src/math/expminus-f32-neonfma-rr2-p5.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 <arm_neon.h>

#include <xnnpack/math-stubs.h>


void xnn_math_f32_expminus__neonfma_rr2_p5(
    size_t n,
    const float* input,
    float* output)
{
  assert(n % (4 * sizeof(float)) == 0);

  // Large number such that ulp(magic bias) == 1 and magic bias === 127 mod 2**22.
  const float32x4_t vmagic_bias = vmovq_n_f32(0x1.8000FEp23f);
  const float32x4_t vlog2e = vmovq_n_f32(0x1.715476p+0f);
  const float32x4_t vminus_ln2_hi = vmovq_n_f32(-0x1.62E43p-1f);
  const float32x4_t vminus_ln2_lo = vmovq_n_f32(0x1.05C61p-29f);
  // Coefficient of polynomial approximation
  //   exp(t) ~ 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
  // on [-log(2)/2, log(2)/2]
  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);
  // The smallest x for which expf(x) is normalized.
  const float32x4_t vdenorm_cutoff = vmovq_n_f32(-0x1.5D589Ep6f);

  for (; n != 0; n -= 4 * sizeof(float)) {
    const float32x4_t vx = vld1q_f32(input); input += 4;

    // Compute reduced argument n := round(x / log(2)).
    // We do it by adding a large number (magic bias) to the product x * (1/log(2)), 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 / log(2)| <= 2**22, i.e. |x| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because inputs outside
    // of [-87.336540, 0.0] underflow expf(x) anyway. We fixup the result for such inputs at the very end of the
    // algorithm.
    float32x4_t vn = vfmaq_f32(vmagic_bias, vx, vlog2e);

    // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
    // -87.33642 <= x <= 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(x / log(2)) as a floating-point number.
    vn = vsubq_f32(vn, vmagic_bias);

    // Compute reduced argument t := x - n * log(2).
    // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
    float32x4_t vt = vfmaq_f32(vx, vn, vminus_ln2_hi);
    vt = vfmaq_f32(vt, vn, vminus_ln2_lo);

    // Compute degree-5 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2]:
    //   P(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) = 1 + t * p
    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(x) value:
    //   exp(x) = 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 vf = vfmaq_f32(vs, vp, vt);

    // 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), vcltq_f32(vx, vdenorm_cutoff)));
    vst1q_f32(output, vf); output += 4;
  }
}