/* Correctly-rounded 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 atan2f.c:
   at input we convert the inputs to _Float16 (exactly),
   we then use the same code than for atan2f,
   and at output we round to _Float16.
   The changes with respect to atan2f.c are marked with a comment
   "specific atan2f16". */

#include <stdint.h>
#ifdef CORE_MATH_SUPPORT_ERRNO
#include <errno.h>
#endif
#include <math.h> // used only in perf.sh

// 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;
}

/* for y/x tiny, use Taylor approximation z - z^3/3 where z=y/x */
static _Float16 // specific atan2f16: changed return type to _Float16
cr_atan2f_tiny (float y, float x)
{
  double dy = y, dx = x;
  double z = dy / dx;
  double e = __builtin_fma (-z, x, y);
  /* z * x + e = y thus y/x = z + e/x */
  static const double c = -0x1.5555555555555p-2; /* -1/3 rounded to nearest */
  double zz = z * z;
  double cz = c * z;
  e = e / (double) x + cz * zz;
  b64u64_u t = {.f = z};
  if ((t.u & 0xfffffffull) == 0) /* boundary case */
  {
    /* If z and e are of same sign (resp. of different signs), we increase
       (resp. decrease) the significant of t by 1 to avoid a double-rounding
       issue when rounding t.f to binary32. */
    if (z * e > 0)
      t.u += 1;
    else
      t.u -= 1;
  }
  _Float16 res = t.f; // specific atan2f16
#ifdef CORE_MATH_SUPPORT_ERRNO
  // specific atan2f16: changed underflow threshold from 0x1p-126 to 0x1p-14
  if (__builtin_fabsf (res) < 0x1p-14f)
    errno = ERANGE; // underflow
#endif
  return res;
}

// specific atan2f16: inputs name changed from y,x to yf16,xf16
_Float16 cr_atan2f16(_Float16 yf16, _Float16 xf16){
  float y = yf16, x = xf16; // specific atan2f16
  static const double cn[] =
    {0x1p+0, 0x1.40e0698f94c35p+1, 0x1.248c5da347f0dp+1, 0x1.d873386572976p-1, 0x1.46fa40b20f1dp-3,
     0x1.33f5e041eed0fp-7, 0x1.546bbf28667c5p-14};
  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};
#define pi 0x1.921fb54442d18p+1
#define pi2 0x1.921fb54442d18p+0
#define pi2l 0x1.1a62633145c07p-54
  static const double off[] = {0.0f, pi2, pi, pi2, -0.0f, -pi2, -pi, -pi2};
  static const double offl[] = {0.0f, pi2l, 2*pi2l, pi2l, -0.0f, -pi2l, -2*pi2l, -pi2l};
  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)){ // x or y is nan or inf
    /* we use x+y below so that the invalid exception is set
       for (x,y) = (qnan,snan) or (snan,qnan) */
    if(ay > (0xff<<23)) return x + y; // case y nan
    if(ax > (0xff<<23)) return x + y; // case x nan
    uint32_t yinf = ay==(0xff<<23), xinf = ax==(0xff<<23);
    if(yinf&xinf){
      if(ux>>31)
	return 0x1.2d97c7f3321d2p+1*sgn[uy>>31]; // +/-3pi/4
      else
	return 0x1.921fb54442d18p-1*sgn[uy>>31]; // +/-pi/4
    }
    if(xinf){
      if(ux>>31)
	return pi*sgn[uy>>31];
      else
	return 0.0*sgn[uy>>31];
    }
    if(yinf){
      return pi2*sgn[uy>>31];
    }
  }
  if(__builtin_expect(ay==0, 0)){
    if(__builtin_expect(!ax,0)){
      uint32_t i = (uy>>31)*4 + (ux>>31)*2;
      if(ux>>31)
	return off[i] + offl[i];
      else
	return off[i];
    }
    if(!(ux>>31)) return 0.0*sgn[uy>>31];
  }
  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);
  // z = x/y if |y| > |x|, and z = y/x otherwise
  double r;
  int d = (int)ax-(int)ay;
  if (__builtin_expect(d<(27<<23)&&d>(-(27<<23)),1)){
    double z2 = z*z, z4 = z2*z2, z8 = z4*z4;
    /* z2 cannot underflow, since for |y|=0x1p-149 and |x|=0x1.fffffep+127
       we get |z| > 2^-277 thus z2 > 2^-554, but z4 and z8 might underflow,
       which might give spurious underflow exceptions. */
    double cn0 = cn[0] + 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;
  } else {
    r = 1;
  }
  z *= sgn[gt];
  r = z*r + off[i];
  b64u64_u res = {.f = r};
  if(__builtin_expect(((res.u + 8)&0xfffffff) <= 16, 0)){
    /* check tiny y/x */
    if (ay < ax && ((ax - ay) >> 23 >= 25))
      return cr_atan2f_tiny (y, x);
    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[32][2] =
      {{0x1p+0, -0x1.8c1dac5492248p-87}, {-0x1.5555555555555p-2, -0x1.55553bf3a2abep-56},
       {0x1.999999999999ap-3, -0x1.99deed1ec9071p-57}, {-0x1.2492492492492p-3, -0x1.fd99c8d18269ap-58},
       {0x1.c71c71c71c717p-4, -0x1.651eee4c4d9dp-61}, {-0x1.745d1745d1649p-4, -0x1.632683d6c44a6p-58},
       {0x1.3b13b13b11c63p-4, 0x1.bf69c1f8af41dp-58}, {-0x1.11111110e6338p-4, 0x1.3c3e431e8bb68p-61},
       {0x1.e1e1e1dc45c4ap-5, -0x1.be2db05c77bbfp-59}, {-0x1.af286b8164b4fp-5, 0x1.a4673491f0942p-61},
       {0x1.86185e9ad4846p-5, 0x1.e12e32d79fceep-59}, {-0x1.642c6d5161faep-5, 0x1.3ce76c1ca03fp-59},
       {0x1.47ad6f277e5bfp-5, -0x1.abd8d85bdb714p-60}, {-0x1.2f64a2ee8896dp-5, 0x1.ef87d4b615323p-61},
       {0x1.1a6a2b31741b5p-5, 0x1.a5d9d973547eep-62}, {-0x1.07fbdad65e0a6p-5, -0x1.65ac07f5d35f4p-61},
       {0x1.ee9932a9a5f8bp-6, 0x1.f8b9623f6f55ap-61}, {-0x1.ce8b5b9584dc6p-6, 0x1.fe5af96e8ea2dp-61},
       {0x1.ac9cb288087b7p-6, -0x1.450cdfceaf5cap-60}, {-0x1.84b025351f3e6p-6, 0x1.579561b0d73dap-61},
       {0x1.52f5b8ecdd52bp-6, 0x1.036bd2c6fba47p-60}, {-0x1.163a8c44909dcp-6, 0x1.18f735ffb9f16p-60},
       {0x1.a400dce3eea6fp-7, -0x1.c90569c0c1b5cp-61}, {-0x1.1caa78ae6db3ap-7, -0x1.4c60f8161ea09p-61},
       {0x1.52672453c0731p-8, 0x1.834efb598c338p-62}, {-0x1.5850c5be137cfp-9, -0x1.445fc150ca7f5p-63},
       {0x1.23eb98d22e1cap-10, -0x1.388fbaf1d783p-64}, {-0x1.8f4e974a40741p-12, 0x1.271198a97da34p-66},
       {0x1.a5cf2e9cf76e5p-14, -0x1.887eb4a63b665p-68}, {-0x1.420c270719e32p-16, 0x1.efd595b27888bp-71},
       {0x1.3ba2d69b51677p-19, -0x1.4fb06829cdfc7p-73}, {-0x1.29b7e6f676385p-23, -0x1.a783b6de718fbp-77}};
    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 + offl[i];
    float rf = sh;
    double th = rf, dh = sh - th, tm = dh + sl;
    b64u64_u tth = {.f = th};
    if(th + th*0x1p-60 == th - th*0x1p-60){
      tth.u &= (uint64_t) 0x7ff<<52;
      tth.u -= (uint64_t) 24<<52;
      if(__builtin_fabs(tm)>tth.f)
	tm *= 1.25;
      else
	tm *= 0.75;
    }
    r = th + tm;
  }
  _Float16 rf = r; /* specific atan2f16: we round to _Float16 directly to
                      avoid a double-rounding issue. For example for
                      y,x=0x1.8p+0,-0x1.c4p-18 and rndn, we have
                      r=0x1.922000999826dp+0: if we first round to float,
                      we get rf=0x1.922p+0, which is incorrectly rounded
                      to 0x1.92p+0 instead of the correct result 0x1.924p+0 */
#ifdef CORE_MATH_SUPPORT_ERRNO
  // if rf underflows, and x != 0, set errno=ERANGE
  // specific atan2f16: 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
  underflow |= __builtin_fabs (r) < 0x1.ffcp-15;
  if (__builtin_expect (underflow, 0))
    errno = ERANGE; // underflow
#endif
  return rf;
}

// dummy function since GNU libc does not provide it
_Float16 atan2f16 (_Float16 y, _Float16 x) {
  return (_Float16) atan2f ((float) y, (float) x);
}
