Marat Dukhan | c60742b | 2020-11-23 12:33:27 -0800 | [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 <xnnpack/common.h> |
| 10 | #include <xnnpack/math-stubs.h> |
| 11 | |
| 12 | #include <fp16/bitcasts.h> |
| 13 | |
| 14 | |
| 15 | // Table of exp2(k / 16) values decremented (as integer) by (k << 19), k = 0..15 |
| 16 | extern XNN_INTERNAL const uint32_t xnn_table_exp2minus_k_over_16[16]; |
| 17 | |
| 18 | void xnn_math_f32_expm1minus__scalar_rr2_lut16_p4( |
| 19 | size_t n, |
| 20 | const float* input, |
| 21 | float* output) |
| 22 | { |
| 23 | assert(n % (4 * sizeof(float)) == 0); |
| 24 | |
| 25 | // Large number such that ulp(magic bias) == exp2(-4) |
| 26 | const float vmagic_bias = 0x1.800000p19f; |
| 27 | const float vlog2e = 0x1.715476p+0f; |
| 28 | // Mask for the lowest 4 bits |
| 29 | const uint32_t vindex_mask = UINT32_C(0xF); |
| 30 | // The largest x for which expm1f(x) is saturated at -1.0f. |
| 31 | const float vsat_cutoff = -0x1.154246p+4f; |
Marat Dukhan | de390d4 | 2020-11-29 19:32:18 -0800 | [diff] [blame^] | 32 | // Last 9 bits are zeroes |
| 33 | const float vminus_ln2_hi = -0x1.62E400p-1f; |
| 34 | const float vminus_ln2_lo = -0x1.7F7D1Cp-20f; |
Marat Dukhan | c60742b | 2020-11-23 12:33:27 -0800 | [diff] [blame] | 35 | // Coefficient of polynomial approximation |
| 36 | // exp(t) - 1 ~ t * (1 + t * (c2 + t * (c3 + t * c4))) |
| 37 | // on [-log(2)/32, log(2)/32] |
| 38 | const float vc4 = 0x1.55563Ap-5f; |
| 39 | const float vc3 = 0x1.555708p-3f; |
| 40 | const float vc2 = 0x1.000000p-1f; |
| 41 | const float vone = 1.0f; |
| 42 | |
| 43 | for (; n != 0; n -= sizeof(float)) { |
| 44 | float vx = *input++; |
| 45 | |
| 46 | // Compute reduced argument n := round(x / log(2), 4). |
| 47 | // We do it by adding a large number (magic bias), which cause rounding of the result to 4 fractional bits, then |
| 48 | // subtracing the large number back. The trick with adding large number is valid only within certain bounds |
| 49 | // (|x / log(2)| <= 2**18, i.e. |x| <= 0x1.62E43p+17 = 181704.375), but that is acceptable, because inputs x are |
| 50 | // restricted to [-17.328680, 0]. |
| 51 | // Note that addition-subtraction of the large number doesn't cause overflow for inputs in this range. |
| 52 | float vn = vx * vlog2e + vmagic_bias; |
| 53 | |
| 54 | // Create a floating-point number s (scale) such that s := 2**n for valid inputs, i.e. -17.328680 <= x <= 0.0. As n |
| 55 | // has 4 fractional bits, we split s == 2**n = 2**int(n) * 2**frac(n). We create s in two steps: |
| 56 | // 1. Fetch 2**frac(n) from the table using the 4 low bits of n, as integer. Note that the fetched values are in |
| 57 | // the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| 58 | // 2. Adjust fecthed value by addition of int(n) to its floating-point exponent. The result is always a normalized |
| 59 | // number, because for -17.328680 <= x <= 0.0 we have -25 <= int(n) <= 0, and thus the adjusted exponent is not |
| 60 | // lower than -25. |
| 61 | // |
| 62 | // Shift bits 4:12 into 23:31 (position of floating-point exponent). |
| 63 | const uint32_t ve = fp32_to_bits(vn) << 19; |
| 64 | |
| 65 | // Use bits 0:4 bits of n, as integer, as an index for table lookup of l := 2**frac(n). |
| 66 | const uint32_t vidx = fp32_to_bits(vn) & vindex_mask; |
| 67 | // Adjust exponent of the value l fetched from the table to get the final s value. |
| 68 | float vs = fp32_from_bits(xnn_table_exp2minus_k_over_16[vidx] + ve); |
| 69 | |
| 70 | // Subtract the large number back to get final n := round(x / log(2), 4). |
| 71 | vn -= vmagic_bias; |
| 72 | |
| 73 | // The function saturates at -1 for large negative inputs: expm1f(x) == -1.0f for x <= sat_cutoff ~= -17.328680. |
| 74 | // To guarantee this behaviour, we zero out s (scale) for x <= sat_cutoff. |
| 75 | if XNN_UNPREDICTABLE(vx <= vsat_cutoff) { |
| 76 | vs = 0.0f; |
| 77 | } |
| 78 | |
| 79 | // Compute reduced argument t := x - n * log(2). |
| 80 | // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy. |
| 81 | float vt = vn * vminus_ln2_hi + vx; |
| 82 | vt = vn * vminus_ln2_lo + vt; |
| 83 | |
| 84 | // Compute degree-4 polynomial approximation for exp(t) - 1 on [-log(2)/32, log(2)/32]. |
| 85 | // P(t) = t * (1 + t * (c2 + t * (c3 + t * c4))) = t + t * (t * (c2 + t * (c3 + t * c4))) = t + t * p |
| 86 | float vp = vc4 * vt + vc3; |
| 87 | vp = vp * vt + vc2; |
| 88 | vp *= vt; |
| 89 | |
| 90 | // Reconstruct the exp(x) - 1 value: |
Marat Dukhan | de390d4 | 2020-11-29 19:32:18 -0800 | [diff] [blame^] | 91 | // exp(x) - 1 = s * (1 + t * (1 + t * (c2 + t * (c3 + t * c4)))) - 1 |
Marat Dukhan | c60742b | 2020-11-23 12:33:27 -0800 | [diff] [blame] | 92 | // = (s - 1) + s * (t + t * p) |
| 93 | // = ((t * s) + (t * s) * p) + (s - 1) |
| 94 | vt *= vs; |
| 95 | const float vsm1 = vs - vone; |
| 96 | vp = vp * vt + vt; |
| 97 | const float vf = vp + vsm1; |
| 98 | |
| 99 | *output++ = vf; |
| 100 | } |
| 101 | } |