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 <math.h>
8 #include <stddef.h>
9
10 #include <arm_neon.h>
11
12 #include <xnnpack/math-stubs.h>
13
14
xnn_math_f32_exp__neonfma_rr2_p5(size_t n,const float * input,float * output)15 void xnn_math_f32_exp__neonfma_rr2_p5(
16 size_t n,
17 const float* input,
18 float* output)
19 {
20 assert(n % (4 * sizeof(float)) == 0);
21
22 const float32x4_t vmagic_bias = vmovq_n_f32(0x1.800000p+23f);
23 // The smallest x for which expf(x) is non-zero.
24 const float32x4_t vzero_cutoff = vmovq_n_f32(-0x1.9FE368p+6f);
25 // The largest x for which expf(x) is finite.
26 const float32x4_t vinf_cutoff = vmovq_n_f32(0x1.62E42Ep+6f);
27 const float32x4_t vlog2e = vmovq_n_f32(0x1.715476p+0f);
28 const float32x4_t vminus_ln2_hi = vmovq_n_f32(-0x1.62E43p-1f);
29 const float32x4_t vminus_ln2_lo = vmovq_n_f32(0x1.05C61p-29f);
30 const float32x4_t vplus_inf = vmovq_n_f32(INFINITY);
31
32 const float32x4_t vc1 = vmovq_n_f32(0x1.FFFFF6p-1f);
33 const float32x4_t vc2 = vmovq_n_f32(0x1.FFFDC6p-2f);
34 const float32x4_t vc3 = vmovq_n_f32(0x1.555A80p-3f);
35 const float32x4_t vc4 = vmovq_n_f32(0x1.573A1Ap-5f);
36 const float32x4_t vc5 = vmovq_n_f32(0x1.0F9F9Cp-7f);
37
38 const int32x4_t vmin_exponent = vmovq_n_s32(INT32_C(0xC1000000));
39 const int32x4_t vmax_exponent = vmovq_n_s32(INT32_C(0x3F800000));
40 const int32x4_t vdefault_exponent = vmax_exponent;
41
42 for (; n != 0; n -= 4 * sizeof(float)) {
43 const float32x4_t vx = vld1q_f32(input); input += 4;
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 result to an integer, then subtracing the
47 // large number back. The first addition is combined with multiplication by log2e into a single FMA instruction.
48 // The trick with adding large number is valid only within certain bounds (|x| <= 2**22), but thats ok, because
49 // inputs outside of [-103.97207, 88.72283] underflow or overflow expf(x) anyway. We fixup the result for such
50 // inputs at the very end of the algorithm.
51 float32x4_t vn = vfmaq_f32(vmagic_bias, vx, vlog2e);
52
53 // Create two floating-point numbers, sn (scale, normal) and so (scale, overflow) such that sn * so == 2**n
54 // for inputs which don't cause overflow, i.e. -103.97207 <= x <= 88.72283, and -150 <= n <= 128 accordingly.
55 // We need to use two numbers rather than one because a normalized single-precision exponent must be in [-127, 126]
56 // range, which is insufficient to cover [-150, 128] range of n.
57 // - When n is within [-127, 126], sn == 2**n and so == 1.0.
58 // - When n < -127, sn == 2**(-127) and so == 2**(n + 127).
59 // - When n > 126, sn == 2**126 and so == 2**(n - 126).
60 int32x4_t veo = vshlq_n_s32(vreinterpretq_s32_f32(vn), 23);
61 int32x4_t ven = vmaxq_s32(veo, vmin_exponent);
62 ven = vminq_s32(ven, vmax_exponent);
63 veo = vsubq_s32(veo, ven);
64 const float32x4_t vsn = vreinterpretq_f32_s32(vaddq_s32(ven, vdefault_exponent));
65 const float32x4_t vso = vreinterpretq_f32_s32(vaddq_s32(veo, vdefault_exponent));
66
67 // Subtract the large number back to get final n := round(x / log(2)).
68 vn = vsubq_f32(vn, vmagic_bias);
69
70 // Compute reduced argument t := x - n * log(2).
71 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
72 float32x4_t vt = vfmaq_f32(vx, vn, vminus_ln2_hi);
73 vt = vfmaq_f32(vt, vn, vminus_ln2_lo);
74
75 // Compute degree-5 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2].
76 float32x4_t vp = vfmaq_f32(vc4, vc5, vt);
77 vp = vfmaq_f32(vc3, vp, vt);
78 vp = vfmaq_f32(vc2, vp, vt);
79 vp = vfmaq_f32(vc1, vp, vt);
80
81 // Reconstruct the final f value:
82 // f = so * sn * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
83 // = sn * (so + (t * so) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))))
84 // = sn * (so + (t * so) * p)
85 vt = vmulq_f32(vt, vso);
86 float32x4_t vf = vmulq_f32(vsn, vfmaq_f32(vso, vt, vp));
87
88 // For inputs below zero cutoff, replace output with +0.0f.
89 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
90 vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcltq_f32(vx, vzero_cutoff)));
91 // For inputs above inf cutoff, replace output with +inf.
92 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
93 vf = vbslq_f32(vcgtq_f32(vx, vinf_cutoff), vplus_inf, vf);
94 vst1q_f32(output, vf); output += 4;
95 }
96 }
97