src/f32-sigmoid/neon-p5.c.in - platform/external/XNNPACK - Gitiles

 // 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>


 void xnn_f32_sigmoid_ukernel__${"neonfma" if FMA else "neon"}_rr${RR_STEPS}_p5_${DIV_ALGO}_x${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(0x1.8000FEp23f);
   // 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 = vmovq_n_f32(-0x1.715476p+0f);
   $if RR_STEPS == 1:
     const float32x4_t vln2 = vmovq_n_f32(0x1.62E43p-1f);
   $else:
     $if FMA:
       const float32x4_t vln2_hi = vmovq_n_f32(0x1.62E43p-1f);
       const float32x4_t vln2_lo = vmovq_n_f32(-0x1.05C61p-29f);
     $else:
       // Last 7 bits are zeroes
       const float32x4_t vln2_hi = vmovq_n_f32(0x1.62E400p-1f);
       const float32x4_t vln2_lo = vmovq_n_f32(0x1.7F7D1Cp-20f);
   const float32x4_t vone = vmovq_n_f32(1.0f);

   const float32x4_t vc1 = vmovq_n_f32(-0x1.FFFFF6p-1f);
   const float32x4_t vc2 = vmovq_n_f32(0x1.FFFDC6p-2f);
   const float32x4_t vc3 = vmovq_n_f32(-0x1.555A80p-3f);
   const float32x4_t vc4 = vmovq_n_f32(0x1.573A1Ap-5f);
   const float32x4_t vc5 = vmovq_n_f32(-0x1.0F9F9Cp-7f);

   $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 / log(2)).
       // We do it by adding a large number (magic bias), which cause rounding of 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| <= 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.
       $for N in range(0, BATCH_TILE, 4):
         float32x4_t vn${ABC[N:N+4]} = ${VMULADDQ_F32}(vmagic_bias, vz${ABC[N:N+4]}, vminus_log2e);

       // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
       // -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
       $for N in range(0, BATCH_TILE, 4):
         const float32x4_t vs${ABC[N:N+4]} = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn${ABC[N:N+4]}), 23));

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

       // Compute reduced argument -t := -z - n * log(2) = -(z + n * log(2)).
       $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);
       $else:
         // Use Cody-Waite range reduction method (note two constants to represent log(2)) 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_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_lo);

       // Compute degree-5 polynomial approxiatmion for exp(-t) on [-log(2)/2, log(2)/2].
       $for N in range(0, BATCH_TILE, 4):
         float32x4_t vp${ABC[N:N+4]} = ${VMULADDQ_F32}(vc4, vc5, vt${ABC[N:N+4]});

       $for N in range(0, BATCH_TILE, 4):
         vp${ABC[N:N+4]} = ${VMULADDQ_F32}(vc3, vp${ABC[N:N+4]}, vt${ABC[N:N+4]});

       $for N in range(0, BATCH_TILE, 4):
         vp${ABC[N:N+4]} = ${VMULADDQ_F32}(vc2, vp${ABC[N:N+4]}, vt${ABC[N:N+4]});

       $for N in range(0, BATCH_TILE, 4):
         vp${ABC[N:N+4]} = ${VMULADDQ_F32}(vc1, vp${ABC[N:N+4]}, vt${ABC[N:N+4]});

       // 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
       $for N in range(0, BATCH_TILE, 4):
         vt${ABC[N:N+4]} = vmulq_f32(vt${ABC[N:N+4]}, vs${ABC[N:N+4]});

       $for N in range(0, BATCH_TILE, 4):
         float32x4_t ve${ABC[N:N+4]} = ${VMULADDQ_F32}(vs${ABC[N:N+4]}, vp${ABC[N:N+4]}, vt${ABC[N:N+4]});

       // Denominator of the sigmoid fraction: 1.0 + exp(-z)
       $for N in range(0, BATCH_TILE, 4):
         float32x4_t vd${ABC[N:N+4]} = vaddq_f32(ve${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(ve${ABC[N:N+4]}, vd${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(ve${ABC[N:N+4]}, vr${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 / log(2)).
     // We do it by adding a large number (magic bias), which cause rounding of 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| <= 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.
     float32x4_t vn = ${VMULADDQ_F32}(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.
     // -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
     const float32x4_t vs = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn), 23));

     // Subtract the large number back to get final n := round(-z / log(2)).
     vn = vsubq_f32(vn, vmagic_bias);

     // Compute reduced argument -t := -z - n * log(2) = -(z + n * log(2)).
     $if RR_STEPS == 1:
       float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2);
     $else:
       // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
       float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_hi);
       vt = ${VMULADDQ_F32}(vt, vn, vln2_lo);

     // Compute degree-5 polynomial approxiatmion for exp(-t) on [-log(2)/2, log(2)/2].
     float32x4_t vp = ${VMULADDQ_F32}(vc4, vc5, vt);
     vp = ${VMULADDQ_F32}(vc3, vp, vt);
     vp = ${VMULADDQ_F32}(vc2, vp, vt);
     vp = ${VMULADDQ_F32}(vc1, vp, vt);

     // 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 = vmulq_f32(vt, vs);
     float32x4_t ve = ${VMULADDQ_F32}(vs, vp, vt);

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

     $if DIV_ALGO == "div":
       // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
       float32x4_t vf = vdivq_f32(ve, 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(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_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 / log(2)).
     // We do it by adding a large number (magic bias), which cause rounding of 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| <= 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.
     float32x4_t vn = ${VMULADDQ_F32}(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.
     // -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
     const float32x4_t vs = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn), 23));

     // Subtract the large number back to get final n := round(-z / log(2)).
     vn = vsubq_f32(vn, vmagic_bias);

     // Compute reduced argument -t := -z - n * log(2) = -(z + n * log(2)).
     $if RR_STEPS == 1:
       float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2);
     $else:
       // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
       float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_hi);
       vt = ${VMULADDQ_F32}(vt, vn, vln2_lo);

     // Compute degree-5 polynomial approxiatmion for exp(-t) on [-log(2)/2, log(2)/2].
     float32x4_t vp = ${VMULADDQ_F32}(vc4, vc5, vt);
     vp = ${VMULADDQ_F32}(vc3, vp, vt);
     vp = ${VMULADDQ_F32}(vc2, vp, vt);
     vp = ${VMULADDQ_F32}(vc1, vp, vt);

     // 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 = vmulq_f32(vt, vs);
     float32x4_t ve = ${VMULADDQ_F32}(vs, vp, vt);

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

     $if DIV_ALGO == "div":
       // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
       float32x4_t vf = vdivq_f32(ve, 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(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_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>


	void xnn_f32_sigmoid_ukernel__${"neonfma" if FMA else "neon"}_rr${RR_STEPS}_p5_${DIV_ALGO}_x${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(0x1.8000FEp23f);
	// 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 = vmovq_n_f32(-0x1.715476p+0f);
	$if RR_STEPS == 1:
	const float32x4_t vln2 = vmovq_n_f32(0x1.62E43p-1f);
	$else:
	$if FMA:
	const float32x4_t vln2_hi = vmovq_n_f32(0x1.62E43p-1f);
	const float32x4_t vln2_lo = vmovq_n_f32(-0x1.05C61p-29f);
	$else:
	// Last 7 bits are zeroes
	const float32x4_t vln2_hi = vmovq_n_f32(0x1.62E400p-1f);
	const float32x4_t vln2_lo = vmovq_n_f32(0x1.7F7D1Cp-20f);
	const float32x4_t vone = vmovq_n_f32(1.0f);

	const float32x4_t vc1 = vmovq_n_f32(-0x1.FFFFF6p-1f);
	const float32x4_t vc2 = vmovq_n_f32(0x1.FFFDC6p-2f);
	const float32x4_t vc3 = vmovq_n_f32(-0x1.555A80p-3f);
	const float32x4_t vc4 = vmovq_n_f32(0x1.573A1Ap-5f);
	const float32x4_t vc5 = vmovq_n_f32(-0x1.0F9F9Cp-7f);

	$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 / log(2)).
	// We do it by adding a large number (magic bias), which cause rounding of 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\| <= 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.
	$for N in range(0, BATCH_TILE, 4):
	float32x4_t vn${ABC[N:N+4]} = ${VMULADDQ_F32}(vmagic_bias, vz${ABC[N:N+4]}, vminus_log2e);

	// Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
	// -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
	$for N in range(0, BATCH_TILE, 4):
	const float32x4_t vs${ABC[N:N+4]} = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn${ABC[N:N+4]}), 23));

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

	// Compute reduced argument -t := -z - n * log(2) = -(z + n * log(2)).
	$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);
	$else:
	// Use Cody-Waite range reduction method (note two constants to represent log(2)) 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_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_lo);

	// Compute degree-5 polynomial approxiatmion for exp(-t) on [-log(2)/2, log(2)/2].
	$for N in range(0, BATCH_TILE, 4):
	float32x4_t vp${ABC[N:N+4]} = ${VMULADDQ_F32}(vc4, vc5, vt${ABC[N:N+4]});

	$for N in range(0, BATCH_TILE, 4):
	vp${ABC[N:N+4]} = ${VMULADDQ_F32}(vc3, vp${ABC[N:N+4]}, vt${ABC[N:N+4]});

	$for N in range(0, BATCH_TILE, 4):
	vp${ABC[N:N+4]} = ${VMULADDQ_F32}(vc2, vp${ABC[N:N+4]}, vt${ABC[N:N+4]});

	$for N in range(0, BATCH_TILE, 4):
	vp${ABC[N:N+4]} = ${VMULADDQ_F32}(vc1, vp${ABC[N:N+4]}, vt${ABC[N:N+4]});

	// 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
	$for N in range(0, BATCH_TILE, 4):
	vt${ABC[N:N+4]} = vmulq_f32(vt${ABC[N:N+4]}, vs${ABC[N:N+4]});

	$for N in range(0, BATCH_TILE, 4):
	float32x4_t ve${ABC[N:N+4]} = ${VMULADDQ_F32}(vs${ABC[N:N+4]}, vp${ABC[N:N+4]}, vt${ABC[N:N+4]});

	// Denominator of the sigmoid fraction: 1.0 + exp(-z)
	$for N in range(0, BATCH_TILE, 4):
	float32x4_t vd${ABC[N:N+4]} = vaddq_f32(ve${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(ve${ABC[N:N+4]}, vd${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(ve${ABC[N:N+4]}, vr${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 / log(2)).
	// We do it by adding a large number (magic bias), which cause rounding of 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\| <= 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.
	float32x4_t vn = ${VMULADDQ_F32}(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.
	// -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
	const float32x4_t vs = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn), 23));

	// Subtract the large number back to get final n := round(-z / log(2)).
	vn = vsubq_f32(vn, vmagic_bias);

	// Compute reduced argument -t := -z - n * log(2) = -(z + n * log(2)).
	$if RR_STEPS == 1:
	float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2);
	$else:
	// Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
	float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_hi);
	vt = ${VMULADDQ_F32}(vt, vn, vln2_lo);

	// Compute degree-5 polynomial approxiatmion for exp(-t) on [-log(2)/2, log(2)/2].
	float32x4_t vp = ${VMULADDQ_F32}(vc4, vc5, vt);
	vp = ${VMULADDQ_F32}(vc3, vp, vt);
	vp = ${VMULADDQ_F32}(vc2, vp, vt);
	vp = ${VMULADDQ_F32}(vc1, vp, vt);

	// 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 = vmulq_f32(vt, vs);
	float32x4_t ve = ${VMULADDQ_F32}(vs, vp, vt);

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

	$if DIV_ALGO == "div":
	// Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
	float32x4_t vf = vdivq_f32(ve, 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(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_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 / log(2)).
	// We do it by adding a large number (magic bias), which cause rounding of 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\| <= 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.
	float32x4_t vn = ${VMULADDQ_F32}(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.
	// -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
	const float32x4_t vs = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn), 23));

	// Subtract the large number back to get final n := round(-z / log(2)).
	vn = vsubq_f32(vn, vmagic_bias);

	// Compute reduced argument -t := -z - n * log(2) = -(z + n * log(2)).
	$if RR_STEPS == 1:
	float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2);
	$else:
	// Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
	float32x4_t vt = ${VMULADDQ_F32}(vz, vn, vln2_hi);
	vt = ${VMULADDQ_F32}(vt, vn, vln2_lo);

	// Compute degree-5 polynomial approxiatmion for exp(-t) on [-log(2)/2, log(2)/2].
	float32x4_t vp = ${VMULADDQ_F32}(vc4, vc5, vt);
	vp = ${VMULADDQ_F32}(vc3, vp, vt);
	vp = ${VMULADDQ_F32}(vc2, vp, vt);
	vp = ${VMULADDQ_F32}(vc1, vp, vt);

	// 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 = vmulq_f32(vt, vs);
	float32x4_t ve = ${VMULADDQ_F32}(vs, vp, vt);

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

	$if DIV_ALGO == "div":
	// Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
	float32x4_t vf = vdivq_f32(ve, 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(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_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);
	}
	}
	}