| // Copyright 2019 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 <xnnpack/common.h> |
| #include <xnnpack/math-stubs.h> |
| |
| #include <fp16/bitcasts.h> |
| |
| |
| // Table of exp2(k / 64) values, k = 0..63 |
| static const uint32_t exp2_k_over_64_table[64] = { |
| 0x3F800000, 0x3F8164D2, 0x3F82CD87, 0x3F843A29, |
| 0x3F85AAC3, 0x3F871F62, 0x3F88980F, 0x3F8A14D5, |
| 0x3F8B95C2, 0x3F8D1ADF, 0x3F8EA43A, 0x3F9031DC, |
| 0x3F91C3D3, 0x3F935A2B, 0x3F94F4F0, 0x3F96942D, |
| 0x3F9837F0, 0x3F99E046, 0x3F9B8D3A, 0x3F9D3EDA, |
| 0x3F9EF532, 0x3FA0B051, 0x3FA27043, 0x3FA43516, |
| 0x3FA5FED7, 0x3FA7CD94, 0x3FA9A15B, 0x3FAB7A3A, |
| 0x3FAD583F, 0x3FAF3B79, 0x3FB123F6, 0x3FB311C4, |
| 0x3FB504F3, 0x3FB6FD92, 0x3FB8FBAF, 0x3FBAFF5B, |
| 0x3FBD08A4, 0x3FBF179A, 0x3FC12C4D, 0x3FC346CD, |
| 0x3FC5672A, 0x3FC78D75, 0x3FC9B9BE, 0x3FCBEC15, |
| 0x3FCE248C, 0x3FD06334, 0x3FD2A81E, 0x3FD4F35B, |
| 0x3FD744FD, 0x3FD99D16, 0x3FDBFBB8, 0x3FDE60F5, |
| 0x3FE0CCDF, 0x3FE33F89, 0x3FE5B907, 0x3FE8396A, |
| 0x3FEAC0C7, 0x3FED4F30, 0x3FEFE4BA, 0x3FF28177, |
| 0x3FF5257D, 0x3FF7D0DF, 0x3FFA83B3, 0x3FFD3E0C, |
| }; |
| |
| void xnn_math_f32_expminus__scalar_lut64_p2( |
| size_t n, |
| const float* input, |
| float* output) |
| { |
| assert(n % sizeof(float) == 0); |
| |
| const float vmagic_bias = 0x1.800000p23f; |
| // The smallest x for which expf(x) is normalized. |
| const float vdenorm_cutoff = -0x1.5D589Ep6f; |
| const float vlog2e_x64 = 0x1.715476p6f; |
| // Last 13 bits are zeroes |
| const float vminus_ln2_o64_hi = -0x1.630000p-7f; |
| const float vminus_ln2_o64_lo = 0x1.BD0106p-19f; |
| |
| const float vc2 = 0x1.FFFF0Ap-2f; |
| |
| const uint32_t vindex_mask = UINT32_C(0x3F); |
| |
| for (; n != 0; n -= sizeof(float)) { |
| const float vx = *input++; |
| |
| // Compute reduced argument n := round(x * 64 / log(2)). |
| // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing |
| // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction. |
| // The trick with adding large number is valid only within certain bounds (|x * 64 / log(2)| <= 2**22, i.e. |
| // |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs outside of [-87.336540, 0.0] |
| // result in denormalized or underflown expf(x). We fixup the result for such inputs at the very end of the |
| // algorithm. |
| float vn = vx * vlog2e_x64 + vmagic_bias; |
| |
| // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that expf(x) is normalized, |
| // 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 |
| // e := int(n / 64). We create s in two steps: |
| // 1. Fetch 2**(n / 64 - e) = 2**(n % 64) from exp2_k_over_64_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 e to its floating-point exponent. The result is always a normalized |
| // number, because for -87.33642 <= x <= 0.0 (inputs for which expf(x) is normalized) we have -126 <= e <= 0, |
| // and thus the adjusted exponent is not lower than -126. |
| // |
| // Extract e from bits 6:14 of n and shift it into bits 23:31 (position of floating-point exponent). |
| const uint32_t ve = (fp32_to_bits(vn) & UINT32_C(0xFFFFFFC0)) << 17; |
| |
| // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64). |
| const uint32_t vidx = fp32_to_bits(vn) & vindex_mask; |
| // Adjust exponent of the value l fetched from the exp2_k_over_64_table to get the final s value. |
| const float vs = fp32_from_bits(exp2_k_over_64_table[vidx] + ve); |
| |
| // Subtract the large number back to get final n := round(x * 64 / log(2)) as a floating-point number. |
| vn -= vmagic_bias; |
| |
| // Compute reduced argument t := x - n * log(2) / 64. |
| // Use Cody-Waite range reduction method (note the two constants representing log(2) / 64) to improve accuracy. |
| float vt = vn * vminus_ln2_o64_hi + vx; |
| vt = vn * vminus_ln2_o64_lo + vt; |
| |
| // Compute degree-2 polynomial approxiatmion for exp(t) on [-log(2)/128, log(2)/128]. |
| float vp = vt * vc2; |
| vp = vp * vt + vt; |
| |
| // Reconstruct the final f value: |
| // f = s * (1 + t * (1 + t * c2)) |
| // = s * (1 + t + t * (t * c2)) |
| // = s + s * (t + t * (t * c2)) |
| // = s + s * p |
| float vf = vp * vs + vs; |
| |
| // For inputs below denormal cutoff, replace output with +0.0f. |
| // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| if XNN_UNPREDICTABLE(vx < vdenorm_cutoff) { |
| vf = 0.0f; |
| } |
| |
| *output++ = vf; |
| } |
| } |