text: // 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.
  
  #include <assert.h>
  #include <stddef.h>
  
  #include <immintrin.h>
  
  #include <xnnpack/math-stubs.h>
  
  
  void xnn_math_f32_sigmoid__avx2_rr2_p5_nr2fma(
      size_t n,
      const float* input,
      float* output)
  {
    assert(n % (8 * sizeof(float)) == 0);
  
    const __m256 vmagic_bias = _mm256_set1_ps(0x1.8000FEp23f);
    // The smallest x for which sigmoidf(x) is normalized.
    // This number is also the smallest x for which expf(x) is normalized.
    const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
    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);
    const __m256 vone = _mm256_set1_ps(1.0f);
    const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
  
    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 (; n != 0; n -= 8 * sizeof(float)) {
      const __m256 vx = _mm256_loadu_ps(input);
  
      // General structure of the algorithm:
      //           / exp(x) / (1 + exp(x)) if x <= 0
      //   f[x] := 
      //           \ 1 - f[-x] if x >= 0
      //
      // First we compute f[z] := exp(z) / (1 + exp(z)) where z = -abs(x),
      // then replace result with 1 - f[z] if x >= 0.
      const __m256 vz = _mm256_or_ps(vx, vsign_mask);
  
      // Compute reduced argument n := round(z / log(2)).
      // We do it by adding a large number (magic bias) to the product z * (1/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 thats ok, because inputs x outside of [-87.336544, 17.328678] (i.e. z outsize
      // [0, 87.336544]) underflow or saturate sigmoidf(x) anyway. We fixup the result for such inputs at the very end of
      // the algorithm.
      __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
  
      // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
      // -87.33642 <= z <= 0.0, and -126 <= n <= 0 accordingly.
      const __m256 vs = _mm256_castsi256_ps(_mm256_slli_epi32(_mm256_castps_si256(vn), 23));
  
      // Subtract the large number back to get final n := round(z / log(2)).
      vn = _mm256_sub_ps(vn, vmagic_bias);
  
      // Compute reduced argument t := z - n * 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, vz);
      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);
  
      // Reconstruct the exp(z) value:
      //   e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
      //     = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
      //     = s + (t * s) * p
      vt = _mm256_mul_ps(vt, vs);
      const __m256 ve = _mm256_fmadd_ps(vt, vp, vs);
  
      // Denominator of the sigmoid fraction: 1.0 + exp(z)
      const __m256 vd = _mm256_add_ps(ve, vone);
  
      // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
      // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
      // Thus the reciprocal of the denominator never overflows.
      __m256 vr = _mm256_rcp_ps(vd);
      vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
      vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
  
      // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
      __m256 vf = _mm256_mul_ps(ve, vr);
  
      // For inputs below denormal cutoff, replace output with +0.0f.
      // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
      vf = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
  
      // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
      vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
  
      _mm256_storeu_ps(output, vf);
  
      input += 8;
      output += 8;
    }
  }
  