glibc/sysdeps/aarch64/fpu/atan2_advsimd.c

/* Double-precision AdvSIMD atan2

   Copyright (C) 2023 Free Software Foundation, Inc.
   This file is part of the GNU C Library.

   The GNU C Library is free software; you can redistribute it and/or
   modify it under the terms of the GNU Lesser General Public
   License as published by the Free Software Foundation; either
   version 2.1 of the License, or (at your option) any later version.

   The GNU C Library 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
   Lesser General Public License for more details.

   You should have received a copy of the GNU Lesser General Public
   License along with the GNU C Library; if not, see
   <https://www.gnu.org/licenses/>.  */

#include "v_math.h"
#include "poly_advsimd_f64.h"

static const struct data
{
  float64x2_t pi_over_2;
  float64x2_t poly[20];
} data = {
  /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
     the interval [2**-1022, 1.0].  */
  .poly = { V2 (-0x1.5555555555555p-2),	 V2 (0x1.99999999996c1p-3),
	    V2 (-0x1.2492492478f88p-3),	 V2 (0x1.c71c71bc3951cp-4),
	    V2 (-0x1.745d160a7e368p-4),	 V2 (0x1.3b139b6a88ba1p-4),
	    V2 (-0x1.11100ee084227p-4),	 V2 (0x1.e1d0f9696f63bp-5),
	    V2 (-0x1.aebfe7b418581p-5),	 V2 (0x1.842dbe9b0d916p-5),
	    V2 (-0x1.5d30140ae5e99p-5),	 V2 (0x1.338e31eb2fbbcp-5),
	    V2 (-0x1.00e6eece7de8p-5),	 V2 (0x1.860897b29e5efp-6),
	    V2 (-0x1.0051381722a59p-6),	 V2 (0x1.14e9dc19a4a4ep-7),
	    V2 (-0x1.d0062b42fe3bfp-9),	 V2 (0x1.17739e210171ap-10),
	    V2 (-0x1.ab24da7be7402p-13), V2 (0x1.358851160a528p-16), },
  .pi_over_2 = V2 (0x1.921fb54442d18p+0),
};

#define SignMask v_u64 (0x8000000000000000)

/* Special cases i.e. 0, infinity, NaN (fall back to scalar calls).  */
static float64x2_t VPCS_ATTR NOINLINE
special_case (float64x2_t y, float64x2_t x, float64x2_t ret, uint64x2_t cmp)
{
  return v_call2_f64 (atan2, y, x, ret, cmp);
}

/* Returns 1 if input is the bit representation of 0, infinity or nan.  */
static inline uint64x2_t
zeroinfnan (uint64x2_t i)
{
  /* (2 * i - 1) >= (2 * asuint64 (INFINITY) - 1).  */
  return vcgeq_u64 (vsubq_u64 (vaddq_u64 (i, i), v_u64 (1)),
		    v_u64 (2 * asuint64 (INFINITY) - 1));
}

/* Fast implementation of vector atan2.
   Maximum observed error is 2.8 ulps:
   _ZGVnN2vv_atan2 (0x1.9651a429a859ap+5, 0x1.953075f4ee26p+5)
	got 0x1.92d628ab678ccp-1
       want 0x1.92d628ab678cfp-1.  */
float64x2_t VPCS_ATTR V_NAME_D2 (atan2) (float64x2_t y, float64x2_t x)
{
  const struct data *data_ptr = ptr_barrier (&data);

  uint64x2_t ix = vreinterpretq_u64_f64 (x);
  uint64x2_t iy = vreinterpretq_u64_f64 (y);

  uint64x2_t special_cases = vorrq_u64 (zeroinfnan (ix), zeroinfnan (iy));

  uint64x2_t sign_x = vandq_u64 (ix, SignMask);
  uint64x2_t sign_y = vandq_u64 (iy, SignMask);
  uint64x2_t sign_xy = veorq_u64 (sign_x, sign_y);

  float64x2_t ax = vabsq_f64 (x);
  float64x2_t ay = vabsq_f64 (y);

  uint64x2_t pred_xlt0 = vcltzq_f64 (x);
  uint64x2_t pred_aygtax = vcgtq_f64 (ay, ax);

  /* Set up z for call to atan.  */
  float64x2_t n = vbslq_f64 (pred_aygtax, vnegq_f64 (ax), ay);
  float64x2_t d = vbslq_f64 (pred_aygtax, ay, ax);
  float64x2_t z = vdivq_f64 (n, d);

  /* Work out the correct shift.  */
  float64x2_t shift = vreinterpretq_f64_u64 (
      vandq_u64 (pred_xlt0, vreinterpretq_u64_f64 (v_f64 (-2.0))));
  shift = vbslq_f64 (pred_aygtax, vaddq_f64 (shift, v_f64 (1.0)), shift);
  shift = vmulq_f64 (shift, data_ptr->pi_over_2);

  /* Calculate the polynomial approximation.
     Use split Estrin scheme for P(z^2) with deg(P)=19. Use split instead of
     full scheme to avoid underflow in x^16.
     The order 19 polynomial P approximates
     (atan(sqrt(x))-sqrt(x))/x^(3/2).  */
  float64x2_t z2 = vmulq_f64 (z, z);
  float64x2_t x2 = vmulq_f64 (z2, z2);
  float64x2_t x4 = vmulq_f64 (x2, x2);
  float64x2_t x8 = vmulq_f64 (x4, x4);
  float64x2_t ret
      = vfmaq_f64 (v_estrin_7_f64 (z2, x2, x4, data_ptr->poly),
		   v_estrin_11_f64 (z2, x2, x4, x8, data_ptr->poly + 8), x8);

  /* Finalize. y = shift + z + z^3 * P(z^2).  */
  ret = vfmaq_f64 (z, ret, vmulq_f64 (z2, z));
  ret = vaddq_f64 (ret, shift);

  /* Account for the sign of x and y.  */
  ret = vreinterpretq_f64_u64 (
      veorq_u64 (vreinterpretq_u64_f64 (ret), sign_xy));

  if (__glibc_unlikely (v_any_u64 (special_cases)))
    return special_case (y, x, ret, special_cases);

  return ret;
}