/* Correctly-rounded half revolution arctangent function of two binary16 values.

Copyright (c) 2022-2025 Alexei Sibidanov and Paul Zimmermann

This file is part of the CORE-MATH project
(https://core-math.gitlabpages.inria.fr/).

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.

*/

/* This code is based on the binary32 code atan2pif.c:
   at input we convert the inputs to _Float16 (exactly),
   we then use the same code than for atan2pif,
   and at output we round to _Float16.
   The changes with respect to atan2pif.c are marked with a comment
   "specific atan2pif16". */

#include <stdint.h>
#ifdef CORE_MATH_SUPPORT_ERRNO
#include <errno.h>
#endif
#include <math.h> // only used for performance tests

// Warning: clang also defines __GNUC__
#if defined(__GNUC__) && !defined(__clang__)
#pragma GCC diagnostic ignored "-Wunknown-pragmas"
#endif

#pragma STDC FENV_ACCESS ON

typedef union {float f; uint32_t u;} b32u32_u;
typedef union {double f; uint64_t u;} b64u64_u;

static inline double muldd(double xh, double xl, double ch, double cl, double *l){
  double ahlh = ch*xl, alhh = cl*xh, ahhh = ch*xh, ahhl = __builtin_fma(ch, xh, -ahhh);
  ahhl += alhh + ahlh;
  ch = ahhh + ahhl;
  *l = (ahhh - ch) + ahhl;
  return ch;
}

static double polydd(double xh, double xl, int n, const double c[][2], double *l){
  int i = n-1;
  double ch = c[i][0], cl = c[i][1];
  while(--i>=0){
    ch = muldd(xh,xl,ch,cl,&cl);
    double th = ch + c[i][0], tl = (c[i][0] - th) + ch;
    ch = th;
    cl += tl + c[i][1];
  }
  *l = cl;
  return ch;
}

// specific atan2pif16: inputs name changed from y,x to yf16,xf16
_Float16 cr_atan2pif16(_Float16 yf16, _Float16 xf16){
  float y = yf16, x = xf16; // specific atan2pif16
  static const double cn[] =
    {0x1.45f306dc9c883p-2, 0x1.988d83a142adap-1, 0x1.747bebf492057p-1, 0x1.2cc5645094ff3p-2,
     0x1.a0521c711ab66p-5, 0x1.881b8058b9a0dp-9, 0x1.b16ff514a0afp-16};
  static const double cd[] =
    {0x1p+0, 0x1.6b8b143a3f6dap+1, 0x1.8421201d18ed5p+1, 0x1.8221d086914ebp+0, 0x1.670657e3a07bap-2,
     0x1.0f4951fd1e72dp-5, 0x1.b3874b8798286p-11};
  static const double m[] = {0, 1};
  static const double off[] = {0.0f, 0.5f, 1.0f, 0.5f, -0.0f, -0.5f, -1.0f, -0.5f};
  static const float sgnf[] = {1,-1};
  static const double sgn[] = {1,-1};
  b32u32_u tx = {.f = x}, ty = {.f = y};
  uint32_t ux = tx.u, uy = ty.u, ax = ux&(~0u>>1), ay = uy&(~0u>>1);
  if(__builtin_expect(ay >= (0xff<<23)||ax >= (0xff<<23), 0)){
    if(ay > (0xff<<23)) return x + y; // nan
    if(ax > (0xff<<23)) return x + y; // nan
    uint32_t yinf = ay==(0xff<<23), xinf = ax==(0xff<<23);
    if(yinf&xinf){
      if(ux>>31)
	return 0.75f*sgnf[uy>>31];
      else
	return 0.25f*sgnf[uy>>31];
    }
    if(xinf){
      if(ux>>31)
	return sgnf[uy>>31];
      else
	return 0.0f*sgnf[uy>>31];
    }
    if(yinf){
      return 0.5f*sgnf[uy>>31];
    }
  }
  if(__builtin_expect(ay==0, 0)){
    if(__builtin_expect(!(ay|ax),0)){
      uint32_t i = (uy>>31)*4 + (ux>>31)*2;
      return off[i];
    }
    if(!(ux>>31)) return 0.0f*sgnf[uy>>31];
  }
  if(__builtin_expect(ax==ay, 0)){
    static const float s[] = {0.25,0.75,-0.25,-0.75};
    uint32_t i = (uy>>31)*2 + (ux>>31);
    return s[i];
  }
  uint32_t gt = ay>ax, i = (uy>>31)*4 + (ux>>31)*2 + gt;
  
  double zx = x, zy = y;
  double z = (m[gt]*zx + m[1-gt]*zy)/(m[gt]*zy + m[1-gt]*zx);
  double r = cn[0], z2 = z*z;
  z *= sgn[gt];
  // avoid spurious underflow in the polynomial evaluation excluding extremely small arguments
  if(__builtin_expect(z2>0x1p-54, 1)){
    double z4 = z2*z2, z8 = z4*z4;
    double cn0 =     r + z2*cn[1];
    double cn2 = cn[2] + z2*cn[3];
    double cn4 = cn[4] + z2*cn[5];
    double cn6 = cn[6];
    cn0 += z4*cn2;
    cn4 += z4*cn6;
    cn0 += z8*cn4;
    double cd0 = cd[0] + z2*cd[1];
    double cd2 = cd[2] + z2*cd[3];
    double cd4 = cd[4] + z2*cd[5];
    double cd6 = cd[6];
    cd0 += z4*cd2;
    cd4 += z4*cd6;
    cd0 += z8*cd4;
    r = cn0/cd0;
  }
  r = z*r + off[i];
  b64u64_u res = {.f = r};
  if(__builtin_expect((res.u<<1) > 0x6d40000000000000 && ((res.u + 8)&0xfffffff) <= 16, 0)){
    // |res| > 0x1p-149
    if(ax==ay) {
      static const double off2[] = {0.25, 0.75, -0.25, -0.75};
      r = off2[(uy>>31)*2 + (ux>>31)];
    } else {
      double zh,zl;
      if(!gt){
	zh = zy/zx;
	zl = __builtin_fma(zh,-zx,zy)/zx;
      } else {
	zh = zx/zy;
	zl = __builtin_fma(zh,-zy,zx)/zy;
      }
      double z2l, z2h = muldd(zh,zl,zh,zl,&z2l);
      static const double c[][2] =
	{{0x1.45f306dc9c883p-2, -0x1.6b01ec5513324p-56}, {-0x1.b2995e7b7b604p-4, 0x1.e402b0c13eedcp-58},
	 {0x1.04c26be3b06cfp-4, -0x1.571d178a53efp-60}, {-0x1.7483758e69c03p-5, 0x1.819a6ed7aaf38p-63},
	 {0x1.21bb9452523ffp-5, -0x1.234d866fb9807p-60}, {-0x1.da1bace3cc54ep-6, -0x1.c84f6ada49294p-64},
	 {0x1.912b1c23345ddp-6, -0x1.534890fbc165p-60}, {-0x1.5bade52f5f52ap-6, 0x1.f783bafc832f6p-60},
	 {0x1.32c69d084c5cp-6, 0x1.042d155953025p-60}, {-0x1.127bcfb3e8c7dp-6, -0x1.85aae199a7b6bp-60},
	 {0x1.f0af43b11a731p-7, 0x1.8f0356356663p-61}, {-0x1.c57e86801029ep-7, 0x1.dcdf3e3b38eb4p-61},
	 {0x1.a136408617ea1p-7, 0x1.a71affb36c6c4p-63}, {-0x1.824ac7814ba37p-7, 0x1.8928b295c0898p-61},
	 {0x1.6794e32ea5471p-7, 0x1.0b4334fb41e63p-61}, {-0x1.501d57f643d97p-7, 0x1.516785bf1376ep-61},
	 {0x1.3adf02ff2400ap-7, -0x1.b0e30bb8c8076p-62}, {-0x1.267702f94faap-7, -0x1.7a4d3a1850cc6p-62},
	 {0x1.10dce97099686p-7, 0x1.fcc208eee2571p-61}, {-0x1.eee49cdad8002p-8, -0x1.9109b3f1bab82p-64},
	 {0x1.af93bc191a929p-8, 0x1.069fd3b47d7bp-62}, {-0x1.6240751b54675p-8, -0x1.72dc8cfd03b6fp-62},
	 {0x1.0b61e84080884p-8, 0x1.825824c80941bp-63}, {-0x1.6a72a8a74e3a5p-9, 0x1.8786a82fd117ep-63},
	 {0x1.aede3217d939dp-10, -0x1.93b626982e1fep-68}, {-0x1.b66568f09ebeep-11, -0x1.704a39121d0a5p-66},
	 {0x1.73af3977fa973p-12, -0x1.aa050e2244ea3p-68}, {-0x1.fc69d85ed28c9p-14, 0x1.867f17b764cap-68},
	 {0x1.0c883a9270162p-15, -0x1.6842833896dd9p-70}, {-0x1.9a0b27b6dfe15p-18, 0x1.427fc2f4e1327p-73},
	 {0x1.91e15e7ab5bdcp-21, -0x1.730dbc6279d0dp-77}, {-0x1.7b1119c1ff867p-25, 0x1.145f9980759c4p-79}};
      double pl, ph = polydd(z2h, z2l, 32, c, &pl);
      zh *= sgn[gt];
      zl *= sgn[gt];
      ph = muldd(zh,zl,ph,pl,&pl);
      double sh = ph + off[i], sl = ((off[i] - sh) + ph) + pl;
      float rf = sh;
      double th = rf, dh = sh - th, tm = dh + sl;
      r = th + tm;
      b64u64_u d = {.f = r - th};
      if( !(d.u<<12) ) {
	double ad = __builtin_fabs(d.f), am = __builtin_fabs(tm);
	if(ad > am) r -= d.f*0x1p-10;
	if(ad < am) r += d.f*0x1p-10;
      }
    }
  }
  _Float16 rf = r; // specific atan2pif16
#ifdef CORE_MATH_SUPPORT_ERRNO
  // specific atan2pif16: changed underflow threshold from 0x1p-126 to 0x1p-14
  int underflow = __builtin_fabsf (rf) < 0x1p-14f;
  // we also have underflow when |r| <= 0x1.ffcp-15 (for all roundings)
  if (__builtin_fabs (r) <= 0x1.ffcp-15)
    underflow = 1;
  /* and when |r| < 0x1.ffep-15 for rounding to nearest (for rounding
     toward zero we have |rf| < 0x1p-14 in that case thus underflow was already
     set above).
     Since the only missed case above is when |rf| = 0x1p-14, it suffices to
     set errno in that case. Then rf*2^-24+rf will round to rf when
     rounding to nearest, and to a larger absolute value when rounding away. */
  if (__builtin_fabs (r) < 0x1.ffep-15 &&
      __builtin_fmaf (rf, 0x1p-24f, rf) == rf)
    underflow = 1;
  if (underflow)
    errno = ERANGE; // underflow
#endif
  return rf;
}

// dummy function since GNU libc does not provide it
_Float16 atan2pif16 (_Float16 y, _Float16 x) {
  return (_Float16) (atan2f ((float) y, (float) x) / (float) M_PI);
}
