sysdeps/aarch64/fpu/acosf_advsimd.c

   1 /* Single-precision AdvSIMD inverse cos
   2
   3    Copyright (C) 2023-2024 Free Software Foundation, Inc.
   4    This file is part of the GNU C Library.
   5
   6    The GNU C Library is free software; you can redistribute it and/or
   7    modify it under the terms of the GNU Lesser General Public
   8    License as published by the Free Software Foundation; either
   9    version 2.1 of the License, or (at your option) any later version.
  10
  11    The GNU C Library is distributed in the hope that it will be useful,
  12    but WITHOUT ANY WARRANTY; without even the implied warranty of
  13    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  14    Lesser General Public License for more details.
  15
  16    You should have received a copy of the GNU Lesser General Public
  17    License along with the GNU C Library; if not, see
  18    <https://www.gnu.org/licenses/>.  */
  19
  20 #include "v_math.h"
  21 #include "poly_advsimd_f32.h"
  22
  23 static const struct data
  24 {
  25   float32x4_t poly[5];
  26   float32x4_t pi_over_2f, pif;
  27 } data = {
  28   /* Polynomial approximation of  (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x))  on
  29      [ 0x1p-24 0x1p-2 ] order = 4 rel error: 0x1.00a23bbp-29 .  */
  30   .poly = { V4 (0x1.55555ep-3), V4 (0x1.33261ap-4), V4 (0x1.70d7dcp-5),
  31             V4 (0x1.b059dp-6), V4 (0x1.3af7d8p-5) },
  32   .pi_over_2f = V4 (0x1.921fb6p+0f),
  33   .pif = V4 (0x1.921fb6p+1f),
  34 };
  35
  36 #define AbsMask 0x7fffffff
  37 #define Half 0x3f000000
  38 #define One 0x3f800000
  39 #define Small 0x32800000 /* 2^-26.  */
  40
  41 #if WANT_SIMD_EXCEPT
  42 static float32x4_t VPCS_ATTR NOINLINE
  43 special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
  44 {
  45   return v_call_f32 (acosf, x, y, special);
  46 }
  47 #endif
  48
  49 /* Single-precision implementation of vector acos(x).
  50
  51    For |x| < Small, approximate acos(x) by pi/2 - x. Small = 2^-26 for correct
  52    rounding.
  53    If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the following
  54    approximation.
  55
  56    For |x| in [Small, 0.5], use order 4 polynomial P such that the final
  57    approximation of asin is an odd polynomial:
  58
  59      acos(x) ~ pi/2 - (x + x^3 P(x^2)).
  60
  61     The largest observed error in this region is 1.26 ulps,
  62       _ZGVnN4v_acosf (0x1.843bfcp-2) got 0x1.2e934cp+0 want 0x1.2e934ap+0.
  63
  64     For |x| in [0.5, 1.0], use same approximation with a change of variable
  65
  66       acos(x) = y + y * z * P(z), with  z = (1-x)/2 and y = sqrt(z).
  67
  68    The largest observed error in this region is 1.32 ulps,
  69    _ZGVnN4v_acosf (0x1.15ba56p-1) got 0x1.feb33p-1
  70                            want 0x1.feb32ep-1.  */
  71 float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (acos) (float32x4_t x)
  72 {
  73   const struct data *d = ptr_barrier (&data);
  74
  75   uint32x4_t ix = vreinterpretq_u32_f32 (x);
  76   uint32x4_t ia = vandq_u32 (ix, v_u32 (AbsMask));
  77
  78 #if WANT_SIMD_EXCEPT
  79   /* A single comparison for One, Small and QNaN.  */
  80   uint32x4_t special
  81       = vcgtq_u32 (vsubq_u32 (ia, v_u32 (Small)), v_u32 (One - Small));
  82   if (__glibc_unlikely (v_any_u32 (special)))
  83     return special_case (x, x, v_u32 (0xffffffff));
  84 #endif
  85
  86   float32x4_t ax = vreinterpretq_f32_u32 (ia);
  87   uint32x4_t a_le_half = vcleq_u32 (ia, v_u32 (Half));
  88
  89   /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
  90      z2 = x ^ 2         and z = |x|     , if |x| < 0.5
  91      z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5.  */
  92   float32x4_t z2 = vbslq_f32 (a_le_half, vmulq_f32 (x, x),
  93                               vfmsq_n_f32 (v_f32 (0.5), ax, 0.5));
  94   float32x4_t z = vbslq_f32 (a_le_half, ax, vsqrtq_f32 (z2));
  95
  96   /* Use a single polynomial approximation P for both intervals.  */
  97   float32x4_t p = v_horner_4_f32 (z2, d->poly);
  98   /* Finalize polynomial: z + z * z2 * P(z2).  */
  99   p = vfmaq_f32 (z, vmulq_f32 (z, z2), p);
 100
 101   /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for  |x| < 0.5
 102                = 2 Q(|x|)               , for  0.5 < x < 1.0
 103                = pi - 2 Q(|x|)          , for -1.0 < x < -0.5.  */
 104   float32x4_t y = vbslq_f32 (v_u32 (AbsMask), p, x);
 105
 106   uint32x4_t is_neg = vcltzq_f32 (x);
 107   float32x4_t off = vreinterpretq_f32_u32 (
 108       vandq_u32 (vreinterpretq_u32_f32 (d->pif), is_neg));
 109   float32x4_t mul = vbslq_f32 (a_le_half, v_f32 (-1.0), v_f32 (2.0));
 110   float32x4_t add = vbslq_f32 (a_le_half, d->pi_over_2f, off);
 111
 112   return vfmaq_f32 (add, mul, y);
 113 }
 114 libmvec_hidden_def (V_NAME_F1(acos))
 115 HALF_WIDTH_ALIAS_F1 (acos)