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 <emmintrin.h>
10
11 #include <xnnpack/math-stubs.h>
12
13
xnn_math_f32_expminus__sse2_rr2_p5(size_t n,const float * input,float * output)14 void xnn_math_f32_expminus__sse2_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 __m128 vmagic_bias = _mm_set1_ps(0x1.8000FEp23f);
23 const __m128 vlog2e = _mm_set1_ps(0x1.715476p+0f);
24 // Last 7 bits are zeroes
25 const __m128 vminus_ln2_hi = _mm_set1_ps(-0x1.62E400p-1f);
26 const __m128 vminus_ln2_lo = _mm_set1_ps(-0x1.7F7D1Cp-20f);
27 // Coefficient of polynomial approximation
28 // exp(t) ~ 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
29 // on [-log(2)/2, log(2)/2]
30 const __m128 vc1 = _mm_set1_ps(0x1.FFFFF6p-1f);
31 const __m128 vc2 = _mm_set1_ps(0x1.FFFDC6p-2f);
32 const __m128 vc3 = _mm_set1_ps(0x1.555A80p-3f);
33 const __m128 vc4 = _mm_set1_ps(0x1.573A1Ap-5f);
34 const __m128 vc5 = _mm_set1_ps(0x1.0F9F9Cp-7f);
35 // The smallest x for which expf(x) is normalized.
36 const __m128 vdenorm_cutoff = _mm_set1_ps(-0x1.5D589Ep6f);
37
38 for (; n != 0; n -= 4 * sizeof(float)) {
39 const __m128 vx = _mm_loadu_ps(input);
40
41 // Compute reduced argument n := round(x / log(2)).
42 // We do it by adding a large number (magic bias) to the product x * (1/log(2)), which cause rounding of the result
43 // to an integer, then subtracing the large number back. The trick with adding large number is valid only within
44 // certain bounds (|x / log(2)| <= 2**22, i.e. |x| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because
45 // inputs outside of [-87.336540, 0.0] underflow expf(x) anyway. We fixup the result for such inputs at the very
46 // end of the algorithm.
47 __m128 vn = _mm_add_ps(_mm_mul_ps(vx, vlog2e), vmagic_bias);
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 __m128 vs = _mm_castsi128_ps(_mm_slli_epi32(_mm_castps_si128(vn), 23));
52
53 // Subtract the large number back to get final n := round(x / log(2)) as a floating-point number.
54 vn = _mm_sub_ps(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 __m128 vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_hi), vx);
59 vt = _mm_add_ps(_mm_mul_ps(vn, vminus_ln2_lo), vt);
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 __m128 vp = _mm_add_ps(_mm_mul_ps(vc5, vt), vc4);
64 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc3);
65 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc2);
66 vp = _mm_add_ps(_mm_mul_ps(vp, vt), vc1);
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 = _mm_mul_ps(vt, vs);
73 __m128 vf = _mm_add_ps(_mm_mul_ps(vt, vp), vs);
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 = _mm_andnot_ps(_mm_cmplt_ps(vx, vdenorm_cutoff), vf);
78 _mm_storeu_ps(output, vf);
79
80 input += 4;
81 output += 4;
82 }
83 }
84