1 // Copyright 2019 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_expminus__neonfma_rr2_p5(size_t n,const float * input,float * output)14 void xnn_math_f32_expminus__neonfma_rr2_p5(
15 size_t n,
16 const float* input,
17 float* output)
18 {
19 assert(n % (4 * sizeof(float)) == 0);
20
21 // Large number such that ulp(magic bias) == 1 and magic bias === 127 mod 2**22.
22 const float32x4_t vmagic_bias = vmovq_n_f32(0x1.8000FEp23f);
23 const float32x4_t vlog2e = vmovq_n_f32(0x1.715476p+0f);
24 const float32x4_t vminus_ln2_hi = vmovq_n_f32(-0x1.62E43p-1f);
25 const float32x4_t vminus_ln2_lo = vmovq_n_f32(0x1.05C61p-29f);
26 // Coefficient of polynomial approximation
27 // exp(t) ~ 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
28 // on [-log(2)/2, log(2)/2]
29 const float32x4_t vc1 = vmovq_n_f32(0x1.FFFFF6p-1f);
30 const float32x4_t vc2 = vmovq_n_f32(0x1.FFFDC6p-2f);
31 const float32x4_t vc3 = vmovq_n_f32(0x1.555A80p-3f);
32 const float32x4_t vc4 = vmovq_n_f32(0x1.573A1Ap-5f);
33 const float32x4_t vc5 = vmovq_n_f32(0x1.0F9F9Cp-7f);
34 // The smallest x for which expf(x) is normalized.
35 const float32x4_t vdenorm_cutoff = vmovq_n_f32(-0x1.5D589Ep6f);
36
37 for (; n != 0; n -= 4 * sizeof(float)) {
38 const float32x4_t vx = vld1q_f32(input); input += 4;
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**22, i.e. |x| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because inputs outside
45 // of [-87.336540, 0.0] underflow expf(x) anyway. We fixup the result for such inputs at the very end of the
46 // algorithm.
47 float32x4_t vn = vfmaq_f32(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 // -87.33642 <= x <= 0.0, and -126 <= n <= 0 accordingly.
51 const float32x4_t vs = vreinterpretq_f32_s32(vshlq_n_s32(vreinterpretq_s32_f32(vn), 23));
52
53 // Subtract the large number back to get final n := round(x / log(2)) as a floating-point number.
54 vn = vsubq_f32(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 float32x4_t vt = vfmaq_f32(vx, vn, vminus_ln2_hi);
59 vt = vfmaq_f32(vt, vn, vminus_ln2_lo);
60
61 // Compute degree-5 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2]:
62 // P(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) = 1 + t * p
63 float32x4_t vp = vfmaq_f32(vc4, vc5, vt);
64 vp = vfmaq_f32(vc3, vp, vt);
65 vp = vfmaq_f32(vc2, vp, vt);
66 vp = vfmaq_f32(vc1, vp, vt);
67
68 // Reconstruct the exp(x) value:
69 // exp(x) = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
70 // = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
71 // = s + (t * s) * p
72 vt = vmulq_f32(vt, vs);
73 float32x4_t vf = vfmaq_f32(vs, vp, vt);
74
75 // For inputs below denormal cutoff, replace output with +0.0f.
76 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
77 vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcltq_f32(vx, vdenorm_cutoff)));
78 vst1q_f32(output, vf); output += 4;
79 }
80 }
81