Marat Dukhan | 8d3c07e | 2020-01-02 01:20:59 -0800 | [diff] [blame] | 1 | // Auto-generated file. Do not edit! |
| 2 | // Template: src/f32-sigmoid/neon-lut2048-p1.c.in |
| 3 | // Generator: tools/xngen |
| 4 | // |
| 5 | // Copyright 2019 Google LLC |
| 6 | // |
| 7 | // This source code is licensed under the BSD-style license found in the |
| 8 | // LICENSE file in the root directory of this source tree. |
| 9 | |
| 10 | #include <assert.h> |
| 11 | |
| 12 | #include <arm_neon.h> |
| 13 | |
| 14 | #include <xnnpack/common.h> |
| 15 | #include <xnnpack/vunary.h> |
| 16 | |
| 17 | |
| 18 | extern XNN_INTERNAL const float xnn_table_exp2_k_over_2048[2048]; |
| 19 | |
Marat Dukhan | 4a24a58 | 2020-01-06 13:30:00 -0800 | [diff] [blame] | 20 | void xnn_f32_sigmoid_ukernel__neon_rr2_lut2048_p1_nr2recps_x16( |
Marat Dukhan | 8d3c07e | 2020-01-02 01:20:59 -0800 | [diff] [blame] | 21 | size_t n, |
| 22 | const float* x, |
| 23 | float* y, |
| 24 | const void* params) |
| 25 | { |
| 26 | assert(n % sizeof(float) == 0); |
| 27 | |
| 28 | const float32x4_t vmagic_bias = vmovq_n_f32(0x1.800000p23f); |
| 29 | // The largest z for which sigmoidf(-z) is normalized. |
| 30 | // This number is also the largest z for which expf(-z) is normalized. |
| 31 | const float32x4_t vdenorm_cutoff = vmovq_n_f32(0x1.5D589Ep+6f); |
| 32 | const float32x4_t vminus_log2e_x2048 = vmovq_n_f32(-0x1.715476p11f); |
Marat Dukhan | 68b3b45 | 2020-01-02 10:11:15 -0800 | [diff] [blame] | 33 | // Last 18 bits are zeroes |
| 34 | const float32x4_t vln2_o2048_hi = vmovq_n_f32(0x1.600000p-12f); |
| 35 | const float32x4_t vln2_o2048_lo = vmovq_n_f32(0x1.7217F8p-19f); |
Marat Dukhan | 8d3c07e | 2020-01-02 01:20:59 -0800 | [diff] [blame] | 36 | const float32x4_t vone = vmovq_n_f32(1.0f); |
| 37 | |
| 38 | const float32x4_t vc1 = vmovq_n_f32(-0x1.FFFFFEp-1f); |
| 39 | |
| 40 | const int32x4_t vindex_mask = vmovq_n_s32(INT32_C(0x7FF)); |
| 41 | |
| 42 | for (; n >= 16 * sizeof(float); n -= 16 * sizeof(float)) { |
| 43 | const float32x4_t vx0123 = vld1q_f32(x); x += 4; |
| 44 | const float32x4_t vx4567 = vld1q_f32(x); x += 4; |
| 45 | const float32x4_t vx89AB = vld1q_f32(x); x += 4; |
| 46 | const float32x4_t vxCDEF = vld1q_f32(x); x += 4; |
| 47 | |
| 48 | // General structure of the algorithm: |
| 49 | // / exp(x) / (1 + exp(x)) if x <= 0 |
| 50 | // f[x] := |
| 51 | // \ 1 - f[-x] if x >= 0 |
| 52 | // |
| 53 | // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x), |
| 54 | // then replace result with 1 - f[-z] if x >= 0. |
| 55 | const float32x4_t vz0123 = vabsq_f32(vx0123); |
| 56 | const float32x4_t vz4567 = vabsq_f32(vx4567); |
| 57 | const float32x4_t vz89AB = vabsq_f32(vx89AB); |
| 58 | const float32x4_t vzCDEF = vabsq_f32(vxCDEF); |
| 59 | |
| 60 | // Compute reduced argument n := round(-z * 2048 / log(2)). |
| 61 | // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing |
| 62 | // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction. |
| 63 | // The trick with adding large number is valid only within certain bounds (|z * 2048 / log(2)| <= 2**22, i.e. |
| 64 | // |z| <= 0x1.62E43p+10 = 1419.5654296875), but that is acceptable, because inputs x outside of |
| 65 | // [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup the result |
| 66 | // for such inputs at the very end of the algorithm. |
| 67 | float32x4_t vn0123 = vmlaq_f32(vmagic_bias, vz0123, vminus_log2e_x2048); |
| 68 | float32x4_t vn4567 = vmlaq_f32(vmagic_bias, vz4567, vminus_log2e_x2048); |
| 69 | float32x4_t vn89AB = vmlaq_f32(vmagic_bias, vz89AB, vminus_log2e_x2048); |
| 70 | float32x4_t vnCDEF = vmlaq_f32(vmagic_bias, vzCDEF, vminus_log2e_x2048); |
| 71 | |
| 72 | // Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is |
| 73 | // normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) = |
| 74 | // = 2**e * 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps: |
| 75 | // 1. Fetch 2**(n / 2048 - e) = 2**(n % 2048) from the table using the 6 low bits of n, as integer. Note that the |
| 76 | // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| 77 | // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized |
| 78 | // number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(-z) is normalized) we have -126 <= e <= 0, |
| 79 | // and thus the adjusted exponent is not lower than -126. |
| 80 | // |
| 81 | // Extract e from bits 11:19 of n and shift it into bits 23:31 (position of floating-point exponent). |
| 82 | const int32x4_t ve0123 = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn0123), vmovq_n_s32(INT32_C(0x7FF))), 12); |
| 83 | const int32x4_t ve4567 = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn4567), vmovq_n_s32(INT32_C(0x7FF))), 12); |
| 84 | const int32x4_t ve89AB = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn89AB), vmovq_n_s32(INT32_C(0x7FF))), 12); |
| 85 | const int32x4_t veCDEF = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vnCDEF), vmovq_n_s32(INT32_C(0x7FF))), 12); |
| 86 | |
| 87 | // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048). |
| 88 | const uint64x2_t vidx0123 = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn0123), vindex_mask)); |
| 89 | const uint64x2_t vidx4567 = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn4567), vindex_mask)); |
| 90 | const uint64x2_t vidx89AB = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn89AB), vindex_mask)); |
| 91 | const uint64x2_t vidxCDEF = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vnCDEF), vindex_mask)); |
| 92 | |
| 93 | const uint64_t vidx01 = vgetq_lane_u64(vidx0123, 0); |
| 94 | const uint64_t vidx23 = vgetq_lane_u64(vidx0123, 1); |
| 95 | float32x2_t vl01 = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx01]); |
| 96 | float32x2_t vl23 = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx23]); |
| 97 | const uint64_t vidx45 = vgetq_lane_u64(vidx4567, 0); |
| 98 | const uint64_t vidx67 = vgetq_lane_u64(vidx4567, 1); |
| 99 | float32x2_t vl45 = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx45]); |
| 100 | float32x2_t vl67 = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx67]); |
| 101 | const uint64_t vidx89 = vgetq_lane_u64(vidx89AB, 0); |
| 102 | const uint64_t vidxAB = vgetq_lane_u64(vidx89AB, 1); |
| 103 | float32x2_t vl89 = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx89]); |
| 104 | float32x2_t vlAB = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidxAB]); |
| 105 | const uint64_t vidxCD = vgetq_lane_u64(vidxCDEF, 0); |
| 106 | const uint64_t vidxEF = vgetq_lane_u64(vidxCDEF, 1); |
| 107 | float32x2_t vlCD = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidxCD]); |
| 108 | float32x2_t vlEF = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidxEF]); |
| 109 | |
| 110 | vl01 = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx01 >> 32)], vl01, 1); |
| 111 | vl23 = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx23 >> 32)], vl23, 1); |
| 112 | const float32x4_t vl0123 = vcombine_f32(vl01, vl23); |
| 113 | vl45 = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx45 >> 32)], vl45, 1); |
| 114 | vl67 = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx67 >> 32)], vl67, 1); |
| 115 | const float32x4_t vl4567 = vcombine_f32(vl45, vl67); |
| 116 | vl89 = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx89 >> 32)], vl89, 1); |
| 117 | vlAB = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidxAB >> 32)], vlAB, 1); |
| 118 | const float32x4_t vl89AB = vcombine_f32(vl89, vlAB); |
| 119 | vlCD = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidxCD >> 32)], vlCD, 1); |
| 120 | vlEF = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidxEF >> 32)], vlEF, 1); |
| 121 | const float32x4_t vlCDEF = vcombine_f32(vlCD, vlEF); |
| 122 | |
| 123 | // Adjust exponent of the value l fetched from the table to get the final s value. |
| 124 | const float32x4_t vs0123 = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl0123), ve0123)); |
| 125 | const float32x4_t vs4567 = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl4567), ve4567)); |
| 126 | const float32x4_t vs89AB = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl89AB), ve89AB)); |
| 127 | const float32x4_t vsCDEF = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vlCDEF), veCDEF)); |
| 128 | |
| 129 | // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number. |
| 130 | vn0123 = vsubq_f32(vn0123, vmagic_bias); |
| 131 | vn4567 = vsubq_f32(vn4567, vmagic_bias); |
| 132 | vn89AB = vsubq_f32(vn89AB, vmagic_bias); |
| 133 | vnCDEF = vsubq_f32(vnCDEF, vmagic_bias); |
| 134 | |
| 135 | // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048. |
| 136 | // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy. |
| 137 | float32x4_t vt0123 = vmlaq_f32(vz0123, vn0123, vln2_o2048_hi); |
| 138 | float32x4_t vt4567 = vmlaq_f32(vz4567, vn4567, vln2_o2048_hi); |
| 139 | float32x4_t vt89AB = vmlaq_f32(vz89AB, vn89AB, vln2_o2048_hi); |
| 140 | float32x4_t vtCDEF = vmlaq_f32(vzCDEF, vnCDEF, vln2_o2048_hi); |
| 141 | |
| 142 | vt0123 = vmlaq_f32(vt0123, vn0123, vln2_o2048_lo); |
| 143 | vt4567 = vmlaq_f32(vt4567, vn4567, vln2_o2048_lo); |
| 144 | vt89AB = vmlaq_f32(vt89AB, vn89AB, vln2_o2048_lo); |
| 145 | vtCDEF = vmlaq_f32(vtCDEF, vnCDEF, vln2_o2048_lo); |
| 146 | |
| 147 | // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]: |
| 148 | // P1(t) = 1 + t * c1 |
| 149 | const float32x4_t vp0123 = vmulq_f32(vt0123, vc1); |
| 150 | const float32x4_t vp4567 = vmulq_f32(vt4567, vc1); |
| 151 | const float32x4_t vp89AB = vmulq_f32(vt89AB, vc1); |
| 152 | const float32x4_t vpCDEF = vmulq_f32(vtCDEF, vc1); |
| 153 | |
| 154 | // Reconstruct the exp(-z) value: |
| 155 | // y = s * (1 + t * c1) |
| 156 | // = s + s * (t * c1)) |
| 157 | // = s + s * p |
| 158 | const float32x4_t vy0123 = vmlaq_f32(vs0123, vs0123, vp0123); |
| 159 | const float32x4_t vy4567 = vmlaq_f32(vs4567, vs4567, vp4567); |
| 160 | const float32x4_t vy89AB = vmlaq_f32(vs89AB, vs89AB, vp89AB); |
| 161 | const float32x4_t vyCDEF = vmlaq_f32(vsCDEF, vsCDEF, vpCDEF); |
| 162 | |
| 163 | // Denominator of the sigmoid fraction: 1.0 + exp(-z) |
| 164 | const float32x4_t vd0123 = vaddq_f32(vy0123, vone); |
| 165 | const float32x4_t vd4567 = vaddq_f32(vy4567, vone); |
| 166 | const float32x4_t vd89AB = vaddq_f32(vy89AB, vone); |
| 167 | const float32x4_t vdCDEF = vaddq_f32(vyCDEF, vone); |
| 168 | |
| 169 | // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator. |
| 170 | // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0. |
| 171 | // Thus the reciprocal of the denominator never overflows. |
| 172 | float32x4_t vr0123 = vrecpeq_f32(vd0123); |
| 173 | float32x4_t vr4567 = vrecpeq_f32(vd4567); |
| 174 | float32x4_t vr89AB = vrecpeq_f32(vd89AB); |
| 175 | float32x4_t vrCDEF = vrecpeq_f32(vdCDEF); |
| 176 | |
| 177 | vr0123 = vmulq_f32(vr0123, vrecpsq_f32(vr0123, vd0123)); |
| 178 | vr4567 = vmulq_f32(vr4567, vrecpsq_f32(vr4567, vd4567)); |
| 179 | vr89AB = vmulq_f32(vr89AB, vrecpsq_f32(vr89AB, vd89AB)); |
| 180 | vrCDEF = vmulq_f32(vrCDEF, vrecpsq_f32(vrCDEF, vdCDEF)); |
| 181 | |
| 182 | vr0123 = vmulq_f32(vr0123, vrecpsq_f32(vr0123, vd0123)); |
| 183 | vr4567 = vmulq_f32(vr4567, vrecpsq_f32(vr4567, vd4567)); |
| 184 | vr89AB = vmulq_f32(vr89AB, vrecpsq_f32(vr89AB, vd89AB)); |
| 185 | vrCDEF = vmulq_f32(vrCDEF, vrecpsq_f32(vrCDEF, vdCDEF)); |
| 186 | |
| 187 | // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z)) |
| 188 | float32x4_t vf0123 = vmulq_f32(vy0123, vr0123); |
| 189 | float32x4_t vf4567 = vmulq_f32(vy4567, vr4567); |
| 190 | float32x4_t vf89AB = vmulq_f32(vy89AB, vr89AB); |
| 191 | float32x4_t vfCDEF = vmulq_f32(vyCDEF, vrCDEF); |
| 192 | |
| 193 | // For inputs below denormal cutoff, replace output with +0.0f. |
| 194 | // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| 195 | vf0123 = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf0123), vcagtq_f32(vx0123, vdenorm_cutoff))); |
| 196 | vf4567 = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf4567), vcagtq_f32(vx4567, vdenorm_cutoff))); |
| 197 | vf89AB = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf89AB), vcagtq_f32(vx89AB, vdenorm_cutoff))); |
| 198 | vfCDEF = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vfCDEF), vcagtq_f32(vxCDEF, vdenorm_cutoff))); |
| 199 | |
| 200 | // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z) |
Marat Dukhan | 26cda6d | 2020-01-09 13:54:32 -0800 | [diff] [blame^] | 201 | const uint32x4_t vm0123 = vcltq_f32(vx0123, vmovq_n_f32(0.0f)); |
| 202 | const uint32x4_t vm4567 = vcltq_f32(vx4567, vmovq_n_f32(0.0f)); |
| 203 | const uint32x4_t vm89AB = vcltq_f32(vx89AB, vmovq_n_f32(0.0f)); |
| 204 | const uint32x4_t vmCDEF = vcltq_f32(vxCDEF, vmovq_n_f32(0.0f)); |
Marat Dukhan | 8d3c07e | 2020-01-02 01:20:59 -0800 | [diff] [blame] | 205 | |
| 206 | vf0123 = vbslq_f32(vm0123, vf0123, vsubq_f32(vone, vf0123)); |
| 207 | vf4567 = vbslq_f32(vm4567, vf4567, vsubq_f32(vone, vf4567)); |
| 208 | vf89AB = vbslq_f32(vm89AB, vf89AB, vsubq_f32(vone, vf89AB)); |
| 209 | vfCDEF = vbslq_f32(vmCDEF, vfCDEF, vsubq_f32(vone, vfCDEF)); |
| 210 | |
| 211 | vst1q_f32(y, vf0123); y += 4; |
| 212 | vst1q_f32(y, vf4567); y += 4; |
| 213 | vst1q_f32(y, vf89AB); y += 4; |
| 214 | vst1q_f32(y, vfCDEF); y += 4; |
| 215 | } |
| 216 | for (; n >= 4 * sizeof(float); n -= 4 * sizeof(float)) { |
| 217 | const float32x4_t vx = vld1q_f32(x); x += 4; |
| 218 | |
| 219 | // General structure of the algorithm: |
| 220 | // / exp(x) / (1 + exp(x)) if x <= 0 |
| 221 | // f[x] := |
| 222 | // \ 1 - f[-x] if x >= 0 |
| 223 | // |
| 224 | // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x), |
| 225 | // then replace result with 1 - f[-z] if x >= 0. |
| 226 | const float32x4_t vz = vabsq_f32(vx); |
| 227 | |
| 228 | // Compute reduced argument n := round(-z * 2048 / log(2)). |
| 229 | // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing |
| 230 | // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction. |
| 231 | // The trick with adding large number is valid only within certain bounds (|z * 2048 / log(2)| <= 2**22, i.e. |
| 232 | // |z| <= 0x1.62E43p+10 = 1419.5654296875), but that is acceptable, because inputs x outside of |
| 233 | // [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup the result |
| 234 | // for such inputs at the very end of the algorithm. |
| 235 | float32x4_t vn = vmlaq_f32(vmagic_bias, vz, vminus_log2e_x2048); |
| 236 | |
| 237 | // Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is |
| 238 | // normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) = |
| 239 | // = 2**e * 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps: |
| 240 | // 1. Fetch 2**(n / 2048 - e) = 2**(n % 2048) from exp2_k_over_2048_table using the 6 low bits of n, as integer. Note that the |
| 241 | // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| 242 | // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized |
| 243 | // number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(-z) is normalized) we have -126 <= e <= 0, |
| 244 | // and thus the adjusted exponent is not lower than -126. |
| 245 | // |
| 246 | // Extract e from bits 11:19 of n and shift it into bits 23:31 (position of floating-point exponent). |
| 247 | const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x7FF))), 12); |
| 248 | |
| 249 | // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048). |
| 250 | const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask)); |
| 251 | const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0); |
| 252 | const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1); |
| 253 | float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_lo]); |
| 254 | float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_hi]); |
| 255 | vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_lo >> 32)], vl_lo, 1); |
| 256 | vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_hi >> 32)], vl_hi, 1); |
| 257 | const float32x4_t vl = vcombine_f32(vl_lo, vl_hi); |
| 258 | // Adjust exponent of the value l fetched from the exp2_k_over_2048_table to get the final s value. |
| 259 | const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve)); |
| 260 | |
| 261 | // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number. |
| 262 | vn = vsubq_f32(vn, vmagic_bias); |
| 263 | |
| 264 | // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048. |
| 265 | // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy. |
| 266 | float32x4_t vt = vmlaq_f32(vz, vn, vln2_o2048_hi); |
| 267 | vt = vmlaq_f32(vt, vn, vln2_o2048_lo); |
| 268 | |
| 269 | // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]: |
| 270 | // P1(t) = 1 + t * c1 |
| 271 | const float32x4_t vp = vmulq_f32(vt, vc1); |
| 272 | |
| 273 | // Reconstruct the exp(-z) value: |
| 274 | // y = s * (1 + t * c1) |
| 275 | // = s + s * (t * c1)) |
| 276 | // = s + s * p |
| 277 | const float32x4_t vy = vmlaq_f32(vs, vs, vp); |
| 278 | |
| 279 | // Denominator of the sigmoid fraction: 1.0 + exp(-z) |
| 280 | const float32x4_t vd = vaddq_f32(vy, vone); |
| 281 | |
| 282 | // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator. |
| 283 | // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0. |
| 284 | // Thus the reciprocal of the denominator never overflows. |
| 285 | float32x4_t vr = vrecpeq_f32(vd); |
| 286 | |
| 287 | vr = vmulq_f32(vr, vrecpsq_f32(vr, vd)); |
| 288 | |
| 289 | vr = vmulq_f32(vr, vrecpsq_f32(vr, vd)); |
| 290 | |
| 291 | // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z)) |
| 292 | float32x4_t vf = vmulq_f32(vy, vr); |
| 293 | |
| 294 | // For inputs below denormal cutoff, replace output with +0.0f. |
| 295 | // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| 296 | vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcagtq_f32(vx, vdenorm_cutoff))); |
| 297 | |
| 298 | // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z) |
Marat Dukhan | 26cda6d | 2020-01-09 13:54:32 -0800 | [diff] [blame^] | 299 | const uint32x4_t vm = vcltq_f32(vx, vmovq_n_f32(0.0f)); |
Marat Dukhan | 8d3c07e | 2020-01-02 01:20:59 -0800 | [diff] [blame] | 300 | vf = vbslq_f32(vm, vf, vsubq_f32(vone, vf)); |
| 301 | |
| 302 | vst1q_f32(y, vf); y += 4; |
| 303 | } |
| 304 | if XNN_UNLIKELY(n != 0) { |
| 305 | const float32x4_t vx = vld1q_f32(x); |
| 306 | |
| 307 | // General structure of the algorithm: |
| 308 | // / exp(x) / (1 + exp(x)) if x <= 0 |
| 309 | // f[x] := |
| 310 | // \ 1 - f[-x] if x >= 0 |
| 311 | // |
| 312 | // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x), |
| 313 | // then replace result with 1 - f[-z] if x >= 0. |
| 314 | const float32x4_t vz = vabsq_f32(vx); |
| 315 | |
| 316 | // Compute reduced argument n := round(-z * 2048 / log(2)). |
| 317 | // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing |
| 318 | // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction. |
| 319 | // The trick with adding large number is valid only within certain bounds (|z * 2048 / log(2)| <= 2**22, i.e. |
| 320 | // |z| <= 0x1.62E43p+10 = 1419.5654296875), but that is acceptable, because inputs x outside of |
| 321 | // [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup the result |
| 322 | // for such inputs at the very end of the algorithm. |
| 323 | float32x4_t vn = vmlaq_f32(vmagic_bias, vz, vminus_log2e_x2048); |
| 324 | |
| 325 | // Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is |
| 326 | // normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) = |
| 327 | // = 2**e * 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps: |
| 328 | // 1. Fetch 2**(n / 2048 - e) = 2**(n % 2048) from exp2_k_over_2048_table using the 6 low bits of n, as integer. Note that the |
| 329 | // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| 330 | // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized |
| 331 | // number, because for 0 <= z <= 87.33642 (inputs for which sigmoidf(-z) is normalized) we have -126 <= e <= 0, |
| 332 | // and thus the adjusted exponent is not lower than -126. |
| 333 | // |
| 334 | // Extract e from bits 11:19 of n and shift it into bits 23:31 (position of floating-point exponent). |
| 335 | const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x7FF))), 12); |
| 336 | |
| 337 | // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048). |
| 338 | const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask)); |
| 339 | const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0); |
| 340 | const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1); |
| 341 | float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_lo]); |
| 342 | float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_2048[(uint32_t) vidx_hi]); |
| 343 | vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_lo >> 32)], vl_lo, 1); |
| 344 | vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_2048[(uint32_t) (vidx_hi >> 32)], vl_hi, 1); |
| 345 | const float32x4_t vl = vcombine_f32(vl_lo, vl_hi); |
| 346 | // Adjust exponent of the value l fetched from the exp2_k_over_2048_table to get the final s value. |
| 347 | const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve)); |
| 348 | |
| 349 | // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number. |
| 350 | vn = vsubq_f32(vn, vmagic_bias); |
| 351 | |
| 352 | // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048. |
| 353 | // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy. |
| 354 | float32x4_t vt = vmlaq_f32(vz, vn, vln2_o2048_hi); |
| 355 | vt = vmlaq_f32(vt, vn, vln2_o2048_lo); |
| 356 | |
| 357 | // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/2048, log(2)/2048]: |
| 358 | // P1(t) = 1 + t * c1 |
| 359 | const float32x4_t vp = vmulq_f32(vt, vc1); |
| 360 | |
| 361 | // Reconstruct the exp(-z) value: |
| 362 | // y = s * (1 + t * c1) |
| 363 | // = s + s * (t * c1)) |
| 364 | // = s + s * p |
| 365 | const float32x4_t vy = vmlaq_f32(vs, vs, vp); |
| 366 | |
| 367 | // Denominator of the sigmoid fraction: 1.0 + exp(-z) |
| 368 | const float32x4_t vd = vaddq_f32(vy, vone); |
| 369 | |
| 370 | // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator. |
| 371 | // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0. |
| 372 | // Thus the reciprocal of the denominator never overflows. |
| 373 | float32x4_t vr = vrecpeq_f32(vd); |
| 374 | |
| 375 | vr = vmulq_f32(vr, vrecpsq_f32(vr, vd)); |
| 376 | |
| 377 | vr = vmulq_f32(vr, vrecpsq_f32(vr, vd)); |
| 378 | |
| 379 | // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z)) |
| 380 | float32x4_t vf = vmulq_f32(vy, vr); |
| 381 | |
| 382 | // For inputs below denormal cutoff, replace output with +0.0f. |
| 383 | // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| 384 | vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcagtq_f32(vx, vdenorm_cutoff))); |
| 385 | |
| 386 | // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z) |
Marat Dukhan | 26cda6d | 2020-01-09 13:54:32 -0800 | [diff] [blame^] | 387 | const uint32x4_t vm = vcltq_f32(vx, vmovq_n_f32(0.0f)); |
Marat Dukhan | 8d3c07e | 2020-01-02 01:20:59 -0800 | [diff] [blame] | 388 | vf = vbslq_f32(vm, vf, vsubq_f32(vone, vf)); |
| 389 | |
| 390 | float32x2_t vf_lo = vget_low_f32(vf); |
| 391 | if (n & (2 * sizeof(float))) { |
| 392 | vst1_f32(y, vf_lo); y += 2; |
| 393 | vf_lo = vget_high_f32(vf); |
| 394 | } |
| 395 | if (n & (1 * sizeof(float))) { |
| 396 | vst1_lane_f32(y, vf_lo, 0); |
| 397 | } |
| 398 | } |
| 399 | } |