text: // Copyright 2022 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/math-stubs.h>
  
  
  void xnn_math_f16_sigmoid__neonfp16arith_rr2_p2_nr1recps(
      size_t n,
      const void* input,
      void* output)
  {
    assert(n % (8 * sizeof(__fp16)) == 0);
  
    // Large number such that ulp(magic bias) == 1 and magic bias === 15 mod 2**9.
    const float16x8_t vmagic_bias = vmovq_n_f16(0x1.83Cp+10f);
    const float16x8_t vminus_log2e = vmovq_n_f16(-0x1.714p+0f);
    const float16x8_t vln2_hi = vmovq_n_f16(0x1.630p-1f);
    const float16x8_t vln2_lo = vmovq_n_f16(-0x1.BD0p-13f);
    // Coefficient of polynomial approximation
    //   exp(-t) ~ 1 + t * (c1 + t * c2)
    // on [-log(2)/2, log(2)/2]
    const float16x8_t vc2 = vmovq_n_f16(0x1.FE4p-2f);
    const float16x8_t vc1 = vmovq_n_f16(-0x1.038p+0f);
    const float16x8_t vone = vmovq_n_f16(1.0f);
    // The largest z for which sigmoidh(-z) is normalized.
    // This number is also the largest z for which exph(-z) is normalized.
    const float16x8_t vdenorm_cutoff = vmovq_n_f16(-0x1.368p+3f);
  
    const __fp16* i = (const __fp16*) input;
    __fp16* o = (__fp16*) output;
    for (; n != 0; n -= 8 * sizeof(__fp16)) {
      const float16x8_t vx = vld1q_f16(i); i += 8;
  
      // 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 float16x8_t vz = vabsq_f16(vx);
  
      // 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 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
      // (|-x / log(2)| <= 2**9, i.e. |z| <= 0x1.630p+8 = 355.0), but that is acceptable, because inputs outside
      // of [-9.703125, 8.3125] (i.e. z outside [0, 9.703125]) underflow or saturate sigmoidh(x). We fixup the result for
      // such inputs at the very end of the algorithm.
      float16x8_t vn = vfmaq_f16(vmagic_bias, vz, vminus_log2e);
  
      // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
      // -9.703125 <= -z <= 0.0, and -14 <= n <= 0 accordingly.
      const float16x8_t vs = vreinterpretq_f16_s16(vshlq_n_s16(vreinterpretq_s16_f16(vn), 10));
  
      // Subtract the large number back to get the final n := round(-z / log(2)) as a floating-point number.
      vn = vsubq_f16(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.
      float16x8_t vt = vfmaq_f16(vz, vn, vln2_hi);
      vt = vfmaq_f16(vt, vn, vln2_lo);
  
      // Compute degree-2 polynomial approximation for exp(-t) on [-log(2)/2, log(2)/2]:
      //   P(t) = 1 + t * (c1 + t * c2) = 1 + t * p
      float16x8_t vp = vfmaq_f16(vc1, vc2, vt);
  
      // Reconstruct the exp(-z) value:
      //   e = s * (1 + t * (c1 + t * c2)
      //     = s * (1 + t * p)
      //     = s + (t * s) * p
      vt = vmulq_f16(vt, vs);
      float16x8_t ve = vfmaq_f16(vs, vp, vt);
  
      // Denominator of the sigmoid fraction: 1.0 + exp(-z)
      float16x8_t vd = vaddq_f16(ve, vone);
  
      // Use Newton-Raphson method (1 iteration) 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.
      float16x8_t vr = vrecpeq_f16(vd);
      vr = vmulq_f16(vr, vrecpsq_f16(vr, vd));
  
      // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
      float16x8_t vf = vmulq_f16(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 = vreinterpretq_f16_u16(vbicq_u16(vreinterpretq_u16_f16(vf), vcagtq_f16(vx, vdenorm_cutoff)));
  
      // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
      const uint16x8_t vm = vcltq_f16(vx, vmovq_n_f16(0.0f));
      vf = vbslq_f16(vm, vf, vsubq_f16(vone, vf));
  
      vst1q_f16(o, vf); o += 8;
    }
  }
  