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_p3(size_t n,const void * input,void * output)14 void xnn_math_f16_expminus__neonfp16arith_rr2_p3(
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 vc3 = vmovq_n_f16(0x1.558p-3f);
30 const float16x8_t vc2 = vmovq_n_f16(0x1.020p-1f);
31 const float16x8_t vone = vmovq_n_f16(1.0f);
32 // The smallest x for which exph(x) is normalized.
33 const float16x8_t vdenorm_cutoff = vmovq_n_f16(-0x1.368p3f);
34
35 const __fp16* i = (const __fp16*) input;
36 __fp16* o = (__fp16*) output;
37 for (; n != 0; n -= 8 * sizeof(__fp16)) {
38 const float16x8_t vx = vld1q_f16(i); i += 8;
39
40 // Compute reduced argument n := round(x / log(2)).
41 // We do it by adding a large number (magic bias) to the product x * (1/log(2)), which cause rounding of the result
42 // to an integer, then subtracing the large number back. The first addition is combined with multiplication by
43 // log2e into a single FMA instruction. The trick with adding large number is valid only within certain bounds
44 // (|x / log(2)| <= 2**9, i.e. |x| <= 0x1.630p+8 = 355.0), but that is acceptable, because inputs outside
45 // of [-9.703125, 0.0] underflow exph(x) anyway. We fixup the result for such inputs at the very end of the
46 // algorithm.
47 float16x8_t vn = vfmaq_f16(vmagic_bias, vx, vlog2e);
48
49 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
50 // -9.703125 <= x <= 0.0, and -14 <= n <= 0 accordingly.
51 const float16x8_t vs = vreinterpretq_f16_s16(vshlq_n_s16(vreinterpretq_s16_f16(vn), 10));
52
53 // Subtract the large number back to get final n := round(x / log(2)) as a floating-point number.
54 vn = vsubq_f16(vn, vmagic_bias);
55
56 // Compute reduced argument t := x - n * log(2).
57 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
58 float16x8_t vt = vfmaq_f16(vx, vn, vminus_ln2_hi);
59 vt = vfmaq_f16(vt, vn, vminus_ln2_lo);
60
61 // Compute degree-3 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2]:
62 // P(t) = 1 + t * (1 + t * (c2 + t * c3)) = 1 + t * p
63 float16x8_t vp = vfmaq_f16(vc2, vc3, vt);
64 vp = vfmaq_f16(vone, vp, vt);
65
66 // Reconstruct the exp(x) value:
67 // exp(x) = s * (1 + t * (1 + t * (c2 + t * c3)))
68 // = s + (t * s) * (1 + t * (c2 + t * c3))
69 // = s + (t * s) * p
70 vt = vmulq_f16(vt, vs);
71 float16x8_t vf = vfmaq_f16(vs, vp, vt);
72
73 // For inputs below denormal cutoff, replace output with +0.0f.
74 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
75 vf = vreinterpretq_f16_u16(vbicq_u16(vreinterpretq_u16_f16(vf), vcltq_f16(vx, vdenorm_cutoff)));
76 vst1q_f16(o, vf); o += 8;
77 }
78 }
79