Marat Dukhan | 36173d2 | 2020-10-15 17:14:26 -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 <stddef.h> |
| 8 | |
| 9 | #include <immintrin.h> |
| 10 | |
| 11 | #include <xnnpack/common.h> |
| 12 | #include <xnnpack/math-stubs.h> |
| 13 | |
| 14 | |
| 15 | // Table of exp2(k / 64) values decremented (as integer) by (k << 17), k = 0..63 |
| 16 | extern XNN_INTERNAL const float xnn_table_exp2minus_k_over_64[64]; |
| 17 | |
| 18 | void xnn_math_f32_sigmoid__avx2_rr2_lut64_p2_gather_nr2fma( |
| 19 | size_t n, |
| 20 | const float* input, |
| 21 | float* output) |
| 22 | { |
| 23 | assert(n % (8 * sizeof(float)) == 0); |
| 24 | |
| 25 | // Floating-point mask with only the sign bit set |
| 26 | const __m256 vsign_mask = _mm256_set1_ps(-0.0f); |
| 27 | // Large number such that ulp(magic bias) == exp2(-6) |
| 28 | const __m256 vmagic_bias = _mm256_set1_ps(0x1.800000p17f); |
| 29 | const __m256 vlog2e = _mm256_set1_ps(0x1.715476p0f); |
| 30 | // Mask for the lowest 6 bits |
| 31 | const __m256 vindex_mask = _mm256_castsi256_ps(_mm256_set1_epi32(INT32_C(0x3F))); |
| 32 | const __m256 vminus_ln2_hi = _mm256_set1_ps(-0x1.62E43p-1f); |
| 33 | const __m256 vminus_ln2_lo = _mm256_set1_ps(0x1.05C61p-29f); |
| 34 | // Coefficient of polynomial approximation of exp(t) ~ 1 + t * (1 + t * c2) on [-log(2)/128, log(2)/128] |
| 35 | const __m256 vc2 = _mm256_set1_ps(0x1.FFFF0Ap-2f); |
| 36 | const __m256 vone = _mm256_set1_ps(1.0f); |
| 37 | // The smallest x for which sigmoidf(x) is normalized. |
| 38 | // This number is also the smallest x for which expf(x) is normalized. |
| 39 | const __m256 vdenorm_cutoff = _mm256_set1_ps(-0x1.5D589Ep+6f); |
| 40 | |
| 41 | for (; n != 0; n -= 8 * sizeof(float)) { |
| 42 | const __m256 vx = _mm256_loadu_ps(input); |
| 43 | |
| 44 | // General structure of the algorithm: |
| 45 | // |
| 46 | // / exp(x) / (1 + exp(x)) if x <= 0 |
| 47 | // f[x] := |
| 48 | // \ 1 - f[-x] if x >= 0 |
| 49 | // |
| 50 | // First we compute f[z] := exp(z) / (1 + exp(z)) where z = -abs(x), then replace result with 1 - f[z] if x >= 0. |
| 51 | const __m256 vz = _mm256_or_ps(vx, vsign_mask); |
| 52 | |
| 53 | // Compute reduced argument n := round(z / log(2), 6). |
| 54 | // We do it by adding a large number (magic bias), which cause rounding of the result to 6 fractional bits, then |
| 55 | // subtracing the large number back. The addition is combined with multiplication by log2e into a single FMA |
| 56 | // instruction. The trick with adding large number is valid only within certain bounds (|z / log(2)| <= 2**16, i.e. |
| 57 | // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of [-87.336544, 17.328678] |
| 58 | // (i.e. z outsize [87.336544, 0]) underflow or saturate sigmoidf(x). We fixup the result for such inputs at the |
| 59 | // very end of the algorithm. |
| 60 | __m256 vn = _mm256_fmadd_ps(vz, vlog2e, vmagic_bias); |
| 61 | |
| 62 | // Create a floating-point number s (scale) such that s := 2**n for such inputs that sigmoidf(z) is normalized, |
| 63 | // 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 |
| 64 | // in two steps: |
| 65 | // 1. Fetch 2**frac(n) from the table using the 6 low bits of n, as integer. Note that the fetched values are in |
| 66 | // the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| 67 | // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized |
| 68 | // number, because for -87.33642 <= z <= 0 (inputs for which sigmoidf(z) is normalized) we have |
| 69 | // -126 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126. |
| 70 | // |
| 71 | // Shift bits 6:14 into 23:31 (position of floating-point exponent). |
| 72 | __m256i ve = _mm256_slli_epi32(_mm256_castps_si256(vn), 17); |
| 73 | |
Marat Dukhan | b3fa13c | 2020-11-21 12:51:55 -0800 | [diff] [blame^] | 74 | // Use bits 0:6 of n, as integer, as an index for table lookup of l := 2**frac(n). |
Marat Dukhan | 36173d2 | 2020-10-15 17:14:26 -0700 | [diff] [blame] | 75 | const __m256i vidx = _mm256_castps_si256(_mm256_and_ps(vn, vindex_mask)); |
| 76 | const __m256i vl = _mm256_i32gather_epi32((const int*) xnn_table_exp2minus_k_over_64, vidx, sizeof(float)); |
| 77 | // Adjust exponent of the value l fetched from the table to get the final s value. |
| 78 | const __m256 vs = _mm256_castsi256_ps(_mm256_add_epi32(vl, ve)); |
| 79 | |
| 80 | // Subtract the large number back to get the final n := round(z / log(2), 6) as a floating-point number. |
| 81 | vn = _mm256_sub_ps(vn, vmagic_bias); |
| 82 | |
| 83 | // Compute reduced argument t := z - n * log(2). |
| 84 | // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy. |
| 85 | __m256 vt = _mm256_fmadd_ps(vn, vminus_ln2_hi, vz); |
| 86 | vt = _mm256_fmadd_ps(vn, vminus_ln2_lo, vt); |
| 87 | |
| 88 | // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128]. |
| 89 | // P(t) = 1 + t * (1 + t * c2) = 1 + (t + t * (t * c2)) = 1 + p |
| 90 | __m256 vp = _mm256_mul_ps(vt, vc2); |
| 91 | vp = _mm256_fmadd_ps(vt, vp, vt); |
| 92 | |
| 93 | // Reconstruct the exp(z) value: |
| 94 | // e = s * (1 + t * (1 + t * c2)) |
| 95 | // = s * (1 + p) |
| 96 | // = s + s * p |
| 97 | const __m256 vy = _mm256_fmadd_ps(vs, vp, vs); |
| 98 | |
| 99 | // Denominator of the sigmoid fraction: 1.0 + exp(z) |
| 100 | const __m256 vd = _mm256_add_ps(vy, vone); |
| 101 | |
| 102 | // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator. |
| 103 | // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0. |
| 104 | // Thus the reciprocal of the denominator never overflows. |
| 105 | __m256 vr = _mm256_rcp_ps(vd); |
| 106 | vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr); |
| 107 | vr = _mm256_fmadd_ps(_mm256_fnmadd_ps(vr, vd, vone), vr, vr); |
| 108 | |
| 109 | // Reconstruct sigmoid(z) = exp(z) / (1.0 + exp(z)) |
| 110 | __m256 vf = _mm256_mul_ps(vy, vr); |
| 111 | |
| 112 | // For inputs below denormal cutoff, replace output with +0.0f. |
| 113 | // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| 114 | vf = _mm256_andnot_ps(_mm256_cmp_ps(vz, vdenorm_cutoff, _CMP_LT_OS), vf); |
| 115 | |
| 116 | // Reconstruct sigmoid(x) = x < 0 ? sigmoid(z) : 1.0 - sigmoid(z) |
| 117 | vf = _mm256_blendv_ps(_mm256_sub_ps(vone, vf), vf, vx); |
| 118 | |
| 119 | _mm256_storeu_ps(output, vf); |
| 120 | |
| 121 | input += 8; |
| 122 | output += 8; |
| 123 | } |
| 124 | } |