src/f32-raddstoreexpminusmax/scalar-lut64-p2.c.in

// Copyright 2020 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 ELEMENTS_TILE >= 1
$ABC = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
#include <assert.h>

#include <xnnpack/common.h>
#include <xnnpack/raddstoreexpminusmax.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_raddstoreexpminusmax_ukernel__scalar_lut64_p2_x${ELEMENTS_TILE}${"" if ACCUMULATORS == 1 else "_acc%d" % ACCUMULATORS}(
    size_t elements,
    const float* input,
    float* output,
    float* sum,
    float vi_max)
{
  assert(elements % sizeof(float) == 0);

  const float vmagic_bias = 0x1.800000p23f;
  // The smallest x for which expf(x) is normalized.
  const float vdenorm_cutoff = -0x1.5D589Ep6f;
  const float vlog2e_x64  = 0x1.715476p6f;
  // Last 13 bits are zeroes
  const float vminus_ln2_o64_hi = -0x1.630000p-7f;
  const float vminus_ln2_o64_lo =  0x1.BD0106p-19f;

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

  const uint32_t vindex_mask = UINT32_C(0x3F);

  $if ELEMENTS_TILE > 1:
    $for K in range(ACCUMULATORS):
      float vacc${K} = 0.0f;
    for (; elements >= ${ELEMENTS_TILE} * sizeof(float); elements -= ${ELEMENTS_TILE} * sizeof(float)) {
      // Load ${ELEMENTS_TILE} inputs at a time.
      $for N in range(ELEMENTS_TILE):
        const float vi${N} = input[${N}];
      input += ${ELEMENTS_TILE};

      // Subtract maximum input x := i - i_max. This implies x <= 0.
      $for N in range(ELEMENTS_TILE):
        const float vx${N} = vi${N} - vi_max;

      // Compute reduced argument n := round(x * 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 (|x * 64 / log(2)| <= 2**22, i.e.
      // |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs outside of [-87.336540, 0.0]
      // result in denormalized or underflown expf(x). We fixup the result for such inputs at the very end of the
      // algorithm.
      $for N in range(ELEMENTS_TILE):
        float vn${N} = vx${N} * vlog2e_x64 + vmagic_bias;

      // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that expf(x) is normalized,
      // i.e. -87.33642 <= x <= 0.0. 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 -87.33642 <= x <= 0.0 (inputs for which expf(x) 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).
      $for N in range(ELEMENTS_TILE):
        const uint32_t ve${N} = (fp32_to_bits(vn${N}) & UINT32_C(0xFFFFFFC0)) << 17;

      // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64).
      $for N in range(ELEMENTS_TILE):
        const uint32_t vidx${N} = fp32_to_bits(vn${N}) & vindex_mask;
      // Adjust exponent of the value l fetched from the table to get the final s value.
      $for N in range(ELEMENTS_TILE):
        const float vs${N} = fp32_from_bits(xnn_table_exp2_k_over_64[vidx${N}] + ve${N});

      // Subtract the large number back to get final n := round(x * 64 / log(2)) as a floating-point number.
      $for N in range(ELEMENTS_TILE):
        vn${N} -= vmagic_bias;

      // Compute reduced argument t := x - n * log(2) / 64.
      // Use Cody-Waite range reduction method (note the two constants representing log(2) / 64) to improve accuracy.
      $for N in range(ELEMENTS_TILE):
        float vt${N} = vn${N} * vminus_ln2_o64_hi + vx${N};

      $for N in range(ELEMENTS_TILE):
        vt${N} = vn${N} * vminus_ln2_o64_lo + vt${N};

      // Compute degree-2 polynomial approxiatmion for exp(t) on [-log(2)/128, log(2)/128].
      $for N in range(ELEMENTS_TILE):
        float vp${N} = vt${N} * vc2;

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

      // 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
      $for N in range(ELEMENTS_TILE):
        float vf${N} = vp${N} * vs${N} + vs${N};

      // For inputs below denormal cutoff, replace output with +0.0f.
      // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
      $for N in range(ELEMENTS_TILE):
        if XNN_UNPREDICTABLE(vx${N} < vdenorm_cutoff) {
          vf${N} = 0.0f;
        }

      // Store ${ELEMENTS_TILE} outputs at a time.
      $for N in range(ELEMENTS_TILE):
        output[${N}] = vf${N};
      output += ${ELEMENTS_TILE};

      // Accumulate computed exponents.
      $for N in range(ELEMENTS_TILE):
        vacc${N % ACCUMULATORS} += vf${N};
    }
    $if ACCUMULATORS > 1:
      // Add up all accumulators to vacc0
      $ACC_SLICE = 1
      $while ACC_SLICE < ACCUMULATORS:
        $for A in range(0, ACCUMULATORS, ACC_SLICE * 2):
          $if A + ACC_SLICE < ACCUMULATORS:
            vacc${A} += vacc${A + ACC_SLICE};
        $ACC_SLICE *= 2

    float vacc = vacc0;
  $else:
    float vacc = 0.0f;
  for (; elements >= sizeof(float); elements -= sizeof(float)) {
    // Load 1 input at a time.
    const float vi = *input++;

    // Subtract maximum input x := i - i_max. This implies x <= 0.
    const float vx = vi - vi_max;

    // Compute reduced argument n := round(x * 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 (|x * 64 / log(2)| <= 2**22, i.e.
    // |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs outside of [-87.336540, 0.0]
    // result in denormalized or underflown expf(x). We fixup the result for such inputs at the very end of the
    // algorithm.
    float vn = vx * vlog2e_x64 + vmagic_bias;

    // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that expf(x) is normalized,
    // i.e. -87.33642 <= x <= 0.0. 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 -87.33642 <= x <= 0.0 (inputs for which expf(x) 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) & UINT32_C(0xFFFFFFC0)) << 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 final n := round(x * 64 / log(2)) as a floating-point number.
    vn -= vmagic_bias;

    // Compute reduced argument t := x - n * log(2) / 64.
    // Use Cody-Waite range reduction method (note the two constants representing log(2) / 64) to improve accuracy.
    float vt = vn * vminus_ln2_o64_hi + vx;
    vt = vn * vminus_ln2_o64_lo + vt;

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

    // 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(vx < vdenorm_cutoff) {
      vf = 0.0f;
    }

    // Store 1 output at a time.
    *output++ = vf;

    // Accumulate computed exponents.
    vacc += vf;
  }
  *sum = vacc;
}