atan.c

/*
 * Double-precision vector atan(x) function.
 *
 * Copyright (c) 2021-2024, Arm Limited.
 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
 */

#include "v_math.h"
#include "test_sig.h"
#include "test_defs.h"

static const struct data
{
  float64x2_t c0, c2, c4, c6, c8, c10, c12, c14, c16, c18;
  float64x2_t pi_over_2;
  double c1, c3, c5, c7, c9, c11, c13, c15, c17, c19;
} data = {
  /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
	      [2**-1022, 1.0].  */
  .c0 = V2 (-0x1.5555555555555p-2),	  .c1 = 0x1.99999999996c1p-3,
  .c2 = V2 (-0x1.2492492478f88p-3),	  .c3 = 0x1.c71c71bc3951cp-4,
  .c4 = V2 (-0x1.745d160a7e368p-4),	  .c5 = 0x1.3b139b6a88ba1p-4,
  .c6 = V2 (-0x1.11100ee084227p-4),	  .c7 = 0x1.e1d0f9696f63bp-5,
  .c8 = V2 (-0x1.aebfe7b418581p-5),	  .c9 = 0x1.842dbe9b0d916p-5,
  .c10 = V2 (-0x1.5d30140ae5e99p-5),	  .c11 = 0x1.338e31eb2fbbcp-5,
  .c12 = V2 (-0x1.00e6eece7de8p-5),	  .c13 = 0x1.860897b29e5efp-6,
  .c14 = V2 (-0x1.0051381722a59p-6),	  .c15 = 0x1.14e9dc19a4a4ep-7,
  .c16 = V2 (-0x1.d0062b42fe3bfp-9),	  .c17 = 0x1.17739e210171ap-10,
  .c18 = V2 (-0x1.ab24da7be7402p-13),	  .c19 = 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);
  float64x2_t c13 = vld1q_f64 (&d->c1);
  float64x2_t c57 = vld1q_f64 (&d->c5);
  float64x2_t c911 = vld1q_f64 (&d->c9);
  float64x2_t c1315 = vld1q_f64 (&d->c13);
  float64x2_t c1719 = vld1q_f64 (&d->c17);

  /* 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 (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);

  /* estrin_7.  */
  float64x2_t p01 = vfmaq_laneq_f64 (d->c0, z2, c13, 0);
  float64x2_t p23 = vfmaq_laneq_f64 (d->c2, z2, c13, 1);
  float64x2_t p03 = vfmaq_f64 (p01, x2, p23);

  float64x2_t p45 = vfmaq_laneq_f64 (d->c4, z2, c57, 0);
  float64x2_t p67 = vfmaq_laneq_f64 (d->c6, z2, c57, 1);
  float64x2_t p47 = vfmaq_f64 (p45, x2, p67);

  float64x2_t p07 = vfmaq_f64 (p03, x4, p47);

  /* estrin_11.  */
  float64x2_t p89 = vfmaq_laneq_f64 (d->c8, z2, c911, 0);
  float64x2_t p1011 = vfmaq_laneq_f64 (d->c10, z2, c911, 1);
  float64x2_t p811 = vfmaq_f64 (p89, x2, p1011);

  float64x2_t p1213 = vfmaq_laneq_f64 (d->c12, z2, c1315, 0);
  float64x2_t p1415 = vfmaq_laneq_f64 (d->c14, z2, c1315, 1);
  float64x2_t p1215 = vfmaq_f64 (p1213, x2, p1415);

  float64x2_t p1617 = vfmaq_laneq_f64 (d->c16, z2, c1719, 0);
  float64x2_t p1819 = vfmaq_laneq_f64 (d->c18, z2, c1719, 1);
  float64x2_t p1619 = vfmaq_f64 (p1617, x2, p1819);

  float64x2_t p815 = vfmaq_f64 (p811, x4, p1215);
  float64x2_t p819 = vfmaq_f64 (p815, x8, p1619);

  float64x2_t y = vfmaq_f64 (p07, p819, 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;
}

TEST_SIG (V, D, 1, atan, -10.0, 10.0)
TEST_ULP (V_NAME_D1 (atan), 1.78)
TEST_DISABLE_FENV_IF_NOT (V_NAME_D1 (atan), WANT_SIMD_EXCEPT)
TEST_INTERVAL (V_NAME_D1 (atan), 0, 0x1p-30, 10000)
TEST_INTERVAL (V_NAME_D1 (atan), -0, -0x1p-30, 1000)
TEST_INTERVAL (V_NAME_D1 (atan), 0x1p-30, 0x1p53, 900000)
TEST_INTERVAL (V_NAME_D1 (atan), -0x1p-30, -0x1p53, 90000)
TEST_INTERVAL (V_NAME_D1 (atan), 0x1p53, inf, 10000)
TEST_INTERVAL (V_NAME_D1 (atan), -0x1p53, -inf, 1000)