f32-sigmoid/gen/scalar-lut64-p2-div-x4.c

// Auto-generated file. Do not edit!
//   Template: src/f32-sigmoid/scalar-lut64-p2-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_64[64];

void xnn_f32_sigmoid_ukernel__scalar_lut64_p2_div_x4(
    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_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 >= 4 * sizeof(float); n -= 4 * sizeof(float)) {
    const float vx0 = x[0];
    const float vx1 = x[1];
    const float vx2 = x[2];
    const float vx3 = x[3];
    x += 4;

    // 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 vz0 = fabsf(vx0);
    const float vz1 = fabsf(vx1);
    const float vz2 = fabsf(vx2);
    const float vz3 = fabsf(vx3);

    // 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 vn0 = vz0 * vminus_log2e_x64 + vmagic_bias;
    float vn1 = vz1 * vminus_log2e_x64 + vmagic_bias;
    float vn2 = vz2 * vminus_log2e_x64 + vmagic_bias;
    float vn3 = vz3 * 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 the 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 ve0 = (fp32_to_bits(vn0) & ~vindex_mask) << 17;
    const uint32_t ve1 = (fp32_to_bits(vn1) & ~vindex_mask) << 17;
    const uint32_t ve2 = (fp32_to_bits(vn2) & ~vindex_mask) << 17;
    const uint32_t ve3 = (fp32_to_bits(vn3) & ~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 vidx0 = fp32_to_bits(vn0) & vindex_mask;
    const uint32_t vidx1 = fp32_to_bits(vn1) & vindex_mask;
    const uint32_t vidx2 = fp32_to_bits(vn2) & vindex_mask;
    const uint32_t vidx3 = fp32_to_bits(vn3) & vindex_mask;
    // Adjust exponent of the value l fetched from the table to get the final s value.
    const float vs0 = fp32_from_bits(xnn_table_exp2_k_over_64[vidx0] + ve0);
    const float vs1 = fp32_from_bits(xnn_table_exp2_k_over_64[vidx1] + ve1);
    const float vs2 = fp32_from_bits(xnn_table_exp2_k_over_64[vidx2] + ve2);
    const float vs3 = fp32_from_bits(xnn_table_exp2_k_over_64[vidx3] + ve3);

    // Subtract the large number back to get the final n := round(-z * 64 / log(2)) as a floating-point number.
    vn0 -= vmagic_bias;
    vn1 -= vmagic_bias;
    vn2 -= vmagic_bias;
    vn3 -= 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 vt0 = vn0 * vln2_o64_hi + vz0;
    float vt1 = vn1 * vln2_o64_hi + vz1;
    float vt2 = vn2 * vln2_o64_hi + vz2;
    float vt3 = vn3 * vln2_o64_hi + vz3;

    vt0 = vn0 * vln2_o64_lo + vt0;
    vt1 = vn1 * vln2_o64_lo + vt1;
    vt2 = vn2 * vln2_o64_lo + vt2;
    vt3 = vn3 * vln2_o64_lo + vt3;

    // Compute degree-2 polynomial approxiatmion for exp(-t) on [-log(2)/128, log(2)/128].
    //   P1(t) = 1 + t * (-1 + t * c2)
    float vp0 = vt0 * vc2;
    float vp1 = vt1 * vc2;
    float vp2 = vt2 * vc2;
    float vp3 = vt3 * vc2;

    vp0 = vt0 - vp0 * vt0;
    vp1 = vt1 - vp1 * vt1;
    vp2 = vt2 - vp2 * vt2;
    vp3 = vt3 - vp3 * vt3;

    // 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 vy0 = vs0 - vs0 * vp0;
    const float vy1 = vs1 - vs1 * vp1;
    const float vy2 = vs2 - vs2 * vp2;
    const float vy3 = vs3 - vs3 * vp3;

    // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
    float vf0 = vy0 / (vy0 + vone);
    float vf1 = vy1 / (vy1 + vone);
    float vf2 = vy2 / (vy2 + vone);
    float vf3 = vy3 / (vy3 + 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(vz0 > vdenorm_cutoff) {
      vf0 = 0.0f;
    }
    if XNN_UNPREDICTABLE(vz1 > vdenorm_cutoff) {
      vf1 = 0.0f;
    }
    if XNN_UNPREDICTABLE(vz2 > vdenorm_cutoff) {
      vf2 = 0.0f;
    }
    if XNN_UNPREDICTABLE(vz3 > vdenorm_cutoff) {
      vf3 = 0.0f;
    }

    // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
    if XNN_UNPREDICTABLE(vx0 > 0.0f) {
      vf0 = vone - vf0;
    }
    if XNN_UNPREDICTABLE(vx1 > 0.0f) {
      vf1 = vone - vf1;
    }
    if XNN_UNPREDICTABLE(vx2 > 0.0f) {
      vf2 = vone - vf2;
    }
    if XNN_UNPREDICTABLE(vx3 > 0.0f) {
      vf3 = vone - vf3;
    }

    y[0] = vf0;
    y[1] = vf1;
    y[2] = vf2;
    y[3] = vf3;
    y += 4;
  }
  if XNN_UNLIKELY(n != 0) {
    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 * 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 the 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 table to get the final s value.
      const float vs = fp32_from_bits(xnn_table_exp2_k_over_64[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;
      }

      *y++ = vf;

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