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_expm1minus__neonfp16arith_rr2_p3(size_t n,const void * input,void * output)14 void xnn_math_f16_expm1minus__neonfp16arith_rr2_p3(
15 size_t n,
16 const void* input,
17 void* output)
18 {
19 assert(n % (8 * sizeof(__fp16)) == 0);
20
21 // The largest x for which expm1f(x) is saturated at -1.0f.
22 const float16x8_t vsat_cutoff = vmovq_n_f16(-0x1.0A4p+3f);
23 // Large number such that ulp(magic bias) == 1 and magic bias === 15 mod 2**9.
24 const float16x8_t vmagic_bias = vmovq_n_f16(0x1.83Cp+10f);
25 const float16x8_t vlog2e = vmovq_n_f16(0x1.714p+0f);
26 const float16x8_t vminus_ln2_hi = vmovq_n_f16(-0x1.630p-1f);
27 const float16x8_t vminus_ln2_lo = vmovq_n_f16(0x1.BD0p-13f);
28 // Coefficient of polynomial approximation
29 // exp(t) - 1 ~ t * (1 + t * (c2 + t * c3))
30 // on [-log(2)/2, log(2)/2]
31 const float16x8_t vc3 = vmovq_n_f16(0x1.56Cp-3f);
32 const float16x8_t vc2 = vmovq_n_f16(0x1.020p-1f);
33 const float16x8_t vone = vmovq_n_f16(1.0f);
34
35 const __fp16* i = (const __fp16*) input;
36 __fp16* o = (__fp16*) output;
37 for (; n != 0; n -= 8 * sizeof(__fp16)) {
38 float16x8_t vx = vld1q_f16(i); i += 8;
39
40 // The function saturates at -1 for large negative inputs: expm1h(x) == -1.0h for x <= sat_cutoff ~= -8.3203125.
41 // To guarantee this behaviour, we clip input at sat_cutoff, and leverage the fact that for our implementation
42 // expm1m(sat_cutoff) == -1.0f. NaN inputs are passed unchanged.
43 vx = vmaxq_f16(vx, vsat_cutoff);
44
45 // Compute reduced argument n := round(x / log(2)).
46 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
47 // the large number back. The addition is combined with multiplication by log2e into a single FMA instruction. The
48 // trick with adding large number is valid only within certain bounds (|x / log(2)| <= 2**9, i.e.
49 // |x| <= 0x1.630p+8 = 355.0), but that is acceptable, because inputs x are restricted to [-8.3203125, 0].
50 // Note that addition-subtraction of the large number doesn't cause overflow for inputs in this range.
51 float16x8_t vn = vfmaq_f16(vmagic_bias, vx, vlog2e);
52
53 // Create a floating-point number s (scale) such that s == 2**n for valid inputs, i.e.
54 // -8.3203125 <= x <= 0.0, and -12 <= n <= 0 accordingly.
55 // For NaN inputs, s would have zero mantissa and can have arbitrary sign and exponent, depending on the input
56 // NaN payload. In these cases, n and t are NaNs with the same payload as input while s is non-NaN, and thus
57 // input payload would be propagated in all computations.
58 const float16x8_t vs = vreinterpretq_f16_s16(vshlq_n_s16(vreinterpretq_s16_f16(vn), 10));
59
60 // Subtract the large number back to get final n := round(x / log(2)).
61 vn = vsubq_f16(vn, vmagic_bias);
62
63 // Compute reduced argument t := x - n * log(2).
64 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
65 float16x8_t vt = vfmaq_f16(vx, vn, vminus_ln2_hi);
66 vt = vfmaq_f16(vt, vn, vminus_ln2_lo);
67
68 // Compute degree-3 polynomial approximation for exp(t) - 1 on [-log(2)/2, log(2)/2].
69 // P(t) = t * (1 + t * (c2 + t * c3))
70 // = t + t * (t * (c2 + t * c3)) = t + t * p
71 float16x8_t vp = vfmaq_f16(vc2, vc3, vt);
72 vp = vmulq_f16(vp, vt);
73
74 // Reconstruct the exp(x) - 1 value:
75 // exp(x) - 1 = s * (1 + t * (1 + t * (c2 + t * c3))) - 1
76 // = (s - 1) + s * (t + t * p)
77 // = ((t * s) + (t * s) * p) + (s - 1)
78 vt = vmulq_f16(vt, vs);
79 const float16x8_t vsm1 = vsubq_f16(vs, vone);
80 vp = vfmaq_f16(vt, vp, vt);
81 const float16x8_t vf = vaddq_f16(vp, vsm1);
82
83 vst1q_f16(o, vf); o += 8;
84 }
85 }
86