/*************************************************************************/ /* math_funcs.h */ /*************************************************************************/ /* This file is part of: */ /* GODOT ENGINE */ /* https://godotengine.org */ /*************************************************************************/ /* Copyright (c) 2007-2022 Juan Linietsky, Ariel Manzur. */ /* Copyright (c) 2014-2022 Godot Engine contributors (cf. AUTHORS.md). */ /* */ /* Permission is hereby granted, free of charge, to any person obtaining */ /* a copy of this software and associated documentation files (the */ /* "Software"), to deal in the Software without restriction, including */ /* without limitation the rights to use, copy, modify, merge, publish, */ /* distribute, sublicense, and/or sell copies of the Software, and to */ /* permit persons to whom the Software is furnished to do so, subject to */ /* the following conditions: */ /* */ /* The above copyright notice and this permission notice shall be */ /* included in all copies or substantial portions of the Software. */ /* */ /* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, */ /* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF */ /* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.*/ /* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY */ /* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, */ /* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE */ /* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. */ /*************************************************************************/ #ifndef MATH_FUNCS_H #define MATH_FUNCS_H #include "core/math/math_defs.h" #include "core/math/random_pcg.h" #include "core/typedefs.h" #include "thirdparty/misc/pcg.h" #include #include class Math { static RandomPCG default_rand; public: Math() {} // useless to instance // Not using 'RANDOM_MAX' to avoid conflict with system headers on some OSes (at least NetBSD). static const uint64_t RANDOM_32BIT_MAX = 0xFFFFFFFF; static _ALWAYS_INLINE_ double sin(double p_x) { return ::sin(p_x); } static _ALWAYS_INLINE_ float sin(float p_x) { return ::sinf(p_x); } static _ALWAYS_INLINE_ double cos(double p_x) { return ::cos(p_x); } static _ALWAYS_INLINE_ float cos(float p_x) { return ::cosf(p_x); } static _ALWAYS_INLINE_ double tan(double p_x) { return ::tan(p_x); } static _ALWAYS_INLINE_ float tan(float p_x) { return ::tanf(p_x); } static _ALWAYS_INLINE_ double sinh(double p_x) { return ::sinh(p_x); } static _ALWAYS_INLINE_ float sinh(float p_x) { return ::sinhf(p_x); } static _ALWAYS_INLINE_ float sinc(float p_x) { return p_x == 0 ? 1 : ::sin(p_x) / p_x; } static _ALWAYS_INLINE_ double sinc(double p_x) { return p_x == 0 ? 1 : ::sin(p_x) / p_x; } static _ALWAYS_INLINE_ float sincn(float p_x) { return sinc((float)Math_PI * p_x); } static _ALWAYS_INLINE_ double sincn(double p_x) { return sinc(Math_PI * p_x); } static _ALWAYS_INLINE_ double cosh(double p_x) { return ::cosh(p_x); } static _ALWAYS_INLINE_ float cosh(float p_x) { return ::coshf(p_x); } static _ALWAYS_INLINE_ double tanh(double p_x) { return ::tanh(p_x); } static _ALWAYS_INLINE_ float tanh(float p_x) { return ::tanhf(p_x); } static _ALWAYS_INLINE_ double asin(double p_x) { return ::asin(p_x); } static _ALWAYS_INLINE_ float asin(float p_x) { return ::asinf(p_x); } static _ALWAYS_INLINE_ double acos(double p_x) { return ::acos(p_x); } static _ALWAYS_INLINE_ float acos(float p_x) { return ::acosf(p_x); } static _ALWAYS_INLINE_ double atan(double p_x) { return ::atan(p_x); } static _ALWAYS_INLINE_ float atan(float p_x) { return ::atanf(p_x); } static _ALWAYS_INLINE_ double atan2(double p_y, double p_x) { return ::atan2(p_y, p_x); } static _ALWAYS_INLINE_ float atan2(float p_y, float p_x) { return ::atan2f(p_y, p_x); } static _ALWAYS_INLINE_ double sqrt(double p_x) { return ::sqrt(p_x); } static _ALWAYS_INLINE_ float sqrt(float p_x) { return ::sqrtf(p_x); } static _ALWAYS_INLINE_ double fmod(double p_x, double p_y) { return ::fmod(p_x, p_y); } static _ALWAYS_INLINE_ float fmod(float p_x, float p_y) { return ::fmodf(p_x, p_y); } static _ALWAYS_INLINE_ double floor(double p_x) { return ::floor(p_x); } static _ALWAYS_INLINE_ float floor(float p_x) { return ::floorf(p_x); } static _ALWAYS_INLINE_ double ceil(double p_x) { return ::ceil(p_x); } static _ALWAYS_INLINE_ float ceil(float p_x) { return ::ceilf(p_x); } static _ALWAYS_INLINE_ double pow(double p_x, double p_y) { return ::pow(p_x, p_y); } static _ALWAYS_INLINE_ float pow(float p_x, float p_y) { return ::powf(p_x, p_y); } static _ALWAYS_INLINE_ double log(double p_x) { return ::log(p_x); } static _ALWAYS_INLINE_ float log(float p_x) { return ::logf(p_x); } static _ALWAYS_INLINE_ double log1p(double p_x) { return ::log1p(p_x); } static _ALWAYS_INLINE_ float log1p(float p_x) { return ::log1pf(p_x); } static _ALWAYS_INLINE_ double log2(double p_x) { return ::log2(p_x); } static _ALWAYS_INLINE_ float log2(float p_x) { return ::log2f(p_x); } static _ALWAYS_INLINE_ double exp(double p_x) { return ::exp(p_x); } static _ALWAYS_INLINE_ float exp(float p_x) { return ::expf(p_x); } static _ALWAYS_INLINE_ bool is_nan(double p_val) { #ifdef _MSC_VER return _isnan(p_val); #elif defined(__GNUC__) && __GNUC__ < 6 union { uint64_t u; double f; } ieee754; ieee754.f = p_val; // (unsigned)(0x7ff0000000000001 >> 32) : 0x7ff00000 return ((((unsigned)(ieee754.u >> 32) & 0x7fffffff) + ((unsigned)ieee754.u != 0)) > 0x7ff00000); #else return isnan(p_val); #endif } static _ALWAYS_INLINE_ bool is_nan(float p_val) { #ifdef _MSC_VER return _isnan(p_val); #elif defined(__GNUC__) && __GNUC__ < 6 union { uint32_t u; float f; } ieee754; ieee754.f = p_val; // ----------------------------------- // (single-precision floating-point) // NaN : s111 1111 1xxx xxxx xxxx xxxx xxxx xxxx // : (> 0x7f800000) // where, // s : sign // x : non-zero number // ----------------------------------- return ((ieee754.u & 0x7fffffff) > 0x7f800000); #else return isnan(p_val); #endif } static _ALWAYS_INLINE_ bool is_inf(double p_val) { #ifdef _MSC_VER return !_finite(p_val); // use an inline implementation of isinf as a workaround for problematic libstdc++ versions from gcc 5.x era #elif defined(__GNUC__) && __GNUC__ < 6 union { uint64_t u; double f; } ieee754; ieee754.f = p_val; return ((unsigned)(ieee754.u >> 32) & 0x7fffffff) == 0x7ff00000 && ((unsigned)ieee754.u == 0); #else return isinf(p_val); #endif } static _ALWAYS_INLINE_ bool is_inf(float p_val) { #ifdef _MSC_VER return !_finite(p_val); // use an inline implementation of isinf as a workaround for problematic libstdc++ versions from gcc 5.x era #elif defined(__GNUC__) && __GNUC__ < 6 union { uint32_t u; float f; } ieee754; ieee754.f = p_val; return (ieee754.u & 0x7fffffff) == 0x7f800000; #else return isinf(p_val); #endif } static _ALWAYS_INLINE_ double abs(double g) { return absd(g); } static _ALWAYS_INLINE_ float abs(float g) { return absf(g); } static _ALWAYS_INLINE_ int abs(int g) { return g > 0 ? g : -g; } static _ALWAYS_INLINE_ double fposmod(double p_x, double p_y) { double value = Math::fmod(p_x, p_y); if (((value < 0) && (p_y > 0)) || ((value > 0) && (p_y < 0))) { value += p_y; } value += 0.0; return value; } static _ALWAYS_INLINE_ float fposmod(float p_x, float p_y) { float value = Math::fmod(p_x, p_y); if (((value < 0) && (p_y > 0)) || ((value > 0) && (p_y < 0))) { value += p_y; } value += 0.0f; return value; } static _ALWAYS_INLINE_ float fposmodp(float p_x, float p_y) { float value = Math::fmod(p_x, p_y); if (value < 0) { value += p_y; } value += 0.0f; return value; } static _ALWAYS_INLINE_ double fposmodp(double p_x, double p_y) { double value = Math::fmod(p_x, p_y); if (value < 0) { value += p_y; } value += 0.0; return value; } static _ALWAYS_INLINE_ int64_t posmod(int64_t p_x, int64_t p_y) { int64_t value = p_x % p_y; if (((value < 0) && (p_y > 0)) || ((value > 0) && (p_y < 0))) { value += p_y; } return value; } static _ALWAYS_INLINE_ double deg_to_rad(double p_y) { return p_y * (Math_PI / 180.0); } static _ALWAYS_INLINE_ float deg_to_rad(float p_y) { return p_y * (float)(Math_PI / 180.0); } static _ALWAYS_INLINE_ double rad_to_deg(double p_y) { return p_y * (180.0 / Math_PI); } static _ALWAYS_INLINE_ float rad_to_deg(float p_y) { return p_y * (float)(180.0 / Math_PI); } static _ALWAYS_INLINE_ double lerp(double p_from, double p_to, double p_weight) { return p_from + (p_to - p_from) * p_weight; } static _ALWAYS_INLINE_ float lerp(float p_from, float p_to, float p_weight) { return p_from + (p_to - p_from) * p_weight; } static _ALWAYS_INLINE_ double cubic_interpolate(double p_from, double p_to, double p_pre, double p_post, double p_weight) { return 0.5 * ((p_from * 2.0) + (-p_pre + p_to) * p_weight + (2.0 * p_pre - 5.0 * p_from + 4.0 * p_to - p_post) * (p_weight * p_weight) + (-p_pre + 3.0 * p_from - 3.0 * p_to + p_post) * (p_weight * p_weight * p_weight)); } static _ALWAYS_INLINE_ float cubic_interpolate(float p_from, float p_to, float p_pre, float p_post, float p_weight) { return 0.5f * ((p_from * 2.0f) + (-p_pre + p_to) * p_weight + (2.0f * p_pre - 5.0f * p_from + 4.0f * p_to - p_post) * (p_weight * p_weight) + (-p_pre + 3.0f * p_from - 3.0f * p_to + p_post) * (p_weight * p_weight * p_weight)); } static _ALWAYS_INLINE_ double cubic_interpolate_in_time(double p_from, double p_to, double p_pre, double p_post, double p_weight, double p_to_t, double p_pre_t, double p_post_t) { /* Barry-Goldman method */ double t = Math::lerp(0.0, p_to_t, p_weight); double a1 = Math::lerp(p_pre, p_from, p_pre_t == 0 ? 0.0 : (t - p_pre_t) / -p_pre_t); double a2 = Math::lerp(p_from, p_to, p_to_t == 0 ? 0.5 : t / p_to_t); double a3 = Math::lerp(p_to, p_post, p_post_t - p_to_t == 0 ? 1.0 : (t - p_to_t) / (p_post_t - p_to_t)); double b1 = Math::lerp(a1, a2, p_to_t - p_pre_t == 0 ? 0.0 : (t - p_pre_t) / (p_to_t - p_pre_t)); double b2 = Math::lerp(a2, a3, p_post_t == 0 ? 1.0 : t / p_post_t); return Math::lerp(b1, b2, p_to_t == 0 ? 0.5 : t / p_to_t); } static _ALWAYS_INLINE_ float cubic_interpolate_in_time(float p_from, float p_to, float p_pre, float p_post, float p_weight, float p_to_t, float p_pre_t, float p_post_t) { /* Barry-Goldman method */ float t = Math::lerp(0.0f, p_to_t, p_weight); float a1 = Math::lerp(p_pre, p_from, p_pre_t == 0 ? 0.0f : (t - p_pre_t) / -p_pre_t); float a2 = Math::lerp(p_from, p_to, p_to_t == 0 ? 0.5f : t / p_to_t); float a3 = Math::lerp(p_to, p_post, p_post_t - p_to_t == 0 ? 1.0f : (t - p_to_t) / (p_post_t - p_to_t)); float b1 = Math::lerp(a1, a2, p_to_t - p_pre_t == 0 ? 0.0f : (t - p_pre_t) / (p_to_t - p_pre_t)); float b2 = Math::lerp(a2, a3, p_post_t == 0 ? 1.0f : t / p_post_t); return Math::lerp(b1, b2, p_to_t == 0 ? 0.5f : t / p_to_t); } static _ALWAYS_INLINE_ double bezier_interpolate(double p_start, double p_control_1, double p_control_2, double p_end, double p_t) { /* Formula from Wikipedia article on Bezier curves. */ double omt = (1.0 - p_t); double omt2 = omt * omt; double omt3 = omt2 * omt; double t2 = p_t * p_t; double t3 = t2 * p_t; return p_start * omt3 + p_control_1 * omt2 * p_t * 3.0 + p_control_2 * omt * t2 * 3.0 + p_end * t3; } static _ALWAYS_INLINE_ float bezier_interpolate(float p_start, float p_control_1, float p_control_2, float p_end, float p_t) { /* Formula from Wikipedia article on Bezier curves. */ float omt = (1.0f - p_t); float omt2 = omt * omt; float omt3 = omt2 * omt; float t2 = p_t * p_t; float t3 = t2 * p_t; return p_start * omt3 + p_control_1 * omt2 * p_t * 3.0f + p_control_2 * omt * t2 * 3.0f + p_end * t3; } static _ALWAYS_INLINE_ double lerp_angle(double p_from, double p_to, double p_weight) { double difference = fmod(p_to - p_from, Math_TAU); double distance = fmod(2.0 * difference, Math_TAU) - difference; return p_from + distance * p_weight; } static _ALWAYS_INLINE_ float lerp_angle(float p_from, float p_to, float p_weight) { float difference = fmod(p_to - p_from, (float)Math_TAU); float distance = fmod(2.0f * difference, (float)Math_TAU) - difference; return p_from + distance * p_weight; } static _ALWAYS_INLINE_ double inverse_lerp(double p_from, double p_to, double p_value) { return (p_value - p_from) / (p_to - p_from); } static _ALWAYS_INLINE_ float inverse_lerp(float p_from, float p_to, float p_value) { return (p_value - p_from) / (p_to - p_from); } static _ALWAYS_INLINE_ double range_lerp(double p_value, double p_istart, double p_istop, double p_ostart, double p_ostop) { return Math::lerp(p_ostart, p_ostop, Math::inverse_lerp(p_istart, p_istop, p_value)); } static _ALWAYS_INLINE_ float range_lerp(float p_value, float p_istart, float p_istop, float p_ostart, float p_ostop) { return Math::lerp(p_ostart, p_ostop, Math::inverse_lerp(p_istart, p_istop, p_value)); } static _ALWAYS_INLINE_ double smoothstep(double p_from, double p_to, double p_s) { if (is_equal_approx(p_from, p_to)) { return p_from; } double s = CLAMP((p_s - p_from) / (p_to - p_from), 0.0, 1.0); return s * s * (3.0 - 2.0 * s); } static _ALWAYS_INLINE_ float smoothstep(float p_from, float p_to, float p_s) { if (is_equal_approx(p_from, p_to)) { return p_from; } float s = CLAMP((p_s - p_from) / (p_to - p_from), 0.0f, 1.0f); return s * s * (3.0f - 2.0f * s); } static _ALWAYS_INLINE_ double move_toward(double p_from, double p_to, double p_delta) { return abs(p_to - p_from) <= p_delta ? p_to : p_from + SIGN(p_to - p_from) * p_delta; } static _ALWAYS_INLINE_ float move_toward(float p_from, float p_to, float p_delta) { return abs(p_to - p_from) <= p_delta ? p_to : p_from + SIGN(p_to - p_from) * p_delta; } static _ALWAYS_INLINE_ double linear_to_db(double p_linear) { return Math::log(p_linear) * 8.6858896380650365530225783783321; } static _ALWAYS_INLINE_ float linear_to_db(float p_linear) { return Math::log(p_linear) * (float)8.6858896380650365530225783783321; } static _ALWAYS_INLINE_ double db_to_linear(double p_db) { return Math::exp(p_db * 0.11512925464970228420089957273422); } static _ALWAYS_INLINE_ float db_to_linear(float p_db) { return Math::exp(p_db * (float)0.11512925464970228420089957273422); } static _ALWAYS_INLINE_ double round(double p_val) { return ::round(p_val); } static _ALWAYS_INLINE_ float round(float p_val) { return ::roundf(p_val); } static _ALWAYS_INLINE_ int64_t wrapi(int64_t value, int64_t min, int64_t max) { int64_t range = max - min; return range == 0 ? min : min + ((((value - min) % range) + range) % range); } static _ALWAYS_INLINE_ double wrapf(double value, double min, double max) { double range = max - min; double result = is_zero_approx(range) ? min : value - (range * Math::floor((value - min) / range)); if (is_equal_approx(result, max)) { return min; } return result; } static _ALWAYS_INLINE_ float wrapf(float value, float min, float max) { float range = max - min; float result = is_zero_approx(range) ? min : value - (range * Math::floor((value - min) / range)); if (is_equal_approx(result, max)) { return min; } return result; } static _ALWAYS_INLINE_ float fract(float value) { return value - floor(value); } static _ALWAYS_INLINE_ double fract(double value) { return value - floor(value); } static _ALWAYS_INLINE_ float pingpong(float value, float length) { return (length != 0.0f) ? abs(fract((value - length) / (length * 2.0f)) * length * 2.0f - length) : 0.0f; } static _ALWAYS_INLINE_ double pingpong(double value, double length) { return (length != 0.0) ? abs(fract((value - length) / (length * 2.0)) * length * 2.0 - length) : 0.0; } // double only, as these functions are mainly used by the editor and not performance-critical, static double ease(double p_x, double p_c); static int step_decimals(double p_step); static int range_step_decimals(double p_step); // For editor use only. static double snapped(double p_value, double p_step); static uint32_t larger_prime(uint32_t p_val); static void seed(uint64_t x); static void randomize(); static uint32_t rand_from_seed(uint64_t *seed); static uint32_t rand(); static _ALWAYS_INLINE_ double randd() { return (double)rand() / (double)Math::RANDOM_32BIT_MAX; } static _ALWAYS_INLINE_ float randf() { return (float)rand() / (float)Math::RANDOM_32BIT_MAX; } static double randfn(double mean, double deviation); static double random(double from, double to); static float random(float from, float to); static int random(int from, int to); static _ALWAYS_INLINE_ bool is_equal_approx(float a, float b) { // Check for exact equality first, required to handle "infinity" values. if (a == b) { return true; } // Then check for approximate equality. float tolerance = (float)CMP_EPSILON * abs(a); if (tolerance < (float)CMP_EPSILON) { tolerance = (float)CMP_EPSILON; } return abs(a - b) < tolerance; } static _ALWAYS_INLINE_ bool is_equal_approx(float a, float b, float tolerance) { // Check for exact equality first, required to handle "infinity" values. if (a == b) { return true; } // Then check for approximate equality. return abs(a - b) < tolerance; } static _ALWAYS_INLINE_ bool is_zero_approx(float s) { return abs(s) < (float)CMP_EPSILON; } static _ALWAYS_INLINE_ bool is_equal_approx(double a, double b) { // Check for exact equality first, required to handle "infinity" values. if (a == b) { return true; } // Then check for approximate equality. double tolerance = CMP_EPSILON * abs(a); if (tolerance < CMP_EPSILON) { tolerance = CMP_EPSILON; } return abs(a - b) < tolerance; } static _ALWAYS_INLINE_ bool is_equal_approx(double a, double b, double tolerance) { // Check for exact equality first, required to handle "infinity" values. if (a == b) { return true; } // Then check for approximate equality. return abs(a - b) < tolerance; } static _ALWAYS_INLINE_ bool is_zero_approx(double s) { return abs(s) < CMP_EPSILON; } static _ALWAYS_INLINE_ float absf(float g) { union { float f; uint32_t i; } u; u.f = g; u.i &= 2147483647u; return u.f; } static _ALWAYS_INLINE_ double absd(double g) { union { double d; uint64_t i; } u; u.d = g; u.i &= (uint64_t)9223372036854775807ll; return u.d; } // This function should be as fast as possible and rounding mode should not matter. static _ALWAYS_INLINE_ int fast_ftoi(float a) { // Assuming every supported compiler has `lrint()`. return lrintf(a); } static _ALWAYS_INLINE_ uint32_t halfbits_to_floatbits(uint16_t h) { uint16_t h_exp, h_sig; uint32_t f_sgn, f_exp, f_sig; h_exp = (h & 0x7c00u); f_sgn = ((uint32_t)h & 0x8000u) << 16; switch (h_exp) { case 0x0000u: /* 0 or subnormal */ h_sig = (h & 0x03ffu); /* Signed zero */ if (h_sig == 0) { return f_sgn; } /* Subnormal */ h_sig <<= 1; while ((h_sig & 0x0400u) == 0) { h_sig <<= 1; h_exp++; } f_exp = ((uint32_t)(127 - 15 - h_exp)) << 23; f_sig = ((uint32_t)(h_sig & 0x03ffu)) << 13; return f_sgn + f_exp + f_sig; case 0x7c00u: /* inf or NaN */ /* All-ones exponent and a copy of the significand */ return f_sgn + 0x7f800000u + (((uint32_t)(h & 0x03ffu)) << 13); default: /* normalized */ /* Just need to adjust the exponent and shift */ return f_sgn + (((uint32_t)(h & 0x7fffu) + 0x1c000u) << 13); } } static _ALWAYS_INLINE_ float halfptr_to_float(const uint16_t *h) { union { uint32_t u32; float f32; } u; u.u32 = halfbits_to_floatbits(*h); return u.f32; } static _ALWAYS_INLINE_ float half_to_float(const uint16_t h) { return halfptr_to_float(&h); } static _ALWAYS_INLINE_ uint16_t make_half_float(float f) { union { float fv; uint32_t ui; } ci; ci.fv = f; uint32_t x = ci.ui; uint32_t sign = (unsigned short)(x >> 31); uint32_t mantissa; uint32_t exponent; uint16_t hf; // get mantissa mantissa = x & ((1 << 23) - 1); // get exponent bits exponent = x & (0xFF << 23); if (exponent >= 0x47800000) { // check if the original single precision float number is a NaN if (mantissa && (exponent == (0xFF << 23))) { // we have a single precision NaN mantissa = (1 << 23) - 1; } else { // 16-bit half-float representation stores number as Inf mantissa = 0; } hf = (((uint16_t)sign) << 15) | (uint16_t)((0x1F << 10)) | (uint16_t)(mantissa >> 13); } // check if exponent is <= -15 else if (exponent <= 0x38000000) { /* // store a denorm half-float value or zero exponent = (0x38000000 - exponent) >> 23; mantissa >>= (14 + exponent); hf = (((uint16_t)sign) << 15) | (uint16_t)(mantissa); */ hf = 0; //denormals do not work for 3D, convert to zero } else { hf = (((uint16_t)sign) << 15) | (uint16_t)((exponent - 0x38000000) >> 13) | (uint16_t)(mantissa >> 13); } return hf; } static _ALWAYS_INLINE_ float snap_scalar(float p_offset, float p_step, float p_target) { return p_step != 0 ? Math::snapped(p_target - p_offset, p_step) + p_offset : p_target; } static _ALWAYS_INLINE_ float snap_scalar_separation(float p_offset, float p_step, float p_target, float p_separation) { if (p_step != 0) { float a = Math::snapped(p_target - p_offset, p_step + p_separation) + p_offset; float b = a; if (p_target >= 0) { b -= p_separation; } else { b += p_step; } return (Math::abs(p_target - a) < Math::abs(p_target - b)) ? a : b; } return p_target; } }; #endif // MATH_FUNCS_H