blob: d2643f363f465c51a0b9680276ece8873731b991 [file] [log] [blame] [edit]
// Copyright 2022 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_f16_expminus__neonfp16arith_rr2_p2(
size_t n,
const void* input,
void* output)
{
assert(n % (8 * sizeof(__fp16)) == 0);
// Large number such that ulp(magic bias) == 1 and magic bias === 15 mod 2**9.
const float16x8_t vmagic_bias = vmovq_n_f16(0x1.83Cp+10f);
const float16x8_t vlog2e = vmovq_n_f16(0x1.714p+0f);
const float16x8_t vminus_ln2_hi = vmovq_n_f16(-0x1.630p-1f);
const float16x8_t vminus_ln2_lo = vmovq_n_f16(0x1.BD0p-13f);
// Coefficient of polynomial approximation
// exp(t) ~ 1 + t * (1 + t * (c2 + t * c3))
// on [-log(2)/2, log(2)/2]
const float16x8_t vc2 = vmovq_n_f16(0x1.FE4p-2f);
const float16x8_t vc1 = vmovq_n_f16(0x1.038p0f);
// The smallest x for which exph(x) is normalized.
const float16x8_t vdenorm_cutoff = vmovq_n_f16(-0x1.368p3f);
const __fp16* i = (const __fp16*) input;
__fp16* o = (__fp16*) output;
for (; n != 0; n -= 8 * sizeof(__fp16)) {
const float16x8_t vx = vld1q_f16(i); i += 8;
// 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**9, i.e. |x| <= 0x1.630p+8 = 355.0), but that is acceptable, because inputs outside
// of [-9.703125, 0.0] underflow exph(x) anyway. We fixup the result for such inputs at the very end of the
// algorithm.
float16x8_t vn = vfmaq_f16(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.
// -9.703125 <= x <= 0.0, and -14 <= n <= 0 accordingly.
const float16x8_t vs = vreinterpretq_f16_s16(vshlq_n_s16(vreinterpretq_s16_f16(vn), 10));
// Subtract the large number back to get final n := round(x / log(2)) as a floating-point number.
vn = vsubq_f16(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.
float16x8_t vt = vfmaq_f16(vx, vn, vminus_ln2_hi);
vt = vfmaq_f16(vt, vn, vminus_ln2_lo);
// Compute degree-2 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2]:
// P(t) = 1 + t * (c1 + t * c2) = 1 + t * p
const float16x8_t vp = vfmaq_f16(vc1, vc2, vt);
// Reconstruct the exp(x) value:
// exp(x) = s * (1 + t * (c1 + t * c2))
// = s + (t * s) * (c1 + t * c2)
// = s + (t * s) * p
vt = vmulq_f16(vt, vs);
float16x8_t vf = vfmaq_f16(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_f16_u16(vbicq_u16(vreinterpretq_u16_f16(vf), vcltq_f16(vx, vdenorm_cutoff)));
vst1q_f16(o, vf); o += 8;
}
}