1 // Copyright 2022 Google LLC
2 //
3 // This source code is licensed under the BSD-style license found in the
4 // LICENSE file in the root directory of this source tree.
5
6 #include <assert.h>
7 #include <stddef.h>
8
9 #include <arm_neon.h>
10
11 #include <xnnpack/math-stubs.h>
12
13
xnn_math_f16_expminus__neonfp16arith_rr2_p2(size_t n,const void * input,void * output)14 void xnn_math_f16_expminus__neonfp16arith_rr2_p2(
15 size_t n,
16 const void* input,
17 void* output)
18 {
19 assert(n % (8 * sizeof(__fp16)) == 0);
20
21 // Large number such that ulp(magic bias) == 1 and magic bias === 15 mod 2**9.
22 const float16x8_t vmagic_bias = vmovq_n_f16(0x1.83Cp+10f);
23 const float16x8_t vlog2e = vmovq_n_f16(0x1.714p+0f);
24 const float16x8_t vminus_ln2_hi = vmovq_n_f16(-0x1.630p-1f);
25 const float16x8_t vminus_ln2_lo = vmovq_n_f16(0x1.BD0p-13f);
26 // Coefficient of polynomial approximation
27 // exp(t) ~ 1 + t * (1 + t * (c2 + t * c3))
28 // on [-log(2)/2, log(2)/2]
29 const float16x8_t vc2 = vmovq_n_f16(0x1.FE4p-2f);
30 const float16x8_t vc1 = vmovq_n_f16(0x1.038p0f);
31 // The smallest x for which exph(x) is normalized.
32 const float16x8_t vdenorm_cutoff = vmovq_n_f16(-0x1.368p3f);
33
34 const __fp16* i = (const __fp16*) input;
35 __fp16* o = (__fp16*) output;
36 for (; n != 0; n -= 8 * sizeof(__fp16)) {
37 const float16x8_t vx = vld1q_f16(i); i += 8;
38
39 // Compute reduced argument n := round(x / log(2)).
40 // We do it by adding a large number (magic bias) to the product x * (1/log(2)), which cause rounding of the result
41 // to an integer, then subtracing the large number back. The first addition is combined with multiplication by
42 // log2e into a single FMA instruction. The trick with adding large number is valid only within certain bounds
43 // (|x / log(2)| <= 2**9, i.e. |x| <= 0x1.630p+8 = 355.0), but that is acceptable, because inputs outside
44 // of [-9.703125, 0.0] underflow exph(x) anyway. We fixup the result for such inputs at the very end of the
45 // algorithm.
46 float16x8_t vn = vfmaq_f16(vmagic_bias, vx, vlog2e);
47
48 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
49 // -9.703125 <= x <= 0.0, and -14 <= n <= 0 accordingly.
50 const float16x8_t vs = vreinterpretq_f16_s16(vshlq_n_s16(vreinterpretq_s16_f16(vn), 10));
51
52 // Subtract the large number back to get final n := round(x / log(2)) as a floating-point number.
53 vn = vsubq_f16(vn, vmagic_bias);
54
55 // Compute reduced argument t := x - n * log(2).
56 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
57 float16x8_t vt = vfmaq_f16(vx, vn, vminus_ln2_hi);
58 vt = vfmaq_f16(vt, vn, vminus_ln2_lo);
59
60 // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2]:
61 // P(t) = 1 + t * (c1 + t * c2) = 1 + t * p
62 const float16x8_t vp = vfmaq_f16(vc1, vc2, vt);
63
64 // Reconstruct the exp(x) value:
65 // exp(x) = s * (1 + t * (c1 + t * c2))
66 // = s + (t * s) * (c1 + t * c2)
67 // = s + (t * s) * p
68 vt = vmulq_f16(vt, vs);
69 float16x8_t vf = vfmaq_f16(vs, vp, vt);
70
71 // For inputs below denormal cutoff, replace output with +0.0f.
72 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
73 vf = vreinterpretq_f16_u16(vbicq_u16(vreinterpretq_u16_f16(vf), vcltq_f16(vx, vdenorm_cutoff)));
74 vst1q_f16(o, vf); o += 8;
75 }
76 }
77