approx_atan2.h

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#ifndef DataFormatsMathAPPROX_ATAN2_H
#define DataFormatsMathAPPROX_ATAN2_H

/*
 * approximate atan2 evaluations
 *
 * Polynomials were obtained using Sollya scripts (in comments below)
 *
 *
*/

/*
f= atan((1-x)/(1+x))-atan(1);
I=[-1+10^(-4);1.0];
filename="atan.txt";
print("") > filename;
for deg from 3 to 11 do begin
  p = fpminimax(f, deg,[|1,23...|],I, floating, absolute); 
  display=decimal;
  acc=floor(-log2(sup(supnorm(p, f, I, absolute, 2^(-20)))));
  print( "   // degree = ", deg, 
         "  => absolute accuracy is ",  acc, "bits" ) >> filename;
  print("template<> constexpr float approx_atan2f_P<", deg, ">(float x){") >> filename;
  display=hexadecimal;
  print(" return ", horner(p) , ";") >> filename;
  print("}") >> filename;
end;
*/

#include <cstdint>
#include <cmath>
#include <limits>
#include <algorithm>

// float

template <int DEGREE>
constexpr float approx_atan2f_P(float x);

// degree =  3   => absolute accuracy is  7 bits
template <>
constexpr float approx_atan2f_P<3>(float x) {
  return x * (float(-0xf.8eed2p-4) + x * x * float(0x3.1238p-4));
}

// degree =  5   => absolute accuracy is  10 bits
template <>
constexpr float approx_atan2f_P<5>(float x) {
  auto z = x * x;
  return x * (float(-0xf.ecfc8p-4) + z * (float(0x4.9e79dp-4) + z * float(-0x1.44f924p-4)));
}

// degree =  7   => absolute accuracy is  13 bits
template <>
constexpr float approx_atan2f_P<7>(float x) {
  auto z = x * x;
  return x * (float(-0xf.fcc7ap-4) + z * (float(0x5.23886p-4) + z * (float(-0x2.571968p-4) + z * float(0x9.fb05p-8))));
}

// degree =  9   => absolute accuracy is  16 bits
template <>
constexpr float approx_atan2f_P<9>(float x) {
  auto z = x * x;
  return x * (float(-0xf.ff73ep-4) +
              z * (float(0x5.48ee1p-4) +
                   z * (float(-0x2.e1efe8p-4) + z * (float(0x1.5cce54p-4) + z * float(-0x5.56245p-8)))));
}

// degree =  11   => absolute accuracy is  19 bits
template <>
constexpr float approx_atan2f_P<11>(float x) {
  auto z = x * x;
  return x * (float(-0xf.ffe82p-4) +
              z * (float(0x5.526c8p-4) +
                   z * (float(-0x3.18bea8p-4) +
                        z * (float(0x1.dce3bcp-4) + z * (float(-0xd.7a64ap-8) + z * float(0x3.000eap-8))))));
}

// degree =  13   => absolute accuracy is  21 bits
template <>
constexpr float approx_atan2f_P<13>(float x) {
  auto z = x * x;
  return x * (float(-0xf.fffbep-4) +
              z * (float(0x5.54adp-4) +
                   z * (float(-0x3.2b4df8p-4) +
                        z * (float(0x2.1df79p-4) +
                             z * (float(-0x1.46081p-4) + z * (float(0x8.99028p-8) + z * float(-0x1.be0bc4p-8)))))));
}

// degree =  15   => absolute accuracy is  24 bits
template <>
constexpr float approx_atan2f_P<15>(float x) {
  auto z = x * x;
  return x * (float(-0xf.ffff4p-4) +
              z * (float(0x5.552f9p-4 + z * (float(-0x3.30f728p-4) +
                                             z * (float(0x2.39826p-4) +
                                                  z * (float(-0x1.8a880cp-4) +
                                                       z * (float(0xe.484d6p-8) +
                                                            z * (float(-0x5.93d5p-8) + z * float(0x1.0875dcp-8)))))))));
}

template <int DEGREE>
constexpr float unsafe_atan2f_impl(float y, float x) {
  constexpr float pi4f = 3.1415926535897932384626434 / 4;
  constexpr float pi34f = 3.1415926535897932384626434 * 3 / 4;

  auto r = (std::abs(x) - std::abs(y)) / (std::abs(x) + std::abs(y));
  if (x < 0)
    r = -r;

  auto angle = (x >= 0) ? pi4f : pi34f;
  angle += approx_atan2f_P<DEGREE>(r);

  return ((y < 0)) ? -angle : angle;
}

template <int DEGREE>
constexpr float unsafe_atan2f(float y, float x) {
  return unsafe_atan2f_impl<DEGREE>(y, x);
}

template <int DEGREE>
constexpr float safe_atan2f(float y, float x) {
  return unsafe_atan2f_impl<DEGREE>(y, ((y == 0.f) & (x == 0.f)) ? 0.2f : x);
  // return (y==0.f)&(x==0.f) ? 0.f :  unsafe_atan2f_impl<DEGREE>( y, x);
}

// integer...
/*
  f= (2^31/pi)*(atan((1-x)/(1+x))-atan(1));
  I=[-1+10^(-4);1.0];
  p = fpminimax(f, [|1,3,5,7,9,11|],[|23...|],I, floating, absolute);
 */

template <int DEGREE>
constexpr float approx_atan2i_P(float x);

// degree =  3   => absolute accuracy is  6*10^6
template <>
constexpr float approx_atan2i_P<3>(float x) {
  auto z = x * x;
  return x * (-664694912.f + z * 131209024.f);
}

// degree =  5   => absolute accuracy is  4*10^5
template <>
constexpr float approx_atan2i_P<5>(float x) {
  auto z = x * x;
  return x * (-680392064.f + z * (197338400.f + z * (-54233256.f)));
}

// degree =  7   => absolute accuracy is  6*10^4
template <>
constexpr float approx_atan2i_P<7>(float x) {
  auto z = x * x;
  return x * (-683027840.f + z * (219543904.f + z * (-99981040.f + z * 26649684.f)));
}

// degree =  9   => absolute accuracy is  8000
template <>
constexpr float approx_atan2i_P<9>(float x) {
  auto z = x * x;
  return x * (-683473920.f + z * (225785056.f + z * (-123151184.f + z * (58210592.f + z * (-14249276.f)))));
}

// degree =  11   => absolute accuracy is  1000
template <>
constexpr float approx_atan2i_P<11>(float x) {
  auto z = x * x;
  return x *
         (-683549696.f + z * (227369312.f + z * (-132297008.f + z * (79584144.f + z * (-35987016.f + z * 8010488.f)))));
}

// degree =  13   => absolute accuracy is  163
template <>
constexpr float approx_atan2i_P<13>(float x) {
  auto z = x * x;
  return x * (-683562624.f +
              z * (227746080.f +
                   z * (-135400128.f + z * (90460848.f + z * (-54431464.f + z * (22973256.f + z * (-4657049.f)))))));
}

template <>
constexpr float approx_atan2i_P<15>(float x) {
  auto z = x * x;
  return x * (-683562624.f +
              z * (227746080.f +
                   z * (-135400128.f + z * (90460848.f + z * (-54431464.f + z * (22973256.f + z * (-4657049.f)))))));
}

template <int DEGREE>
constexpr int unsafe_atan2i_impl(float y, float x) {
  constexpr long long maxint = (long long)(std::numeric_limits<int>::max()) + 1LL;
  constexpr int pi4 = int(maxint / 4LL);
  constexpr int pi34 = int(3LL * maxint / 4LL);

  auto r = (std::abs(x) - std::abs(y)) / (std::abs(x) + std::abs(y));
  if (x < 0)
    r = -r;

  auto angle = (x >= 0) ? pi4 : pi34;
  angle += int(approx_atan2i_P<DEGREE>(r));
  // angle += int(std::round(approx_atan2i_P<DEGREE>(r)));

  return (y < 0) ? -angle : angle;
}

template <int DEGREE>
constexpr int unsafe_atan2i(float y, float x) {
  return unsafe_atan2i_impl<DEGREE>(y, x);
}

// short (16bits)

template <int DEGREE>
constexpr float approx_atan2s_P(float x);

// degree =  3   => absolute accuracy is  53
template <>
constexpr float approx_atan2s_P<3>(float x) {
  auto z = x * x;
  return x * ((-10142.439453125f) + z * 2002.0908203125f);
}
// degree =  5   => absolute accuracy is  7
template <>
constexpr float approx_atan2s_P<5>(float x) {
  auto z = x * x;
  return x * ((-10381.9609375f) + z * ((3011.1513671875f) + z * (-827.538330078125f)));
}
// degree =  7   => absolute accuracy is  2
template <>
constexpr float approx_atan2s_P<7>(float x) {
  auto z = x * x;
  return x * ((-10422.177734375f) + z * (3349.97412109375f + z * ((-1525.589599609375f) + z * 406.64190673828125f)));
}
// degree =  9   => absolute accuracy is 1
template <>
constexpr float approx_atan2s_P<9>(float x) {
  auto z = x * x;
  return x * ((-10428.984375f) + z * (3445.20654296875f + z * ((-1879.137939453125f) +
                                                               z * (888.22314453125f + z * (-217.42669677734375f)))));
}

template <int DEGREE>
constexpr short unsafe_atan2s_impl(float y, float x) {
  constexpr int maxshort = (int)(std::numeric_limits<short>::max()) + 1;
  constexpr short pi4 = short(maxshort / 4);
  constexpr short pi34 = short(3 * maxshort / 4);

  auto r = (std::abs(x) - std::abs(y)) / (std::abs(x) + std::abs(y));
  if (x < 0)
    r = -r;

  auto angle = (x >= 0) ? pi4 : pi34;
  angle += short(approx_atan2s_P<DEGREE>(r));

  return (y < 0) ? -angle : angle;
}

template <int DEGREE>
constexpr short unsafe_atan2s(float y, float x) {
  return unsafe_atan2s_impl<DEGREE>(y, x);
}

constexpr int phi2int(float x) {
  constexpr float p2i = ((long long)(std::numeric_limits<int>::max()) + 1LL) / M_PI;
  return std::round(x * p2i);
}

constexpr float int2phi(int x) {
  constexpr float i2p = M_PI / ((long long)(std::numeric_limits<int>::max()) + 1LL);
  return float(x) * i2p;
}

constexpr double int2dphi(int x) {
  constexpr double i2p = M_PI / ((long long)(std::numeric_limits<int>::max()) + 1LL);
  return x * i2p;
}

constexpr short phi2short(float x) {
  constexpr float p2i = ((int)(std::numeric_limits<short>::max()) + 1) / M_PI;
  return std::round(x * p2i);
}

constexpr float short2phi(short x) {
  constexpr float i2p = M_PI / ((int)(std::numeric_limits<short>::max()) + 1);
  return float(x) * i2p;
}

#endif