exp2.c

/*
 * Double-precision vector 2^x function.
 *
 * Copyright (c) 2019-2024, Arm Limited.
 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
 */

#include "v_math.h"
#include "v_poly_f64.h"
#include "test_sig.h"
#include "test_defs.h"

#define N (1 << V_EXP_TABLE_BITS)
#define IndexMask (N - 1)
#define BigBound 1022.0
#define UOFlowBound 1280.0
#define TinyBound 0x2000000000000000 /* asuint64(0x1p-511).  */

static const struct data
{
  float64x2_t poly[4];
  float64x2_t shift, scale_big_bound, scale_uoflow_bound;
} data = {
  /* Coefficients are computed using Remez algorithm with
     minimisation of the absolute error.  */
  .poly = { V2 (0x1.62e42fefa3686p-1), V2 (0x1.ebfbdff82c241p-3),
	    V2 (0x1.c6b09b16de99ap-5), V2 (0x1.3b2abf5571ad8p-7) },
  .shift = V2 (0x1.8p52 / N),
  .scale_big_bound = V2 (BigBound),
  .scale_uoflow_bound = V2 (UOFlowBound),
};

static inline uint64x2_t
lookup_sbits (uint64x2_t i)
{
  return (uint64x2_t){ __v_exp_data[i[0] & IndexMask],
		       __v_exp_data[i[1] & IndexMask] };
}

#if WANT_SIMD_EXCEPT

# define Thres 0x2080000000000000     /* asuint64(512.0) - TinyBound.  */

/* Call scalar exp2 as a fallback.  */
static float64x2_t VPCS_ATTR NOINLINE
special_case (float64x2_t x, float64x2_t y, uint64x2_t is_special)
{
  return v_call_f64 (exp2, x, y, is_special);
}

#else

# define SpecialOffset 0x6000000000000000 /* 0x1p513.  */
/* SpecialBias1 + SpecialBias1 = asuint(1.0).  */
# define SpecialBias1 0x7000000000000000 /* 0x1p769.  */
# define SpecialBias2 0x3010000000000000 /* 0x1p-254.  */

static inline float64x2_t VPCS_ATTR
special_case (float64x2_t s, float64x2_t y, float64x2_t n,
	      const struct data *d)
{
  /* 2^(n/N) may overflow, break it up into s1*s2.  */
  uint64x2_t b = vandq_u64 (vclezq_f64 (n), v_u64 (SpecialOffset));
  float64x2_t s1 = vreinterpretq_f64_u64 (vsubq_u64 (v_u64 (SpecialBias1), b));
  float64x2_t s2 = vreinterpretq_f64_u64 (vaddq_u64 (
      vsubq_u64 (vreinterpretq_u64_f64 (s), v_u64 (SpecialBias2)), b));
  uint64x2_t cmp = vcagtq_f64 (n, d->scale_uoflow_bound);
  float64x2_t r1 = vmulq_f64 (s1, s1);
  float64x2_t r0 = vmulq_f64 (vfmaq_f64 (s2, s2, y), s1);
  return vbslq_f64 (cmp, r1, r0);
}

#endif

/* Fast vector implementation of exp2.
   Maximum measured error is 1.65 ulp.
   _ZGVnN2v_exp2(-0x1.4c264ab5b559bp-6) got 0x1.f8db0d4df721fp-1
				       want 0x1.f8db0d4df721dp-1.  */
VPCS_ATTR
float64x2_t V_NAME_D1 (exp2) (float64x2_t x)
{
  const struct data *d = ptr_barrier (&data);
  uint64x2_t cmp;
#if WANT_SIMD_EXCEPT
  uint64x2_t ia = vreinterpretq_u64_f64 (vabsq_f64 (x));
  cmp = vcgeq_u64 (vsubq_u64 (ia, v_u64 (TinyBound)), v_u64 (Thres));
  /* Mask special lanes and retain a copy of x for passing to special-case
     handler.  */
  float64x2_t xc = x;
  x = v_zerofy_f64 (x, cmp);
#else
  cmp = vcagtq_f64 (x, d->scale_big_bound);
#endif

  /* n = round(x/N).  */
  float64x2_t z = vaddq_f64 (d->shift, x);
  uint64x2_t u = vreinterpretq_u64_f64 (z);
  float64x2_t n = vsubq_f64 (z, d->shift);

  /* r = x - n/N.  */
  float64x2_t r = vsubq_f64 (x, n);

  /* s = 2^(n/N).  */
  uint64x2_t e = vshlq_n_u64 (u, 52 - V_EXP_TABLE_BITS);
  u = lookup_sbits (u);
  float64x2_t s = vreinterpretq_f64_u64 (vaddq_u64 (u, e));

  /* y ~ exp2(r) - 1.  */
  float64x2_t r2 = vmulq_f64 (r, r);
  float64x2_t y = v_pairwise_poly_3_f64 (r, r2, d->poly);
  y = vmulq_f64 (r, y);

  if (unlikely (v_any_u64 (cmp)))
#if !WANT_SIMD_EXCEPT
    return special_case (s, y, n, d);
#else
    return special_case (xc, vfmaq_f64 (s, s, y), cmp);
#endif
  return vfmaq_f64 (s, s, y);
}

TEST_SIG (V, D, 1, exp2, -9.9, 9.9)
TEST_ULP (V_NAME_D1 (exp2), 1.15)
TEST_DISABLE_FENV_IF_NOT (V_NAME_D1 (exp2), WANT_SIMD_EXCEPT)
TEST_SYM_INTERVAL (V_NAME_D1 (exp2), 0, TinyBound, 5000)
TEST_SYM_INTERVAL (V_NAME_D1 (exp2), TinyBound, BigBound, 10000)
TEST_SYM_INTERVAL (V_NAME_D1 (exp2), BigBound, UOFlowBound, 5000)
TEST_SYM_INTERVAL (V_NAME_D1 (exp2), UOFlowBound, inf, 10000)