Marat Dukhan | 8e17294 | 2020-09-20 23:56:23 -0700 | [diff] [blame] | 1 | // Copyright 2020 Google LLC |
| 2 | // |
| 3 | // This source code is licensed under the BSD-style license found in the |
| 4 | // LICENSE file in the root directory of this source tree. |
| 5 | |
| 6 | #include <assert.h> |
| 7 | #include <math.h> |
| 8 | #include <stddef.h> |
| 9 | |
| 10 | #include <emmintrin.h> |
| 11 | |
| 12 | #include <xnnpack/common.h> |
| 13 | #include <xnnpack/math-stubs.h> |
| 14 | |
| 15 | |
| 16 | // Table of exp2(k / 64) values, k = 0..63 |
| 17 | extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64]; |
| 18 | |
Marat Dukhan | b7633f2 | 2020-11-20 16:34:56 -0800 | [diff] [blame^] | 19 | void xnn_math_f32_exp__sse2_rr2_lut64_p2( |
Marat Dukhan | 8e17294 | 2020-09-20 23:56:23 -0700 | [diff] [blame] | 20 | size_t n, |
| 21 | const float* input, |
| 22 | float* output) |
| 23 | { |
| 24 | assert(n % (4 * sizeof(float)) == 0); |
| 25 | |
| 26 | const __m128 vmagic_bias = _mm_set1_ps(0x1.800000p+23f); |
| 27 | // The smallest x for which expf(x) is non-zero. |
| 28 | const __m128 vzero_cutoff = _mm_set1_ps(-0x1.9FE368p+6f); |
| 29 | // The largest x for which expf(x) is finite. |
| 30 | const __m128 vinf_cutoff = _mm_set1_ps(0x1.62E42Ep+6f); |
| 31 | const __m128 vlog2e_x64 = _mm_set1_ps(0x1.715476p+6f); |
| 32 | // Last 13 bits are zeroes |
| 33 | const __m128 vminus_ln2_o64_hi = _mm_set1_ps(-0x1.630000p-7f); |
| 34 | const __m128 vminus_ln2_o64_lo = _mm_set1_ps(0x1.BD0106p-19f); |
| 35 | const __m128 vplus_inf = _mm_set1_ps(INFINITY); |
| 36 | |
| 37 | const __m128 vc2 = _mm_set1_ps(0x1.FFFF0Ap-2f); |
| 38 | |
| 39 | const __m128i vmin_exponent = _mm_set1_epi32(0xC1000000); |
| 40 | const __m128i vmax_exponent = _mm_set1_epi32(0x3F800000); |
| 41 | const __m128i vdefault_exponent = vmax_exponent; |
| 42 | const __m128i vindex_mask = _mm_set1_epi32(0x3F); |
| 43 | |
| 44 | for (; n != 0; n -= 4 * sizeof(float)) { |
| 45 | const __m128 vx = _mm_loadu_ps(input); |
| 46 | |
| 47 | // Compute reduced argument n := round(x * 64 / log(2)). |
| 48 | // We do it by adding a large number (magic bias) to the product x * (64/log(2)), which cause rounding of the |
| 49 | // result to an integer, then subtracing the large number back. The trick with adding large number is valid only |
| 50 | // within certain bounds (|x| <= 2**22), but thats ok, because inputs outside of [-103.97207, 88.72283] underflow |
| 51 | // or overflow expf(x) anyway. We fixup the result for such inputs at the very end of the algorithm. |
| 52 | __m128 vn = _mm_add_ps(_mm_mul_ps(vx, vlog2e_x64), vmagic_bias); |
| 53 | |
| 54 | // Create two floating-point numbers, sn (scale, normal) and so (scale, overflow) such that sn * so == 2**n |
| 55 | // for inputs which don't cause overflow, i.e. -103.97207 <= x <= 88.72283, and -150 <= n <= 128 accordingly. |
| 56 | // We need to use two numbers rather than one because a normalized single-precision exponent must be in [-127, 126] |
| 57 | // range, which is insufficient to cover [-150, 128] range of n. |
| 58 | // - When n is within [-127, 126], sn == 2**n and so == 1.0. |
| 59 | // - When n < -127, sn == 2**(-127) and so == 2**(n + 127). |
| 60 | // - When n > 126, sn == 2**126 and so == 2**(n - 126). |
| 61 | // While we explicitly compute sn, the so is fused into the value l fetched from a table by adjusting its exponential. |
| 62 | __m128i veo = _mm_slli_epi32(_mm_andnot_si128(vindex_mask, _mm_castps_si128(vn)), 17); |
| 63 | __m128i ven = _mm_max_epi16(veo, vmin_exponent); |
| 64 | ven = _mm_min_epi16(ven, vmax_exponent); |
| 65 | veo = _mm_sub_epi32(veo, ven); |
| 66 | const __m128 vsn = _mm_castsi128_ps(_mm_add_epi32(ven, vdefault_exponent)); |
| 67 | |
| 68 | // Use the low 6 bits of n (as integer) for table lookup. |
Marat Dukhan | b32b018 | 2020-09-21 01:21:04 -0700 | [diff] [blame] | 69 | const __m128i vidx = _mm_slli_epi32(_mm_and_si128(_mm_castps_si128(vn), vindex_mask), 2); |
Marat Dukhan | 8e17294 | 2020-09-20 23:56:23 -0700 | [diff] [blame] | 70 | #if XNN_ARCH_X86_64 |
| 71 | const uint64_t vidx01 = (uint64_t) _mm_cvtsi128_si64(vidx); |
| 72 | const uint64_t vidx23 = (uint64_t) _mm_cvtsi128_si64(_mm_unpackhi_epi64(vidx, vidx)); |
Marat Dukhan | b32b018 | 2020-09-21 01:21:04 -0700 | [diff] [blame] | 73 | const __m128i vl0 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) &xnn_table_exp2_k_over_64 + (uint32_t) vidx01))); |
| 74 | const __m128i vl2 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + (uint32_t) vidx23))); |
| 75 | const __m128i vl1 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + (uint32_t) (vidx01 >> 32)))); |
| 76 | const __m128i vl3 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + (uint32_t) (vidx23 >> 32)))); |
Marat Dukhan | 8e17294 | 2020-09-20 23:56:23 -0700 | [diff] [blame] | 77 | #else |
| 78 | const uint32_t vidx0 = (uint32_t) _mm_cvtsi128_si32(vidx); |
| 79 | const uint32_t vidx1 = (uint32_t) _mm_extract_epi16(vidx, 2); |
| 80 | const uint32_t vidx2 = (uint32_t) _mm_extract_epi16(vidx, 4); |
| 81 | const uint32_t vidx3 = (uint32_t) _mm_extract_epi16(vidx, 6); |
Marat Dukhan | b32b018 | 2020-09-21 01:21:04 -0700 | [diff] [blame] | 82 | const __m128i vl0 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + vidx0))); |
| 83 | const __m128i vl2 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + vidx2))); |
| 84 | const __m128i vl1 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + vidx1))); |
| 85 | const __m128i vl3 = _mm_cvtsi32_si128(*((const int*) ((uintptr_t) xnn_table_exp2_k_over_64 + vidx3))); |
Marat Dukhan | 8e17294 | 2020-09-20 23:56:23 -0700 | [diff] [blame] | 86 | #endif |
| 87 | // Fuse so into the value l fetched from a table by adjusting its exponential. |
| 88 | const __m128 vl = _mm_castsi128_ps(_mm_add_epi32(_mm_unpacklo_epi64(_mm_unpacklo_epi32(vl0, vl1), _mm_unpacklo_epi32(vl2, vl3)), veo)); |
| 89 | |
| 90 | // Subtract the large number back to get final n := round(x * 64 / log(2)). |
| 91 | vn = _mm_sub_ps(vn, vmagic_bias); |
| 92 | |
| 93 | // Compute reduced argument t := x - n * log(2). |
| 94 | // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy. |
| 95 | __m128 vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_o64_hi), vx); |
| 96 | vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_o64_lo), vt); |
| 97 | |
Marat Dukhan | 102a739 | 2020-11-20 01:18:10 -0800 | [diff] [blame] | 98 | // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128]. |
Marat Dukhan | 8e17294 | 2020-09-20 23:56:23 -0700 | [diff] [blame] | 99 | __m128 vp = _mm_mul_ps(vt, vc2); |
| 100 | vp = _mm_add_ps(vt, _mm_mul_ps(vt, vp)); |
| 101 | |
| 102 | // Reconstruct the final f value: |
| 103 | // f = sn * (so * l) * (1 + t * (1 + t * c2)) |
| 104 | // = sn * (so * l) * (1 + t + t * (t * c2)) |
| 105 | // = sn * ((so * l) + (so * l) * (t + t * (t * c2))) |
| 106 | __m128 vf = _mm_add_ps(vl, _mm_mul_ps(vl, vp)); |
| 107 | vf = _mm_mul_ps(vf, vsn); |
| 108 | |
| 109 | // For inputs below zero cutoff, replace output with +0.0f. |
| 110 | // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| 111 | vf = _mm_andnot_ps(_mm_cmplt_ps(vx, vzero_cutoff), vf); |
| 112 | // For inputs above inf cutoff, replace output with +inf. |
| 113 | // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| 114 | const __m128 vm = _mm_cmpgt_ps(vx, vinf_cutoff); |
| 115 | vf = _mm_or_ps(_mm_and_ps(vplus_inf, vm), _mm_andnot_ps(vm, vf)); |
| 116 | _mm_storeu_ps(output, vf); |
| 117 | |
| 118 | input += 4; |
| 119 | output += 4; |
| 120 | } |
| 121 | } |