1 // Copyright 2020 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_f32_expm1minus__neon_rr2_p6(size_t n,const float * input,float * output)14 void xnn_math_f32_expm1minus__neon_rr2_p6(
15 size_t n,
16 const float* input,
17 float* output)
18 {
19 assert(n % (4 * sizeof(float)) == 0);
20
21 // The largest x for which expm1f(x) is saturated at -1.0f.
22 const float32x4_t vsat_cutoff = vmovq_n_f32(-0x1.154246p+4f);
23 // Large number such that ulp(magic bias) == 1 and magic bias === 127 mod 2**22.
24 const float32x4_t vmagic_bias = vmovq_n_f32(0x1.8000FEp23f);
25 const float32x4_t vlog2e = vmovq_n_f32(0x1.715476p+0f);
26 // Last 5 bits are zeroes
27 const float32x4_t vminus_ln2_hi = vmovq_n_f32(-0x1.62E440p-1f);
28 const float32x4_t vminus_ln2_lo = vmovq_n_f32(0x1.0105C6p-21f);
29 // Coefficient of polynomial approximation
30 // exp(t) - 1 ~ t * (1 + t * (c2 + t * (c3 + t * (c4 + t * (c5 + t * c6)))))
31 // on [-log(2)/2, log(2)/2]
32 const float32x4_t vc6 = vmovq_n_f32(0x1.6b7338p-10f);
33 const float32x4_t vc5 = vmovq_n_f32(0x1.12278Ep-7f);
34 const float32x4_t vc4 = vmovq_n_f32(0x1.555716p-5f);
35 const float32x4_t vc3 = vmovq_n_f32(0x1.5554B0p-3f);
36 const float32x4_t vc2 = vmovq_n_f32(0x1.FFFFFEp-2f);
37 const float32x4_t vone = vmovq_n_f32(1.0f);
38
39 for (; n != 0; n -= 4 * sizeof(float)) {
40 float32x4_t vx = vld1q_f32(input); input += 4;
41
42 // The function saturates at -1 for large negative inputs: expm1f(x) == -1.0f for x <= sat_cutoff ~= -17.328680.
43 // To guarantee this behaviour, we clip input at sat_cutoff, and leverage the fact that for our implementation
44 // expm1f(sat_cutoff) == -1.0f. NaN inputs are passed unchanged.
45 vx = vmaxq_f32(vx, vsat_cutoff);
46
47 // Compute reduced argument n := round(x / log(2)).
48 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
49 // the large number back. The trick with adding large number is valid only within certain bounds
50 // (|x / log(2)| <= 2**22, i.e. |x| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because inputs x are
51 // restricted to [-17.328680, 0].
52 // Note that addition-subtraction of the large number doesn't cause overflow for inputs in this range.
53 float32x4_t vn = vmlaq_f32(vmagic_bias, vx, vlog2e);
54
55 // Create a floating-point number s (scale) such that s == 2**n for valid inputs, i.e.
56 // -17.328680 <= x <= 0.0, and -25 <= n <= 0 accordingly.
57 // For NaN inputs, s would have zero mantissa and can have arbitrary sign and exponent, depending on the input
58 // NaN payload. In these cases, n and t are NaNs with the same payload as input while s is non-NaN, and thus
59 // input payload would be propagated in all computations.
60 const float32x4_t vs = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn), 23));
61
62 // Subtract the large number back to get final n := round(x / log(2)).
63 vn = vsubq_f32(vn, vmagic_bias);
64
65 // Compute reduced argument t := x - n * log(2).
66 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
67 float32x4_t vt = vmlaq_f32(vx, vn, vminus_ln2_hi);
68 vt = vmlaq_f32(vt, vn, vminus_ln2_lo);
69
70 // Compute degree-6 polynomial approximation for exp(t) - 1 on [-log(2)/2, log(2)/2].
71 // P(t) = t * (1 + t * (c2 + t * (c3 + t * (c4 + t * (c5 + t * c6)))))
72 // = t + t * (t * (c2 + t * (c3 + t * (c4 + t * (c5 + t * c6))))) = t + t * p
73 float32x4_t vp = vmlaq_f32(vc5, vc6, vt);
74 vp = vmlaq_f32(vc4, vp, vt);
75 vp = vmlaq_f32(vc3, vp, vt);
76 vp = vmlaq_f32(vc2, vp, vt);
77 vp = vmulq_f32(vp, vt);
78
79 // Reconstruct the exp(x) - 1 value:
80 // exp(x) - 1 = s * (1 + t * (1 + t * (c2 + t * (c3 + t * (c4 + t * (c5 + t * c6)))))) - 1
81 // = (s - 1) + s * (t + t * p)
82 // = ((t * s) + (t * s) * p) + (s - 1)
83 vt = vmulq_f32(vt, vs);
84 const float32x4_t vsm1 = vsubq_f32(vs, vone);
85 vp = vmlaq_f32(vt, vp, vt);
86 const float32x4_t vf = vaddq_f32(vp, vsm1);
87
88 vst1q_f32(output, vf); output += 4;
89 }
90 }
91