src/math/sigmoid-scalar-rr2-lut2048-p1-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 / 2048) values decremented (as integer) by (k << 12), k = 0..2048
extern XNN_INTERNAL const uint32_t xnn_table_exp2minus_k_over_2048[2048];

void xnn_math_f32_sigmoid__scalar_rr2_lut2048_p1_div(
    size_t n,
    const float* input,
    float* output)
{
  assert(n % sizeof(float) == 0);

  // Large number such that ulp(magic bias) == exp2(-11)
  const float vmagic_bias = 0x1.800000p12f;
  const float vminus_log2e = -0x1.715476p0f;
  // Mask for the lowest 11 bits
  const uint32_t vindex_mask = UINT32_C(0x7FF);
  // Last 13 bits are zeroes
  const float vln2_hi = 0x1.600000p-1f;
  const float vln2_lo = 0x1.7217F8p-8f;
  // Coefficient of polynomial approximation of exp(-t) ~ 1 + t * c1 on [-log(2)/4096, log(2)/4096]
  const float vc1 = -0x1.FFFFFEp-1f;
  const float vone = 1.0f;
  // 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;

  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 / log(2), 11).
    // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
    // the large number back. The trick with adding large number is valid only within certain bounds
    // (|-z / log(2)| <= 2**11, 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 + vmagic_bias;

    // Create a floating-point number s (scale) such that s := 2**n 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 = 2**int(n) * 2**frac(n). We create s
    // in two steps:
    // 1. Fetch 2**frac(n) from the table using the 11 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 int(n) 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 <= int(n) <= 0, and thus the adjusted exponent is not lower than -126.
    //
    // Shift bits 11:19 into 23:31 (position of floating-point exponent).
    const uint32_t ve = fp32_to_bits(vn) << 12;

    // Use bits 0:11 bits of n, as integer, as an index for table lookup of l := 2**frac(n).
    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_exp2minus_k_over_2048[vidx] + ve);

    // Subtract the large number back to get the final n := round(-z / log(2), 11) as a floating-point number.
    vn -= vmagic_bias;

    // Compute reduced argument t := (z + n * log(2)). Note that -t = -z - n * log(2).
    // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
    float vt = vn * vln2_hi + vz;
    vt = vn * vln2_lo + vt;

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

    // Reconstruct the exp(-z) value:
    //   e = s * (1 + t * c1)
    //     = s * (1 + p)
    //     = 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 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;
  }
}