src/f32-sigmoid/neon-lut2048-p1.c.in - platform/external/XNNPACK - Git at Google

 // 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.

 $assert BATCH_TILE % 4 == 0
 $assert BATCH_TILE >= 4
 $assert RR_STEPS in [1, 2]
 $assert DIV_ALGO in ["div", "nr2fma", "nr2recps", "nr1recps1fma"]
 $ABC = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
 $VMULADDQ_F32 = "vfmaq_f32" if FMA else "vmlaq_f32"
 #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_2048[2048];

 void xnn_f32_sigmoid_ukernel__${"neonfma" if FMA else "neon"}_rr${RR_STEPS}_lut2048_p1_${DIV_ALGO}_x${BATCH_TILE}(
     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_x2048  = vmovq_n_f32(-0x1.715476p11f);
   $if RR_STEPS == 1:
     const float32x4_t vln2_o2048 = vmovq_n_f32(0x1.62E43p-12f);
   $else:
     $if FMA:
       const float32x4_t vln2_o2048_hi = vmovq_n_f32(0x1.62E43p-12f);
       const float32x4_t vln2_o2048_lo = vmovq_n_f32(-0x1.05C61p-40f);
     $else:
       // Last 18 bits are zeroes
       const float32x4_t vln2_o2048_hi = vmovq_n_f32(0x1.600000p-12f);
       const float32x4_t vln2_o2048_lo = vmovq_n_f32(0x1.7217F8p-19f);
   const float32x4_t vone = vmovq_n_f32(1.0f);

   const float32x4_t vc1 = vmovq_n_f32(-0x1.FFFFFEp-1f);

   const int32x4_t vindex_mask = vmovq_n_s32(INT32_C(0x7FF));

   $if BATCH_TILE > 4:
     for (; n >= ${BATCH_TILE} * sizeof(float); n -= ${BATCH_TILE} * sizeof(float)) {
       $for N in range(0, BATCH_TILE, 4):
         const float32x4_t vx${ABC[N:N+4]} = 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.
       $for N in range(0, BATCH_TILE, 4):
         const float32x4_t vz${ABC[N:N+4]} = vabsq_f32(vx${ABC[N:N+4]});

       // Compute reduced argument n := round(-z * 2048 / 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 * 2048 / log(2)| <= 2**22, i.e.
       // |z| <= 0x1.62E43p+10 = 1419.5654296875), 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.
       $for N in range(0, BATCH_TILE, 4):
         float32x4_t vn${ABC[N:N+4]} = ${VMULADDQ_F32}(vmagic_bias, vz${ABC[N:N+4]}, vminus_log2e_x2048);

       // Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is
       // normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) =
       // = 2**e * 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps:
       // 1. Fetch 2**(n / 2048 - e) = 2**(n % 2048) 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 11:19 of n and shift it into bits 23:31 (position of floating-point exponent).
       $for N in range(0, BATCH_TILE, 4):
         const int32x4_t ve${ABC[N:N+4]} = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn${ABC[N:N+4]}), vmovq_n_s32(INT32_C(0x7FF))), 12);

       // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
       $for N in range(0, BATCH_TILE, 4):
         const uint64x2_t vidx${ABC[N:N+4]} = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn${ABC[N:N+4]}), vindex_mask));

       $for N in range(0, BATCH_TILE, 4):
         const uint64_t vidx${ABC[N:N+2]} = vgetq_lane_u64(vidx${ABC[N:N+4]}, 0);
         const uint64_t vidx${ABC[N+2:N+4]} = vgetq_lane_u64(vidx${ABC[N:N+4]}, 1);
         float32x2_t vl${ABC[N:N+2]} = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx${ABC[N:N+2]}]);
         float32x2_t vl${ABC[N+2:N+4]} = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx${ABC[N+2:N+4]}]);

       $for N in range(0, BATCH_TILE, 4):
         vl${ABC[N:N+2]} = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx${ABC[N:N+2]} >> 32)], vl${ABC[N:N+2]}, 1);
         vl${ABC[N+2:N+4]} = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx${ABC[N+2:N+4]} >> 32)], vl${ABC[N+2:N+4]}, 1);
         const float32x4_t vl${ABC[N:N+4]} = vcombine_f32(vl${ABC[N:N+2]}, vl${ABC[N+2:N+4]});

       // Adjust exponent of the value l fetched from the table to get the final s value.
       $for N in range(0, BATCH_TILE, 4):
         const float32x4_t vs${ABC[N:N+4]} = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl${ABC[N:N+4]}), ve${ABC[N:N+4]}));

       // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
       $for N in range(0, BATCH_TILE, 4):
         vn${ABC[N:N+4]} = vsubq_f32(vn${ABC[N:N+4]}, vmagic_bias);

       // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
       $if RR_STEPS == 1:
         $for N in range(0, BATCH_TILE, 4):
           float32x4_t vt${ABC[N:N+4]} = ${VMULADDQ_F32}(vz${ABC[N:N+4]}, vn${ABC[N:N+4]}, vln2_o2048);
       $else:
         // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
         $for N in range(0, BATCH_TILE, 4):
           float32x4_t vt${ABC[N:N+4]} = ${VMULADDQ_F32}(vz${ABC[N:N+4]}, vn${ABC[N:N+4]}, vln2_o2048_hi);

         $for N in range(0, BATCH_TILE, 4):
           vt${ABC[N:N+4]} = ${VMULADDQ_F32}(vt${ABC[N:N+4]}, vn${ABC[N:N+4]}, vln2_o2048_lo);

       // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]:
       //   P1(t) = 1 + t * c1
       $for N in range(0, BATCH_TILE, 4):
         const float32x4_t vp${ABC[N:N+4]} = vmulq_f32(vt${ABC[N:N+4]}, vc1);

       // Reconstruct the exp(-z) value:
       //   y = s * (1 + t * c1)
       //     = s + s * (t * c1))
       //     = s + s * p
       $for N in range(0, BATCH_TILE, 4):
         const float32x4_t vy${ABC[N:N+4]} = ${VMULADDQ_F32}(vs${ABC[N:N+4]}, vs${ABC[N:N+4]}, vp${ABC[N:N+4]});

       // Denominator of the sigmoid fraction: 1.0 + exp(-z)
       $for N in range(0, BATCH_TILE, 4):
         const float32x4_t vd${ABC[N:N+4]} = vaddq_f32(vy${ABC[N:N+4]}, vone);

       $if DIV_ALGO == "div":
         // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
         $for N in range(0, BATCH_TILE, 4):
           float32x4_t vf${ABC[N:N+4]} = vdivq_f32(vy${ABC[N:N+4]}, vd${ABC[N:N+4]});
       $else:
         // 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.
         $for N in range(0, BATCH_TILE, 4):
           float32x4_t vr${ABC[N:N+4]} = vrecpeq_f32(vd${ABC[N:N+4]});

         $if DIV_ALGO == "nr2fma":
           $for N in range(0, BATCH_TILE, 4):
             vr${ABC[N:N+4]} = vfmaq_f32(vr${ABC[N:N+4]}, vr${ABC[N:N+4]}, vfmsq_f32(vone, vr${ABC[N:N+4]}, vd${ABC[N:N+4]}));
         $else:
           $for N in range(0, BATCH_TILE, 4):
             vr${ABC[N:N+4]} = vmulq_f32(vr${ABC[N:N+4]}, vrecpsq_f32(vr${ABC[N:N+4]}, vd${ABC[N:N+4]}));

         $if DIV_ALGO == "nr2recps":
           $for N in range(0, BATCH_TILE, 4):
             vr${ABC[N:N+4]} = vmulq_f32(vr${ABC[N:N+4]}, vrecpsq_f32(vr${ABC[N:N+4]}, vd${ABC[N:N+4]}));
         $else:
           $for N in range(0, BATCH_TILE, 4):
             vr${ABC[N:N+4]} = vfmaq_f32(vr${ABC[N:N+4]}, vr${ABC[N:N+4]}, vfmsq_f32(vone, vr${ABC[N:N+4]}, vd${ABC[N:N+4]}));

         // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
         $for N in range(0, BATCH_TILE, 4):
           float32x4_t vf${ABC[N:N+4]} = vmulq_f32(vy${ABC[N:N+4]}, vr${ABC[N:N+4]});

       // For inputs below denormal cutoff, replace output with +0.0f.
       // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
       $for N in range(0, BATCH_TILE, 4):
         vf${ABC[N:N+4]} = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf${ABC[N:N+4]}), vcagtq_f32(vx${ABC[N:N+4]}, vdenorm_cutoff)));

       // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
       $for N in range(0, BATCH_TILE, 4):
         const uint32x4_t vm${ABC[N:N+4]} = vcltq_f32(vx${ABC[N:N+4]}, vmovq_n_f32(0.0f));

       $for N in range(0, BATCH_TILE, 4):
         vf${ABC[N:N+4]} = vbslq_f32(vm${ABC[N:N+4]}, vf${ABC[N:N+4]}, vsubq_f32(vone, vf${ABC[N:N+4]}));

       $for N in range(0, BATCH_TILE, 4):
         vst1q_f32(y, vf${ABC[N:N+4]}); y += 4;
     }
   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 * 2048 / 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 * 2048 / log(2)| <= 2**22, i.e.
     // |z| <= 0x1.62E43p+10 = 1419.5654296875), 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 = ${VMULADDQ_F32}(vmagic_bias, vz, vminus_log2e_x2048);

     // Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is
     // normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) =
     // = 2**e * 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps:
     // 1. Fetch 2**(n / 2048 - e) = 2**(n % 2048) from exp2_k_over_2048_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 11:19 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(0x7FF))), 12);

     // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
     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_2048[(uint32_t) vidx_lo]);
     float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_hi]);
     vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_lo >> 32)], vl_lo, 1);
     vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(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 exp2_k_over_2048_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 * 2048 / log(2)) as a floating-point number.
     vn = vsubq_f32(vn, vmagic_bias);

     // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
     $if RR_STEPS == 1:
       float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_o2048);
     $else:
       // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
       float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_o2048_hi);
       vt = ${VMULADDQ_F32}(vt, vn, vln2_o2048_lo);

     // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]:
     //   P1(t) = 1 + t * c1
     const float32x4_t vp = vmulq_f32(vt, vc1);

     // Reconstruct the exp(-z) value:
     //   y = s * (1 + t * c1)
     //     = s + s * (t * c1))
     //     = s + s * p
     const float32x4_t vy = ${VMULADDQ_F32}(vs, vs, vp);

     // Denominator of the sigmoid fraction: 1.0 + exp(-z)
     const float32x4_t vd = vaddq_f32(vy, vone);

     $if DIV_ALGO == "div":
       // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
       float32x4_t vf = vdivq_f32(vy, vd);
     $else:
       // 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);

       $if DIV_ALGO == "nr2fma":
         vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, vr, vd));
       $else:
         vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));

       $if DIV_ALGO == "nr2recps":
         vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
       $else:
         vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, 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 * 2048 / 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 * 2048 / log(2)| <= 2**22, i.e.
     // |z| <= 0x1.62E43p+10 = 1419.5654296875), 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 = ${VMULADDQ_F32}(vmagic_bias, vz, vminus_log2e_x2048);

     // Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is
     // normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) =
     // = 2**e * 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps:
     // 1. Fetch 2**(n / 2048 - e) = 2**(n % 2048) from exp2_k_over_2048_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 11:19 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(0x7FF))), 12);

     // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
     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_2048[(uint32_t) vidx_lo]);
     float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_hi]);
     vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_lo >> 32)], vl_lo, 1);
     vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(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 exp2_k_over_2048_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 * 2048 / log(2)) as a floating-point number.
     vn = vsubq_f32(vn, vmagic_bias);

     // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
     $if RR_STEPS == 1:
       float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_o2048);
     $else:
       // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
       float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_o2048_hi);
       vt = ${VMULADDQ_F32}(vt, vn, vln2_o2048_lo);

     // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]:
     //   P1(t) = 1 + t * c1
     const float32x4_t vp = vmulq_f32(vt, vc1);

     // Reconstruct the exp(-z) value:
     //   y = s * (1 + t * c1)
     //     = s + s * (t * c1))
     //     = s + s * p
     const float32x4_t vy = ${VMULADDQ_F32}(vs, vs, vp);

     // Denominator of the sigmoid fraction: 1.0 + exp(-z)
     const float32x4_t vd = vaddq_f32(vy, vone);

     $if DIV_ALGO == "div":
       // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
       float32x4_t vf = vdivq_f32(vy, vd);
     $else:
       // 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);

       $if DIV_ALGO == "nr2fma":
         vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, vr, vd));
       $else:
         vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));

       $if DIV_ALGO == "nr2recps":
         vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
       $else:
         vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, 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);
     }
   }
 }
	// 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.

	$assert BATCH_TILE % 4 == 0
	$assert BATCH_TILE >= 4
	$assert RR_STEPS in [1, 2]
	$assert DIV_ALGO in ["div", "nr2fma", "nr2recps", "nr1recps1fma"]
	$ABC = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
	$VMULADDQ_F32 = "vfmaq_f32" if FMA else "vmlaq_f32"
	#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_2048[2048];

	void xnn_f32_sigmoid_ukernel__${"neonfma" if FMA else "neon"}_rr${RR_STEPS}_lut2048_p1_${DIV_ALGO}_x${BATCH_TILE}(
	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_x2048 = vmovq_n_f32(-0x1.715476p11f);
	$if RR_STEPS == 1:
	const float32x4_t vln2_o2048 = vmovq_n_f32(0x1.62E43p-12f);
	$else:
	$if FMA:
	const float32x4_t vln2_o2048_hi = vmovq_n_f32(0x1.62E43p-12f);
	const float32x4_t vln2_o2048_lo = vmovq_n_f32(-0x1.05C61p-40f);
	$else:
	// Last 18 bits are zeroes
	const float32x4_t vln2_o2048_hi = vmovq_n_f32(0x1.600000p-12f);
	const float32x4_t vln2_o2048_lo = vmovq_n_f32(0x1.7217F8p-19f);
	const float32x4_t vone = vmovq_n_f32(1.0f);

	const float32x4_t vc1 = vmovq_n_f32(-0x1.FFFFFEp-1f);

	const int32x4_t vindex_mask = vmovq_n_s32(INT32_C(0x7FF));

	$if BATCH_TILE > 4:
	for (; n >= ${BATCH_TILE} * sizeof(float); n -= ${BATCH_TILE} * sizeof(float)) {
	$for N in range(0, BATCH_TILE, 4):
	const float32x4_t vx${ABC[N:N+4]} = 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.
	$for N in range(0, BATCH_TILE, 4):
	const float32x4_t vz${ABC[N:N+4]} = vabsq_f32(vx${ABC[N:N+4]});

	// Compute reduced argument n := round(-z * 2048 / 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 * 2048 / log(2)\| <= 2**22, i.e.
	// \|z\| <= 0x1.62E43p+10 = 1419.5654296875), 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.
	$for N in range(0, BATCH_TILE, 4):
	float32x4_t vn${ABC[N:N+4]} = ${VMULADDQ_F32}(vmagic_bias, vz${ABC[N:N+4]}, vminus_log2e_x2048);

	// Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is
	// normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) =
	// = 2*e 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps:
	// 1. Fetch 2(n / 2048 - e) = 2(n % 2048) 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 11:19 of n and shift it into bits 23:31 (position of floating-point exponent).
	$for N in range(0, BATCH_TILE, 4):
	const int32x4_t ve${ABC[N:N+4]} = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn${ABC[N:N+4]}), vmovq_n_s32(INT32_C(0x7FF))), 12);

	// Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
	$for N in range(0, BATCH_TILE, 4):
	const uint64x2_t vidx${ABC[N:N+4]} = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn${ABC[N:N+4]}), vindex_mask));

	$for N in range(0, BATCH_TILE, 4):
	const uint64_t vidx${ABC[N:N+2]} = vgetq_lane_u64(vidx${ABC[N:N+4]}, 0);
	const uint64_t vidx${ABC[N+2:N+4]} = vgetq_lane_u64(vidx${ABC[N:N+4]}, 1);
	float32x2_t vl${ABC[N:N+2]} = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx${ABC[N:N+2]}]);
	float32x2_t vl${ABC[N+2:N+4]} = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx${ABC[N+2:N+4]}]);

	$for N in range(0, BATCH_TILE, 4):
	vl${ABC[N:N+2]} = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx${ABC[N:N+2]} >> 32)], vl${ABC[N:N+2]}, 1);
	vl${ABC[N+2:N+4]} = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx${ABC[N+2:N+4]} >> 32)], vl${ABC[N+2:N+4]}, 1);
	const float32x4_t vl${ABC[N:N+4]} = vcombine_f32(vl${ABC[N:N+2]}, vl${ABC[N+2:N+4]});

	// Adjust exponent of the value l fetched from the table to get the final s value.
	$for N in range(0, BATCH_TILE, 4):
	const float32x4_t vs${ABC[N:N+4]} = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl${ABC[N:N+4]}), ve${ABC[N:N+4]}));

	// Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
	$for N in range(0, BATCH_TILE, 4):
	vn${ABC[N:N+4]} = vsubq_f32(vn${ABC[N:N+4]}, vmagic_bias);

	// Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
	$if RR_STEPS == 1:
	$for N in range(0, BATCH_TILE, 4):
	float32x4_t vt${ABC[N:N+4]} = ${VMULADDQ_F32}(vz${ABC[N:N+4]}, vn${ABC[N:N+4]}, vln2_o2048);
	$else:
	// Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
	$for N in range(0, BATCH_TILE, 4):
	float32x4_t vt${ABC[N:N+4]} = ${VMULADDQ_F32}(vz${ABC[N:N+4]}, vn${ABC[N:N+4]}, vln2_o2048_hi);

	$for N in range(0, BATCH_TILE, 4):
	vt${ABC[N:N+4]} = ${VMULADDQ_F32}(vt${ABC[N:N+4]}, vn${ABC[N:N+4]}, vln2_o2048_lo);

	// Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]:
	// P1(t) = 1 + t * c1
	$for N in range(0, BATCH_TILE, 4):
	const float32x4_t vp${ABC[N:N+4]} = vmulq_f32(vt${ABC[N:N+4]}, vc1);

	// Reconstruct the exp(-z) value:
	// y = s * (1 + t * c1)
	// = s + s * (t * c1))
	// = s + s * p
	$for N in range(0, BATCH_TILE, 4):
	const float32x4_t vy${ABC[N:N+4]} = ${VMULADDQ_F32}(vs${ABC[N:N+4]}, vs${ABC[N:N+4]}, vp${ABC[N:N+4]});

	// Denominator of the sigmoid fraction: 1.0 + exp(-z)
	$for N in range(0, BATCH_TILE, 4):
	const float32x4_t vd${ABC[N:N+4]} = vaddq_f32(vy${ABC[N:N+4]}, vone);

	$if DIV_ALGO == "div":
	// Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
	$for N in range(0, BATCH_TILE, 4):
	float32x4_t vf${ABC[N:N+4]} = vdivq_f32(vy${ABC[N:N+4]}, vd${ABC[N:N+4]});
	$else:
	// 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.
	$for N in range(0, BATCH_TILE, 4):
	float32x4_t vr${ABC[N:N+4]} = vrecpeq_f32(vd${ABC[N:N+4]});

	$if DIV_ALGO == "nr2fma":
	$for N in range(0, BATCH_TILE, 4):
	vr${ABC[N:N+4]} = vfmaq_f32(vr${ABC[N:N+4]}, vr${ABC[N:N+4]}, vfmsq_f32(vone, vr${ABC[N:N+4]}, vd${ABC[N:N+4]}));
	$else:
	$for N in range(0, BATCH_TILE, 4):
	vr${ABC[N:N+4]} = vmulq_f32(vr${ABC[N:N+4]}, vrecpsq_f32(vr${ABC[N:N+4]}, vd${ABC[N:N+4]}));

	$if DIV_ALGO == "nr2recps":
	$for N in range(0, BATCH_TILE, 4):
	vr${ABC[N:N+4]} = vmulq_f32(vr${ABC[N:N+4]}, vrecpsq_f32(vr${ABC[N:N+4]}, vd${ABC[N:N+4]}));
	$else:
	$for N in range(0, BATCH_TILE, 4):
	vr${ABC[N:N+4]} = vfmaq_f32(vr${ABC[N:N+4]}, vr${ABC[N:N+4]}, vfmsq_f32(vone, vr${ABC[N:N+4]}, vd${ABC[N:N+4]}));

	// Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
	$for N in range(0, BATCH_TILE, 4):
	float32x4_t vf${ABC[N:N+4]} = vmulq_f32(vy${ABC[N:N+4]}, vr${ABC[N:N+4]});

	// For inputs below denormal cutoff, replace output with +0.0f.
	// Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
	$for N in range(0, BATCH_TILE, 4):
	vf${ABC[N:N+4]} = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf${ABC[N:N+4]}), vcagtq_f32(vx${ABC[N:N+4]}, vdenorm_cutoff)));

	// Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
	$for N in range(0, BATCH_TILE, 4):
	const uint32x4_t vm${ABC[N:N+4]} = vcltq_f32(vx${ABC[N:N+4]}, vmovq_n_f32(0.0f));

	$for N in range(0, BATCH_TILE, 4):
	vf${ABC[N:N+4]} = vbslq_f32(vm${ABC[N:N+4]}, vf${ABC[N:N+4]}, vsubq_f32(vone, vf${ABC[N:N+4]}));

	$for N in range(0, BATCH_TILE, 4):
	vst1q_f32(y, vf${ABC[N:N+4]}); y += 4;
	}
	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 * 2048 / 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 * 2048 / log(2)\| <= 2**22, i.e.
	// \|z\| <= 0x1.62E43p+10 = 1419.5654296875), 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 = ${VMULADDQ_F32}(vmagic_bias, vz, vminus_log2e_x2048);

	// Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is
	// normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) =
	// = 2*e 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps:
	// 1. Fetch 2(n / 2048 - e) = 2(n % 2048) from exp2_k_over_2048_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 11:19 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(0x7FF))), 12);

	// Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
	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_2048[(uint32_t) vidx_lo]);
	float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_hi]);
	vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_lo >> 32)], vl_lo, 1);
	vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(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 exp2_k_over_2048_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 * 2048 / log(2)) as a floating-point number.
	vn = vsubq_f32(vn, vmagic_bias);

	// Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
	$if RR_STEPS == 1:
	float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_o2048);
	$else:
	// Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
	float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_o2048_hi);
	vt = ${VMULADDQ_F32}(vt, vn, vln2_o2048_lo);

	// Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]:
	// P1(t) = 1 + t * c1
	const float32x4_t vp = vmulq_f32(vt, vc1);

	// Reconstruct the exp(-z) value:
	// y = s * (1 + t * c1)
	// = s + s * (t * c1))
	// = s + s * p
	const float32x4_t vy = ${VMULADDQ_F32}(vs, vs, vp);

	// Denominator of the sigmoid fraction: 1.0 + exp(-z)
	const float32x4_t vd = vaddq_f32(vy, vone);

	$if DIV_ALGO == "div":
	// Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
	float32x4_t vf = vdivq_f32(vy, vd);
	$else:
	// 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);

	$if DIV_ALGO == "nr2fma":
	vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, vr, vd));
	$else:
	vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));

	$if DIV_ALGO == "nr2recps":
	vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
	$else:
	vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, 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 * 2048 / 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 * 2048 / log(2)\| <= 2**22, i.e.
	// \|z\| <= 0x1.62E43p+10 = 1419.5654296875), 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 = ${VMULADDQ_F32}(vmagic_bias, vz, vminus_log2e_x2048);

	// Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is
	// normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) =
	// = 2*e 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps:
	// 1. Fetch 2(n / 2048 - e) = 2(n % 2048) from exp2_k_over_2048_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 11:19 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(0x7FF))), 12);

	// Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
	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_2048[(uint32_t) vidx_lo]);
	float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_hi]);
	vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_lo >> 32)], vl_lo, 1);
	vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(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 exp2_k_over_2048_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 * 2048 / log(2)) as a floating-point number.
	vn = vsubq_f32(vn, vmagic_bias);

	// Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
	$if RR_STEPS == 1:
	float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_o2048);
	$else:
	// Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
	float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_o2048_hi);
	vt = ${VMULADDQ_F32}(vt, vn, vln2_o2048_lo);

	// Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]:
	// P1(t) = 1 + t * c1
	const float32x4_t vp = vmulq_f32(vt, vc1);

	// Reconstruct the exp(-z) value:
	// y = s * (1 + t * c1)
	// = s + s * (t * c1))
	// = s + s * p
	const float32x4_t vy = ${VMULADDQ_F32}(vs, vs, vp);

	// Denominator of the sigmoid fraction: 1.0 + exp(-z)
	const float32x4_t vd = vaddq_f32(vy, vone);

	$if DIV_ALGO == "div":
	// Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
	float32x4_t vf = vdivq_f32(vy, vd);
	$else:
	// 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);

	$if DIV_ALGO == "nr2fma":
	vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, vr, vd));
	$else:
	vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));

	$if DIV_ALGO == "nr2recps":
	vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));
	$else:
	vr = vfmaq_f32(vr, vr, vfmsq_f32(vone, 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);
	}
	}
	}