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 <arm_neon.h>
  
  #include <xnnpack/common.h>
  #include <xnnpack/math-stubs.h>
  
  
  // Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
  extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64];
  
  void xnn_math_f32_sigmoid__neonfma_rr2_lut64_p2_nr2recps(
      size_t n,
      const float* input,
      float* output)
  {
    assert(n % (4 * sizeof(float)) == 0);
  
    // Large number such that ulp(magic bias) == exp2(-6)
    const float32x4_t vmagic_bias = vmovq_n_f32(0x1.800000p17f);
    const float32x4_t vminus_log2e = vmovq_n_f32(-0x1.715476p0f);
    // Mask for the lowest 6 bits
    const int32x4_t vindex_mask = vmovq_n_s32(INT32_C(0x3F));
    const float32x4_t vln2_hi = vmovq_n_f32(0x1.62E43p-1f);
    const float32x4_t vln2_lo = vmovq_n_f32(-0x1.05C61p-29f);
    // Coefficient of polynomial approximation of exp(-t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
    const float32x4_t vc2 = vmovq_n_f32(0x1.FFFF0Ap-2f);
    const float32x4_t vone = vmovq_n_f32(1.0f);
    // 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);
  
    for (; n != 0; n -= 4 * sizeof(float)) {
      const float32x4_t vx = vld1q_f32(input); input += 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 / log(2), 6).
      // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
      // the large number back. The trick with adding large number is valid only within certain bounds
      // (|-z / log(2)| <= 2**16, i.e. |z| <= 0x1.62E43p+15 = 5814540.0), 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 = vfmaq_f32(vmagic_bias, vz, vminus_log2e);
  
      // Create a floating-point number s (scale) such that s := 2**n 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 = 2**int(n) * 2**frac(n). We create s
      // in two steps:
      // 1. Fetch 2**frac(n) 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 int(n) 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 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
      //
      // Shift bits 6:14 into 23:31 (position of floating-point exponent).
      const int32x4_t ve = vshlq_n_s32(vreinterpretq_s32_f32(vn), 17);
  
      // Use bits 0:6 of n, as integer, as an index for table lookup of l := 2**frac(n).
      const uint64x2_t vidx = vreinterpretq_u64_s32(vshlq_n_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask), 2));
      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((const float*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) vidx_lo));
      float32x2_t vl_hi = vld1_dup_f32((const float*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) vidx_hi));
      vl_lo = vld1_lane_f32((const float*) ((uintptr_t) xnn_table_exp2minus_k_over_64 + (uint32_t) (vidx_lo >> 32)), vl_lo, 1);
      vl_hi = vld1_lane_f32((const float*) ((uintptr_t) xnn_table_exp2minus_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 / log(2), 6) as a floating-point number.
      vn = vsubq_f32(vn, vmagic_bias);
  
      // Compute reduced argument t := (z + n * log(2)). Note that -t = -z - n * log(2).
      // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
      float32x4_t vt = vfmaq_f32(vz, vn, vln2_hi);
      vt = vfmaq_f32(vt, vn, vln2_lo);
  
      // Compute degree-2 polynomial approximation for exp(-t) on [-log(2)/128, log(2)/128].
      //   P(t) = 1 + t * (-1 + t * c2) = 1 - (t - t * (t * c2)) = 1 - p
      float32x4_t vp = vmulq_f32(vt, vc2);
      vp = vfmsq_f32(vt, vp, vt);
  
      // Reconstruct the exp(-z) value:
      //   e = s * (1 + t * (-1 + t * c2))
      //     = s * (1 - p)
      //     = s - s * p
      const float32x4_t vy = vfmsq_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(output, vf); output += 4;
    }
  }
  