src/f32-sigmoid/scalar-lut2048-p1-div.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 >= 1
 $ABC = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
 #include <assert.h>
 #include <math.h>

 #include <xnnpack/common.h>
 #include <xnnpack/vunary.h>

 #include <fp16/bitcasts.h>


 // Note redefine as uint32[] to avoid redundant bitcasts.
 extern XNN_INTERNAL const uint32_t xnn_table_exp2_k_over_2048[2048];

 void xnn_f32_sigmoid_ukernel__scalar_lut2048_p1_div_x${BATCH_TILE}(
     size_t n,
     const float* x,
     float* y,
     const void* params)
 {
   assert(n % sizeof(float) == 0);

   const float vmagic_bias = 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 float vdenorm_cutoff = 0x1.5D589Ep+6f;
   const float vminus_log2e_x2048 = -0x1.715476p11f;
   // Last 18 bits are zeroes
   const float vln2_o2048_hi = 0x1.600000p-12f;
   const float vln2_o2048_lo = 0x1.7217F8p-19f;
   const float vone = 1.0f;

   const float vc1 = -0x1.FFFFFEp-1f;

   const uint32_t vindex_mask = UINT32_C(0x7FF);

   $if BATCH_TILE > 1:
     for (; n >= ${BATCH_TILE} * sizeof(float); n -= ${BATCH_TILE} * sizeof(float)) {
       $for N in range(BATCH_TILE):
         const float vx${N} = x[${N}];
       x += ${BATCH_TILE};

       // 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.
       $for N in range(BATCH_TILE):
         const float vz${N} = fabsf(vx${N});

       // 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.
       $for N in range(BATCH_TILE):
         float vn${N} = vz${N} * vminus_log2e_x2048 + vmagic_bias;

       // 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 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).
       $for N in range(BATCH_TILE):
         const uint32_t ve${N} = (fp32_to_bits(vn${N}) & ~vindex_mask) << 12;

       // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
       $for N in range(BATCH_TILE):
         const uint32_t vidx${N} = fp32_to_bits(vn${N}) & vindex_mask;
       // Adjust exponent of the value l fetched from the table to get the final s value.
       $for N in range(BATCH_TILE):
         const float vs${N} = fp32_from_bits(xnn_table_exp2_k_over_2048[vidx${N}] + ve${N});

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

       // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
       // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
       $for N in range(BATCH_TILE):
         float vt${N} = vn${N} * vln2_o2048_hi + vz${N};

       $for N in range(BATCH_TILE):
         vt${N} = vn${N} * vln2_o2048_lo + vt${N};

       // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/4096, log(2)/4096]:
       //   P1(t) = 1 + t * c1
       $for N in range(BATCH_TILE):
         const float vp${N} = vt${N} * vc1;

       // Reconstruct the exp(-z) value:
       //   y = s * (1 + t * c1)
       //     = s + s * (t * c1))
       //     = s + s * p
       $for N in range(BATCH_TILE):
         const float vy${N} = vp${N} * vs${N} + vs${N};

       // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
       $for N in range(BATCH_TILE):
         float vf${N} = vy${N} / (vy${N} + vone);

       // For inputs above denormal cutoff, replace output with +0.0f.
       // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
       $for N in range(BATCH_TILE):
         if XNN_UNPREDICTABLE(vz${N} > vdenorm_cutoff) {
           vf${N} = 0.0f;
         }

       // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
       $for N in range(BATCH_TILE):
         if XNN_UNPREDICTABLE(vx${N} > 0.0f) {
           vf${N} = vone - vf${N};
         }

       $for N in range(BATCH_TILE):
         y[${N}] = vf${N};
       y += ${BATCH_TILE};
     }
   $if BATCH_TILE == 1:
     do {
       const float vx = *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 float vz = fabsf(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.
       float vn = vz * vminus_log2e_x2048 + vmagic_bias;

       // 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 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 uint32_t ve = (fp32_to_bits(vn) & ~vindex_mask) << 12;

       // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
       const uint32_t vidx = fp32_to_bits(vn) & vindex_mask;
       // Adjust exponent of the value l fetched from the table to get the final s value.
       const float vs = fp32_from_bits(xnn_table_exp2_k_over_2048[vidx] + ve);

       // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
       vn -= vmagic_bias;

       // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
       // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
       float vt = vn * vln2_o2048_hi + vz;
       vt = vn * vln2_o2048_lo + vt;

       // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/4096, log(2)/4096]:
       //   P1(t) = 1 + t * c1
       const float vp = vt * vc1;

       // Reconstruct the exp(-z) value:
       //   y = s * (1 + t * c1)
       //     = s + s * (t * c1))
       //     = s + s * p
       const float vy = vp * vs + vs;

       // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
       float vf = vy / (vy + vone);

       // For inputs above denormal cutoff, replace output with +0.0f.
       // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
       if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
         vf = 0.0f;
       }

       // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
       if XNN_UNPREDICTABLE(vx > 0.0f) {
         vf = vone - vf;
       }

       *y++ = vf;

       n -= sizeof(float);
     } while (n != 0);
   $elif BATCH_TILE == 2:
     if XNN_UNLIKELY(n != 0) {
       const float vx = *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 float vz = fabsf(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.
       float vn = vz * vminus_log2e_x2048 + vmagic_bias;

       // 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 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 uint32_t ve = (fp32_to_bits(vn) & ~vindex_mask) << 12;

       // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
       const uint32_t vidx = fp32_to_bits(vn) & vindex_mask;
       // Adjust exponent of the value l fetched from the table to get the final s value.
       const float vs = fp32_from_bits(xnn_table_exp2_k_over_2048[vidx] + ve);

       // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
       vn -= vmagic_bias;

       // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
       // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
       float vt = vn * vln2_o2048_hi + vz;
       vt = vn * vln2_o2048_lo + vt;

       // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/4096, log(2)/4096]:
       //   P1(t) = 1 + t * c1
       const float vp = vt * vc1;

       // Reconstruct the exp(-z) value:
       //   y = s * (1 + t * c1)
       //     = s + s * (t * c1))
       //     = s + s * p
       const float vy = vp * vs + vs;

       // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
       float vf = vy / (vy + vone);

       // For inputs above denormal cutoff, replace output with +0.0f.
       // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
       if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
         vf = 0.0f;
       }

       // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
       if XNN_UNPREDICTABLE(vx > 0.0f) {
         vf = vone - vf;
       }

       *y = vf;
     }
   $else:
     if XNN_UNLIKELY(n != 0) {
       do {
         const float vx = *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 float vz = fabsf(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.
         float vn = vz * vminus_log2e_x2048 + vmagic_bias;

         // 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 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 uint32_t ve = (fp32_to_bits(vn) & ~vindex_mask) << 12;

         // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
         const uint32_t vidx = fp32_to_bits(vn) & vindex_mask;
         // Adjust exponent of the value l fetched from the table to get the final s value.
         const float vs = fp32_from_bits(xnn_table_exp2_k_over_2048[vidx] + ve);

         // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
         vn -= vmagic_bias;

         // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
         // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
         float vt = vn * vln2_o2048_hi + vz;
         vt = vn * vln2_o2048_lo + vt;

         // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/4096, log(2)/4096]:
         //   P1(t) = 1 + t * c1
         const float vp = vt * vc1;

         // Reconstruct the exp(-z) value:
         //   y = s * (1 + t * c1)
         //     = s + s * (t * c1))
         //     = s + s * p
         const float vy = vp * vs + vs;

         // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
         float vf = vy / (vy + vone);

         // For inputs above denormal cutoff, replace output with +0.0f.
         // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
         if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
           vf = 0.0f;
         }

         // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
         if XNN_UNPREDICTABLE(vx > 0.0f) {
           vf = vone - vf;
         }

         *y++ = vf;

         n -= sizeof(float);
       } while (n != 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 >= 1
	$ABC = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
	#include <assert.h>
	#include <math.h>

	#include <xnnpack/common.h>
	#include <xnnpack/vunary.h>

	#include <fp16/bitcasts.h>


	// Note redefine as uint32[] to avoid redundant bitcasts.
	extern XNN_INTERNAL const uint32_t xnn_table_exp2_k_over_2048[2048];

	void xnn_f32_sigmoid_ukernel__scalar_lut2048_p1_div_x${BATCH_TILE}(
	size_t n,
	const float* x,
	float* y,
	const void* params)
	{
	assert(n % sizeof(float) == 0);

	const float vmagic_bias = 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 float vdenorm_cutoff = 0x1.5D589Ep+6f;
	const float vminus_log2e_x2048 = -0x1.715476p11f;
	// Last 18 bits are zeroes
	const float vln2_o2048_hi = 0x1.600000p-12f;
	const float vln2_o2048_lo = 0x1.7217F8p-19f;
	const float vone = 1.0f;

	const float vc1 = -0x1.FFFFFEp-1f;

	const uint32_t vindex_mask = UINT32_C(0x7FF);

	$if BATCH_TILE > 1:
	for (; n >= ${BATCH_TILE} * sizeof(float); n -= ${BATCH_TILE} * sizeof(float)) {
	$for N in range(BATCH_TILE):
	const float vx${N} = x[${N}];
	x += ${BATCH_TILE};

	// 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.
	$for N in range(BATCH_TILE):
	const float vz${N} = fabsf(vx${N});

	// 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.
	$for N in range(BATCH_TILE):
	float vn${N} = vz${N} * vminus_log2e_x2048 + vmagic_bias;

	// 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 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).
	$for N in range(BATCH_TILE):
	const uint32_t ve${N} = (fp32_to_bits(vn${N}) & ~vindex_mask) << 12;

	// Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
	$for N in range(BATCH_TILE):
	const uint32_t vidx${N} = fp32_to_bits(vn${N}) & vindex_mask;
	// Adjust exponent of the value l fetched from the table to get the final s value.
	$for N in range(BATCH_TILE):
	const float vs${N} = fp32_from_bits(xnn_table_exp2_k_over_2048[vidx${N}] + ve${N});

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

	// Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
	// Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
	$for N in range(BATCH_TILE):
	float vt${N} = vn${N} * vln2_o2048_hi + vz${N};

	$for N in range(BATCH_TILE):
	vt${N} = vn${N} * vln2_o2048_lo + vt${N};

	// Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/4096, log(2)/4096]:
	// P1(t) = 1 + t * c1
	$for N in range(BATCH_TILE):
	const float vp${N} = vt${N} * vc1;

	// Reconstruct the exp(-z) value:
	// y = s * (1 + t * c1)
	// = s + s * (t * c1))
	// = s + s * p
	$for N in range(BATCH_TILE):
	const float vy${N} = vp${N} * vs${N} + vs${N};

	// Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
	$for N in range(BATCH_TILE):
	float vf${N} = vy${N} / (vy${N} + vone);

	// For inputs above denormal cutoff, replace output with +0.0f.
	// Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
	$for N in range(BATCH_TILE):
	if XNN_UNPREDICTABLE(vz${N} > vdenorm_cutoff) {
	vf${N} = 0.0f;
	}

	// Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
	$for N in range(BATCH_TILE):
	if XNN_UNPREDICTABLE(vx${N} > 0.0f) {
	vf${N} = vone - vf${N};
	}

	$for N in range(BATCH_TILE):
	y[${N}] = vf${N};
	y += ${BATCH_TILE};
	}
	$if BATCH_TILE == 1:
	do {
	const float vx = *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 float vz = fabsf(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.
	float vn = vz * vminus_log2e_x2048 + vmagic_bias;

	// 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 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 uint32_t ve = (fp32_to_bits(vn) & ~vindex_mask) << 12;

	// Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
	const uint32_t vidx = fp32_to_bits(vn) & vindex_mask;
	// Adjust exponent of the value l fetched from the table to get the final s value.
	const float vs = fp32_from_bits(xnn_table_exp2_k_over_2048[vidx] + ve);

	// Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
	vn -= vmagic_bias;

	// Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
	// Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
	float vt = vn * vln2_o2048_hi + vz;
	vt = vn * vln2_o2048_lo + vt;

	// Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/4096, log(2)/4096]:
	// P1(t) = 1 + t * c1
	const float vp = vt * vc1;

	// Reconstruct the exp(-z) value:
	// y = s * (1 + t * c1)
	// = s + s * (t * c1))
	// = s + s * p
	const float vy = vp * vs + vs;

	// Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
	float vf = vy / (vy + vone);

	// For inputs above denormal cutoff, replace output with +0.0f.
	// Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
	if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
	vf = 0.0f;
	}

	// Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
	if XNN_UNPREDICTABLE(vx > 0.0f) {
	vf = vone - vf;
	}

	*y++ = vf;

	n -= sizeof(float);
	} while (n != 0);
	$elif BATCH_TILE == 2:
	if XNN_UNLIKELY(n != 0) {
	const float vx = *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 float vz = fabsf(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.
	float vn = vz * vminus_log2e_x2048 + vmagic_bias;

	// 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 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 uint32_t ve = (fp32_to_bits(vn) & ~vindex_mask) << 12;

	// Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
	const uint32_t vidx = fp32_to_bits(vn) & vindex_mask;
	// Adjust exponent of the value l fetched from the table to get the final s value.
	const float vs = fp32_from_bits(xnn_table_exp2_k_over_2048[vidx] + ve);

	// Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
	vn -= vmagic_bias;

	// Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
	// Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
	float vt = vn * vln2_o2048_hi + vz;
	vt = vn * vln2_o2048_lo + vt;

	// Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/4096, log(2)/4096]:
	// P1(t) = 1 + t * c1
	const float vp = vt * vc1;

	// Reconstruct the exp(-z) value:
	// y = s * (1 + t * c1)
	// = s + s * (t * c1))
	// = s + s * p
	const float vy = vp * vs + vs;

	// Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
	float vf = vy / (vy + vone);

	// For inputs above denormal cutoff, replace output with +0.0f.
	// Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
	if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
	vf = 0.0f;
	}

	// Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
	if XNN_UNPREDICTABLE(vx > 0.0f) {
	vf = vone - vf;
	}

	*y = vf;
	}
	$else:
	if XNN_UNLIKELY(n != 0) {
	do {
	const float vx = *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 float vz = fabsf(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.
	float vn = vz * vminus_log2e_x2048 + vmagic_bias;

	// 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 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 uint32_t ve = (fp32_to_bits(vn) & ~vindex_mask) << 12;

	// Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
	const uint32_t vidx = fp32_to_bits(vn) & vindex_mask;
	// Adjust exponent of the value l fetched from the table to get the final s value.
	const float vs = fp32_from_bits(xnn_table_exp2_k_over_2048[vidx] + ve);

	// Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
	vn -= vmagic_bias;

	// Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
	// Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
	float vt = vn * vln2_o2048_hi + vz;
	vt = vn * vln2_o2048_lo + vt;

	// Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/4096, log(2)/4096]:
	// P1(t) = 1 + t * c1
	const float vp = vt * vc1;

	// Reconstruct the exp(-z) value:
	// y = s * (1 + t * c1)
	// = s + s * (t * c1))
	// = s + s * p
	const float vy = vp * vs + vs;

	// Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
	float vf = vy / (vy + vone);

	// For inputs above denormal cutoff, replace output with +0.0f.
	// Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
	if XNN_UNPREDICTABLE(vz > vdenorm_cutoff) {
	vf = 0.0f;
	}

	// Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
	if XNN_UNPREDICTABLE(vx > 0.0f) {
	vf = vone - vf;
	}

	*y++ = vf;

	n -= sizeof(float);
	} while (n != 0);
	}
	}