Marat Dukhan | 8137e4c | 2020-01-25 12:56:58 -0800 | [diff] [blame] | 1 | // Auto-generated file. Do not edit! |
| 2 | // Template: src/f32-raddstoreexpminusmax/neon-lut64-p2.c.in |
| 3 | // Generator: tools/xngen |
| 4 | // |
| 5 | // Copyright 2020 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/raddstoreexpminusmax.h> |
| 16 | |
| 17 | |
| 18 | extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64]; |
| 19 | |
| 20 | void xnn_f32_raddstoreexpminusmax_ukernel__neonfma_lut64_p2_x16_acc4( |
| 21 | size_t elements, |
| 22 | const float* input, |
| 23 | float* output, |
| 24 | float* sum, |
Marat Dukhan | b2217dd | 2020-05-28 17:30:28 -0700 | [diff] [blame] | 25 | float max) XNN_DISABLE_TSAN |
Marat Dukhan | 8137e4c | 2020-01-25 12:56:58 -0800 | [diff] [blame] | 26 | { |
| 27 | assert(elements % sizeof(float) == 0); |
| 28 | |
| 29 | const float32x4_t vmagic_bias = vmovq_n_f32(0x1.800000p23f); |
| 30 | // The smallest x for which expf(x) is normalized. |
| 31 | const float32x4_t vdenorm_cutoff = vmovq_n_f32(-0x1.5D589Ep6f); |
| 32 | const float32x4_t vlog2e_x64 = vmovq_n_f32(0x1.715476p6f); |
| 33 | const float32x4_t vminus_ln2_o64_hi = vmovq_n_f32(-0x1.62e43p-7f); |
| 34 | const float32x4_t vminus_ln2_o64_lo = vmovq_n_f32(0x1.05c61p-35f); |
| 35 | |
| 36 | const float32x4_t vc2 = vmovq_n_f32(0x1.FFFF0Ap-2f); |
| 37 | |
| 38 | const int32x4_t vindex_mask = vmovq_n_s32(INT32_C(0x3F)); |
| 39 | |
| 40 | const float32x4_t vi_max = vdupq_n_f32(max); |
| 41 | |
| 42 | float32x4_t vacc0 = vmovq_n_f32(0.0f); |
| 43 | float32x4_t vacc1 = vmovq_n_f32(0.0f); |
| 44 | float32x4_t vacc2 = vmovq_n_f32(0.0f); |
| 45 | float32x4_t vacc3 = vmovq_n_f32(0.0f); |
| 46 | for (; elements >= 16 * sizeof(float); elements -= 16 * sizeof(float)) { |
| 47 | // Load 16 (4x4) inputs at a time. |
| 48 | const float32x4_t vi0123 = vld1q_f32(input); input += 4; |
| 49 | const float32x4_t vi4567 = vld1q_f32(input); input += 4; |
| 50 | const float32x4_t vi89AB = vld1q_f32(input); input += 4; |
| 51 | const float32x4_t viCDEF = vld1q_f32(input); input += 4; |
| 52 | |
| 53 | // Subtract maximum input x := i - i_max. This implies x <= 0. |
| 54 | const float32x4_t vx0123 = vsubq_f32(vi0123, vi_max); |
| 55 | const float32x4_t vx4567 = vsubq_f32(vi4567, vi_max); |
| 56 | const float32x4_t vx89AB = vsubq_f32(vi89AB, vi_max); |
| 57 | const float32x4_t vxCDEF = vsubq_f32(viCDEF, vi_max); |
| 58 | |
| 59 | // Compute reduced argument n := round(x * 64 / log(2)). |
| 60 | // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing |
| 61 | // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction. |
| 62 | // The trick with adding large number is valid only within certain bounds (|x * 64 / log(2)| <= 2**22, i.e. |
| 63 | // |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs outside of [-87.336540, 0.0] |
| 64 | // result in denormalized or underflown expf(x). We fixup the result for such inputs at the very end of the |
| 65 | // algorithm. |
| 66 | float32x4_t vn0123 = vfmaq_f32(vmagic_bias, vx0123, vlog2e_x64); |
| 67 | float32x4_t vn4567 = vfmaq_f32(vmagic_bias, vx4567, vlog2e_x64); |
| 68 | float32x4_t vn89AB = vfmaq_f32(vmagic_bias, vx89AB, vlog2e_x64); |
| 69 | float32x4_t vnCDEF = vfmaq_f32(vmagic_bias, vxCDEF, vlog2e_x64); |
| 70 | |
| 71 | // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that expf(x) is normalized, |
| 72 | // i.e. -87.33642 <= x <= 0.0. As n has 6 fractional bits, we split s == 2**(n / 64) = 2**e * 2**(n / 64 - e), where |
| 73 | // e := int(n / 64). We create s in two steps: |
| 74 | // 1. Fetch 2**(n / 64 - e) = 2**(n % 64) from the table using the 6 low bits of n, as integer. Note that the |
| 75 | // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| 76 | // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized |
| 77 | // number, because for -87.33642 <= x <= 0.0 (inputs for which expf(x) is normalized) we have -126 <= e <= 0, |
| 78 | // and thus the adjusted exponent is not lower than -126. |
| 79 | // |
| 80 | // Extract e from bits 6:14 of n and shift it into bits 23:31 (position of floating-point exponent). |
| 81 | const int32x4_t ve0123 = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn0123), vmovq_n_s32(INT32_C(0x3F))), 17); |
| 82 | const int32x4_t ve4567 = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn4567), vmovq_n_s32(INT32_C(0x3F))), 17); |
| 83 | const int32x4_t ve89AB = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn89AB), vmovq_n_s32(INT32_C(0x3F))), 17); |
| 84 | const int32x4_t veCDEF = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vnCDEF), vmovq_n_s32(INT32_C(0x3F))), 17); |
| 85 | |
| 86 | // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64). |
| 87 | const uint64x2_t vidx0123 = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn0123), vindex_mask)); |
| 88 | const uint64_t vidx01 = vgetq_lane_u64(vidx0123, 0); |
| 89 | const uint64_t vidx23 = vgetq_lane_u64(vidx0123, 1); |
| 90 | const uint64x2_t vidx4567 = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn4567), vindex_mask)); |
| 91 | const uint64_t vidx45 = vgetq_lane_u64(vidx4567, 0); |
| 92 | const uint64_t vidx67 = vgetq_lane_u64(vidx4567, 1); |
| 93 | const uint64x2_t vidx89AB = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn89AB), vindex_mask)); |
| 94 | const uint64_t vidx89 = vgetq_lane_u64(vidx89AB, 0); |
| 95 | const uint64_t vidxAB = vgetq_lane_u64(vidx89AB, 1); |
| 96 | const uint64x2_t vidxCDEF = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vnCDEF), vindex_mask)); |
| 97 | const uint64_t vidxCD = vgetq_lane_u64(vidxCDEF, 0); |
| 98 | const uint64_t vidxEF = vgetq_lane_u64(vidxCDEF, 1); |
| 99 | |
| 100 | float32x2_t vl01 = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx01]); |
| 101 | float32x2_t vl23 = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx23]); |
| 102 | float32x2_t vl45 = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx45]); |
| 103 | float32x2_t vl67 = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx67]); |
| 104 | float32x2_t vl89 = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx89]); |
| 105 | float32x2_t vlAB = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidxAB]); |
| 106 | float32x2_t vlCD = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidxCD]); |
| 107 | float32x2_t vlEF = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidxEF]); |
| 108 | |
| 109 | vl01 = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx01 >> 32)], vl01, 1); |
| 110 | vl23 = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx23 >> 32)], vl23, 1); |
| 111 | const float32x4_t vl0123 = vcombine_f32(vl01, vl23); |
| 112 | vl45 = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx45 >> 32)], vl45, 1); |
| 113 | vl67 = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx67 >> 32)], vl67, 1); |
| 114 | const float32x4_t vl4567 = vcombine_f32(vl45, vl67); |
| 115 | vl89 = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx89 >> 32)], vl89, 1); |
| 116 | vlAB = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidxAB >> 32)], vlAB, 1); |
| 117 | const float32x4_t vl89AB = vcombine_f32(vl89, vlAB); |
| 118 | vlCD = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidxCD >> 32)], vlCD, 1); |
| 119 | vlEF = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidxEF >> 32)], vlEF, 1); |
| 120 | const float32x4_t vlCDEF = vcombine_f32(vlCD, vlEF); |
| 121 | |
| 122 | // Adjust exponent of the value l fetched from the table to get the final s value. |
| 123 | const float32x4_t vs0123 = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl0123), ve0123)); |
| 124 | const float32x4_t vs4567 = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl4567), ve4567)); |
| 125 | const float32x4_t vs89AB = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl89AB), ve89AB)); |
| 126 | const float32x4_t vsCDEF = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vlCDEF), veCDEF)); |
| 127 | |
| 128 | // Subtract the large number back to get final n := round(x * 64 / log(2)) as a floating-point number. |
| 129 | vn0123 = vsubq_f32(vn0123, vmagic_bias); |
| 130 | vn4567 = vsubq_f32(vn4567, vmagic_bias); |
| 131 | vn89AB = vsubq_f32(vn89AB, vmagic_bias); |
| 132 | vnCDEF = vsubq_f32(vnCDEF, vmagic_bias); |
| 133 | |
| 134 | // Compute reduced argument t := x - n * log(2) / 64. |
| 135 | // Use Cody-Waite range reduction method (note the two constants representing log(2) / 64) to improve accuracy. |
| 136 | float32x4_t vt0123 = vfmaq_f32(vx0123, vn0123, vminus_ln2_o64_hi); |
| 137 | float32x4_t vt4567 = vfmaq_f32(vx4567, vn4567, vminus_ln2_o64_hi); |
| 138 | float32x4_t vt89AB = vfmaq_f32(vx89AB, vn89AB, vminus_ln2_o64_hi); |
| 139 | float32x4_t vtCDEF = vfmaq_f32(vxCDEF, vnCDEF, vminus_ln2_o64_hi); |
| 140 | |
| 141 | vt0123 = vfmaq_f32(vt0123, vn0123, vminus_ln2_o64_lo); |
| 142 | vt4567 = vfmaq_f32(vt4567, vn4567, vminus_ln2_o64_lo); |
| 143 | vt89AB = vfmaq_f32(vt89AB, vn89AB, vminus_ln2_o64_lo); |
| 144 | vtCDEF = vfmaq_f32(vtCDEF, vnCDEF, vminus_ln2_o64_lo); |
| 145 | |
Marat Dukhan | 102a739 | 2020-11-20 01:18:10 -0800 | [diff] [blame^] | 146 | // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128]. |
Marat Dukhan | 8137e4c | 2020-01-25 12:56:58 -0800 | [diff] [blame] | 147 | float32x4_t vp0123 = vmulq_f32(vt0123, vc2); |
| 148 | float32x4_t vp4567 = vmulq_f32(vt4567, vc2); |
| 149 | float32x4_t vp89AB = vmulq_f32(vt89AB, vc2); |
| 150 | float32x4_t vpCDEF = vmulq_f32(vtCDEF, vc2); |
| 151 | |
| 152 | vp0123 = vfmaq_f32(vt0123, vt0123, vp0123); |
| 153 | vp4567 = vfmaq_f32(vt4567, vt4567, vp4567); |
| 154 | vp89AB = vfmaq_f32(vt89AB, vt89AB, vp89AB); |
| 155 | vpCDEF = vfmaq_f32(vtCDEF, vtCDEF, vpCDEF); |
| 156 | |
| 157 | // Reconstruct the final f value: |
| 158 | // f = s * (1 + t * (1 + t * c2)) |
| 159 | // = s * (1 + t + t * (t * c2)) |
| 160 | // = s + s * (t + t * (t * c2)) |
| 161 | // = s + s * p |
| 162 | float32x4_t vf0123 = vfmaq_f32(vs0123, vs0123, vp0123); |
| 163 | float32x4_t vf4567 = vfmaq_f32(vs4567, vs4567, vp4567); |
| 164 | float32x4_t vf89AB = vfmaq_f32(vs89AB, vs89AB, vp89AB); |
| 165 | float32x4_t vfCDEF = vfmaq_f32(vsCDEF, vsCDEF, vpCDEF); |
| 166 | |
| 167 | // For inputs below denormal cutoff, replace output with +0.0f. |
| 168 | // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| 169 | vf0123 = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf0123), vcltq_f32(vx0123, vdenorm_cutoff))); |
| 170 | vf4567 = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf4567), vcltq_f32(vx4567, vdenorm_cutoff))); |
| 171 | vf89AB = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf89AB), vcltq_f32(vx89AB, vdenorm_cutoff))); |
| 172 | vfCDEF = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vfCDEF), vcltq_f32(vxCDEF, vdenorm_cutoff))); |
| 173 | |
| 174 | // Store 16 (4x4) outputs at a time. |
| 175 | vst1q_f32(output, vf0123); output += 4; |
| 176 | vst1q_f32(output, vf4567); output += 4; |
| 177 | vst1q_f32(output, vf89AB); output += 4; |
| 178 | vst1q_f32(output, vfCDEF); output += 4; |
| 179 | |
| 180 | // Accumulate computed exponents. |
| 181 | vacc0 = vaddq_f32(vacc0, vf0123); |
| 182 | vacc0 = vaddq_f32(vacc0, vf4567); |
| 183 | vacc0 = vaddq_f32(vacc0, vf89AB); |
| 184 | vacc0 = vaddq_f32(vacc0, vfCDEF); |
| 185 | } |
| 186 | // Add up all accumulators to vacc0 |
| 187 | vacc0 = vaddq_f32(vacc0, vacc1); |
| 188 | vacc2 = vaddq_f32(vacc2, vacc3); |
| 189 | vacc0 = vaddq_f32(vacc0, vacc2); |
| 190 | |
| 191 | float32x4_t vacc = vacc0; |
| 192 | for (; elements >= 4 * sizeof(float); elements -= 4 * sizeof(float)) { |
| 193 | // Load 4 inputs at a time. |
| 194 | const float32x4_t vi = vld1q_f32(input); input += 4; |
| 195 | |
| 196 | // Subtract maximum input x := i - i_max. This implies x <= 0. |
| 197 | const float32x4_t vx = vsubq_f32(vi, vi_max); |
| 198 | |
| 199 | // Compute reduced argument n := round(x * 64 / log(2)). |
| 200 | // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing |
| 201 | // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction. |
| 202 | // The trick with adding large number is valid only within certain bounds (|x * 64 / log(2)| <= 2**22, i.e. |
| 203 | // |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs outside of [-87.336540, 0.0] |
| 204 | // result in denormalized or underflown expf(x). We fixup the result for such inputs at the very end of the |
| 205 | // algorithm. |
| 206 | float32x4_t vn = vfmaq_f32(vmagic_bias, vx, vlog2e_x64); |
| 207 | |
| 208 | // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that expf(x) is normalized, |
| 209 | // i.e. -87.33642 <= x <= 0.0. As n has 6 fractional bits, we split s == 2**(n / 64) = 2**e * 2**(n / 64 - e), where |
| 210 | // e := int(n / 64). We create s in two steps: |
| 211 | // 1. Fetch 2**(n / 64 - e) = 2**(n % 64) from the table using the 6 low bits of n, as integer. Note that the |
| 212 | // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| 213 | // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized |
| 214 | // number, because for -87.33642 <= x <= 0.0 (inputs for which expf(x) is normalized) we have -126 <= e <= 0, |
| 215 | // and thus the adjusted exponent is not lower than -126. |
| 216 | // |
| 217 | // Extract e from bits 6:14 of n and shift it into bits 23:31 (position of floating-point exponent). |
| 218 | const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x3F))), 17); |
| 219 | |
| 220 | // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64). |
| 221 | const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask)); |
| 222 | const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0); |
| 223 | const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1); |
| 224 | float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_lo]); |
| 225 | float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_hi]); |
| 226 | vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_lo >> 32)], vl_lo, 1); |
| 227 | vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_hi >> 32)], vl_hi, 1); |
| 228 | const float32x4_t vl = vcombine_f32(vl_lo, vl_hi); |
| 229 | // Adjust exponent of the value l fetched from the table to get the final s value. |
| 230 | const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve)); |
| 231 | |
| 232 | // Subtract the large number back to get final n := round(x * 64 / log(2)) as a floating-point number. |
| 233 | vn = vsubq_f32(vn, vmagic_bias); |
| 234 | |
| 235 | // Compute reduced argument t := x - n * log(2) / 64. |
| 236 | // Use Cody-Waite range reduction method (note the two constants representing log(2) / 64) to improve accuracy. |
| 237 | float32x4_t vt = vfmaq_f32(vx, vn, vminus_ln2_o64_hi); |
| 238 | vt = vfmaq_f32(vt, vn, vminus_ln2_o64_lo); |
| 239 | |
Marat Dukhan | 102a739 | 2020-11-20 01:18:10 -0800 | [diff] [blame^] | 240 | // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128]. |
Marat Dukhan | 8137e4c | 2020-01-25 12:56:58 -0800 | [diff] [blame] | 241 | float32x4_t vp = vmulq_f32(vt, vc2); |
| 242 | vp = vfmaq_f32(vt, vt, vp); |
| 243 | |
| 244 | // Reconstruct the final f value: |
| 245 | // f = s * (1 + t * (1 + t * c2)) |
| 246 | // = s * (1 + t + t * (t * c2)) |
| 247 | // = s + s * (t + t * (t * c2)) |
| 248 | // = s + s * p |
| 249 | float32x4_t vf = vfmaq_f32(vs, vs, vp); |
| 250 | |
| 251 | // For inputs below denormal cutoff, replace output with +0.0f. |
| 252 | // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| 253 | vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcltq_f32(vx, vdenorm_cutoff))); |
| 254 | |
| 255 | // Store 4 outputs at a time. |
| 256 | vst1q_f32(output, vf); output += 4; |
| 257 | |
| 258 | // Accumulate computed exponents. |
| 259 | vacc = vaddq_f32(vacc, vf); |
| 260 | } |
| 261 | #if XNN_ARCH_ARM64 |
| 262 | float vacc_lo = vaddvq_f32(vacc); |
| 263 | #else |
| 264 | float32x2_t vacc_lo = vadd_f32(vget_high_f32(vacc), vget_low_f32(vacc)); |
| 265 | #endif |
| 266 | if (elements != 0) { |
| 267 | assert(elements >= 1 * sizeof(float)); |
| 268 | assert(elements <= 3 * sizeof(float)); |
| 269 | // Load 4 inputs at a time. |
| 270 | const float32x4_t vi = vld1q_f32(input); input += 4; |
| 271 | |
| 272 | // Subtract maximum input x := i - i_max. This implies x <= 0. |
| 273 | const float32x4_t vx = vsubq_f32(vi, vi_max); |
| 274 | |
| 275 | // Compute reduced argument n := round(x * 64 / log(2)). |
| 276 | // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing |
| 277 | // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction. |
| 278 | // The trick with adding large number is valid only within certain bounds (|x * 64 / log(2)| <= 2**22, i.e. |
| 279 | // |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs outside of [-87.336540, 0.0] |
| 280 | // result in denormalized or underflown expf(x). We fixup the result for such inputs at the very end of the |
| 281 | // algorithm. |
| 282 | float32x4_t vn = vfmaq_f32(vmagic_bias, vx, vlog2e_x64); |
| 283 | |
| 284 | // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that expf(x) is normalized, |
| 285 | // i.e. -87.33642 <= x <= 0.0. As n has 6 fractional bits, we split s == 2**(n / 64) = 2**e * 2**(n / 64 - e), where |
| 286 | // e := int(n / 64). We create s in two steps: |
| 287 | // 1. Fetch 2**(n / 64 - e) = 2**(n % 64) from the table using the 6 low bits of n, as integer. Note that the |
| 288 | // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| 289 | // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized |
| 290 | // number, because for -87.33642 <= x <= 0.0 (inputs for which expf(x) is normalized) we have -126 <= e <= 0, |
| 291 | // and thus the adjusted exponent is not lower than -126. |
| 292 | // |
| 293 | // Extract e from bits 6:14 of n and shift it into bits 23:31 (position of floating-point exponent). |
| 294 | const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x3F))), 17); |
| 295 | |
| 296 | // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64). |
| 297 | const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask)); |
| 298 | const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0); |
| 299 | const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1); |
| 300 | float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_lo]); |
| 301 | float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_hi]); |
| 302 | vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_lo >> 32)], vl_lo, 1); |
| 303 | vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_hi >> 32)], vl_hi, 1); |
| 304 | const float32x4_t vl = vcombine_f32(vl_lo, vl_hi); |
| 305 | // Adjust exponent of the value l fetched from the table to get the final s value. |
| 306 | const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve)); |
| 307 | |
| 308 | // Subtract the large number back to get final n := round(x * 64 / log(2)) as a floating-point number. |
| 309 | vn = vsubq_f32(vn, vmagic_bias); |
| 310 | |
| 311 | // Compute reduced argument t := x - n * log(2) / 64. |
| 312 | // Use Cody-Waite range reduction method (note the two constants representing log(2) / 64) to improve accuracy. |
| 313 | float32x4_t vt = vfmaq_f32(vx, vn, vminus_ln2_o64_hi); |
| 314 | vt = vfmaq_f32(vt, vn, vminus_ln2_o64_lo); |
| 315 | |
Marat Dukhan | 102a739 | 2020-11-20 01:18:10 -0800 | [diff] [blame^] | 316 | // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128]. |
Marat Dukhan | 8137e4c | 2020-01-25 12:56:58 -0800 | [diff] [blame] | 317 | float32x4_t vp = vmulq_f32(vt, vc2); |
| 318 | vp = vfmaq_f32(vt, vt, vp); |
| 319 | |
| 320 | // Reconstruct the final f value: |
| 321 | // f = s * (1 + t * (1 + t * c2)) |
| 322 | // = s * (1 + t + t * (t * c2)) |
| 323 | // = s + s * (t + t * (t * c2)) |
| 324 | // = s + s * p |
| 325 | float32x4_t vf = vfmaq_f32(vs, vs, vp); |
| 326 | |
| 327 | // For inputs below denormal cutoff, replace output with +0.0f. |
| 328 | // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| 329 | vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcltq_f32(vx, vdenorm_cutoff))); |
| 330 | |
| 331 | float32x2_t vf_lo = vget_low_f32(vf); |
| 332 | if (elements & (2 * sizeof(float))) { |
| 333 | // Store 2 outputs at a time. |
| 334 | vst1_f32(output, vf_lo); output += 2; |
| 335 | |
| 336 | // Accumulate 2 computed exponents. |
| 337 | #if XNN_ARCH_ARM64 |
| 338 | vacc_lo += vaddv_f32(vf_lo); |
| 339 | #else |
| 340 | vacc_lo = vadd_f32(vacc_lo, vf_lo); |
| 341 | #endif |
| 342 | |
| 343 | vf_lo = vget_high_f32(vf); |
| 344 | } |
| 345 | if (elements & (1 * sizeof(float))) { |
| 346 | // Store 1 output at a time. |
| 347 | vst1_lane_f32(output, vf_lo, 0); |
| 348 | |
| 349 | // Accumulate 1 computed exponent. |
| 350 | #if XNN_ARCH_ARM64 |
| 351 | vacc_lo += vget_lane_f32(vf_lo, 0); |
| 352 | #else |
| 353 | vacc_lo = vadd_f32(vacc_lo, vreinterpret_f32_u64(vshl_n_u64(vreinterpret_u64_f32(vf_lo), 32))); |
| 354 | #endif |
| 355 | } |
| 356 | } |
| 357 | // Reduce 4 elements in the SIMD register |
| 358 | #if XNN_ARCH_ARM64 |
| 359 | *sum = vacc_lo; |
| 360 | #else |
| 361 | vst1_lane_f32(sum, vpadd_f32(vacc_lo, vacc_lo), 0); |
| 362 | #endif |
| 363 | } |