src/f32-sigmoid/gen/scalar-lut2048-p1-div-x1.c

// Auto-generated file. Do not edit!
//   Template: src/f32-sigmoid/scalar-lut2048-p1-div.c.in
//   Generator: tools/xngen
//
// 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 <math.h>

#include <xnnpack/common.h>
#include <xnnpack/vunary.h>

#include <fp16/bitcasts.h>


// Note redefine as uint32[] to avoid redundant bitcasts.
extern XNN_INTERNAL const uint32_t xnn_table_exp2_k_over_2048[2048];

void xnn_f32_sigmoid_ukernel__scalar_lut2048_p1_div_x1(
    size_t n,
    const float* x,
    float* y,
    const void* params)
{
  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_x2048 = -0x1.715476p11f;
  // Last 18 bits are zeroes
  const float vln2_o2048_hi = 0x1.600000p-12f;
  const float vln2_o2048_lo = 0x1.7217F8p-19f;
  const float vone = 1.0f;

  const float vc1 = -0x1.FFFFFEp-1f;

  const uint32_t vindex_mask = UINT32_C(0x7FF);

  do {
    const float vx = *x++;

    // 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 * 2048 / 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+10 = 1419.5654296875), 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_x2048 + vmagic_bias;

    // Create a floating-point number s (scale) such that s := 2**(n / 2048) for such inputs that sigmoidf(-z) is
    // normalized, i.e. 0 <= z <= 87.33642. As n has 11 fractional bits, we split s == 2**(n / 2048) =
    // = 2**e * 2**(n / 2048 - e), where e := int(n / 2048). We create s in two steps:
    // 1. Fetch 2**(n / 2048 - e) = 2**(n % 2048) from 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 11:19 of n and shift it into bits 23:31 (position of floating-point exponent).
    const uint32_t ve = (fp32_to_bits(vn) & ~vindex_mask) << 12;

    // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**(n % 2048).
    const uint32_t vidx = fp32_to_bits(vn) & vindex_mask;
    // Adjust exponent of the value l fetched from the table to get the final s value.
    const float vs = fp32_from_bits(xnn_table_exp2_k_over_2048[vidx] + ve);

    // Subtract the large number back to get the final n := round(-z * 2048 / log(2)) as a floating-point number.
    vn -= vmagic_bias;

    // Compute reduced argument t := (z + n * log(2) / 2048). Note that -t = -z - n * log(2) / 2048.
    // Use Cody-Waite range reduction method (note two constants to represent log(2) / 2048) to improve accuracy.
    float vt = vn * vln2_o2048_hi + vz;
    vt = vn * vln2_o2048_lo + vt;

    // Compute degree-1 polynomial approximation for exp(-t) on [-log(2)/4096, log(2)/4096]:
    //   P1(t) = 1 + t * c1
    const float vp = vt * vc1;

    // Reconstruct the exp(-z) value:
    //   y = s * (1 + t * c1)
    //     = s + s * (t * c1))
    //     = s + s * p
    const float vy = vp * vs + vs;

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

    // For inputs above 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;
    }

    *y++ = vf;

    n -= sizeof(float);
  } while (n != 0);
}