text: // Auto-generated file. Do not edit!
  //   Template: src/f32-sigmoid/neon-lut64-p2.c.in
  //   Generator: tools/xngen
  //
  // 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 <arm_neon.h>
  
  #include <xnnpack/common.h>
  #include <xnnpack/vunary.h>
  
  
  extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64];
  
  void xnn_f32_sigmoid_ukernel__neon_rr2_lut64_p2_nr2recps_x4(
      size_t n,
      const float* x,
      float* y,
      const void* params)
  {
    assert(n % sizeof(float) == 0);
  
    const float32x4_t vmagic_bias = vmovq_n_f32(0x1.800000p23f);
    // The largest z for which sigmoidf(-z) is normalized.
    // This number is also the largest z for which expf(-z) is normalized.
    const float32x4_t vdenorm_cutoff = vmovq_n_f32(0x1.5D589Ep+6f);
    const float32x4_t vminus_log2e_x64 = vmovq_n_f32(-0x1.715476p6f);
    // Last 13 bits are zeroes
    const float32x4_t vln2_o64_hi = vmovq_n_f32(0x1.630000p-7f);
    const float32x4_t vln2_o64_lo = vmovq_n_f32(-0x1.BD0106p-19f);
    const float32x4_t vone = vmovq_n_f32(1.0f);
  
    const float32x4_t vc2 = vmovq_n_f32(0x1.FFFF0Ap-2f);
  
    const int32x4_t vindex_mask = vmovq_n_s32(INT32_C(0x3F));
  
    for (; n >= 4 * sizeof(float); n -= 4 * sizeof(float)) {
      const float32x4_t vx = vld1q_f32(x); x += 4;
  
      // 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 float32x4_t vz = vabsq_f32(vx);
  
      // Compute reduced argument n := round(-z * 64 / log(2)).
      // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing
      // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction.
      // The trick with adding large number is valid only within certain bounds (|z * 64 / log(2)| <= 2**22, i.e.
      // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
      // (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup the result  for such inputs at the
      // very end of the algorithm.
      float32x4_t vn = vmlaq_f32(vmagic_bias, vz, vminus_log2e_x64);
  
      // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that sigmoidf(-z) is
      // normalized, i.e. 0 <= z <= 87.33642. As n has 6 fractional bits, we split s == 2**(n / 64) =
      // = 2**e * 2**(n / 64 - e), where e := int(n / 64). We create s in two steps:
      // 1. Fetch 2**(n / 64 - e) = 2**(n % 64) from the table using the 6 low bits of n, as integer. Note that the
      //    fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
      // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized
      //    number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(-z) is normalized) we have -126 <= e <= 0,
      //    and thus the adjusted exponent is not lower than -126.
      //
      // Extract e from bits 6:14 of n and shift it into bits 23:31 (position of floating-point exponent).
      const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x3F))), 17);
  
      // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64).
      const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask));
      const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0);
      const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1);
      float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_lo]);
      float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_hi]);
      vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_lo >> 32)], vl_lo, 1);
      vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_hi >> 32)], vl_hi, 1);
      const float32x4_t vl = vcombine_f32(vl_lo, vl_hi);
      // Adjust exponent of the value l fetched from the table to get the final s value.
      const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve));
  
      // Subtract the large number back to get the final n := round(-z * 64 / log(2)) as a floating-point number.
      vn = vsubq_f32(vn, vmagic_bias);
  
      // Compute reduced argument t := (z + n * log(2) / 64). Note that -t = -z - n * log(2) / 64.
      // Use Cody-Waite range reduction method (note two constants to represent log(2) / 64) to improve accuracy.
      float32x4_t vt = vmlaq_f32(vz, vn, vln2_o64_hi);
      vt = vmlaq_f32(vt, vn, vln2_o64_lo);
  
      // Compute degree-2 polynomial approxiatmion for exp(-t) on [-log(2)/128, log(2)/128].
      //   P1(t) = 1 + t * (-1 + t * c2)
      float32x4_t vp = vmulq_f32(vt, vc2);
      vp = vmlsq_f32(vt, vp, vt);
  
      // Reconstruct the exp(-z) value:
      //   f = s * (1 + t * (-1 + t * c2))
      //     = s * (1 - t + t * (t * c2))
      //     = s - s * (t - t * (t * c2))
      //     = s - s * p
      const float32x4_t vy = vmlsq_f32(vs, vs, vp);
  
      // Denominator of the sigmoid fraction: 1.0 + exp(-z)
      const float32x4_t vd = vaddq_f32(vy, 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.
      float32x4_t vr = vrecpeq_f32(vd);
  
      vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
  
      vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
  
      // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
      float32x4_t vf = vmulq_f32(vy, 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 = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcagtq_f32(vx, vdenorm_cutoff)));
  
      // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
      const uint32x4_t vm = vcltq_f32(vx, vmovq_n_f32(0.0f));
      vf = vbslq_f32(vm, vf, vsubq_f32(vone, vf));
  
      vst1q_f32(y, vf); y += 4;
    }
    if XNN_UNLIKELY(n != 0) {
      const float32x4_t vx = vld1q_f32(x);
  
      // 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 float32x4_t vz = vabsq_f32(vx);
  
      // Compute reduced argument n := round(-z * 64 / log(2)).
      // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing
      // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction.
      // The trick with adding large number is valid only within certain bounds (|z * 64 / log(2)| <= 2**22, i.e.
      // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
      // (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup the result  for such inputs at the
      // very end of the algorithm.
      float32x4_t vn = vmlaq_f32(vmagic_bias, vz, vminus_log2e_x64);
  
      // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that sigmoidf(-z) is
      // normalized, i.e. 0 <= z <= 87.33642. As n has 6 fractional bits, we split s == 2**(n / 64) =
      // = 2**e * 2**(n / 64 - e), where e := int(n / 64). We create s in two steps:
      // 1. Fetch 2**(n / 64 - e) = 2**(n % 64) from the table using the 6 low bits of n, as integer. Note that the
      //    fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
      // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized
      //    number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(-z) is normalized) we have -126 <= e <= 0,
      //    and thus the adjusted exponent is not lower than -126.
      //
      // Extract e from bits 6:14 of n and shift it into bits 23:31 (position of floating-point exponent).
      const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x3F))), 17);
  
      // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64).
      const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask));
      const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0);
      const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1);
      float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_lo]);
      float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_hi]);
      vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_lo >> 32)], vl_lo, 1);
      vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_hi >> 32)], vl_hi, 1);
      const float32x4_t vl = vcombine_f32(vl_lo, vl_hi);
      // Adjust exponent of the value l fetched from the table to get the final s value.
      const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve));
  
      // Subtract the large number back to get the final n := round(-z * 64 / log(2)) as a floating-point number.
      vn = vsubq_f32(vn, vmagic_bias);
  
      // Compute reduced argument t := (z + n * log(2) / 64). Note that -t = -z - n * log(2) / 64.
      // Use Cody-Waite range reduction method (note two constants to represent log(2) / 64) to improve accuracy.
      float32x4_t vt = vmlaq_f32(vz, vn, vln2_o64_hi);
      vt = vmlaq_f32(vt, vn, vln2_o64_lo);
  
      // Compute degree-2 polynomial approxiatmion for exp(-t) on [-log(2)/128, log(2)/128].
      //   P1(t) = 1 + t * (-1 + t * c2)
      float32x4_t vp = vmulq_f32(vt, vc2);
      vp = vmlsq_f32(vt, vp, vt);
  
      // Reconstruct the exp(-z) value:
      //   f = s * (1 + t * (-1 + t * c2))
      //     = s * (1 - t + t * (t * c2))
      //     = s - s * (t - t * (t * c2))
      //     = s - s * p
      const float32x4_t vy = vmlsq_f32(vs, vs, vp);
  
      // Denominator of the sigmoid fraction: 1.0 + exp(-z)
      const float32x4_t vd = vaddq_f32(vy, 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.
      float32x4_t vr = vrecpeq_f32(vd);
  
      vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
  
      vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
  
      // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
      float32x4_t vf = vmulq_f32(vy, 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 = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcagtq_f32(vx, vdenorm_cutoff)));
  
      // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
      const uint32x4_t vm = vcltq_f32(vx, vmovq_n_f32(0.0f));
      vf = vbslq_f32(vm, vf, vsubq_f32(vone, vf));
  
      float32x2_t vf_lo = vget_low_f32(vf);
      if (n & (2 * sizeof(float))) {
        vst1_f32(y, vf_lo); y += 2;
        vf_lo = vget_high_f32(vf);
      }
      if (n & (1 * sizeof(float))) {
        vst1_lane_f32(y, vf_lo, 0);
      }
    }
  }
  