src/math/exp-f32-sse2-rr2-lut64-p2.c

// Copyright 2020 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.

#include <assert.h>
#include <math.h>
#include <stddef.h>

#include <emmintrin.h>

#include <xnnpack/common.h>
#include <xnnpack/math-stubs.h>


// Table of exp2(k / 64) values, k = 0..63
extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64];

void xnn_math_f32_exp__sse2_rr2_lut64_p2(
    size_t n,
    const float* input,
    float* output)
{
  assert(n % (4 * sizeof(float)) == 0);

  const __m128 vmagic_bias = _mm_set1_ps(0x1.800000p+23f);
  // The smallest x for which expf(x) is non-zero.
  const __m128 vzero_cutoff = _mm_set1_ps(-0x1.9FE368p+6f);
  // The largest x for which expf(x) is finite.
  const __m128 vinf_cutoff = _mm_set1_ps(0x1.62E42Ep+6f);
  const __m128 vlog2e_x64  = _mm_set1_ps(0x1.715476p+6f);
  // Last 13 bits are zeroes
  const __m128 vminus_ln2_o64_hi = _mm_set1_ps(-0x1.630000p-7f);
  const __m128 vminus_ln2_o64_lo = _mm_set1_ps(0x1.BD0106p-19f);
  const __m128 vplus_inf = _mm_set1_ps(INFINITY);

  const __m128 vc2 = _mm_set1_ps(0x1.FFFF0Ap-2f);

  const __m128i vmin_exponent = _mm_set1_epi32(0xC1000000);
  const __m128i vmax_exponent = _mm_set1_epi32(0x3F800000);
  const __m128i vdefault_exponent = vmax_exponent;
  const __m128i vindex_mask = _mm_set1_epi32(0x3F);

  for (; n != 0; n -= 4 * sizeof(float)) {
    const __m128 vx = _mm_loadu_ps(input);

    // Compute reduced argument n := round(x * 64 / log(2)).
    // We do it by adding a large number (magic bias) to the product x * (64/log(2)), which cause rounding of the
    // result to an integer, then subtracing the large number back. The trick with adding large number is valid only
    // within certain bounds (|x| <= 2**22), but that's ok, because inputs outside of [-103.97207, 88.72283] underflow
    // or overflow expf(x) anyway. We fixup the result for such inputs at the very end of the algorithm.
    __m128 vn = _mm_add_ps(_mm_mul_ps(vx, vlog2e_x64), vmagic_bias);

    // Create two floating-point numbers, sn (scale, normal) and so (scale, overflow) such that sn * so == 2**n
    // for inputs which don't cause overflow, i.e. -103.97207 <= x <= 88.72283, and -150 <= n <= 128 accordingly.
    // We need to use two numbers rather than one because a normalized single-precision exponent must be in [-127, 126]
    // range, which is insufficient to cover [-150, 128] range of n.
    // - When n is within [-127, 126], sn == 2**n and so == 1.0.
    // - When n < -127, sn == 2**(-127) and so == 2**(n + 127).
    // - When n > 126, sn == 2**126 and so == 2**(n - 126).
    // While we explicitly compute sn, the so is fused into the value l fetched from a table by adjusting its exponential.
    __m128i veo = _mm_slli_epi32(_mm_andnot_si128(vindex_mask, _mm_castps_si128(vn)), 17);
    __m128i ven = _mm_max_epi16(veo, vmin_exponent);
    ven = _mm_min_epi16(ven, vmax_exponent);
    veo = _mm_sub_epi32(veo, ven);
    const __m128 vsn = _mm_castsi128_ps(_mm_add_epi32(ven, vdefault_exponent));

    // Use the low 6 bits of n (as integer) for table lookup.
    const __m128i vidx = _mm_slli_epi32(_mm_and_si128(_mm_castps_si128(vn), vindex_mask), 2);
#if XNN_ARCH_X86_64
    const uint64_t vidx01 = (uint64_t) _mm_cvtsi128_si64(vidx);
    const uint64_t vidx23 = (uint64_t) _mm_cvtsi128_si64(_mm_unpackhi_epi64(vidx, vidx));
    const __m128i vl0 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) &xnn_table_exp2_k_over_64 + (uint32_t) vidx01)));
    const __m128i vl2 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + (uint32_t) vidx23)));
    const __m128i vl1 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + (uint32_t) (vidx01 >> 32))));
    const __m128i vl3 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + (uint32_t) (vidx23 >> 32))));
#else
    const uint32_t vidx0 = (uint32_t) _mm_cvtsi128_si32(vidx);
    const uint32_t vidx1 = (uint32_t) _mm_extract_epi16(vidx, 2);
    const uint32_t vidx2 = (uint32_t) _mm_extract_epi16(vidx, 4);
    const uint32_t vidx3 = (uint32_t) _mm_extract_epi16(vidx, 6);
    const __m128i vl0 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + vidx0)));
    const __m128i vl2 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + vidx2)));
    const __m128i vl1 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + vidx1)));
    const __m128i vl3 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + vidx3)));
#endif
    // Fuse so into the value l fetched from a table by adjusting its exponential.
    const __m128 vl = _mm_castsi128_ps(_mm_add_epi32(_mm_unpacklo_epi64(_mm_unpacklo_epi32(vl0, vl1), _mm_unpacklo_epi32(vl2, vl3)), veo));

    // Subtract the large number back to get final n := round(x * 64 / log(2)).
    vn = _mm_sub_ps(vn, vmagic_bias);

    // Compute reduced argument t := x - n * log(2).
    // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
    __m128 vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_o64_hi), vx);
    vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_o64_lo), vt);

    // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
    __m128 vp = _mm_mul_ps(vt, vc2);
    vp = _mm_add_ps(vt, _mm_mul_ps(vt, vp));

    // Reconstruct the final f value:
    //   f = sn * (so * l) * (1 + t * (1 + t * c2))
    //     = sn * (so * l) * (1 + t + t * (t * c2))
    //     = sn * ((so * l) + (so * l) * (t + t * (t * c2)))
    __m128 vf = _mm_add_ps(vl, _mm_mul_ps(vl, vp));
    vf = _mm_mul_ps(vf, vsn);

    // For inputs below zero cutoff, replace output with +0.0f.
    // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
    vf = _mm_andnot_ps(_mm_cmplt_ps(vx, vzero_cutoff), vf);
    // For inputs above inf cutoff, replace output with +inf.
    // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
    const __m128 vm = _mm_cmpgt_ps(vx, vinf_cutoff);
    vf = _mm_or_ps(_mm_and_ps(vplus_inf, vm), _mm_andnot_ps(vm, vf));
    _mm_storeu_ps(output, vf);

    input += 4;
    output += 4;
  }
}