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 <emmintrin.h>
11
12 #include <xnnpack/math-stubs.h>
13
14
xnn_math_f32_exp__sse2_rr2_p5(size_t n,const float * input,float * output)15 void xnn_math_f32_exp__sse2_rr2_p5(
16 size_t n,
17 const float* input,
18 float* output)
19 {
20 assert(n % (4 * sizeof(float)) == 0);
21
22 const __m128 vmagic_bias = _mm_set1_ps(0x1.800000p+23f);
23 // The smallest x for which expf(x) is non-zero.
24 const __m128 vzero_cutoff = _mm_set1_ps(-0x1.9FE368p+6f);
25 // The largest x for which expf(x) is finite.
26 const __m128 vinf_cutoff = _mm_set1_ps(0x1.62E42Ep+6f);
27 const __m128 vlog2e = _mm_set1_ps(0x1.715476p+0f);
28 // Last 8 bits are zeroes
29 const __m128 vminus_ln2_hi = _mm_set1_ps(-0x1.62E400p-1f);
30 const __m128 vminus_ln2_lo = _mm_set1_ps(-0x1.7F7D1Cp-20f);
31 const __m128 vplus_inf = _mm_set1_ps(INFINITY);
32
33 const __m128 vc1 = _mm_set1_ps(0x1.FFFFF6p-1f);
34 const __m128 vc2 = _mm_set1_ps(0x1.FFFDC6p-2f);
35 const __m128 vc3 = _mm_set1_ps(0x1.555A80p-3f);
36 const __m128 vc4 = _mm_set1_ps(0x1.573A1Ap-5f);
37 const __m128 vc5 = _mm_set1_ps(0x1.0F9F9Cp-7f);
38
39 const __m128i vmin_exponent = _mm_set1_epi32(0xC1000000);
40 const __m128i vmax_exponent = _mm_set1_epi32(0x3F800000);
41 const __m128i vdefault_exponent = vmax_exponent;
42
43 for (; n != 0; n -= 4 * sizeof(float)) {
44 const __m128 vx = _mm_loadu_ps(input);
45
46 // Compute reduced argument n := round(x / log(2)).
47 // We do it by adding a large number (magic bias) to the product x * (1/log(2)), which cause rounding of the result
48 // to an integer, then subtracing the large number back. The trick with adding large number is valid only within
49 // certain bounds (|x| <= 2**22), but thats ok, because inputs outside of [-103.97207, 88.72283] underflow or
50 // overflow expf(x) anyway. We fixup the result for such inputs at the very end of the algorithm.
51 __m128 vn = _mm_add_ps(_mm_mul_ps(vx, vlog2e), vmagic_bias);
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 __m128i veo = _mm_slli_epi32(_mm_castps_si128(vn), 23);
61 __m128i ven = _mm_max_epi16(veo, vmin_exponent);
62 ven = _mm_min_epi16(ven, vmax_exponent);
63 veo = _mm_sub_epi32(veo, ven);
64 const __m128 vsn = _mm_castsi128_ps(_mm_add_epi32(ven, vdefault_exponent));
65 const __m128 vso = _mm_castsi128_ps(_mm_add_epi32(veo, vdefault_exponent));
66
67 // Subtract the large number back to get final n := round(x / log(2)).
68 vn = _mm_sub_ps(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 __m128 vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_hi), vx);
73 vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_lo), vt);
74
75 // Compute degree-5 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2].
76 __m128 vp = _mm_add_ps(_mm_mul_ps(vc5, vt), vc4);
77 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc3);
78 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc2);
79 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc1);
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 = _mm_mul_ps(vt, vso);
86 __m128 vf = _mm_mul_ps(vsn, _mm_add_ps(_mm_mul_ps(vt, vp), vso));
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 = _mm_andnot_ps(_mm_cmplt_ps(vx, vzero_cutoff), vf);
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 const __m128 vm = _mm_cmpgt_ps(vx, vinf_cutoff);
94 vf = _mm_or_ps(_mm_and_ps(vplus_inf, vm), _mm_andnot_ps(vm, vf));
95 _mm_storeu_ps(output, vf);
96
97 input += 4;
98 output += 4;
99 }
100 }
101