Tasmota/sonoff/support_float.ino

/*
  support_float.ino - support for Sonoff-Tasmota

  Copyright (C) 2019  Theo Arends

  This program is free software: you can redistribute it and/or modify
  it under the terms of the GNU General Public License as published by
  the Free Software Foundation, either version 3 of the License, or
  (at your option) any later version.

  This program is distributed in the hope that it will be useful,
  but WITHOUT ANY WARRANTY; without even the implied warranty of
  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  GNU General Public License for more details.

  You should have received a copy of the GNU General Public License
  along with this program.  If not, see <http://www.gnu.org/licenses/>.
*/

double FastPrecisePow(double a, double b)
{
  // https://martin.ankerl.com/2012/01/25/optimized-approximative-pow-in-c-and-cpp/
  // calculate approximation with fraction of the exponent
  int e = abs((int)b);
  union {
    double d;
    int x[2];
  } u = { a };
  u.x[1] = (int)((b - e) * (u.x[1] - 1072632447) + 1072632447);
  u.x[0] = 0;
  // exponentiation by squaring with the exponent's integer part
  // double r = u.d makes everything much slower, not sure why
  double r = 1.0;
  while (e) {
    if (e & 1) {
      r *= a;
    }
    a *= a;
    e >>= 1;
  }
  return r * u.d;
}

float FastPrecisePowf(const float x, const float y)
{
//  return (float)(pow((double)x, (double)y));
  return (float)FastPrecisePow(x, y);
}

double TaylorLog(double x)
{
  // https://stackoverflow.com/questions/46879166/finding-the-natural-logarithm-of-a-number-using-taylor-series-in-c

  if (x <= 0.0) { return NAN; }
  double z = (x + 1) / (x - 1);                              // We start from power -1, to make sure we get the right power in each iteration;
  double step = ((x - 1) * (x - 1)) / ((x + 1) * (x + 1));   // Store step to not have to calculate it each time
  double totalValue = 0;
  double powe = 1;
  double y;
  for (uint32_t count = 0; count < 10; count++) {                 // Experimental number of 10 iterations
    z *= step;
    y = (1 / powe) * z;
    totalValue = totalValue + y;
    powe = powe + 2;
  }
  totalValue *= 2;
/*
  char logxs[33];
  dtostrfd(x, 8, logxs);
  double log1 = log(x);
  char log1s[33];
  dtostrfd(log1, 8, log1s);
  char log2s[33];
  dtostrfd(totalValue, 8, log2s);
  AddLog_P2(LOG_LEVEL_DEBUG, PSTR("input %s, log %s, taylor %s"), logxs, log1s, log2s);
*/
  return totalValue;
}

// All code adapted from: http://www.ganssle.com/approx.htm

/// ========================================
//  The following code implements approximations to various trig functions.
//
//  This is demo code to guide developers in implementing their own approximation
// software. This code is merely meant to illustrate algorithms.

inline float sinf(float x) { return sin_52(x); }
inline float cosf(float x) { return cos_52(x); }
inline float tanf(float x) { return tan_56(x); }
inline float atanf(float x) { return atan_66(x); }
inline float asinf(float x) { return asinf1(x); }
inline float acosf(float x) { return acosf1(x); }
inline float sqrtf(float x) { return sqrt1(x); }
inline float powf(float x, float y) { return FastPrecisePow(x, y); }

// Math constants we'll use
double const f_pi=3.1415926535897932384626433;	// f_pi
double const f_twopi=2.0*f_pi;			// f_pi times 2
double const f_two_over_pi= 2.0/f_pi;		// 2/f_pi
double const f_halfpi=f_pi/2.0;			// f_pi divided by 2
double const f_threehalfpi=3.0*f_pi/2.0;  		// f_pi times 3/2, used in tan routines
double const f_four_over_pi=4.0/f_pi;		// 4/f_pi, used in tan routines
double const f_qtrpi=f_pi/4.0;			// f_pi/4.0, used in tan routines
double const f_sixthpi=f_pi/6.0;			// f_pi/6.0, used in atan routines
double const f_tansixthpi=tan(f_sixthpi);		// tan(f_pi/6), used in atan routines
double const f_twelfthpi=f_pi/12.0;			// f_pi/12.0, used in atan routines
double const f_tantwelfthpi=tan(f_twelfthpi);	// tan(f_pi/12), used in atan routines

// *********************************************************
// ***
// ***   Routines to compute sine and cosine to 5.2 digits
// ***  of accuracy.
// ***
// *********************************************************
//
//		cos_52s computes cosine (x)
//
//  Accurate to about 5.2 decimal digits over the range [0, f_pi/2].
//  The input argument is in radians.
//
//  Algorithm:
//		cos(x)= c1 + c2*x**2 + c3*x**4 + c4*x**6
//   which is the same as:
//		cos(x)= c1 + x**2(c2 + c3*x**2 + c4*x**4)
//		cos(x)= c1 + x**2(c2 + x**2(c3 + c4*x**2))
//
float cos_52s(float x)
{
const float c1= 0.9999932946;
const float c2=-0.4999124376;
const float c3= 0.0414877472;
const float c4=-0.0012712095;

float x2;							// The input argument squared

x2=x * x;
return (c1 + x2*(c2 + x2*(c3 + c4*x2)));
}

//
//  This is the main cosine approximation "driver"
// It reduces the input argument's range to [0, f_pi/2],
// and then calls the approximator.
// See the notes for an explanation of the range reduction.
//
float cos_52(float x){
	int quad;						// what quadrant are we in?

	x=fmodf(x, f_twopi);				// Get rid of values > 2* f_pi
	if(x<0)x=-x;					// cos(-x) = cos(x)
	quad=int(x * (float)f_two_over_pi);			// Get quadrant # (0 to 3) we're in
	switch (quad){
	case 0: return  cos_52s(x);
	case 1: return -cos_52s((float)f_pi-x);
	case 2: return -cos_52s(x-(float)f_pi);
	case 3: return  cos_52s((float)f_twopi-x);
	}
}
//
//   The sine is just cosine shifted a half-f_pi, so
// we'll adjust the argument and call the cosine approximation.
//
float sin_52(float x){
	return cos_52((float)f_halfpi-x);
}

// *********************************************************
// ***
// ***   Routines to compute tangent to 5.6 digits
// ***  of accuracy.
// ***
// *********************************************************
//
//		tan_56s computes tan(f_pi*x/4)
//
//  Accurate to about 5.6 decimal digits over the range [0, f_pi/4].
//  The input argument is in radians. Note that the function
//  computes tan(f_pi*x/4), NOT tan(x); it's up to the range
//  reduction algorithm that calls this to scale things properly.
//
//  Algorithm:
//		tan(x)= x(c1 + c2*x**2)/(c3 + x**2)
//
float tan_56s(float x)
{
const float c1=-3.16783027;
const float c2= 0.134516124;
const float c3=-4.033321984;

float x2;							// The input argument squared

x2=x * x;
return (x*(c1 + c2 * x2)/(c3 + x2));
}

//
//  This is the main tangent approximation "driver"
// It reduces the input argument's range to [0, f_pi/4],
// and then calls the approximator.
// See the notes for an explanation of the range reduction.
// Enter with positive angles only.
//
// WARNING: We do not test for the tangent approaching infinity,
// which it will at x=f_pi/2 and x=3*f_pi/2. If this is a problem
// in your application, take appropriate action.
//
float tan_56(float x){
	int octant;						// what octant are we in?

	x=fmodf(x, (float)f_twopi);				// Get rid of values >2 *f_pi
	octant=int(x * (float)f_four_over_pi);			// Get octant # (0 to 7)
	switch (octant){
	case 0: return       tan_56s(x                     *(float)f_four_over_pi);
	case 1: return  1.0f/tan_56s(((float)f_halfpi-x)     *(float)f_four_over_pi);
	case 2: return -1.0f/tan_56s((x-(float)f_halfpi)     *(float)f_four_over_pi);
	case 3: return -     tan_56s(((float)f_pi-x)         *(float)f_four_over_pi);
	case 4: return       tan_56s((x-(float)f_pi)         *(float)f_four_over_pi);
	case 5: return  1.0f/tan_56s(((float)f_threehalfpi-x)*(float)f_four_over_pi);
	case 6: return -1.0f/tan_56s((x-(float)f_threehalfpi)*(float)f_four_over_pi);
	case 7: return -     tan_56s(((float)f_twopi-x)      *(float)f_four_over_pi);
	}
}

// *********************************************************
// ***
// ***   Routines to compute arctangent to 6.6 digits
// ***  of accuracy.
// ***
// *********************************************************
//
//		atan_66s computes atan(x)
//
//  Accurate to about 6.6 decimal digits over the range [0, f_pi/12].
//
//  Algorithm:
//		atan(x)= x(c1 + c2*x**2)/(c3 + x**2)
//
float atan_66s(float x)
{
const float c1=1.6867629106;
const float c2=0.4378497304;
const float c3=1.6867633134;

float x2;							// The input argument squared

x2=x * x;
return (x*(c1 + x2*c2)/(c3 + x2));
}

//
//  This is the main arctangent approximation "driver"
// It reduces the input argument's range to [0, f_pi/12],
// and then calls the approximator.
//
//
float atan_66(float x){
  float y;							// return from atan__s function
  bool complement= false;				// true if arg was >1
  bool region= false;					// true depending on region arg is in
  bool sign= false;					// true if arg was < 0

  if (x <0 ){
  	x=-x;
  	sign=true;						// arctan(-x)=-arctan(x)
  }
  if (x > 1.0){
  	x=1.0/x;						// keep arg between 0 and 1
  	complement=true;
  }
  if (x > (float)f_tantwelfthpi){
  	x = (x-(float)f_tansixthpi)/(1+(float)f_tansixthpi*x);	// reduce arg to under tan(f_pi/12)
  	region=true;
  }

  y=atan_66s(x);						// run the approximation
  if (region) y+=(float)f_sixthpi;				// correct for region we're in
  if (complement)y=(float)f_halfpi-y;			// correct for 1/x if we did that
  if (sign)y=-y;						// correct for negative arg
  return (y);
}

float asinf1(float x) {
	float d = 1.0f - x*x;
	if (d < 0.0f) { return nanf(""); }
	return 2 * atan_66(x / (1 + sqrt1(d)));
}

float acosf1(float x) {
	float d = 1.0f - x*x;
	if (d < 0.0f) { return nanf(""); }
	float y = asinf1(sqrt1(d));
	if (x >= 0.0f) {
		return y;
	} else {
		return (float)f_pi - y;
	}
}

// https://www.codeproject.com/Articles/69941/Best-Square-Root-Method-Algorithm-Function-Precisi
float sqrt1(const float x)
{
  union
  {
    int i;
    float x;
  } u;
  u.x = x;
  u.i = (1<<29) + (u.i >> 1) - (1<<22);

  // Two Babylonian Steps (simplified from:)
  // u.x = 0.5f * (u.x + x/u.x);
  // u.x = 0.5f * (u.x + x/u.x);
  u.x =       u.x + x/u.x;
  u.x = 0.25f*u.x + x/u.x;

  return u.x;
}