src/f32-sigmoid/scalar-p5-div.c.in

// 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.

$assert BATCH_TILE >= 1
$ABC = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
#include <assert.h>
#include <math.h>

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

#include <fp16/bitcasts.h>


void xnn_f32_sigmoid_ukernel__scalar_p5_div_x${BATCH_TILE}(
    size_t n,
    const float* x,
    float* y,
    const void* params)
{
  assert(n % sizeof(float) == 0);

  const float vmagic_bias = 0x1.8000FEp23f;
  // 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 = -0x1.715476p+0f;
  // Last 7 bits are zeroes
  const float vln2_hi = 0x1.62E400p-1f;
  const float vln2_lo = 0x1.7F7D1Cp-20f;
  const float vone = 1.0f;

  const float vc1 = -0x1.FFFFF6p-1f;
  const float vc2 =  0x1.FFFDC6p-2f;
  const float vc3 = -0x1.555A80p-3f;
  const float vc4 =  0x1.573A1Ap-5f;
  const float vc5 = -0x1.0F9F9Cp-7f;

  $if BATCH_TILE > 1:
    for (; n >= ${BATCH_TILE} * sizeof(float); n -= ${BATCH_TILE} * sizeof(float)) {
      $for N in range(BATCH_TILE):
        const float vx${N} = x[${N}];
      x += ${BATCH_TILE};

      // 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.
      $for N in range(BATCH_TILE):
        const float vz${N} = fabsf(vx${N});

      // Compute reduced argument n := round(-z / log(2)).
      // We do it by adding a large number (magic bias), which cause rounding of 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 (|x| <= 2**22), but thats ok, because
      // inputs x outside of [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x)
      // anyway. We fixup the result for such inputs at the very end of the algorithm.
      $for N in range(BATCH_TILE):
        float vn${N} = vz${N} * vminus_log2e + vmagic_bias;

      // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
      // -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
      $for N in range(BATCH_TILE):
        const float vs${N} = fp32_from_bits(fp32_to_bits(vn${N}) << 23);

      // Subtract the large number back to get the final n := round(-z / log(2)) as a floating-point number.
      $for N in range(BATCH_TILE):
        vn${N} -= 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.
      $for N in range(BATCH_TILE):
        float vt${N} = vn${N} * vln2_hi + vz${N};

      $for N in range(BATCH_TILE):
        vt${N} = vn${N} * vln2_lo + vt${N};

      // Compute degree-5 polynomial approximation for exp(-t) on [-log(2)/2, log(2)/2]:
      //   P5(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
      $for N in range(BATCH_TILE):
        float vp${N} = vt${N} * vc5 + vc4;

      $for N in range(BATCH_TILE):
        vp${N} = vt${N} * vp${N} + vc3;

      $for N in range(BATCH_TILE):
        vp${N} = vt${N} * vp${N} + vc2;

      $for N in range(BATCH_TILE):
        vp${N} = vt${N} * vp${N} + vc1;

      // Reconstruct the exp(-z) value:
      //   e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
      //     = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
      //     = s + (t * s) * p
      $for N in range(BATCH_TILE):
        vt${N} *= vs${N};

      $for N in range(BATCH_TILE):
        const float ve${N} = vt${N} * vp${N} + vs${N};

      // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
      $for N in range(BATCH_TILE):
        float vf${N} = ve${N} / (ve${N} + 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.
      $for N in range(BATCH_TILE):
        if XNN_UNPREDICTABLE(vz${N} > vdenorm_cutoff) {
          vf${N} = 0.0f;
        }

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

      $for N in range(BATCH_TILE):
        y[${N}] = vf${N};
      y += ${BATCH_TILE};
    }
  $if BATCH_TILE == 1:
    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 / log(2)).
      // We do it by adding a large number (magic bias), which cause rounding of 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 (|x| <= 2**22), but thats ok, because
      // inputs x outside of [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x)
      // anyway. 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 inputs which don't cause underflow, i.e.
      // -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
      const float vs = fp32_from_bits(fp32_to_bits(vn) << 23);

      // Subtract the large number back to get the final n := round(-z / log(2)) 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-5 polynomial approximation for exp(-t) on [-log(2)/2, log(2)/2]:
      //   P5(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
      float vp = vt * vc5 + vc4;
      vp = vt * vp + vc3;
      vp = vt * vp + vc2;
      vp = vt * vp + vc1;

      // Reconstruct the exp(-z) value:
      //   e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
      //     = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
      //     = s + (t * s) * p
      vt *= vs;
      const float ve = vt * vp + vs;

      // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
      float vf = ve / (ve + 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);
  $elif BATCH_TILE == 2:
    if XNN_UNLIKELY(n != 0) {
      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 / log(2)).
      // We do it by adding a large number (magic bias), which cause rounding of 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 (|x| <= 2**22), but thats ok, because
      // inputs x outside of [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x)
      // anyway. 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 inputs which don't cause underflow, i.e.
      // -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
      const float vs = fp32_from_bits(fp32_to_bits(vn) << 23);

      // Subtract the large number back to get the final n := round(-z / log(2)) 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-5 polynomial approximation for exp(-t) on [-log(2)/2, log(2)/2]:
      //   P5(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
      float vp = vt * vc5 + vc4;
      vp = vt * vp + vc3;
      vp = vt * vp + vc2;
      vp = vt * vp + vc1;

      // Reconstruct the exp(-z) value:
      //   e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
      //     = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
      //     = s + (t * s) * p
      vt *= vs;
      const float ve = vt * vp + vs;

      // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
      float vf = ve / (ve + 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;
    }
  $else:
    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 / log(2)).
        // We do it by adding a large number (magic bias), which cause rounding of 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 (|x| <= 2**22), but thats ok, because
        // inputs x outside of [-87.336544, 17.328678] (i.e. z outsize [0, 87.336544]) underflow or saturate sigmoidf(x)
        // anyway. 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 inputs which don't cause underflow, i.e.
        // -87.336544 <= -z <= 0.0, and -126 <= n <= 0 accordingly.
        const float vs = fp32_from_bits(fp32_to_bits(vn) << 23);

        // Subtract the large number back to get the final n := round(-z / log(2)) 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-5 polynomial approximation for exp(-t) on [-log(2)/2, log(2)/2]:
        //   P5(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
        float vp = vt * vc5 + vc4;
        vp = vt * vp + vc3;
        vp = vt * vp + vc2;
        vp = vt * vp + vc1;

        // Reconstruct the exp(-z) value:
        //   e = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
        //     = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
        //     = s + (t * s) * p
        vt *= vs;
        const float ve = vt * vp + vs;

        // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
        float vf = ve / (ve + 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);
    }
}