glibc/sysdeps/aarch64/fpu/atan_advsimd.c

/* Double-precision AdvSIMD inverse tan

   Copyright (C) 2023-2024 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
	      [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)
#define TinyBound 0x3e10000000000000 /* asuint64(0x1p-30).  */
#define BigBound 0x4340000000000000  /* asuint64(0x1p53).  */

/* Fast implementation of vector atan.
   Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using
   z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps:
   _ZGVnN2v_atan (0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1
				       want 0x1.9225645bdd7c3p-1.  */
float64x2_t VPCS_ATTR V_NAME_D1 (atan) (float64x2_t x)
{
  const struct data *d = ptr_barrier (&data);

  /* Small cases, infs and nans are supported by our approximation technique,
     but do not set fenv flags correctly. Only trigger special case if we need
     fenv.  */
  uint64x2_t ix = vreinterpretq_u64_f64 (x);
  uint64x2_t sign = vandq_u64 (ix, SignMask);

#if WANT_SIMD_EXCEPT
  uint64x2_t ia12 = vandq_u64 (ix, v_u64 (0x7ff0000000000000));
  uint64x2_t special = vcgtq_u64 (vsubq_u64 (ia12, v_u64 (TinyBound)),
				  v_u64 (BigBound - TinyBound));
  /* If any lane is special, fall back to the scalar routine for all lanes.  */
  if (__glibc_unlikely (v_any_u64 (special)))
    return v_call_f64 (atan, x, v_f64 (0), v_u64 (-1));
#endif

  /* Argument reduction:
     y := arctan(x) for x < 1
     y := pi/2 + arctan(-1/x) for x > 1
     Hence, use z=-1/a if x>=1, otherwise z=a.  */
  uint64x2_t red = vcagtq_f64 (x, v_f64 (1.0));
  /* Avoid dependency in abs(x) in division (and comparison).  */
  float64x2_t z = vbslq_f64 (red, vdivq_f64 (v_f64 (1.0), x), x);
  float64x2_t shift = vreinterpretq_f64_u64 (
      vandq_u64 (red, vreinterpretq_u64_f64 (d->pi_over_2)));
  /* Use absolute value only when needed (odd powers of z).  */
  float64x2_t az = vbslq_f64 (
      SignMask, vreinterpretq_f64_u64 (vandq_u64 (SignMask, red)), z);

  /* 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 y
      = vfmaq_f64 (v_estrin_7_f64 (z2, x2, x4, d->poly),
		   v_estrin_11_f64 (z2, x2, x4, x8, d->poly + 8), x8);

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

  /* y = atan(x) if x>0, -atan(-x) otherwise.  */
  y = vreinterpretq_f64_u64 (veorq_u64 (vreinterpretq_u64_f64 (y), sign));
  return y;
}