Evaluation stubs for gather-based AVX2/AVX512 Sigmoid
PiperOrigin-RevId: 337414295
diff --git a/src/math/sigmoid-avx2-rr1-lut64-p2-gather-div.c b/src/math/sigmoid-avx2-rr1-lut64-p2-gather-div.c
new file mode 100644
index 0000000..7743562
--- /dev/null
+++ b/src/math/sigmoid-avx2-rr1-lut64-p2-gather-div.c
@@ -0,0 +1,114 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx2_rr1_lut64_p2_gather_div(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (8 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p17f);
+ const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m256 vindex_mask = _mm256_castsi256_ps(_mm256_set1_epi32(INT32_C(0x3F)));
+ const __m256 vminus_ln2 = _mm256_set1_ps(-0x1.62E43p-1f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m256 vc2 = _mm256_set1_ps(0x1.FFFF0Ap-2f);
+ const __m256 vone = _mm256_set1_ps(1.0f);
+ // The smallest x for which sigmoidf(x) is normalized.
+ // This number is also the smallest x for which expf(x) is normalized.
+ const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
+
+ for (; n != 0; n -= 8 * sizeof(float)) {
+ const __m256 vx = _mm256_loadu_ps(input);
+
+ // 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 __m256 vz = _mm256_or_ps(vx, vsign_mask);
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(z) is normalized,
+ // i.e. -87.33642 <= z <= 0. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
+ // in two steps:
+ // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
+ // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
+ // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
+ // number, because for -87.33642 <= z <= 0 (inputs for which sigmoidf(z) is normalized) we have
+ // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
+ //
+ // Shift bits 6:14 into 23:31 (position of floating-point exponent).
+ __m256i ve = _mm256_slli_epi32(_mm256_castps_si256(vn), 17);
+
+ // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
+ const __m256i vidx = _mm256_castps_si256(_mm256_and_ps(vn, vindex_mask));
+ const __m256i vl = _mm256_i32gather_epi32((const int*) xnn_table_exp2minus_k_over_64, vidx, sizeof(float));
+ // Adjust exponent of the value l fetched from the table to get the final s value.
+ const __m256 vs = _mm256_castsi256_ps(_mm256_add_epi32(vl, ve));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm256_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ const __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2, vz);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
+ __m256 vp = _mm256_mul_ps(vt, vc2);
+ vp = _mm256_fmadd_ps(vt, vp, vt);
+
+ // Reconstruct the exp(z) value:
+ // e = s * (1 + t * (1 + t * c2))
+ // = s * (1 + p)
+ // = s + s * p
+ const __m256 vy = _mm256_fmadd_ps(vs, vp, vs);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m256 vd = _mm256_add_ps(vy, vone);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
+ __m256 vf = _mm256_div_ps(vy, vd);
+
+ // 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 = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
+
+ _mm256_storeu_ps(output, vf);
+
+ input += 8;
+ output += 8;
+ }
+}
diff --git a/src/math/sigmoid-avx2-rr1-lut64-p2-gather-nr1fma.c b/src/math/sigmoid-avx2-rr1-lut64-p2-gather-nr1fma.c
new file mode 100644
index 0000000..0d58579
--- /dev/null
+++ b/src/math/sigmoid-avx2-rr1-lut64-p2-gather-nr1fma.c
@@ -0,0 +1,120 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx2_rr1_lut64_p2_gather_nr1fma(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (8 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p17f);
+ const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m256 vindex_mask = _mm256_castsi256_ps(_mm256_set1_epi32(INT32_C(0x3F)));
+ const __m256 vminus_ln2 = _mm256_set1_ps(-0x1.62E43p-1f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m256 vc2 = _mm256_set1_ps(0x1.FFFF0Ap-2f);
+ const __m256 vone = _mm256_set1_ps(1.0f);
+ // The smallest x for which sigmoidf(x) is normalized.
+ // This number is also the smallest x for which expf(x) is normalized.
+ const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
+
+ for (; n != 0; n -= 8 * sizeof(float)) {
+ const __m256 vx = _mm256_loadu_ps(input);
+
+ // 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 __m256 vz = _mm256_or_ps(vx, vsign_mask);
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(z) is normalized,
+ // i.e. -87.33642 <= z <= 0. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
+ // in two steps:
+ // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
+ // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
+ // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
+ // number, because for -87.33642 <= z <= 0 (inputs for which sigmoidf(z) is normalized) we have
+ // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
+ //
+ // Shift bits 6:14 into 23:31 (position of floating-point exponent).
+ __m256i ve = _mm256_slli_epi32(_mm256_castps_si256(vn), 17);
+
+ // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
+ const __m256i vidx = _mm256_castps_si256(_mm256_and_ps(vn, vindex_mask));
+ const __m256i vl = _mm256_i32gather_epi32((const int*) xnn_table_exp2minus_k_over_64, vidx, sizeof(float));
+ // Adjust exponent of the value l fetched from the table to get the final s value.
+ const __m256 vs = _mm256_castsi256_ps(_mm256_add_epi32(vl, ve));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm256_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ const __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2, vz);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
+ __m256 vp = _mm256_mul_ps(vt, vc2);
+ vp = _mm256_fmadd_ps(vt, vp, vt);
+
+ // Reconstruct the exp(z) value:
+ // e = s * (1 + t * (1 + t * c2))
+ // = s * (1 + p)
+ // = s + s * p
+ const __m256 vy = _mm256_fmadd_ps(vs, vp, vs);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m256 vd = _mm256_add_ps(vy, vone);
+
+ // Use Newton-Raphson method (1 iteration) to compute reciprocal of denominator.
+ // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
+ // Thus the reciprocal of the denominator never overflows.
+ __m256 vr = _mm256_rcp_ps(vd);
+ vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
+ __m256 vf = _mm256_mul_ps(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 = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
+
+ _mm256_storeu_ps(output, vf);
+
+ input += 8;
+ output += 8;
+ }
+}
diff --git a/src/math/sigmoid-avx2-rr1-lut64-p2-gather-nr2fma.c b/src/math/sigmoid-avx2-rr1-lut64-p2-gather-nr2fma.c
new file mode 100644
index 0000000..d09c470
--- /dev/null
+++ b/src/math/sigmoid-avx2-rr1-lut64-p2-gather-nr2fma.c
@@ -0,0 +1,121 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx2_rr1_lut64_p2_gather_nr2fma(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (8 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p17f);
+ const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m256 vindex_mask = _mm256_castsi256_ps(_mm256_set1_epi32(INT32_C(0x3F)));
+ const __m256 vminus_ln2 = _mm256_set1_ps(-0x1.62E43p-1f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m256 vc2 = _mm256_set1_ps(0x1.FFFF0Ap-2f);
+ const __m256 vone = _mm256_set1_ps(1.0f);
+ // The smallest x for which sigmoidf(x) is normalized.
+ // This number is also the smallest x for which expf(x) is normalized.
+ const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
+
+ for (; n != 0; n -= 8 * sizeof(float)) {
+ const __m256 vx = _mm256_loadu_ps(input);
+
+ // 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 __m256 vz = _mm256_or_ps(vx, vsign_mask);
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(z) is normalized,
+ // i.e. -87.33642 <= z <= 0. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
+ // in two steps:
+ // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
+ // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
+ // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
+ // number, because for -87.33642 <= z <= 0 (inputs for which sigmoidf(z) is normalized) we have
+ // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
+ //
+ // Shift bits 6:14 into 23:31 (position of floating-point exponent).
+ __m256i ve = _mm256_slli_epi32(_mm256_castps_si256(vn), 17);
+
+ // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
+ const __m256i vidx = _mm256_castps_si256(_mm256_and_ps(vn, vindex_mask));
+ const __m256i vl = _mm256_i32gather_epi32((const int*) xnn_table_exp2minus_k_over_64, vidx, sizeof(float));
+ // Adjust exponent of the value l fetched from the table to get the final s value.
+ const __m256 vs = _mm256_castsi256_ps(_mm256_add_epi32(vl, ve));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm256_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ const __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2, vz);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
+ __m256 vp = _mm256_mul_ps(vt, vc2);
+ vp = _mm256_fmadd_ps(vt, vp, vt);
+
+ // Reconstruct the exp(z) value:
+ // e = s * (1 + t * (1 + t * c2))
+ // = s * (1 + p)
+ // = s + s * p
+ const __m256 vy = _mm256_fmadd_ps(vs, vp, vs);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m256 vd = _mm256_add_ps(vy, vone);
+
+ // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
+ // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
+ // Thus the reciprocal of the denominator never overflows.
+ __m256 vr = _mm256_rcp_ps(vd);
+ vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
+ vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
+ __m256 vf = _mm256_mul_ps(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 = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
+
+ _mm256_storeu_ps(output, vf);
+
+ input += 8;
+ output += 8;
+ }
+}
diff --git a/src/math/sigmoid-avx2-rr1-lut64-p2-gather-nr2fma1adj.c b/src/math/sigmoid-avx2-rr1-lut64-p2-gather-nr2fma1adj.c
new file mode 100644
index 0000000..5e11d70
--- /dev/null
+++ b/src/math/sigmoid-avx2-rr1-lut64-p2-gather-nr2fma1adj.c
@@ -0,0 +1,122 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx2_rr1_lut64_p2_gather_nr2fma1adj(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (8 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p17f);
+ const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m256 vindex_mask = _mm256_castsi256_ps(_mm256_set1_epi32(INT32_C(0x3F)));
+ const __m256 vminus_ln2 = _mm256_set1_ps(-0x1.62E43p-1f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m256 vc2 = _mm256_set1_ps(0x1.FFFF0Ap-2f);
+ const __m256 vone = _mm256_set1_ps(1.0f);
+ // The smallest x for which sigmoidf(x) is normalized.
+ // This number is also the smallest x for which expf(x) is normalized.
+ const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
+
+ for (; n != 0; n -= 8 * sizeof(float)) {
+ const __m256 vx = _mm256_loadu_ps(input);
+
+ // 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 __m256 vz = _mm256_or_ps(vx, vsign_mask);
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(z) is normalized,
+ // i.e. -87.33642 <= z <= 0. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
+ // in two steps:
+ // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
+ // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
+ // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
+ // number, because for -87.33642 <= z <= 0 (inputs for which sigmoidf(z) is normalized) we have
+ // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
+ //
+ // Shift bits 6:14 into 23:31 (position of floating-point exponent).
+ __m256i ve = _mm256_slli_epi32(_mm256_castps_si256(vn), 17);
+
+ // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
+ const __m256i vidx = _mm256_castps_si256(_mm256_and_ps(vn, vindex_mask));
+ const __m256i vl = _mm256_i32gather_epi32((const int*) xnn_table_exp2minus_k_over_64, vidx, sizeof(float));
+ // Adjust exponent of the value l fetched from the table to get the final s value.
+ const __m256 vs = _mm256_castsi256_ps(_mm256_add_epi32(vl, ve));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm256_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ const __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2, vz);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
+ __m256 vp = _mm256_mul_ps(vt, vc2);
+ vp = _mm256_fmadd_ps(vt, vp, vt);
+
+ // Reconstruct the exp(z) value:
+ // e = s * (1 + t * (1 + t * c2))
+ // = s * (1 + p)
+ // = s + s * p
+ const __m256 vy = _mm256_fmadd_ps(vs, vp, vs);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m256 vd = _mm256_add_ps(vy, vone);
+
+ // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
+ // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
+ // Thus the reciprocal of the denominator never overflows.
+ __m256 vr = _mm256_rcp_ps(vd);
+ vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
+ vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z)) with adjustment to match IEEE division result
+ __m256 vf = _mm256_mul_ps(vy, vr);
+ vf = _mm256_fmadd_ps(_mm256_fnmadd_ps(vf, vd, vy), vr, vf);
+
+ // 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 = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
+
+ _mm256_storeu_ps(output, vf);
+
+ input += 8;
+ output += 8;
+ }
+}
diff --git a/src/math/sigmoid-avx2-rr2-lut64-p2-gather-div.c b/src/math/sigmoid-avx2-rr2-lut64-p2-gather-div.c
new file mode 100644
index 0000000..9425b45
--- /dev/null
+++ b/src/math/sigmoid-avx2-rr2-lut64-p2-gather-div.c
@@ -0,0 +1,117 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx2_rr2_lut64_p2_gather_div(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (8 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p17f);
+ const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m256 vindex_mask = _mm256_castsi256_ps(_mm256_set1_epi32(INT32_C(0x3F)));
+ const __m256 vminus_ln2_hi = _mm256_set1_ps(-0x1.62E43p-1f);
+ const __m256 vminus_ln2_lo = _mm256_set1_ps(0x1.05C61p-29f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m256 vc2 = _mm256_set1_ps(0x1.FFFF0Ap-2f);
+ const __m256 vone = _mm256_set1_ps(1.0f);
+ // The smallest x for which sigmoidf(x) is normalized.
+ // This number is also the smallest x for which expf(x) is normalized.
+ const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
+
+ for (; n != 0; n -= 8 * sizeof(float)) {
+ const __m256 vx = _mm256_loadu_ps(input);
+
+ // 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 __m256 vz = _mm256_or_ps(vx, vsign_mask);
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(z) is normalized,
+ // i.e. -87.33642 <= z <= 0. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
+ // in two steps:
+ // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
+ // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
+ // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
+ // number, because for -87.33642 <= z <= 0 (inputs for which sigmoidf(z) is normalized) we have
+ // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
+ //
+ // Shift bits 6:14 into 23:31 (position of floating-point exponent).
+ __m256i ve = _mm256_slli_epi32(_mm256_castps_si256(vn), 17);
+
+ // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
+ const __m256i vidx = _mm256_castps_si256(_mm256_and_ps(vn, vindex_mask));
+ const __m256i vl = _mm256_i32gather_epi32((const int*) xnn_table_exp2minus_k_over_64, vidx, sizeof(float));
+ // Adjust exponent of the value l fetched from the table to get the final s value.
+ const __m256 vs = _mm256_castsi256_ps(_mm256_add_epi32(vl, ve));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm256_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
+ __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2_hi, vz);
+ vt = _mm256_fmadd_ps(vn, vminus_ln2_lo, vt);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
+ __m256 vp = _mm256_mul_ps(vt, vc2);
+ vp = _mm256_fmadd_ps(vt, vp, vt);
+
+ // Reconstruct the exp(z) value:
+ // e = s * (1 + t * (1 + t * c2))
+ // = s * (1 + p)
+ // = s + s * p
+ const __m256 vy = _mm256_fmadd_ps(vs, vp, vs);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m256 vd = _mm256_add_ps(vy, vone);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
+ __m256 vf = _mm256_div_ps(vy, vd);
+
+ // 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 = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
+
+ _mm256_storeu_ps(output, vf);
+
+ input += 8;
+ output += 8;
+ }
+}
diff --git a/src/math/sigmoid-avx2-rr2-lut64-p2-gather-nr1fma.c b/src/math/sigmoid-avx2-rr2-lut64-p2-gather-nr1fma.c
new file mode 100644
index 0000000..9690bd3
--- /dev/null
+++ b/src/math/sigmoid-avx2-rr2-lut64-p2-gather-nr1fma.c
@@ -0,0 +1,123 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx2_rr2_lut64_p2_gather_nr1fma(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (8 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p17f);
+ const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m256 vindex_mask = _mm256_castsi256_ps(_mm256_set1_epi32(INT32_C(0x3F)));
+ const __m256 vminus_ln2_hi = _mm256_set1_ps(-0x1.62E43p-1f);
+ const __m256 vminus_ln2_lo = _mm256_set1_ps(0x1.05C61p-29f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m256 vc2 = _mm256_set1_ps(0x1.FFFF0Ap-2f);
+ const __m256 vone = _mm256_set1_ps(1.0f);
+ // The smallest x for which sigmoidf(x) is normalized.
+ // This number is also the smallest x for which expf(x) is normalized.
+ const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
+
+ for (; n != 0; n -= 8 * sizeof(float)) {
+ const __m256 vx = _mm256_loadu_ps(input);
+
+ // 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 __m256 vz = _mm256_or_ps(vx, vsign_mask);
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(z) is normalized,
+ // i.e. -87.33642 <= z <= 0. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
+ // in two steps:
+ // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
+ // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
+ // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
+ // number, because for -87.33642 <= z <= 0 (inputs for which sigmoidf(z) is normalized) we have
+ // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
+ //
+ // Shift bits 6:14 into 23:31 (position of floating-point exponent).
+ __m256i ve = _mm256_slli_epi32(_mm256_castps_si256(vn), 17);
+
+ // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
+ const __m256i vidx = _mm256_castps_si256(_mm256_and_ps(vn, vindex_mask));
+ const __m256i vl = _mm256_i32gather_epi32((const int*) xnn_table_exp2minus_k_over_64, vidx, sizeof(float));
+ // Adjust exponent of the value l fetched from the table to get the final s value.
+ const __m256 vs = _mm256_castsi256_ps(_mm256_add_epi32(vl, ve));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm256_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
+ __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2_hi, vz);
+ vt = _mm256_fmadd_ps(vn, vminus_ln2_lo, vt);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
+ __m256 vp = _mm256_mul_ps(vt, vc2);
+ vp = _mm256_fmadd_ps(vt, vp, vt);
+
+ // Reconstruct the exp(z) value:
+ // e = s * (1 + t * (1 + t * c2))
+ // = s * (1 + p)
+ // = s + s * p
+ const __m256 vy = _mm256_fmadd_ps(vs, vp, vs);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m256 vd = _mm256_add_ps(vy, vone);
+
+ // Use Newton-Raphson method (1 iteration) to compute reciprocal of denominator.
+ // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
+ // Thus the reciprocal of the denominator never overflows.
+ __m256 vr = _mm256_rcp_ps(vd);
+ vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
+ __m256 vf = _mm256_mul_ps(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 = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
+
+ _mm256_storeu_ps(output, vf);
+
+ input += 8;
+ output += 8;
+ }
+}
diff --git a/src/math/sigmoid-avx2-rr2-lut64-p2-gather-nr2fma.c b/src/math/sigmoid-avx2-rr2-lut64-p2-gather-nr2fma.c
new file mode 100644
index 0000000..8b5e876
--- /dev/null
+++ b/src/math/sigmoid-avx2-rr2-lut64-p2-gather-nr2fma.c
@@ -0,0 +1,124 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx2_rr2_lut64_p2_gather_nr2fma(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (8 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p17f);
+ const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m256 vindex_mask = _mm256_castsi256_ps(_mm256_set1_epi32(INT32_C(0x3F)));
+ const __m256 vminus_ln2_hi = _mm256_set1_ps(-0x1.62E43p-1f);
+ const __m256 vminus_ln2_lo = _mm256_set1_ps(0x1.05C61p-29f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m256 vc2 = _mm256_set1_ps(0x1.FFFF0Ap-2f);
+ const __m256 vone = _mm256_set1_ps(1.0f);
+ // The smallest x for which sigmoidf(x) is normalized.
+ // This number is also the smallest x for which expf(x) is normalized.
+ const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
+
+ for (; n != 0; n -= 8 * sizeof(float)) {
+ const __m256 vx = _mm256_loadu_ps(input);
+
+ // 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 __m256 vz = _mm256_or_ps(vx, vsign_mask);
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(z) is normalized,
+ // i.e. -87.33642 <= z <= 0. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
+ // in two steps:
+ // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
+ // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
+ // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
+ // number, because for -87.33642 <= z <= 0 (inputs for which sigmoidf(z) is normalized) we have
+ // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
+ //
+ // Shift bits 6:14 into 23:31 (position of floating-point exponent).
+ __m256i ve = _mm256_slli_epi32(_mm256_castps_si256(vn), 17);
+
+ // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
+ const __m256i vidx = _mm256_castps_si256(_mm256_and_ps(vn, vindex_mask));
+ const __m256i vl = _mm256_i32gather_epi32((const int*) xnn_table_exp2minus_k_over_64, vidx, sizeof(float));
+ // Adjust exponent of the value l fetched from the table to get the final s value.
+ const __m256 vs = _mm256_castsi256_ps(_mm256_add_epi32(vl, ve));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm256_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
+ __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2_hi, vz);
+ vt = _mm256_fmadd_ps(vn, vminus_ln2_lo, vt);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
+ __m256 vp = _mm256_mul_ps(vt, vc2);
+ vp = _mm256_fmadd_ps(vt, vp, vt);
+
+ // Reconstruct the exp(z) value:
+ // e = s * (1 + t * (1 + t * c2))
+ // = s * (1 + p)
+ // = s + s * p
+ const __m256 vy = _mm256_fmadd_ps(vs, vp, vs);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m256 vd = _mm256_add_ps(vy, vone);
+
+ // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
+ // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
+ // Thus the reciprocal of the denominator never overflows.
+ __m256 vr = _mm256_rcp_ps(vd);
+ vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
+ vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
+ __m256 vf = _mm256_mul_ps(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 = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
+
+ _mm256_storeu_ps(output, vf);
+
+ input += 8;
+ output += 8;
+ }
+}
diff --git a/src/math/sigmoid-avx2-rr2-lut64-p2-gather-nr2fma1adj.c b/src/math/sigmoid-avx2-rr2-lut64-p2-gather-nr2fma1adj.c
new file mode 100644
index 0000000..b3a7943
--- /dev/null
+++ b/src/math/sigmoid-avx2-rr2-lut64-p2-gather-nr2fma1adj.c
@@ -0,0 +1,125 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx2_rr2_lut64_p2_gather_nr2fma1adj(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (8 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m256 vsign_mask = _mm256_set1_ps(-0.0f);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p17f);
+ const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m256 vindex_mask = _mm256_castsi256_ps(_mm256_set1_epi32(INT32_C(0x3F)));
+ const __m256 vminus_ln2_hi = _mm256_set1_ps(-0x1.62E43p-1f);
+ const __m256 vminus_ln2_lo = _mm256_set1_ps(0x1.05C61p-29f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m256 vc2 = _mm256_set1_ps(0x1.FFFF0Ap-2f);
+ const __m256 vone = _mm256_set1_ps(1.0f);
+ // The smallest x for which sigmoidf(x) is normalized.
+ // This number is also the smallest x for which expf(x) is normalized.
+ const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f);
+
+ for (; n != 0; n -= 8 * sizeof(float)) {
+ const __m256 vx = _mm256_loadu_ps(input);
+
+ // 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 __m256 vz = _mm256_or_ps(vx, vsign_mask);
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(z) is normalized,
+ // i.e. -87.33642 <= z <= 0. As n has 6 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s
+ // in two steps:
+ // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in
+ // the [1.0, 2.0) range, i.e. their floating-point exponent is 0.
+ // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized
+ // number, because for -87.33642 <= z <= 0 (inputs for which sigmoidf(z) is normalized) we have
+ // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
+ //
+ // Shift bits 6:14 into 23:31 (position of floating-point exponent).
+ __m256i ve = _mm256_slli_epi32(_mm256_castps_si256(vn), 17);
+
+ // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
+ const __m256i vidx = _mm256_castps_si256(_mm256_and_ps(vn, vindex_mask));
+ const __m256i vl = _mm256_i32gather_epi32((const int*) xnn_table_exp2minus_k_over_64, vidx, sizeof(float));
+ // Adjust exponent of the value l fetched from the table to get the final s value.
+ const __m256 vs = _mm256_castsi256_ps(_mm256_add_epi32(vl, ve));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm256_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
+ __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2_hi, vz);
+ vt = _mm256_fmadd_ps(vn, vminus_ln2_lo, vt);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p
+ __m256 vp = _mm256_mul_ps(vt, vc2);
+ vp = _mm256_fmadd_ps(vt, vp, vt);
+
+ // Reconstruct the exp(z) value:
+ // e = s * (1 + t * (1 + t * c2))
+ // = s * (1 + p)
+ // = s + s * p
+ const __m256 vy = _mm256_fmadd_ps(vs, vp, vs);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m256 vd = _mm256_add_ps(vy, vone);
+
+ // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
+ // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
+ // Thus the reciprocal of the denominator never overflows.
+ __m256 vr = _mm256_rcp_ps(vd);
+ vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
+ vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z)) with adjustment to match IEEE division result
+ __m256 vf = _mm256_mul_ps(vy, vr);
+ vf = _mm256_fmadd_ps(_mm256_fnmadd_ps(vf, vd, vy), vr, vf);
+
+ // 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 = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx);
+
+ _mm256_storeu_ps(output, vf);
+
+ input += 8;
+ output += 8;
+ }
+}
diff --git a/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-div.c b/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-div.c
index 7f9a4c1..608b675 100644
--- a/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-div.c
+++ b/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-div.c
@@ -54,7 +54,7 @@
const __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
// Compute reduced argument n := round(z / log(2), 5).
- // We do it by adding a large number (magic bias), which cause rounding of the result to 4 fractional bits, then
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 5 fractional bits, then
// subtracing the large number back. The 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 / log(2)| <= 2**17,
// i.e. |z| <= 0x1.62E43p+16 = 90852.1875), but that is acceptable, because inputs x outside of
diff --git a/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-nr1fma.c b/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-nr1fma.c
index 6b0a7c9..44b58e5 100644
--- a/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-nr1fma.c
+++ b/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-nr1fma.c
@@ -54,7 +54,7 @@
const __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
// Compute reduced argument n := round(z / log(2), 5).
- // We do it by adding a large number (magic bias), which cause rounding of the result to 4 fractional bits, then
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 5 fractional bits, then
// subtracing the large number back. The 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 / log(2)| <= 2**17,
// i.e. |z| <= 0x1.62E43p+16 = 90852.1875), but that is acceptable, because inputs x outside of
diff --git a/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-nr1fma1adj.c b/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-nr1fma1adj.c
index 165d735..b33f513 100644
--- a/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-nr1fma1adj.c
+++ b/src/math/sigmoid-avx512f-rr1-lut32-p2-perm2-scalef-nr1fma1adj.c
@@ -54,7 +54,7 @@
const __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
// Compute reduced argument n := round(z / log(2), 5).
- // We do it by adding a large number (magic bias), which cause rounding of the result to 4 fractional bits, then
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 5 fractional bits, then
// subtracing the large number back. The 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 / log(2)| <= 2**17,
// i.e. |z| <= 0x1.62E43p+16 = 90852.1875), but that is acceptable, because inputs x outside of
diff --git a/src/math/sigmoid-avx512f-rr1-lut64-p2-gather-scalef-div.c b/src/math/sigmoid-avx512f-rr1-lut64-p2-gather-scalef-div.c
new file mode 100644
index 0000000..17ec9a8
--- /dev/null
+++ b/src/math/sigmoid-avx512f-rr1-lut64-p2-gather-scalef-div.c
@@ -0,0 +1,93 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values, k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx512f_rr1_lut64_p2_gather_scalef_div(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (16 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m512i vsign_mask = _mm512_set1_epi32(0x80000000);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m512 vmagic_bias = _mm512_set1_ps(0x1.800000p17f);
+ const __m512 vlog2e = _mm512_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m512i vindex_mask = _mm512_set1_epi32(INT32_C(0x3F));
+ const __m512 vminus_ln2 = _mm512_set1_ps(-0x1.62e43p-1f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m512 vc2 = _mm512_set1_ps(0x1.FFFF0Ap-2f);
+ const __m512 vone = _mm512_set1_ps(1.0f);
+
+ for (; n != 0; n -= 16 * sizeof(float)) {
+ const __m512 vx = _mm512_loadu_ps(input);
+
+ // 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 __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m512 vn = _mm512_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Use the low 6 bits of n (as integer) for table lookup.
+ const __m512i vidx = _mm512_and_epi32(_mm512_castps_si512(vn), vindex_mask);
+ const __m512 vl = _mm512_i32gather_ps(vidx, xnn_table_exp2_k_over_64, sizeof(float));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm512_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ const __m512 vt = _mm512_fmadd_ps(vn, vminus_ln2, vz);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2))
+ // p = l * P(t)
+ // = l + l * (t + t * (t * c2))
+ __m512 vp = _mm512_mul_ps(vt, vc2);
+ vp = _mm512_fmadd_ps(vt, vp, vt);
+ vp = _mm512_fmadd_ps(vl, vp, vl);
+
+ // Reconstruct the exp(z) value: e = exp2(floor(n)) * p.
+ const __m512 ve = _mm512_scalef_ps(vp, vn);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m512 vd = _mm512_add_ps(ve, vone);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
+ __m512 vf = _mm512_div_ps(ve, vd);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm512_mask_sub_ps(vf, _mm512_testn_epi32_mask(_mm512_castps_si512(vx), vsign_mask), vone, vf);
+
+ _mm512_storeu_ps(output, vf);
+
+ input += 16;
+ output += 16;
+ }
+}
diff --git a/src/math/sigmoid-avx512f-rr1-lut64-p2-gather-scalef-nr1fma.c b/src/math/sigmoid-avx512f-rr1-lut64-p2-gather-scalef-nr1fma.c
new file mode 100644
index 0000000..c43cfe1
--- /dev/null
+++ b/src/math/sigmoid-avx512f-rr1-lut64-p2-gather-scalef-nr1fma.c
@@ -0,0 +1,99 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values, k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx512f_rr1_lut64_p2_gather_scalef_nr1fma(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (16 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m512i vsign_mask = _mm512_set1_epi32(0x80000000);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m512 vmagic_bias = _mm512_set1_ps(0x1.800000p17f);
+ const __m512 vlog2e = _mm512_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m512i vindex_mask = _mm512_set1_epi32(INT32_C(0x3F));
+ const __m512 vminus_ln2 = _mm512_set1_ps(-0x1.62e43p-1f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m512 vc2 = _mm512_set1_ps(0x1.FFFF0Ap-2f);
+ const __m512 vone = _mm512_set1_ps(1.0f);
+
+ for (; n != 0; n -= 16 * sizeof(float)) {
+ const __m512 vx = _mm512_loadu_ps(input);
+
+ // 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 __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m512 vn = _mm512_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Use the low 6 bits of n (as integer) for table lookup.
+ const __m512i vidx = _mm512_and_epi32(_mm512_castps_si512(vn), vindex_mask);
+ const __m512 vl = _mm512_i32gather_ps(vidx, xnn_table_exp2_k_over_64, sizeof(float));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm512_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ const __m512 vt = _mm512_fmadd_ps(vn, vminus_ln2, vz);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2))
+ // p = l * P(t)
+ // = l + l * (t + t * (t * c2))
+ __m512 vp = _mm512_mul_ps(vt, vc2);
+ vp = _mm512_fmadd_ps(vt, vp, vt);
+ vp = _mm512_fmadd_ps(vl, vp, vl);
+
+ // Reconstruct the exp(z) value: e = exp2(floor(n)) * p.
+ const __m512 ve = _mm512_scalef_ps(vp, vn);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m512 vd = _mm512_add_ps(ve, vone);
+
+ // Use Newton-Raphson method (1 iteration) to compute reciprocal of denominator.
+ // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
+ // Thus the reciprocal of the denominator never overflows.
+ __m512 vr = _mm512_rcp14_ps(vd);
+ vr = _mm512_fmadd_ps(_mm512_fnmadd_ps(vr, vd, vone), vr, vr);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
+ __m512 vf = _mm512_mul_ps(ve, vr);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm512_mask_sub_ps(vf, _mm512_testn_epi32_mask(_mm512_castps_si512(vx), vsign_mask), vone, vf);
+
+ _mm512_storeu_ps(output, vf);
+
+ input += 16;
+ output += 16;
+ }
+}
diff --git a/src/math/sigmoid-avx512f-rr1-lut64-p2-gather-scalef-nr1fma1adj.c b/src/math/sigmoid-avx512f-rr1-lut64-p2-gather-scalef-nr1fma1adj.c
new file mode 100644
index 0000000..2f6a7be
--- /dev/null
+++ b/src/math/sigmoid-avx512f-rr1-lut64-p2-gather-scalef-nr1fma1adj.c
@@ -0,0 +1,100 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values, k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx512f_rr1_lut64_p2_gather_scalef_nr1fma1adj(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (16 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m512i vsign_mask = _mm512_set1_epi32(0x80000000);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m512 vmagic_bias = _mm512_set1_ps(0x1.800000p17f);
+ const __m512 vlog2e = _mm512_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m512i vindex_mask = _mm512_set1_epi32(INT32_C(0x3F));
+ const __m512 vminus_ln2 = _mm512_set1_ps(-0x1.62e43p-1f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m512 vc2 = _mm512_set1_ps(0x1.FFFF0Ap-2f);
+ const __m512 vone = _mm512_set1_ps(1.0f);
+
+ for (; n != 0; n -= 16 * sizeof(float)) {
+ const __m512 vx = _mm512_loadu_ps(input);
+
+ // 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 __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m512 vn = _mm512_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Use the low 6 bits of n (as integer) for table lookup.
+ const __m512i vidx = _mm512_and_epi32(_mm512_castps_si512(vn), vindex_mask);
+ const __m512 vl = _mm512_i32gather_ps(vidx, xnn_table_exp2_k_over_64, sizeof(float));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm512_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ const __m512 vt = _mm512_fmadd_ps(vn, vminus_ln2, vz);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2))
+ // p = l * P(t)
+ // = l + l * (t + t * (t * c2))
+ __m512 vp = _mm512_mul_ps(vt, vc2);
+ vp = _mm512_fmadd_ps(vt, vp, vt);
+ vp = _mm512_fmadd_ps(vl, vp, vl);
+
+ // Reconstruct the exp(z) value: e = exp2(floor(n)) * p.
+ const __m512 ve = _mm512_scalef_ps(vp, vn);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m512 vd = _mm512_add_ps(ve, vone);
+
+ // Use Newton-Raphson method (1 iteration) to compute reciprocal of denominator.
+ // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
+ // Thus the reciprocal of the denominator never overflows.
+ __m512 vr = _mm512_rcp14_ps(vd);
+ vr = _mm512_fmadd_ps(_mm512_fnmadd_ps(vr, vd, vone), vr, vr);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z)) with adjustment to match IEEE division result
+ __m512 vf = _mm512_mul_ps(ve, vr);
+ vf = _mm512_fmadd_ps(_mm512_fnmadd_ps(vf, vd, ve), vr, vf);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm512_mask_sub_ps(vf, _mm512_testn_epi32_mask(_mm512_castps_si512(vx), vsign_mask), vone, vf);
+
+ _mm512_storeu_ps(output, vf);
+
+ input += 16;
+ output += 16;
+ }
+}
diff --git a/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-div.c b/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-div.c
index 5c68c02..bdf978a 100644
--- a/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-div.c
+++ b/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-div.c
@@ -55,7 +55,7 @@
const __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
// Compute reduced argument n := round(z / log(2), 5).
- // We do it by adding a large number (magic bias), which cause rounding of the result to 4 fractional bits, then
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 5 fractional bits, then
// subtracing the large number back. The 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 / log(2)| <= 2**17,
// i.e. |z| <= 0x1.62E43p+16 = 90852.1875), but that is acceptable, because inputs x outside of
diff --git a/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-nr1fma.c b/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-nr1fma.c
index 3e9b70d..4cd12e2 100644
--- a/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-nr1fma.c
+++ b/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-nr1fma.c
@@ -55,7 +55,7 @@
const __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
// Compute reduced argument n := round(z / log(2), 5).
- // We do it by adding a large number (magic bias), which cause rounding of the result to 4 fractional bits, then
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 5 fractional bits, then
// subtracing the large number back. The 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 / log(2)| <= 2**17,
// i.e. |z| <= 0x1.62E43p+16 = 90852.1875), but that is acceptable, because inputs x outside of
diff --git a/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-nr1fma1adj.c b/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-nr1fma1adj.c
index c6e7039..4307820 100644
--- a/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-nr1fma1adj.c
+++ b/src/math/sigmoid-avx512f-rr2-lut32-p2-perm2-scalef-nr1fma1adj.c
@@ -55,7 +55,7 @@
const __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
// Compute reduced argument n := round(z / log(2), 5).
- // We do it by adding a large number (magic bias), which cause rounding of the result to 4 fractional bits, then
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 5 fractional bits, then
// subtracing the large number back. The 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 / log(2)| <= 2**17,
// i.e. |z| <= 0x1.62E43p+16 = 90852.1875), but that is acceptable, because inputs x outside of
diff --git a/src/math/sigmoid-avx512f-rr2-lut64-p2-gather-scalef-div.c b/src/math/sigmoid-avx512f-rr2-lut64-p2-gather-scalef-div.c
new file mode 100644
index 0000000..2bacbc7
--- /dev/null
+++ b/src/math/sigmoid-avx512f-rr2-lut64-p2-gather-scalef-div.c
@@ -0,0 +1,96 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values, k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx512f_rr2_lut64_p2_gather_scalef_div(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (16 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m512i vsign_mask = _mm512_set1_epi32(0x80000000);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m512 vmagic_bias = _mm512_set1_ps(0x1.800000p17f);
+ const __m512 vlog2e = _mm512_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m512i vindex_mask = _mm512_set1_epi32(INT32_C(0x3F));
+ const __m512 vminus_ln2_hi = _mm512_set1_ps(-0x1.62e43p-1f);
+ const __m512 vminus_ln2_lo = _mm512_set1_ps(0x1.05c61p-29f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m512 vc2 = _mm512_set1_ps(0x1.FFFF0Ap-2f);
+ const __m512 vone = _mm512_set1_ps(1.0f);
+
+ for (; n != 0; n -= 16 * sizeof(float)) {
+ const __m512 vx = _mm512_loadu_ps(input);
+
+ // 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 __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m512 vn = _mm512_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Use the low 6 bits of n (as integer) for table lookup.
+ const __m512i vidx = _mm512_and_epi32(_mm512_castps_si512(vn), vindex_mask);
+ const __m512 vl = _mm512_i32gather_ps(vidx, xnn_table_exp2_k_over_64, sizeof(float));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm512_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
+ __m512 vt = _mm512_fmadd_ps(vn, vminus_ln2_hi, vz);
+ vt = _mm512_fmadd_ps(vn, vminus_ln2_lo, vt);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2))
+ // p = l * P(t)
+ // = l + l * (t + t * (t * c2))
+ __m512 vp = _mm512_mul_ps(vt, vc2);
+ vp = _mm512_fmadd_ps(vt, vp, vt);
+ vp = _mm512_fmadd_ps(vl, vp, vl);
+
+ // Reconstruct the exp(z) value: e = exp2(floor(n)) * p.
+ const __m512 ve = _mm512_scalef_ps(vp, vn);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m512 vd = _mm512_add_ps(ve, vone);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
+ __m512 vf = _mm512_div_ps(ve, vd);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm512_mask_sub_ps(vf, _mm512_testn_epi32_mask(_mm512_castps_si512(vx), vsign_mask), vone, vf);
+
+ _mm512_storeu_ps(output, vf);
+
+ input += 16;
+ output += 16;
+ }
+}
diff --git a/src/math/sigmoid-avx512f-rr2-lut64-p2-gather-scalef-nr1fma.c b/src/math/sigmoid-avx512f-rr2-lut64-p2-gather-scalef-nr1fma.c
new file mode 100644
index 0000000..c80a220
--- /dev/null
+++ b/src/math/sigmoid-avx512f-rr2-lut64-p2-gather-scalef-nr1fma.c
@@ -0,0 +1,102 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values, k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx512f_rr2_lut64_p2_gather_scalef_nr1fma(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (16 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m512i vsign_mask = _mm512_set1_epi32(0x80000000);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m512 vmagic_bias = _mm512_set1_ps(0x1.800000p17f);
+ const __m512 vlog2e = _mm512_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m512i vindex_mask = _mm512_set1_epi32(INT32_C(0x3F));
+ const __m512 vminus_ln2_hi = _mm512_set1_ps(-0x1.62e43p-1f);
+ const __m512 vminus_ln2_lo = _mm512_set1_ps(0x1.05c61p-29f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m512 vc2 = _mm512_set1_ps(0x1.FFFF0Ap-2f);
+ const __m512 vone = _mm512_set1_ps(1.0f);
+
+ for (; n != 0; n -= 16 * sizeof(float)) {
+ const __m512 vx = _mm512_loadu_ps(input);
+
+ // 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 __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m512 vn = _mm512_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Use the low 6 bits of n (as integer) for table lookup.
+ const __m512i vidx = _mm512_and_epi32(_mm512_castps_si512(vn), vindex_mask);
+ const __m512 vl = _mm512_i32gather_ps(vidx, xnn_table_exp2_k_over_64, sizeof(float));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm512_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
+ __m512 vt = _mm512_fmadd_ps(vn, vminus_ln2_hi, vz);
+ vt = _mm512_fmadd_ps(vn, vminus_ln2_lo, vt);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2))
+ // p = l * P(t)
+ // = l + l * (t + t * (t * c2))
+ __m512 vp = _mm512_mul_ps(vt, vc2);
+ vp = _mm512_fmadd_ps(vt, vp, vt);
+ vp = _mm512_fmadd_ps(vl, vp, vl);
+
+ // Reconstruct the exp(z) value: e = exp2(floor(n)) * p.
+ const __m512 ve = _mm512_scalef_ps(vp, vn);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m512 vd = _mm512_add_ps(ve, vone);
+
+ // Use Newton-Raphson method (1 iteration) to compute reciprocal of denominator.
+ // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
+ // Thus the reciprocal of the denominator never overflows.
+ __m512 vr = _mm512_rcp14_ps(vd);
+ vr = _mm512_fmadd_ps(_mm512_fnmadd_ps(vr, vd, vone), vr, vr);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z))
+ __m512 vf = _mm512_mul_ps(ve, vr);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm512_mask_sub_ps(vf, _mm512_testn_epi32_mask(_mm512_castps_si512(vx), vsign_mask), vone, vf);
+
+ _mm512_storeu_ps(output, vf);
+
+ input += 16;
+ output += 16;
+ }
+}
diff --git a/src/math/sigmoid-avx512f-rr2-lut64-p2-gather-scalef-nr1fma1adj.c b/src/math/sigmoid-avx512f-rr2-lut64-p2-gather-scalef-nr1fma1adj.c
new file mode 100644
index 0000000..6a96858
--- /dev/null
+++ b/src/math/sigmoid-avx512f-rr2-lut64-p2-gather-scalef-nr1fma1adj.c
@@ -0,0 +1,103 @@
+// Copyright 2020 Google LLC
+//
+// This source code is licensed under the BSD-style license found in the
+// LICENSE file in the root directory of this source tree.
+
+#include <assert.h>
+#include <stddef.h>
+
+#include <immintrin.h>
+
+#include <xnnpack/common.h>
+#include <xnnpack/math-stubs.h>
+
+
+// Table of exp2(k / 64) values, k = 0..63
+extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64];
+
+void xnn_math_f32_sigmoid__avx512f_rr2_lut64_p2_gather_scalef_nr1fma1adj(
+ size_t n,
+ const float* input,
+ float* output)
+{
+ assert(n % (16 * sizeof(float)) == 0);
+
+ // Floating-point mask with only the sign bit set
+ const __m512i vsign_mask = _mm512_set1_epi32(0x80000000);
+ // Large number such that ulp(magic bias) == exp2(-6)
+ const __m512 vmagic_bias = _mm512_set1_ps(0x1.800000p17f);
+ const __m512 vlog2e = _mm512_set1_ps(0x1.715476p0f);
+ // Mask for the lowest 6 bits
+ const __m512i vindex_mask = _mm512_set1_epi32(INT32_C(0x3F));
+ const __m512 vminus_ln2_hi = _mm512_set1_ps(-0x1.62e43p-1f);
+ const __m512 vminus_ln2_lo = _mm512_set1_ps(0x1.05c61p-29f);
+ // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128]
+ const __m512 vc2 = _mm512_set1_ps(0x1.FFFF0Ap-2f);
+ const __m512 vone = _mm512_set1_ps(1.0f);
+
+ for (; n != 0; n -= 16 * sizeof(float)) {
+ const __m512 vx = _mm512_loadu_ps(input);
+
+ // 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 __m512 vz = _mm512_castsi512_ps(_mm512_or_epi32(_mm512_castps_si512(vx), vsign_mask));
+
+ // Compute reduced argument n := round(z / log(2), 6).
+ // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then
+ // subtracing the large number back. The 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 / log(2)| <= 2**16, i.e.
+ // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678]
+ // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the
+ // very end of the algorithm.
+ __m512 vn = _mm512_fmadd_ps(vz, vlog2e, vmagic_bias);
+
+ // Use the low 6 bits of n (as integer) for table lookup.
+ const __m512i vidx = _mm512_and_epi32(_mm512_castps_si512(vn), vindex_mask);
+ const __m512 vl = _mm512_i32gather_ps(vidx, xnn_table_exp2_k_over_64, sizeof(float));
+
+ // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number.
+ vn = _mm512_sub_ps(vn, vmagic_bias);
+
+ // Compute reduced argument t := z - n * log(2).
+ // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
+ __m512 vt = _mm512_fmadd_ps(vn, vminus_ln2_hi, vz);
+ vt = _mm512_fmadd_ps(vn, vminus_ln2_lo, vt);
+
+ // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128].
+ // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2))
+ // p = l * P(t)
+ // = l + l * (t + t * (t * c2))
+ __m512 vp = _mm512_mul_ps(vt, vc2);
+ vp = _mm512_fmadd_ps(vt, vp, vt);
+ vp = _mm512_fmadd_ps(vl, vp, vl);
+
+ // Reconstruct the exp(z) value: e = exp2(floor(n)) * p.
+ const __m512 ve = _mm512_scalef_ps(vp, vn);
+
+ // Denominator of the sigmoid fraction: 1.0 + exp(z)
+ const __m512 vd = _mm512_add_ps(ve, vone);
+
+ // Use Newton-Raphson method (1 iteration) to compute reciprocal of denominator.
+ // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
+ // Thus the reciprocal of the denominator never overflows.
+ __m512 vr = _mm512_rcp14_ps(vd);
+ vr = _mm512_fmadd_ps(_mm512_fnmadd_ps(vr, vd, vone), vr, vr);
+
+ // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z)) with adjustment to match IEEE division result
+ __m512 vf = _mm512_mul_ps(ve, vr);
+ vf = _mm512_fmadd_ps(_mm512_fnmadd_ps(vf, vd, ve), vr, vf);
+
+ // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z)
+ vf = _mm512_mask_sub_ps(vf, _mm512_testn_epi32_mask(_mm512_castps_si512(vx), vsign_mask), vone, vf);
+
+ _mm512_storeu_ps(output, vf);
+
+ input += 16;
+ output += 16;
+ }
+}