src/f32-raddextexp/avx2-p5.c.in

// Copyright 2019 Google LLC
//
// This source code is licensed under the BSD-style license found in the
// LICENSE file in the root directory of this source tree.

$assert ELEMENTS_TILE % 8 == 0
$assert ELEMENTS_TILE >= 8
$SIMD_TILE = ELEMENTS_TILE // 8
$ABC = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
#include <assert.h>
#include <math.h>

#include <immintrin.h>

#include <xnnpack/raddextexp.h>


static const int32_t mask_table[14] = {-1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0};

void xnn_f32_raddextexp_ukernel__avx2_p5_x${ELEMENTS_TILE}${"" if ACCUMULATORS == 1 else "_acc%d" % ACCUMULATORS}(
    size_t elements,
    const float* x,
    float* sum)
{
  assert(elements % sizeof(float) == 0);

  const __m256 vlog2e = _mm256_set1_ps(0x1.715476p+0f);
  const __m256 vminus_ln2_hi = _mm256_set1_ps(-0x1.62E43p-1f);
  const __m256 vminus_ln2_lo = _mm256_set1_ps(0x1.05C61p-29f);

  // The smallest elements such that 2**elements is considered non-negligible.
  // For smaller elements, 2**elements is replaced with zero.
  const __m256 vmin_exponent = _mm256_set1_ps(-127.0f);
  const __m256 vmagic_bias = _mm256_set1_ps(0x1.8000FEp23f);
  const __m256 vminus_inf = _mm256_set1_ps(-INFINITY);

  const __m256 vc0 = _mm256_set1_ps(1.0f);
  const __m256 vc1 = _mm256_set1_ps(0x1.FFFFF6p-1f);
  const __m256 vc2 = _mm256_set1_ps(0x1.FFFDC6p-2f);
  const __m256 vc3 = _mm256_set1_ps(0x1.555A80p-3f);
  const __m256 vc4 = _mm256_set1_ps(0x1.573A1Ap-5f);
  const __m256 vc5 = _mm256_set1_ps(0x1.0F9F9Cp-7f);

  $for K in range(ACCUMULATORS):
    __m256 vaccv${K} = _mm256_setzero_ps();
  $for K in range(ACCUMULATORS):
    __m256 vacce${K} = vminus_inf;
  for (; elements >= ${ELEMENTS_TILE} * sizeof(float); elements -= ${ELEMENTS_TILE} * sizeof(float)) {
    // Load ${ELEMENTS_TILE} (${SIMD_TILE}x8) inputs at a time.
    const __m256 vx0 = _mm256_loadu_ps(x);
    $for N in range(1, SIMD_TILE):
      const __m256 vx${N} = _mm256_loadu_ps(x + ${N * 8});
    x += ${ELEMENTS_TILE};

    // Compute reduced argument elements := round(x / log(2)).
    $for N in range(SIMD_TILE):
      const __m256 vn${N} = _mm256_round_ps(_mm256_mul_ps(vx${N}, vlog2e), _MM_FROUND_TO_NEAREST_INT | _MM_FROUND_NO_EXC);

    // Compute reduced argument t := x - elements * log(2).
    // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
    $for N in range(SIMD_TILE):
      __m256 vt${N} = _mm256_fmadd_ps(vn${N}, vminus_ln2_hi, vx${N});

    $for N in range(SIMD_TILE):
      vt${N} = _mm256_fmadd_ps(vn${N}, vminus_ln2_lo, vt${N});

    // Compute degree-5 polynomial approxiatmion for exp(t) on [-log(2)/2, log(2)/2].
    $for N in range(SIMD_TILE):
      __m256 vp${N} = _mm256_fmadd_ps(vc5, vt${N}, vc4);

    $for N in range(SIMD_TILE):
      vp${N} = _mm256_fmadd_ps(vp${N}, vt${N}, vc3);

    $for N in range(SIMD_TILE):
      vp${N} = _mm256_fmadd_ps(vp${N}, vt${N}, vc2);

    $for N in range(SIMD_TILE):
      vp${N} = _mm256_fmadd_ps(vp${N}, vt${N}, vc1);

    $for N in range(SIMD_TILE):
      vp${N} = _mm256_fmadd_ps(vp${N}, vt${N}, vc0);

    // Accumulate "extended" floating-point numbers in ("mantissa", "exponent") representation where
    //  - vnX is "exponent"
    //  - vpX is "mantissa"
    //   
    // exp2(ae) * av + exp2(be) * bv = 
    //   = exp2(max(ae, be)) * exp2(ae - max(ae, be)) * av + exp2(max(ae, be)) * exp2(be - max(ae, be)) * bv
    //   = exp2(max_e) * (exp2(ae - max_e) * av + exp2(be - max_e) * bv)
    //   = exp2(max_e) * (exp2(delta_ae) * av + exp2(delta_be) * bv)
    //
    // For computational efficiency we may add several "extended" floating-point numbers at a time.
    $for N in range(SIMD_TILE):
      $if N < ACCUMULATORS:
        __m256 vmax_e${N} = _mm256_max_ps(vacce${N}, vn${N});
      $else:
        vmax_e${N % ACCUMULATORS} = _mm256_max_ps(vmax_e${N % ACCUMULATORS}, vn${N});

    // For computational efficiency, replace exp2(delta_e) with 0.0f when delta_e <= -127.0.
    // This replacement is done in two steps:
    // 1. Clamp minimum delta_e at -127.0.
    // 2. Map delta_e to scale factor 0.0 when delta_e == -127.0
    $for K in range(ACCUMULATORS):
      const __m256 vdelta_acce${K} = _mm256_max_ps(_mm256_sub_ps(vacce${K}, vmax_e${K}), vmin_exponent);
    $for N in range(SIMD_TILE):
      const __m256 vdelta_e${N} = _mm256_max_ps(_mm256_sub_ps(vn${N}, vmax_e${N % ACCUMULATORS}), vmin_exponent);

    // Convert delta-exponents into scale factors:
    // - s = exp2(delta_e) when delta_e > -127.0
    // - s = 0.0 when delta_e <= -127.0
    //
    // Note: delta-exponents can not exceed 0.0, thus scale factors can not exceed 1.0.
    $for K in range(ACCUMULATORS):
      const __m256 vaccs${K} = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(_mm256_add_ps(vdelta_acce${K}, vmagic_bias)), 23));
    $for N in range(SIMD_TILE):
      const __m256 vs${N} = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(_mm256_add_ps(vdelta_e${N}, vmagic_bias)), 23));

    // Update accumulated "mantissa" and "exponent" values
    $for K in range(ACCUMULATORS):
      vaccv${K} = _mm256_mul_ps(vaccv${K}, vaccs${K});
    $for N in range(SIMD_TILE):
      vaccv${N % ACCUMULATORS} = _mm256_fmadd_ps(vp${N}, vs${N}, vaccv${N % ACCUMULATORS});

    $for K in range(ACCUMULATORS):
      vacce${K} = vmax_e${K};
  }

  // Reduce partial sums of "extended" floating-point numbers into a single "extended" SIMD vector of sums.
  $if ACCUMULATORS > 1:
    $for A in range(0, ACCUMULATORS, 2):
      $if A + 1 < ACCUMULATORS:
        const __m256 vmax_acce${ABC[A:A+2]} = _mm256_max_ps(vacce${A}, vacce${A+1});
      $else:
        const __m256 vmax_acce${ABC[A]} = vacce${A};
    $ACC_SLICE = 2
    $while ACC_SLICE < ACCUMULATORS:
      $for A in range(0, ACCUMULATORS, ACC_SLICE * 2):
        $if A + ACC_SLICE < ACCUMULATORS:
          const __m256 vmax_acce${ABC[A:min(A+ACC_SLICE*2, ACCUMULATORS)]} = _mm256_max_ps(vmax_acce${ABC[A:A+ACC_SLICE]}, vmax_acce${ABC[A+ACC_SLICE:min(ACCUMULATORS,A+ACC_SLICE*2)]});
      $ACC_SLICE *= 2

    $for K in range(ACCUMULATORS):
      const __m256 vdelta_acce${K} = _mm256_max_ps(_mm256_sub_ps(vacce${K}, vmax_acce${ABC[0:ACCUMULATORS]}), vmin_exponent);

    $for K in range(ACCUMULATORS):
      const __m256 vaccs${K} = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(_mm256_add_ps(vdelta_acce${K}, vmagic_bias)), 23));

    __m256 vaccv = _mm256_mul_ps(vaccv0, vaccs0);
    $for K in range(1, ACCUMULATORS):
      vaccv = _mm256_fmadd_ps(vaccv${K}, vaccs${K}, vaccv);
    __m256 vacce = vmax_acce${ABC[0:ACCUMULATORS]};
  $else:
    __m256 vaccv = vaccv0;
    __m256 vacce = vacce0;

  for (; elements >= 8 * sizeof(float); elements -= 8 * sizeof(float)) {
    // Load 8 inputs at a time.
    const __m256 vx = _mm256_loadu_ps(x);
    x += 8;

    // Compute reduced argument elements := round(x / log(2)).
    const __m256 vn = _mm256_round_ps(_mm256_mul_ps(vx, vlog2e), _MM_FROUND_TO_NEAREST_INT | _MM_FROUND_NO_EXC);

    // Compute reduced argument t := x - elements * log(2).
    // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
    __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2_hi, vx);
    vt = _mm256_fmadd_ps(vn, vminus_ln2_lo, vt);

    // Compute degree-5 polynomial approxiatmion for exp(t) on [-log(2)/2, log(2)/2].
    __m256 vp = _mm256_fmadd_ps(vc5, vt, vc4);
    vp = _mm256_fmadd_ps(vp, vt, vc3);
    vp = _mm256_fmadd_ps(vp, vt, vc2);
    vp = _mm256_fmadd_ps(vp, vt, vc1);
    vp = _mm256_fmadd_ps(vp, vt, vc0);

    // Accumulate "extended" floating-point numbers in ("mantissa", "exponent") representation.
    const __m256 vmax_e = _mm256_max_ps(vacce, vn);

    // For computational efficiency, clamp minimum exp2(delta_e) at -127.0. It will be mapped to 0.0 scale factor later.
    const __m256 vdelta_acce = _mm256_max_ps(_mm256_sub_ps(vacce, vmax_e), vmin_exponent);
    const __m256 vdelta_e = _mm256_max_ps(_mm256_sub_ps(vn, vmax_e), vmin_exponent);

    // Convert exponents into scale factors.
    const __m256 vaccs = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(_mm256_add_ps(vdelta_acce, vmagic_bias)), 23));
    const __m256 vs = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(_mm256_add_ps(vdelta_e, vmagic_bias)), 23));
    
    // Update accumulated "mantissa" and "exponent" values.
    vaccv = _mm256_mul_ps(vaccv, vaccs);
    vaccv = _mm256_fmadd_ps(vp, vs, vaccv);

    vacce = vmax_e;
  }
  if XNN_UNLIKELY(elements != 0) {
    assert(elements >= 1 * sizeof(float));
    assert(elements <= 7 * sizeof(float));
    const __m256i vmask = _mm256_loadu_si256((const __m256i*) ((uintptr_t) &mask_table[7] - elements));

    // Load up to 7 inputs at a time.
    const __m256 vx = _mm256_maskload_ps(x, vmask);

    // Compute reduced argument elements := round(x / log(2)).
    __m256 vn = _mm256_round_ps(_mm256_mul_ps(vx, vlog2e), _MM_FROUND_TO_NEAREST_INT | _MM_FROUND_NO_EXC);

    // Compute reduced argument t := x - elements * log(2).
    // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
    __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2_hi, vx);
    vt = _mm256_fmadd_ps(vn, vminus_ln2_lo, vt);

    // Correct reduced argument elements for masked out elements.
    vn = _mm256_blendv_ps(vacce, vn, _mm256_castsi256_ps(vmask));

    // Compute degree-5 polynomial approxiatmion for exp(t) on [-log(2)/2, log(2)/2].
    __m256 vp = _mm256_fmadd_ps(vc5, vt, vc4);
    vp = _mm256_fmadd_ps(vp, vt, vc3);
    vp = _mm256_fmadd_ps(vp, vt, vc2);
    vp = _mm256_fmadd_ps(vp, vt, vc1);
    vp = _mm256_fmadd_ps(vp, vt, vc0);
    vp = _mm256_and_ps(vp, _mm256_castsi256_ps(vmask));

    // Accumulate "extended" floating-point numbers in ("mantissa", "exponent") representation.
    const __m256 vmax_e = _mm256_max_ps(vacce, vn);

    // For computational efficiency, clamp minimum exp2(delta_e) at -127.0. It will be mapped to 0.0 scale factor later.
    const __m256 vdelta_e = _mm256_max_ps(_mm256_sub_ps(vn, vmax_e), vmin_exponent);
    const __m256 vdelta_acce = _mm256_max_ps(_mm256_sub_ps(vacce, vmax_e), vmin_exponent);

    // Convert exponents into scale factors.
    const __m256 vs = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(_mm256_add_ps(vdelta_e, vmagic_bias)), 23));
    const __m256 vaccs = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(_mm256_add_ps(vdelta_acce, vmagic_bias)), 23));

    // Update accumulated "mantissa" and "exponent" values.
    vaccv = _mm256_mul_ps(vaccv, vaccs);
    vaccv = _mm256_fmadd_ps(vp, vs, vaccv);

    vacce = vmax_e;
  }

  // Reduce partial sums of "extended" floating-point numbers into a single "extended" floating-point sum.
  __m256 vmax_acce = _mm256_max_ps(vacce, _mm256_permute2f128_ps(vacce, vacce, 1));
  vmax_acce = _mm256_max_ps(vmax_acce, _mm256_shuffle_ps(vmax_acce, vmax_acce, _MM_SHUFFLE(1, 0, 3, 2)));
  vmax_acce = _mm256_max_ps(vmax_acce, _mm256_shuffle_ps(vmax_acce, vmax_acce, _MM_SHUFFLE(2, 3, 0, 1)));
  const __m256 vdelta_acce = _mm256_max_ps(_mm256_sub_ps(vacce, vmax_acce), vmin_exponent);
  const __m256 vaccs = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(_mm256_add_ps(vdelta_acce, vmagic_bias)), 23));
  
  vaccv = _mm256_mul_ps(vaccv, vaccs);
  __m128 vaccv_sum = _mm_add_ps(_mm256_castps256_ps128(vaccv), _mm256_extractf128_ps(vaccv, 1));
  vaccv_sum = _mm_add_ps(vaccv_sum, _mm_movehl_ps(vaccv_sum, vaccv_sum));
  vaccv_sum = _mm_add_ss(vaccv_sum, _mm_movehdup_ps(vaccv_sum));

  _mm_store_ss(&sum[0], vaccv_sum);
  _mm_store_ss(&sum[1], _mm256_castps256_ps128(vmax_acce));

  _mm256_zeroupper();
}