f32-sigmoid/gen/neonfma-rr1-p5-nr2recps-x16.c

// Auto-generated file. Do not edit!
//   Template: src/f32-sigmoid/neon-p5.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 <arm_neon.h>

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


void xnn_f32_sigmoid_ukernel__neonfma_rr1_p5_nr2recps_x16(
    size_t n,
    const float* x,
    float* y,
    const void* params)
{
  assert(n % sizeof(float) == 0);

  const float32x4_t vmagic_bias = vmovq_n_f32(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 float32x4_t vdenorm_cutoff = vmovq_n_f32(0x1.5D589Ep+6f);
  const float32x4_t vminus_log2e = vmovq_n_f32(-0x1.715476p+0f);
  const float32x4_t vln2 = vmovq_n_f32(0x1.62E43p-1f);
  const float32x4_t vone = vmovq_n_f32(1.0f);

  const float32x4_t vc1 = vmovq_n_f32(-0x1.FFFFF6p-1f);
  const float32x4_t vc2 = vmovq_n_f32(0x1.FFFDC6p-2f);
  const float32x4_t vc3 = vmovq_n_f32(-0x1.555A80p-3f);
  const float32x4_t vc4 = vmovq_n_f32(0x1.573A1Ap-5f);
  const float32x4_t vc5 = vmovq_n_f32(-0x1.0F9F9Cp-7f);

  for (; n >= 16 * sizeof(float); n -= 16 * sizeof(float)) {
    const float32x4_t vx0123 = vld1q_f32(x); x += 4;
    const float32x4_t vx4567 = vld1q_f32(x); x += 4;
    const float32x4_t vx89AB = vld1q_f32(x); x += 4;
    const float32x4_t vxCDEF = vld1q_f32(x); 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 float32x4_t vz0123 = vabsq_f32(vx0123);
    const float32x4_t vz4567 = vabsq_f32(vx4567);
    const float32x4_t vz89AB = vabsq_f32(vx89AB);
    const float32x4_t vzCDEF = vabsq_f32(vxCDEF);

    // 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.
    float32x4_t vn0123 = vfmaq_f32(vmagic_bias, vz0123, vminus_log2e);
    float32x4_t vn4567 = vfmaq_f32(vmagic_bias, vz4567, vminus_log2e);
    float32x4_t vn89AB = vfmaq_f32(vmagic_bias, vz89AB, vminus_log2e);
    float32x4_t vnCDEF = vfmaq_f32(vmagic_bias, vzCDEF, vminus_log2e);

    // 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 float32x4_t vs0123 = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn0123), 23));
    const float32x4_t vs4567 = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn4567), 23));
    const float32x4_t vs89AB = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn89AB), 23));
    const float32x4_t vsCDEF = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vnCDEF), 23));

    // Subtract the large number back to get final n := round(-z / log(2)).
    vn0123 = vsubq_f32(vn0123, vmagic_bias);
    vn4567 = vsubq_f32(vn4567, vmagic_bias);
    vn89AB = vsubq_f32(vn89AB, vmagic_bias);
    vnCDEF = vsubq_f32(vnCDEF, vmagic_bias);

    // Compute reduced argument -t := -z - n * log(2) = -(z + n * log(2)).
    float32x4_t vt0123 = vfmaq_f32(vz0123, vn0123, vln2);
    float32x4_t vt4567 = vfmaq_f32(vz4567, vn4567, vln2);
    float32x4_t vt89AB = vfmaq_f32(vz89AB, vn89AB, vln2);
    float32x4_t vtCDEF = vfmaq_f32(vzCDEF, vnCDEF, vln2);

    // Compute degree-5 polynomial approxiatmion for exp(-t) on [-log(2)/2, log(2)/2].
    float32x4_t vp0123 = vfmaq_f32(vc4, vc5, vt0123);
    float32x4_t vp4567 = vfmaq_f32(vc4, vc5, vt4567);
    float32x4_t vp89AB = vfmaq_f32(vc4, vc5, vt89AB);
    float32x4_t vpCDEF = vfmaq_f32(vc4, vc5, vtCDEF);

    vp0123 = vfmaq_f32(vc3, vp0123, vt0123);
    vp4567 = vfmaq_f32(vc3, vp4567, vt4567);
    vp89AB = vfmaq_f32(vc3, vp89AB, vt89AB);
    vpCDEF = vfmaq_f32(vc3, vpCDEF, vtCDEF);

    vp0123 = vfmaq_f32(vc2, vp0123, vt0123);
    vp4567 = vfmaq_f32(vc2, vp4567, vt4567);
    vp89AB = vfmaq_f32(vc2, vp89AB, vt89AB);
    vpCDEF = vfmaq_f32(vc2, vpCDEF, vtCDEF);

    vp0123 = vfmaq_f32(vc1, vp0123, vt0123);
    vp4567 = vfmaq_f32(vc1, vp4567, vt4567);
    vp89AB = vfmaq_f32(vc1, vp89AB, vt89AB);
    vpCDEF = vfmaq_f32(vc1, vpCDEF, vtCDEF);

    // 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
    vt0123 = vmulq_f32(vt0123, vs0123);
    vt4567 = vmulq_f32(vt4567, vs4567);
    vt89AB = vmulq_f32(vt89AB, vs89AB);
    vtCDEF = vmulq_f32(vtCDEF, vsCDEF);

    float32x4_t ve0123 = vfmaq_f32(vs0123, vp0123, vt0123);
    float32x4_t ve4567 = vfmaq_f32(vs4567, vp4567, vt4567);
    float32x4_t ve89AB = vfmaq_f32(vs89AB, vp89AB, vt89AB);
    float32x4_t veCDEF = vfmaq_f32(vsCDEF, vpCDEF, vtCDEF);

    // Denominator of the sigmoid fraction: 1.0 + exp(-z)
    float32x4_t vd0123 = vaddq_f32(ve0123, vone);
    float32x4_t vd4567 = vaddq_f32(ve4567, vone);
    float32x4_t vd89AB = vaddq_f32(ve89AB, vone);
    float32x4_t vdCDEF = vaddq_f32(veCDEF, vone);

    // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
    // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
    // Thus the reciprocal of the denominator never overflows.
    float32x4_t vr0123 = vrecpeq_f32(vd0123);
    float32x4_t vr4567 = vrecpeq_f32(vd4567);
    float32x4_t vr89AB = vrecpeq_f32(vd89AB);
    float32x4_t vrCDEF = vrecpeq_f32(vdCDEF);

    vr0123 = vmulq_f32(vr0123, vrecpsq_f32(vr0123, vd0123));
    vr4567 = vmulq_f32(vr4567, vrecpsq_f32(vr4567, vd4567));
    vr89AB = vmulq_f32(vr89AB, vrecpsq_f32(vr89AB, vd89AB));
    vrCDEF = vmulq_f32(vrCDEF, vrecpsq_f32(vrCDEF, vdCDEF));

    vr0123 = vmulq_f32(vr0123, vrecpsq_f32(vr0123, vd0123));
    vr4567 = vmulq_f32(vr4567, vrecpsq_f32(vr4567, vd4567));
    vr89AB = vmulq_f32(vr89AB, vrecpsq_f32(vr89AB, vd89AB));
    vrCDEF = vmulq_f32(vrCDEF, vrecpsq_f32(vrCDEF, vdCDEF));

    // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
    float32x4_t vf0123 = vmulq_f32(ve0123, vr0123);
    float32x4_t vf4567 = vmulq_f32(ve4567, vr4567);
    float32x4_t vf89AB = vmulq_f32(ve89AB, vr89AB);
    float32x4_t vfCDEF = vmulq_f32(veCDEF, vrCDEF);

    // For inputs below denormal cutoff, replace output with +0.0f.
    // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
    vf0123 = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf0123), vcagtq_f32(vx0123, vdenorm_cutoff)));
    vf4567 = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf4567), vcagtq_f32(vx4567, vdenorm_cutoff)));
    vf89AB = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf89AB), vcagtq_f32(vx89AB, vdenorm_cutoff)));
    vfCDEF = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vfCDEF), vcagtq_f32(vxCDEF, vdenorm_cutoff)));

    // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
    const uint32x4_t vm0123 = vcltq_f32(vx0123, vmovq_n_f32(0.0f));
    const uint32x4_t vm4567 = vcltq_f32(vx4567, vmovq_n_f32(0.0f));
    const uint32x4_t vm89AB = vcltq_f32(vx89AB, vmovq_n_f32(0.0f));
    const uint32x4_t vmCDEF = vcltq_f32(vxCDEF, vmovq_n_f32(0.0f));

    vf0123 = vbslq_f32(vm0123, vf0123, vsubq_f32(vone, vf0123));
    vf4567 = vbslq_f32(vm4567, vf4567, vsubq_f32(vone, vf4567));
    vf89AB = vbslq_f32(vm89AB, vf89AB, vsubq_f32(vone, vf89AB));
    vfCDEF = vbslq_f32(vmCDEF, vfCDEF, vsubq_f32(vone, vfCDEF));

    vst1q_f32(y, vf0123); y += 4;
    vst1q_f32(y, vf4567); y += 4;
    vst1q_f32(y, vf89AB); y += 4;
    vst1q_f32(y, vfCDEF); y += 4;
  }
  for (; n >= 4 * sizeof(float); n -= 4 * sizeof(float)) {
    const float32x4_t vx = vld1q_f32(x); 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 float32x4_t vz = vabsq_f32(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.
    float32x4_t vn = vfmaq_f32(vmagic_bias, vz, vminus_log2e);

    // 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 float32x4_t vs = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn), 23));

    // Subtract the large number back to get final n := round(-z / log(2)).
    vn = vsubq_f32(vn, vmagic_bias);

    // Compute reduced argument -t := -z - n * log(2) = -(z + n * log(2)).
    float32x4_t vt = vfmaq_f32(vz, vn, vln2);

    // Compute degree-5 polynomial approxiatmion for exp(-t) on [-log(2)/2, log(2)/2].
    float32x4_t vp = vfmaq_f32(vc4, vc5, vt);
    vp = vfmaq_f32(vc3, vp, vt);
    vp = vfmaq_f32(vc2, vp, vt);
    vp = vfmaq_f32(vc1, vp, vt);

    // 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 = vmulq_f32(vt, vs);
    float32x4_t ve = vfmaq_f32(vs, vp, vt);

    // Denominator of the sigmoid fraction: 1.0 + exp(-z)
    float32x4_t vd = vaddq_f32(ve, vone);

    // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
    // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
    // Thus the reciprocal of the denominator never overflows.
    float32x4_t vr = vrecpeq_f32(vd);

    vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));

    vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));

    // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
    float32x4_t vf = vmulq_f32(ve, vr);

    // For inputs below denormal cutoff, replace output with +0.0f.
    // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
    vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcagtq_f32(vx, vdenorm_cutoff)));

    // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
    const uint32x4_t vm = vcltq_f32(vx, vmovq_n_f32(0.0f));
    vf = vbslq_f32(vm, vf, vsubq_f32(vone, vf));

    vst1q_f32(y, vf); y += 4;
  }
  if XNN_UNLIKELY(n != 0) {
    const float32x4_t vx = vld1q_f32(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 float32x4_t vz = vabsq_f32(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.
    float32x4_t vn = vfmaq_f32(vmagic_bias, vz, vminus_log2e);

    // 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 float32x4_t vs = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn), 23));

    // Subtract the large number back to get final n := round(-z / log(2)).
    vn = vsubq_f32(vn, vmagic_bias);

    // Compute reduced argument -t := -z - n * log(2) = -(z + n * log(2)).
    float32x4_t vt = vfmaq_f32(vz, vn, vln2);

    // Compute degree-5 polynomial approxiatmion for exp(-t) on [-log(2)/2, log(2)/2].
    float32x4_t vp = vfmaq_f32(vc4, vc5, vt);
    vp = vfmaq_f32(vc3, vp, vt);
    vp = vfmaq_f32(vc2, vp, vt);
    vp = vfmaq_f32(vc1, vp, vt);

    // 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 = vmulq_f32(vt, vs);
    float32x4_t ve = vfmaq_f32(vs, vp, vt);

    // Denominator of the sigmoid fraction: 1.0 + exp(-z)
    float32x4_t vd = vaddq_f32(ve, vone);

    // Use Newton-Raphson method (2 iterations) to compute reciprocal of denominator.
    // Note: 1 < d <= 2, because z >= 0.0 and 0 < exp(-z) <= 1.0.
    // Thus the reciprocal of the denominator never overflows.
    float32x4_t vr = vrecpeq_f32(vd);

    vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));

    vr = vmulq_f32(vr, vrecpsq_f32(vr, vd));

    // Reconstruct sigmoid(-z) = exp(-z) / (1.0 + exp(-z))
    float32x4_t vf = vmulq_f32(ve, vr);

    // For inputs below denormal cutoff, replace output with +0.0f.
    // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
    vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcagtq_f32(vx, vdenorm_cutoff)));

    // Reconstruct sigmoid(x) = x < 0 ? sigmoid(-z) : 1.0 - sigmoid(-z)
    const uint32x4_t vm = vcltq_f32(vx, vmovq_n_f32(0.0f));
    vf = vbslq_f32(vm, vf, vsubq_f32(vone, vf));

    float32x2_t vf_lo = vget_low_f32(vf);
    if (n & (2 * sizeof(float))) {
      vst1_f32(y, vf_lo); y += 2;
      vf_lo = vget_high_f32(vf);
    }
    if (n & (1 * sizeof(float))) {
      vst1_lane_f32(y, vf_lo, 0);
    }
  }
}