src/math/sigmoid-scalar-lut64-p2-div.c

// 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 <math.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_sigmoid__scalar_lut64_p2_div(
    size_t n,
    const float* input,
    float* output)
{
  assert(n % sizeof(float) == 0);

  const float vmagic_bias = 0x1.800000p23f;
  // The largest z for which sigmoidf(-z) is normalized.
  // This number is also the largest z for which expf(-z) is normalized.
  const float vdenorm_cutoff = 0x1.5D589Ep+6f;
  const float vminus_log2e_x64 = -0x1.715476p6f;
  // Last 13 bits are zeroes
  const float vln2_o64_hi =  0x1.630000p-7f;
  const float vln2_o64_lo = -0x1.BD0106p-19f;
  const float vone = 1.0f;

  const float vc2 = 0x1.FFFF0Ap-2f;

  const uint32_t vindex_mask = UINT32_C(0x3F);

  for (; n != 0; n -= sizeof(float)) {
    const float vx = *input++;

    // General structure of the algorithm:
    //           / exp(x) / (1 + exp(x)) if x <= 0
    //   f[x] := 
    //           \ 1 - f[-x] if x >= 0
    //
    // First we compute f[-z] := exp(-z) / (1 + exp(-z)) where z = abs(x),
    // then replace result with 1 - f[-z] if x >= 0.
    const float vz = fabsf(vx);

    // Compute reduced argument n := round(-z * 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 (|z * 2048 / log(2)| <= 2**22, i.e.
    // |z| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs x outside of
    // [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x). We fixup the result
    // for such inputs at the very end of the algorithm.
    float vn = vz * vminus_log2e_x64 + vmagic_bias;

    // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that sigmoidf(-z) is
    // normalized, i.e. 0 <= z <= 87.33642. 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 0 <= z <= 87.33642 (inputs for which sigmoidf(-z) 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) & ~vindex_mask) << 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 the final n := round(-z * 64 / log(2)) as a floating-point number.
    vn -= vmagic_bias;

    // Compute reduced argument t := (z + n * log(2) / 64). Note that -t = -z - n * log(2) / 64.
    // Use Cody-Waite range reduction method (note two constants to represent log(2) / 64) to improve accuracy.
    float vt = vn * vln2_o64_hi + vz;
    vt = vn * vln2_o64_lo + vt;

    // Compute degree-2 polynomial approxiatmion for exp(-t) on [-log(2)/128, log(2)/128].
    //   P1(t) = 1 + t * (-1 + t * c2)
    float vp = vt * vc2;
    vp = vt - vp * 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
    const float vy = vs - vs * vp;

    // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
    float vf = vy / (vy + vone);

    // 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(vz > vdenorm_cutoff) {
      vf = 0.0f;
    }

    // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
    if XNN_UNPREDICTABLE(vx > 0.0f) {
      vf = vone - vf;
    }

    *output++ = vf;
  }
}