irrMath.h

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// Copyright (C) 2002-2012 Nikolaus Gebhardt
// This file is part of the "Irrlicht Engine".
// For conditions of distribution and use, see copyright notice in irrlicht.h

#ifndef __IRR_MATH_H_INCLUDED__
#define __IRR_MATH_H_INCLUDED__

#include "IrrCompileConfig.h"
#include "irrTypes.h"
#include <math.h>
#include <float.h>
#include <stdlib.h> // for abs() etc.
#include <limits.h> // For INT_MAX / UINT_MAX

#if defined(_IRR_SOLARIS_PLATFORM_) || defined(__BORLANDC__) || defined (__BCPLUSPLUS__) || defined (_WIN32_WCE)
        #define sqrtf(X) (irr::f32)sqrt((irr::f64)(X))
        #define sinf(X) (irr::f32)sin((irr::f64)(X))
        #define cosf(X) (irr::f32)cos((irr::f64)(X))
        #define asinf(X) (irr::f32)asin((irr::f64)(X))
        #define acosf(X) (irr::f32)acos((irr::f64)(X))
        #define atan2f(X,Y) (irr::f32)atan2((irr::f64)(X),(irr::f64)(Y))
        #define ceilf(X) (irr::f32)ceil((irr::f64)(X))
        #define floorf(X) (irr::f32)floor((irr::f64)(X))
        #define powf(X,Y) (irr::f32)pow((irr::f64)(X),(irr::f64)(Y))
        #define fmodf(X,Y) (irr::f32)fmod((irr::f64)(X),(irr::f64)(Y))
        #define fabsf(X) (irr::f32)fabs((irr::f64)(X))
        #define logf(X) (irr::f32)log((irr::f64)(X))
#endif

#ifndef FLT_MAX
#define FLT_MAX 3.402823466E+38F
#endif

#ifndef FLT_MIN
#define FLT_MIN 1.17549435e-38F
#endif

namespace irr
{
namespace core
{


        const s32 ROUNDING_ERROR_S32 = 0;

#ifdef __IRR_HAS_S64
        const s64 ROUNDING_ERROR_S64 = 0;
#endif
        const f32 ROUNDING_ERROR_f32 = 0.000001f;
        const f64 ROUNDING_ERROR_f64 = 0.00000001;

#ifdef PI // make sure we don't collide with a define
#undef PI
#endif
        const f32 PI = 3.14159265359f;

        const f32 RECIPROCAL_PI = 1.0f/PI;

        const f32 HALF_PI = PI/2.0f;

#ifdef PI64 // make sure we don't collide with a define
#undef PI64
#endif
        const f64 PI64 = 3.1415926535897932384626433832795028841971693993751;

        const f64 RECIPROCAL_PI64 = 1.0/PI64;

        const f32 DEGTORAD = PI / 180.0f;

        const f32 RADTODEG   = 180.0f / PI;

        const f64 DEGTORAD64 = PI64 / 180.0;

        const f64 RADTODEG64 = 180.0 / PI64;


        inline f32 radToDeg(f32 radians)
        {
                return RADTODEG * radians;
        }


        inline f64 radToDeg(f64 radians)
        {
                return RADTODEG64 * radians;
        }


        inline f32 degToRad(f32 degrees)
        {
                return DEGTORAD * degrees;
        }


        inline f64 degToRad(f64 degrees)
        {
                return DEGTORAD64 * degrees;
        }

        template<class T>
        inline const T& min_(const T& a, const T& b)
        {
                return a < b ? a : b;
        }

        template<class T>
        inline const T& min_(const T& a, const T& b, const T& c)
        {
                return a < b ? min_(a, c) : min_(b, c);
        }

        template<class T>
        inline const T& max_(const T& a, const T& b)
        {
                return a < b ? b : a;
        }

        template<class T>
        inline const T& max_(const T& a, const T& b, const T& c)
        {
                return a < b ? max_(b, c) : max_(a, c);
        }

        template<class T>
        inline T abs_(const T& a)
        {
                return a < (T)0 ? -a : a;
        }

        template<class T>
        inline T lerp(const T& a, const T& b, const f32 t)
        {
                return (T)(a*(1.f-t)) + (b*t);
        }

        template <class T>
        inline const T clamp (const T& value, const T& low, const T& high)
        {
                return min_ (max_(value,low), high);
        }

        // Note: We use the same trick as boost and use two template arguments to
        // avoid ambiguity when swapping objects of an Irrlicht type that has not
        // it's own swap overload. Otherwise we get conflicts with some compilers
        // in combination with stl.
        template <class T1, class T2>
        inline void swap(T1& a, T2& b)
        {
                T1 c(a);
                a = b;
                b = c;
        }

        template <class T>
        inline T roundingError();

        template <>
        inline f32 roundingError()
        {
                return ROUNDING_ERROR_f32;
        }

        template <>
        inline f64 roundingError()
        {
                return ROUNDING_ERROR_f64;
        }

        template <>
        inline s32 roundingError()
        {
                return ROUNDING_ERROR_S32;
        }

        template <>
        inline u32 roundingError()
        {
                return ROUNDING_ERROR_S32;
        }

#ifdef __IRR_HAS_S64
        template <>
        inline s64 roundingError()
        {
                return ROUNDING_ERROR_S64;
        }

        template <>
        inline u64 roundingError()
        {
                return ROUNDING_ERROR_S64;
        }
#endif

        template <class T>
        inline T relativeErrorFactor()
        {
                return 1;
        }

        template <>
        inline f32 relativeErrorFactor()
        {
                return 4;
        }

        template <>
        inline f64 relativeErrorFactor()
        {
                return 8;
        }

        template <class T>
        inline bool equals(const T a, const T b, const T tolerance = roundingError<T>()) 
        {
                return (a + tolerance >= b) && (a - tolerance <= b);
        }


        template <class T>
        inline bool equalsRelative( const T a, const T b, const T factor = relativeErrorFactor<T>())
        {
                //https://eagergames.wordpress.com/2017/04/01/fast-parallel-lines-and-vectors-test/

                const T maxi = max_( a, b);
                const T mini = min_( a, b);
                const T maxMagnitude = max_( maxi, -mini);

                return  (maxMagnitude*factor + maxi) == (maxMagnitude*factor + mini); // MAD Wise
        }

        union FloatIntUnion32
        {
                FloatIntUnion32(float f1 = 0.0f) : f(f1) {}
                // Portable sign-extraction
                bool sign() const { return (i >> 31) != 0; }

                irr::s32 i;
                irr::f32 f;
        };

        //\result true when numbers have a ULP <= maxUlpDiff AND have the same sign.
        inline bool equalsByUlp(f32 a, f32 b, int maxUlpDiff)
        {
                // Based on the ideas and code from Bruce Dawson on
                // http://www.altdevblogaday.com/2012/02/22/comparing-floating-point-numbers-2012-edition/
                // When floats are interpreted as integers the two nearest possible float numbers differ just
                // by one integer number. Also works the other way round, an integer of 1 interpreted as float
                // is for example the smallest possible float number.

                const FloatIntUnion32 fa(a);
                const FloatIntUnion32 fb(b);

                // Different signs, we could maybe get difference to 0, but so close to 0 using epsilons is better.
                if ( fa.sign() != fb.sign() )
                {
                        // Check for equality to make sure +0==-0
                        if (fa.i == fb.i)
                                return true;
                        return false;
                }

                // Find the difference in ULPs.
                const int ulpsDiff = abs_(fa.i- fb.i);
                if (ulpsDiff <= maxUlpDiff)
                        return true;

                return false;
        }

        inline bool iszero(const f64 a, const f64 tolerance = ROUNDING_ERROR_f64)
        {
                return fabs(a) <= tolerance;
        }

        inline bool iszero(const f32 a, const f32 tolerance = ROUNDING_ERROR_f32)
        {
                return fabsf(a) <= tolerance;
        }

        inline bool isnotzero(const f32 a, const f32 tolerance = ROUNDING_ERROR_f32)
        {
                return fabsf(a) > tolerance;
        }

        inline bool iszero(const s32 a, const s32 tolerance = 0)
        {
                return ( a & 0x7ffffff ) <= tolerance;
        }

        inline bool iszero(const u32 a, const u32 tolerance = 0)
        {
                return a <= tolerance;
        }

#ifdef __IRR_HAS_S64
        inline bool iszero(const s64 a, const s64 tolerance = 0)
        {
                return abs_(a) <= tolerance;
        }
#endif

        inline s32 s32_min(s32 a, s32 b)
        {
                const s32 mask = (a - b) >> 31;
                return (a & mask) | (b & ~mask);
        }

        inline s32 s32_max(s32 a, s32 b)
        {
                const s32 mask = (a - b) >> 31;
                return (b & mask) | (a & ~mask);
        }

        inline s32 s32_clamp (s32 value, s32 low, s32 high)
        {
                return s32_min(s32_max(value,low), high);
        }

        /*
                float IEEE-754 bit representation

                0      0x00000000
                1.0    0x3f800000
                0.5    0x3f000000
                3      0x40400000
                +inf   0x7f800000
                -inf   0xff800000
                +NaN   0x7fc00000 or 0x7ff00000
                in general: number = (sign ? -1:1) * 2^(exponent) * 1.(mantissa bits)
        */

        typedef union { u32 u; s32 s; f32 f; } inttofloat;

        #define F32_AS_S32(f)           (*((s32 *) &(f)))
        #define F32_AS_U32(f)           (*((u32 *) &(f)))
        #define F32_AS_U32_POINTER(f)   ( ((u32 *) &(f)))

        #define F32_VALUE_0             0x00000000
        #define F32_VALUE_1             0x3f800000
        #define F32_SIGN_BIT            0x80000000U
        #define F32_EXPON_MANTISSA      0x7FFFFFFFU

#ifdef IRRLICHT_FAST_MATH
        #define IR(x)                   ((u32&)(x))
#else
        inline u32 IR(f32 x) {inttofloat tmp; tmp.f=x; return tmp.u;}
#endif

        #define AIR(x)                  (IR(x)&0x7fffffff)

#ifdef IRRLICHT_FAST_MATH
        #define FR(x)                   ((f32&)(x))
#else
        inline f32 FR(u32 x) {inttofloat tmp; tmp.u=x; return tmp.f;}
        inline f32 FR(s32 x) {inttofloat tmp; tmp.s=x; return tmp.f;}
#endif

        #define IEEE_1_0                0x3f800000
        #define IEEE_255_0              0x437f0000

#ifdef IRRLICHT_FAST_MATH
        #define F32_LOWER_0(f)          (F32_AS_U32(f) >  F32_SIGN_BIT)
        #define F32_LOWER_EQUAL_0(f)    (F32_AS_S32(f) <= F32_VALUE_0)
        #define F32_GREATER_0(f)        (F32_AS_S32(f) >  F32_VALUE_0)
        #define F32_GREATER_EQUAL_0(f)  (F32_AS_U32(f) <= F32_SIGN_BIT)
        #define F32_EQUAL_1(f)          (F32_AS_U32(f) == F32_VALUE_1)
        #define F32_EQUAL_0(f)          ( (F32_AS_U32(f) & F32_EXPON_MANTISSA ) == F32_VALUE_0)

        // only same sign
        #define F32_A_GREATER_B(a,b)    (F32_AS_S32((a)) > F32_AS_S32((b)))

#else

        #define F32_LOWER_0(n)          ((n) <  0.0f)
        #define F32_LOWER_EQUAL_0(n)    ((n) <= 0.0f)
        #define F32_GREATER_0(n)        ((n) >  0.0f)
        #define F32_GREATER_EQUAL_0(n)  ((n) >= 0.0f)
        #define F32_EQUAL_1(n)          ((n) == 1.0f)
        #define F32_EQUAL_0(n)          ((n) == 0.0f)
        #define F32_A_GREATER_B(a,b)    ((a) > (b))
#endif


#ifndef REALINLINE
        #ifdef _MSC_VER
                #define REALINLINE __forceinline
        #else
                #define REALINLINE inline
        #endif
#endif

#if defined(__BORLANDC__) || defined (__BCPLUSPLUS__)

        // 8-bit bools in Borland builder

        REALINLINE u32 if_c_a_else_b ( const c8 condition, const u32 a, const u32 b )
        {
                return ( ( -condition >> 7 ) & ( a ^ b ) ) ^ b;
        }

        REALINLINE u32 if_c_a_else_0 ( const c8 condition, const u32 a )
        {
                return ( -condition >> 31 ) & a;
        }
#else

        REALINLINE u32 if_c_a_else_b ( const s32 condition, const u32 a, const u32 b )
        {
                return ( ( -condition >> 31 ) & ( a ^ b ) ) ^ b;
        }

        REALINLINE u16 if_c_a_else_b ( const s16 condition, const u16 a, const u16 b )
        {
                return ( ( -condition >> 15 ) & ( a ^ b ) ) ^ b;
        }

        REALINLINE u32 if_c_a_else_0 ( const s32 condition, const u32 a )
        {
                return ( -condition >> 31 ) & a;
        }
#endif

        /*
                if (condition) state |= m; else state &= ~m;
        */
        REALINLINE void setbit_cond ( u32 &state, s32 condition, u32 mask )
        {
                // 0, or any positive to mask
                //s32 conmask = -condition >> 31;
                state ^= ( ( -condition >> 31 ) ^ state ) & mask;
        }

        // NOTE: This is not as exact as the c99/c++11 round function, especially at high numbers starting with 8388609
        //       (only low number which seems to go wrong is 0.49999997 which is rounded to 1)
        //      Also negative 0.5 is rounded up not down unlike with the standard function (p.E. input -0.5 will be 0 and not -1)
        inline f32 round_( f32 x )
        {
                return floorf( x + 0.5f );
        }

        // calculate: sqrt ( x )
        REALINLINE f32 squareroot(const f32 f)
        {
                return sqrtf(f);
        }

        // calculate: sqrt ( x )
        REALINLINE f64 squareroot(const f64 f)
        {
                return sqrt(f);
        }

        // calculate: sqrt ( x )
        REALINLINE s32 squareroot(const s32 f)
        {
                return static_cast<s32>(squareroot(static_cast<f32>(f)));
        }

#ifdef __IRR_HAS_S64
        // calculate: sqrt ( x )
        REALINLINE s64 squareroot(const s64 f)
        {
                return static_cast<s64>(squareroot(static_cast<f64>(f)));
        }
#endif

        // calculate: 1 / sqrt ( x )
        REALINLINE f64 reciprocal_squareroot(const f64 x)
        {
                return 1.0 / sqrt(x);
        }

        // calculate: 1 / sqrtf ( x )
        REALINLINE f32 reciprocal_squareroot(const f32 f)
        {
#if defined ( IRRLICHT_FAST_MATH )
                // NOTE: Unlike comment below says I found inaccuracies already at 4'th significant bit.
                // p.E: Input 1, expected 1, got 0.999755859

        #if defined(_MSC_VER) && !defined(_WIN64)
                // SSE reciprocal square root estimate, accurate to 12 significant
                // bits of the mantissa
                f32 recsqrt;
                __asm rsqrtss xmm0, f           // xmm0 = rsqrtss(f)
                __asm movss recsqrt, xmm0       // return xmm0
                return recsqrt;

/*
                // comes from Nvidia
                u32 tmp = (u32(IEEE_1_0 << 1) + IEEE_1_0 - *(u32*)&x) >> 1;
                f32 y = *(f32*)&tmp;
                return y * (1.47f - 0.47f * x * y * y);
*/
        #else
                return 1.f / sqrtf(f);
        #endif
#else // no fast math
                return 1.f / sqrtf(f);
#endif
        }

        // calculate: 1 / sqrtf( x )
        REALINLINE s32 reciprocal_squareroot(const s32 x)
        {
                return static_cast<s32>(reciprocal_squareroot(static_cast<f32>(x)));
        }

        // calculate: 1 / x
        REALINLINE f32 reciprocal( const f32 f )
        {
#if defined (IRRLICHT_FAST_MATH)
                // NOTE: Unlike with 1.f / f the values very close to 0 return -nan instead of inf

                // SSE Newton-Raphson reciprocal estimate, accurate to 23 significant
                // bi ts of the mantissa
                // One Newton-Raphson Iteration:
                // f(i+1) = 2 * rcpss(f) - f * rcpss(f) * rcpss(f)
#if defined(_MSC_VER) && !defined(_WIN64)
                f32 rec;
                __asm rcpss xmm0, f               // xmm0 = rcpss(f)
                __asm movss xmm1, f               // xmm1 = f
                __asm mulss xmm1, xmm0            // xmm1 = f * rcpss(f)
                __asm mulss xmm1, xmm0            // xmm2 = f * rcpss(f) * rcpss(f)
                __asm addss xmm0, xmm0            // xmm0 = 2 * rcpss(f)
                __asm subss xmm0, xmm1            // xmm0 = 2 * rcpss(f)
                                                                                  //        - f * rcpss(f) * rcpss(f)
                __asm movss rec, xmm0             // return xmm0
                return rec;
#else // no support yet for other compilers
                return 1.f / f;
#endif
                // instead set f to a high value to get a return value near zero..
                // -1000000000000.f.. is use minus to stay negative..
                // must test's here (plane.normal dot anything ) checks on <= 0.f
                //u32 x = (-(AIR(f) != 0 ) >> 31 ) & ( IR(f) ^ 0xd368d4a5 ) ^ 0xd368d4a5;
                //return 1.f / FR ( x );

#else // no fast math
                return 1.f / f;
#endif
        }

        // calculate: 1 / x
        REALINLINE f64 reciprocal ( const f64 f )
        {
                return 1.0 / f;
        }


        // calculate: 1 / x, low precision allowed
        REALINLINE f32 reciprocal_approxim ( const f32 f )
        {
#if defined( IRRLICHT_FAST_MATH)

                // SSE Newton-Raphson reciprocal estimate, accurate to 23 significant
                // bi ts of the mantissa
                // One Newton-Raphson Iteration:
                // f(i+1) = 2 * rcpss(f) - f * rcpss(f) * rcpss(f)
#if defined(_MSC_VER) && !defined(_WIN64)
                f32 rec;
                __asm rcpss xmm0, f               // xmm0 = rcpss(f)
                __asm movss xmm1, f               // xmm1 = f
                __asm mulss xmm1, xmm0            // xmm1 = f * rcpss(f)
                __asm mulss xmm1, xmm0            // xmm2 = f * rcpss(f) * rcpss(f)
                __asm addss xmm0, xmm0            // xmm0 = 2 * rcpss(f)
                __asm subss xmm0, xmm1            // xmm0 = 2 * rcpss(f)
                                                                                  //        - f * rcpss(f) * rcpss(f)
                __asm movss rec, xmm0             // return xmm0
                return rec;
#else // no support yet for other compilers
                return 1.f / f;
#endif

/*
                // SSE reciprocal estimate, accurate to 12 significant bits of
                f32 rec;
                __asm rcpss xmm0, f             // xmm0 = rcpss(f)
                __asm movss rec , xmm0          // return xmm0
                return rec;
*/
/*
                register u32 x = 0x7F000000 - IR ( p );
                const f32 r = FR ( x );
                return r * (2.0f - p * r);
*/
#else // no fast math
                return 1.f / f;
#endif
        }


        REALINLINE s32 floor32(f32 x)
        {
                return (s32) floorf ( x );
        }

        REALINLINE s32 ceil32 ( f32 x )
        {
                return (s32) ceilf ( x );
        }

        // NOTE: Please check round_ documentation about some inaccuracies in this compared to standard library round function.
        REALINLINE s32 round32(f32 x)
        {
                return (s32) round_(x);
        }

        inline f32 f32_max3(const f32 a, const f32 b, const f32 c)
        {
                return a > b ? (a > c ? a : c) : (b > c ? b : c);
        }

        inline f32 f32_min3(const f32 a, const f32 b, const f32 c)
        {
                return a < b ? (a < c ? a : c) : (b < c ? b : c);
        }

        inline f32 fract ( f32 x )
        {
                return x - floorf ( x );
        }

} // end namespace core
} // end namespace irr

#ifndef IRRLICHT_FAST_MATH
        using irr::core::IR;
        using irr::core::FR;
#endif

#endif