPoint((int)((a).x), (int)((a).y))
#define sign(a) (a)>0?1:((a)==0?0:-1)
#define toSize(a) cvSize((int)a.x, (int)a.y)
#ifndef RND
#define RND(x) ((x)*((double)rand())/((double)(RAND_MAX+1.0)))
#endif
#define epsilon 1e-6
#define DEG(x) ((x)*180/PIf)
#define RAD(x) ((x)*PIf/180)
//#define max(a,b) (a)>(b)?(a):(b)
class Vec2{
public:
float x, y;
// constructors
Vec2() : x(0), y(0) {}
Vec2(int a, int b): x((float)a), y((float)b) {}
Vec2(float a, float b): x(a), y(b) {}
Vec2(double a, double b): x((float)a), y((float)b) {}
//operators
Vec2 operator+(const Vec2 &v) const {
return Vec2(x + v.x, y + v.y);
}
Vec2& operator+=(const Vec2 &v) {
x += v.x;
y += v.y;
return *this;
}
Vec2 operator-(const Vec2 &v) const {
return Vec2(x - v.x, y - v.y);
}
Vec2& operator-=(const Vec2 &v) {
x -= v.x;
y -= v.y;
return *this;
}
Vec2 operator*(const float &d) const {
return Vec2(x * d, y * d);
}
Vec2& operator*=(const float &d) {
x *= d;
y *= d;
return *this;
}
Vec2 operator/(const float &d) const {
return Vec2(x / d, y / d);
}
Vec2 operator/(const int &d) const {
return Vec2(x / d, y / d);
}
Vec2& operator/=(const float &d) {
assert(d!=0);
float inv = 1.f / d;
x *= inv;
y *= inv;
return *this;
}
// element-wise multiplication
Vec2 operator*(const Vec2 &v) const {
return Vec2(x * v.x, y * v.y);
}
Vec2& operator*=(const Vec2 &v) {
x *= v.x;
y *= v.y;
return *this;
}
bool operator==(const Vec2 &v) const {
return x == v.x && y == v.y;
}
bool operator!=(const Vec2 &v) const {
return x != v.x || y != v.y;
}
Vec2 operator-() const {
return Vec2(-x, -y);
}
// other functions
Vec2 normalize() {
assert(length() > 0);
float l = length();
x /= l;
y /= l;
return *this;
}
float lengthSquared() {
return x * x + y * y;
}
float length() {
return sqrt(lengthSquared());
}
float dot(const Vec2 &v) const {
return x * v.x + y * v.y;
}
Vec2 round(){
return Vec2((int)(x+0.5),(int)(y+0.5));
}
float cosAngle(const Vec2 &v)//using cos(a-b) = cosa*cosb+sina*sinb
{ float l=length();float lv=v.length();
if(l*lv<epsilon*epsilon)return 0.0;
return (x*v.x+y*v.y)/(l*lv);}
// void print(ostream& out){out<<x<<" "<<y;}
};
static std::ostream& operator<<(std::ostream& out,const Vec2 v){
out<<v.x<<" "<<v.y;
return out;
}
static std::istream& operator>>(std::istream& in,Vec2& v){
in>>v.x>>v.y;
return in;
}
static std::ostream& operator<<(std::ostream& out,const CvSize s){
out<<s.width<<" "<<s.height;
return out;
}
static float Dot(Vec3 a, Vec3 b)
{
return a.x*b.x + a.y*b.y + a.z*b.z;
}
static Vec3 Mul(Vec3 a, Vec3 b)
{
return vec3(a.y * b.z - a.z * b.y, a.z * b.x - a.x * b.z, a.x * b.y - a.y * b.x);
}
static Vec3 MulS(Vec3 a, float b)
{
return vec3(a.x * b, a.y * b, a.z * b);
}
static Vec3 Add(Vec3 a, Vec3 b)
{
return vec3(a.x + b.x, a.y + b.y, a.z + b.z);
}
static Vec3 AddS(Vec3 a, float b)
{
return vec3(a.x + b, a.y + b, a.z + b);
}
static Vec3 Sub(Vec3 a, Vec3 b)
{
return vec3(a.x - b.x, a.y - b.y, a.z - b.z);
}
static float Dot(Vec2 a, Vec2 b)
{
return a.x*b.x + a.y*b.y;
}
s
static Vec3 Normalize(Vec3 a)
{
float n = Norm(a);
return vec3(a.x/n, a.y/n, a.z/n);
}
static void gram_schmit(Vec3 v1, Vec3 v2, float *m)
{
//v3 = v1 x v2;
//e3 = v3 / |v3|;
//e2 = v2 - e3 * (v2 * e3) / |v2 - e3 * (v2 * e3)|;
//e1 = v1 - e3 * (v1 * e3) - e2*(v1*e2);
Vec3 v3 = Mul(v1, v2);
Vec3 e3 = Normalize(v3);
Vec3 e2 = Normalize(Sub(v2, MulS(e3, Dot(v2, e3))));
Vec3 e1 = Sub(Sub(v1, MulS(e3, Dot(v1, e3))), MulS(e2, Dot(v1, e2)));
m[0] = e1.x; m[1] = e2.x; m[2] = e3.x; m[3] = 0;
m[4] = e1.y; m[5] = e2.y; m[6] = e3.y; m[7] = 0;
m[8] = e1.z; m[9] = e2.z; m[10] = e3.z; m[11] = 0;
m[12] = 0; m[13] = 0; m[14] = 0; m[15] = 1;
}
static float *Transpose(float *m)
{
float *n = new float[16];
n[0] = m[0]; n[1] = m[4]; n[2] = m[8]; n[3] = m[3];
n[4] = m[1]; n[5] = m[5]; n[6] = m[9]; n[7] = m[7];
n[8] = m[2]; n[9] = m[6]; n[10] = m[10]; n[11] = m[11];
n[12] = m[12]; n[13] = m[13]; n[14] = m[14]; n[15] = m[15];
return n;
}
static float Angle(Vec3 a, Vec3 b)
{
// acos( a.b / |a|.|b|)
float phi = Dot(a,b) / (Norm(a)*Norm(b));
return acos(phi);
}
/*!
* Cubic interpolation class adapted from an algorithm by Marino 'Mutilate' Alge
* works with Vec2 and Vec3 types
*/
template <class T>
class CCubic
{
private:
T a, b, c, d;
public:
/*!
Constructor, takes 4 points as input for the interpolation
\param x0-x3 the input points for interpolation
*/
CCubic(T x0, T x1, T x2, T x3)
{
// a = x1;
// b = (x2 - x0)*0.5f;
// c = x0 - x1*2.5f + x2*2.0f - x3*0.5f;
// d = (x3 - x0)*0.5f - (x2 - x1)*1.5f;
a = x1;
b = MulS(Sub(x2,x0),0.5f);
c = Add(Sub(x0,MulS(x1,2.5f)),Sub(MulS(x2,2.0f),MulS(x3,0.5f)));
d = Sub(MulS(Sub(x3,x0),0.5f),MulS(Sub(x2,x1),1.5f));
}
/*!
The interpolation prediction function
\param t the input time
\return returns the value expected at time t
*/
T X(float t) const
{
//return a + (b + (c + d*t)*t)*t;
return Add(a,MulS(Add(b,MulS(Add(c,MulS(d,t)),t)),t));
}
/*!
All in one interpolation function
\param x0-x3 the input points for interpolation
\param t the input time
\return returns the value expected at time t, using points x0-x3 for interpolation
*/
static T X(T x0, T x1, T x2, T x3, float t)
{
//return x1 + ((x2 - x0)*0.5f + (x0 - x1*2.5f + x2*2.0f - x3*0.5f + (x3 - x0)*0.5f - (x2 - x1)*1.5f*t)*t)*t;
T asd7 = Sub(x0, MulS(x1,2.5f));
T asd6 = Sub(MulS(x2,2.0f), MulS(x3,0.5f));
T asd5 = Sub(MulS(Sub(x3,x0),0.5f), MulS(Sub(x2,x1),1.5f*t));
T asd4 = Add(asd5,Add(asd6,asd7));
T asd3 = MulS(asd4,t);
T asd2 = Add(MulS(Sub(x2,x0),0.5f), asd3);
T asd = MulS(asd2,t);
return Add(x1, asd);
}
};
/*!
Random Permutations array
\param length the length of the output array
\param seed random seed (optional) clock used otherwise
\return returns an array of indices between 0 and 'length' randomly sorted
*/
static u32 *randPerm(u32 length, s32 seed=-1)
{
u32 *perm = new u32[length];
memset(perm, 0, length*sizeof(u32));
u32 *usable = new u32[length];
FOR(i, length) usable[i] = i;
if(seed == -1) srand((u32)cvGetTickCount());
else srand((u32)seed);
u32 uLength = length;
for (register u32 i=0; i<length; i++)
{
register u32 r = ((rand()<< 7) + rand()) % uLength;
perm[i] = usable[r];
uLength--;
usable[r] = usable[uLength];
usable[uLength] = 0;
}
delete [] usable;
return perm;
}
// matlab code to generate the table
// erf(x) = (x>0?1:-1) * erftable((int)(min(6,abs(x))*100));
// 0:0.01:6
static const float erftable [] =
{
0.0000000000000000f, 0.0112834155558496f, 0.0225645746918449f, 0.0338412223417354f, 0.0451111061451247f, 0.0563719777970166f, 0.0676215943933084f, 0.0788577197708907f, 0.0900781258410182f, 0.1012805939146269f, 0.1124629160182849f, 0.1236228961994743f, 0.1347583518199201f, 0.1458671148356958f, 0.1569470330628558f, 0.1679959714273635f, 0.1790118131981057f, 0.1899924612018088f, 0.2009358390186958f, 0.2118398921577497f, 0.2227025892104785f, 0.2335219229821036f, 0.2442959115991287f, 0.2550225995922731f, 0.2657000589537920f, 0.2763263901682369f, 0.2868997232157491f, 0.2974182185470128f, 0.3078800680290340f, 0.3182834958609522f, 0.3286267594591273f, 0.3389081503107902f, 0.3491259947955827f, 0.3592786549743590f, 0.3693645293446587f, 0.3793820535623103f, 0.3893297011286642f, 0.3992059840429992f, 0.4090094534196940f, 0.4187387000697961f, 0.4283923550466685f, 0.4379690901554394f, 0.4474676184260253f, 0.4568866945495403f, 0.4662251152779575f, 0.4754817197869237f, 0.4846553900016797f, 0.4937450508860821f, 0.5027496706947650f, 0.5116682611885233f, 0.5204998778130465f, 0.5292436198411704f, 0.5378986304788544f, 0.5464640969351416f, 0.5549392504563904f, 0.5633233663251089f, 0.5716157638237684f, 0.5798158061639961f, 0.5879229003816007f, 0.5959364971979084f, 0.6038560908479259f, 0.6116812188758802f, 0.6194114618987212f, 0.6270464433381957f, 0.6345858291221413f, 0.6420293273556719f, 0.6493766879629542f, 0.6566277023003051f, 0.6637822027413580f, 0.6708400622350779f, 0.6778011938374186f, 0.6846655502174442f, 0.6914331231387512f, 0.6981039429170445f, 0.7046780778547458f, 0.7111556336535152f, 0.7175367528055909f, 0.7238216139648592f, 0.7300104312985789f, 0.7361034538206912f, 0.7421009647076605f, 0.7480032805977895f, 0.7538107508749625f, 0.7595237569377731f, 0.7651427114549946f, 0.7706680576083524f, 0.7761002683235567f, 0.7814398454905507f, 0.7866873191739325f, 0.7918432468144954f, 0.7969082124228322f, 0.8018828257659413f, 0.8067677215477618f, 0.8115635585845578f, 0.8162710189760625f, 0.8208908072732779f, 0.8254236496438183f, 0.8298702930356671f, 0.8342315043402079f, 0.8385080695553697f, 0.8427007929497148f, 0.8468104962282766f, 0.8508380177009420f, 0.8547842114541484f, 0.8586499465266515f, 0.8624361060900967f, 0.8661435866351080f, 0.8697732971635868f, 0.8733261583878896f, 0.8768031019375383f, 0.8802050695740817f, 0.8835330124147180f, 0.8867878901652547f, 0.8899706703629624f, 0.8930823276298567f, 0.8961238429369151f, 0.8990962028797120f, 0.9020003989659357f, 0.9048374269152169f, 0.9076082859716850f, 0.9103139782296355f, 0.9129555079726694f, 0.9155338810266469f, 0.9180501041267614f, 0.9205051842990297f, 0.9229001282564582f, 0.9252359418101295f, 0.9275136292954247f, 0.9297341930135782f, 0.9318986326887336f, 0.9340079449406524f, 0.9360631227731995f, 0.9380651550787114f, 0.9400150261583302f, 0.9419137152583653f, 0.9437621961227241f, 0.9455614365614331f, 0.9473123980352520f, 0.9490160352563626f, 0.9506732958050965f, 0.9522851197626489f, 0.9538524393597054f, 0.9553761786408961f, 0.9568572531449688f, 0.9582965696005648f, 0.9596950256374592f, 0.9610535095131181f, 0.9623728998544057f, 0.9636540654142689f, 0.9648978648432043f, 0.9661051464753108f, 0.9672767481287117f, 0.9684134969201232f, 0.9695162090933357f, 0.9705856898613637f, 0.9716227332620125f, 0.9726281220266002f, 0.9736026274615670f, 0.9745470093426969f, 0.9754620158216676f, 0.9763483833446440f, 0.9772068365826185f, 0.9780380883732035f, 0.9788428396735702f, 0.9796217795242320f, 0.9803755850233603f, 0.9811049213113221f, 0.9818104415651265f, 0.9824927870024649f, 0.9831525868950262f, 0.9837904585907746f, 0.9844070075448683f, 0.9850028273589058f, 0.9855784998281805f, 0.9861345949966329f, 0.9866716712191824f, 0.9871902752311301f, 0.9876909422243223f, 0.9881741959297683f, 0.9886405487064082f, 0.9890905016357308f, 0.9895245446219444f, 0.9899431564974077f, 0.9903468051330306f, 0.9907359475533626f, 0.9911110300560857f, 0.9914724883356396f, 0.9918207476107068f, 0.9921562227552937f, 0.9924793184331480f, 0.9927904292352574f, 0.9930899398201836f, 0.9933782250569847f, 0.9936556501704964f, 0.9939225708887325f, 0.9941793335921891f, 0.9944262754648279f, 0.9946637246465300f, 0.9948920003868136f, 0.9951114131996171f, 0.9953222650189527f, 0.9955248493552482f, 0.9957194514521921f, 0.9959063484439121f, 0.9960858095123195f, 0.9962580960444569f, 0.9964234617896959f, 0.9965821530166383f, 0.9967344086695764f, 0.9968804605243777f, 0.9970205333436670f, 0.9971548450311778f, 0.9972836067851606f, 0.9974070232507333f, 0.9975252926710697f, 0.9976386070373253f, 0.9977471522372077f, 0.9978511082021002f, 0.9979506490526588f, 0.9980459432428015f, 0.9981371537020181f, 0.9982244379759344f, 0.9983079483650648f, 0.9983878320616981f, 0.9984642312848625f, 0.9985372834133188f, 0.9986071211165417f, 0.9986738724836455f, 0.9987376611502190f, 0.9987986064230412f, 0.9988568234026434f, 0.9989124231037001f, 0.9989655125732240f, 0.9990161950065498f, 0.9990645698610920f, 0.9991107329678676f, 0.9991547766407751f, 0.9991967897836264f, 0.9992368579949287f, 0.9992750636704192f, 0.9993114861033550f, 0.9993462015825647f, 0.9993792834882711f, 0.9994108023856942f, 0.9994408261164486f, 0.9994694198877490f, 0.9994966463594419f, 0.9995225657288811f, 0.9995472358136659f, 0.9995707121322661f, 0.9995930479825550f, 0.9996142945182758f, 0.9996345008234653f, 0.9996537139848649f, 0.9996719791623431f, 0.9996893396573607f, 0.9997058369795080f, 0.9997215109111428f, 0.9997363995701628f, 0.9997505394709432f, 0.9997639655834707f, 0.9997767113907082f, 0.9997888089442237f, 0.9998002889181156f, 0.9998111806612684f, 0.9998215122479760f, 0.9998313105269614f, 0.9998406011688324f, 0.9998494087120056f, 0.9998577566071316f, 0.9998656672600594f, 0.9998731620733716f, 0.9998802614865254f, 0.9998869850146334f, 0.9998933512859194f, 0.9998993780778804f, 0.9999050823521898f, 0.9999104802883753f, 0.9999155873163016f, 0.9999204181474947f, 0.9999249868053346f, 0.9999293066541523f, 0.9999333904272598f, 0.9999372502539452f, 0.9999408976854610f, 0.9999443437200386f, 0.9999475988269556f, 0.9999506729696857f, 0.9999535756281590f, 0.9999563158201617f, 0.9999589021219005f, 0.9999613426877595f, 0.9999636452692755f, 0.9999658172333573f, 0.9999678655797740f, 0.9999697969579359f, 0.9999716176829931f, 0.9999733337512747f, 0.9999749508550908f, 0.9999764743969193f, 0.9999779095030015f, 0.9999792610363629f, 0.9999805336092855f, 0.9999817315952467f, 0.9999828591403461f, 0.9999839201742398f, 0.9999849184206001f, 0.9999858574071167f, 0.9999867404750594f, 0.9999875707884177f, 0.9999883513426329f, 0.9999890849729398f, 0.9999897743623336f, 0.9999904220491747f, 0.9999910304344468f, 0.9999916017886847f, 0.9999921382585810f, 0.9999926418732865f, 0.9999931145504183f, 0.9999935581017863f, 0.9999939742388482f, 0.9999943645779092f, 0.9999947306450711f, 0.9999950738809456f, 0.9999953956451422f, 0.9999956972205364f, 0.9999959798173321f, 0.9999962445769250f, 0.9999964925755764f, 0.9999967248279045f, 0.9999969422902035f, 0.9999971458635975f, 0.9999973363970345f, 0.9999975146901312f, 0.9999976814958739f, 0.9999978375231799f, 0.9999979834393308f, 0.9999981198722784f, 0.9999982474128331f, 0.9999983666167385f, 0.9999984780066371f, 0.9999985820739346f, 0.9999986792805644f, 0.9999987700606605f, 0.9999988548221410f, 0.9999989339482065f, 0.9999990077987595f, 0.9999990767117464f, 0.9999991410044279f, 0.9999992009745795f, 0.9999992569016276f, 0.9999993090477226f, 0.9999993576587528f, 0.9999994029653040f, 0.9999994451835634f, 0.9999994845161754f, 0.9999995211530479f, 0.9999995552721144f, 0.9999995870400529f, 0.9999996166129631f, 0.9999996441370069f, 0.9999996697490110f, 0.9999996935770344f, 0.9999997157409060f, 0.9999997363527273f, 0.9999997555173494f, 0.9999997733328196f, 0.9999997898908039f, 0.9999998052769828f, 0.9999998195714259f, 0.9999998328489421f, 0.9999998451794108f, 0.9999998566280922f, 0.9999998672559198f, 0.9999998771197746f, 0.9999998862727435f, 0.9999998947643614f, 0.9999999026408388f, 0.9999999099452765f, 0.9999999167178646f, 0.9999999229960725f, 0.9999999288148247f, 0.9999999342066670f, 0.9999999392019217f, 0.9999999438288334f, 0.9999999481137065f, 0.9999999520810322f, 0.9999999557536089f, 0.9999999591526549f, 0.9999999622979134f, 0.9999999652077514f, 0.9999999678992515f, 0.9999999703882987f, 0.9999999726896611f, 0.9999999748170654f, 0.9999999767832677f, 0.9999999786001196f, 0.9999999802786297f, 0.9999999818290218f, 0.9999999832607887f, 0.9999999845827421f, 0.9999999858030606f, 0.9999999869293328f, 0.9999999879685986f, 0.9999999889273877f, 0.9999999898117551f, 0.9999999906273142f, 0.9999999913792682f, 0.9999999920724392f, 0.9999999927112944f, 0.9999999932999724f, 0.9999999938423057f, 0.9999999943418427f, 0.9999999948018690f, 0.9999999952254246f, 0.9999999956153229f, 0.9999999959741669f, 0.9999999963043638f, 0.9999999966081397f, 0.9999999968875528f, 0.9999999971445058f, 0.9999999973807567f, 0.9999999975979301f, 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0.9999999999700474f, 0.9999999999727929f, 0.9999999999752915f, 0.9999999999775653f, 0.9999999999796336f, 0.9999999999815150f, 0.9999999999832258f, 0.9999999999847813f, 0.9999999999861953f, 0.9999999999874802f, 0.9999999999886479f, 0.9999999999897087f, 0.9999999999906721f, 0.9999999999915470f, 0.9999999999923415f, 0.9999999999930624f, 0.9999999999937168f, 0.9999999999943107f, 0.9999999999948495f, 0.9999999999953380f, 0.9999999999957810f, 0.9999999999961828f, 0.9999999999965470f, 0.9999999999968769f, 0.9999999999971760f, 0.9999999999974469f, 0.9999999999976923f, 0.9999999999979145f, 0.9999999999981156f, 0.9999999999982978f, 0.9999999999984626f, 0.9999999999986117f, 0.9999999999987466f, 0.9999999999988686f, 0.9999999999989789f, 0.9999999999990787f, 0.9999999999991689f, 0.9999999999992504f, 0.9999999999993240f, 0.9999999999993905f, 0.9999999999994507f, 0.9999999999995048f, 0.9999999999995539f, 0.9999999999995981f, 0.9999999999996381f, 0.9999999999996740f, 0.9999999999997065f, 0.9999999999997358f, 0.9999999999997622f, 0.9999999999997861f, 0.9999999999998075f, 0.9999999999998268f, 0.9999999999998444f, 0.9999999999998600f, 0.9999999999998741f, 0.9999999999998870f, 0.9999999999998983f, 0.9999999999999087f, 0.9999999999999181f, 0.9999999999999263f, 0.9999999999999338f, 0.9999999999999407f, 0.9999999999999467f, 0.9999999999999523f, 0.9999999999999572f, 0.9999999999999616f, 0.9999999999999655f, 0.9999999999999691f, 0.9999999999999722f, 0.9999999999999751f, 0.9999999999999778f, 0.9999999999999800f, 0.9999999999999821f, 0.9999999999999840f, 0.9999999999999857f, 0.9999999999999871f, 0.9999999999999885f, 0.9999999999999898f, 0.9999999999999908f, 0.9999999999999918f, 0.9999999999999927f, 0.9999999999999933f, 0.9999999999999941f, 0.9999999999999947f, 0.9999999999999953f, 0.9999999999999958f, 0.9999999999999962f, 0.9999999999999967f, 0.9999999999999970f, 0.9999999999999973f, 0.9999999999999976f, 0.9999999999999979f, 0.9999999999999981f, 0.9999999999999983f, 0.9999999999999985f, 0.9999999999999987f, 0.9999999999999988f, 0.9999999999999989f, 0.9999999999999991f, 0.9999999999999991f, 0.9999999999999992f, 0.9999999999999993f, 0.9999999999999993f, 0.9999999999999994f, 0.9999999999999996f, 0.9999999999999996f, 0.9999999999999996f, 0.9999999999999997f, 0.9999999999999998f, 0.9999999999999998f, 0.9999999999999998f, 0.9999999999999998f, 0.9999999999999998f, 0.9999999999999998f, 0.9999999999999998f, 0.9999999999999999f, 0.9999999999999999f, 0.9999999999999999f, 0.9999999999999999f, 1.0000000000000000f, 1.0000000000000000f, 1.0000000000000000f, 1.0000000000000000f, 1.0000000000000000f, 1.0000000000000000f, 1.0000000000000000f, 1.0000000000000000f, 1.0000000000000000f, 1.0000000000000000f, 1.0000000000000000f, 1.0000000000000000f
};
/*!
Gaussian Error Function
*/
static float erf(const float x)
{
if(x>0) return erftable[(int)(MIN(6,x)*100)];
return -erftable[(int)(MIN(6,-x)*100)];
}
#endif // _BASICMATH_H_

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