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 <xnnpack/common.h>
10 #include <xnnpack/math-stubs.h>
11
12 #include <fp16/bitcasts.h>
13
14
xnn_math_f32_expminus__scalar_rr2_p5(size_t n,const float * input,float * output)15 void xnn_math_f32_expminus__scalar_rr2_p5(
16 size_t n,
17 const float* input,
18 float* output)
19 {
20 assert(n % sizeof(float) == 0);
21
22 // Large number such that ulp(magic bias) == 1 and magic bias === 127 mod 2**22.
23 const float vmagic_bias = 0x1.8000FEp23f;
24 const float vlog2e = 0x1.715476p+0f;
25 // Last 7 bits are zeroes
26 const float vminus_ln2_hi = -0x1.62E400p-1f;
27 const float vminus_ln2_lo = -0x1.7F7D1Cp-20f;
28 // Coefficient of polynomial approximation
29 // exp(t) ~ 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
30 // on [-log(2)/2, log(2)/2]
31 const float vc5 = 0x1.0F9F9Cp-7f;
32 const float vc4 = 0x1.573A1Ap-5f;
33 const float vc3 = 0x1.555A80p-3f;
34 const float vc2 = 0x1.FFFDC6p-2f;
35 const float vc1 = 0x1.FFFFF6p-1f;
36 // The smallest x for which expf(x) is normalized.
37 const float vdenorm_cutoff = -0x1.5D589Ep6f;
38
39 for (; n != 0; n -= sizeof(float)) {
40 const float vx = *input++;
41
42 // Compute reduced argument n := round(x / log(2)).
43 // We do it by adding a large number (magic bias) to the product x * (1/log(2)), which cause rounding of the result
44 // to an integer, then subtracing the large number back. The trick with adding large number is valid only within
45 // certain bounds (|x / log(2)| <= 2**22, i.e. |x| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because
46 // inputs outside of [-87.336540, 0.0] underflow expf(x) anyway. We fixup the result for such inputs at the very
47 // end of the algorithm.
48 float vn = vx * vlog2e + vmagic_bias;
49
50 // Create a floating-point number s (scale) such that s == 2**n for inputs which don't cause underflow, i.e.
51 // -87.33642 <= x <= 0.0, and -126 <= n <= 0 accordingly.
52 const float vs = fp32_from_bits(fp32_to_bits(vn) << 23);
53
54 // Subtract the large number back to get final n := round(x / log(2)) as a floating-point number.
55 vn -= vmagic_bias;
56
57 // Compute reduced argument t := x - n * log(2).
58 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
59 float vt = vn * vminus_ln2_hi + vx;
60 vt = vn * vminus_ln2_lo + vt;
61
62 // Compute degree-5 polynomial approximation for exp(t) on [-log(2)/2, log(2)/2]:
63 // P(t) = 1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))) = 1 + t * p
64 float vp = vc5 * vt + vc4;
65 vp = vp * vt + vc3;
66 vp = vp * vt + vc2;
67 vp = vp * vt + vc1;
68
69 // Reconstruct the exp(x) value:
70 // exp(x) = s * (1 + t * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5)))))
71 // = s + (t * s) * (c1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
72 // = s + (t * s) * p
73 vt *= vs;
74 float vf = vt * vp + vs;
75
76 // For inputs below denormal cutoff, replace output with +0.0f.
77 // Note that for NaN inputs, comparison result is false, and outputs are left unchanged.
78 if XNN_UNPREDICTABLE(vx < vdenorm_cutoff) {
79 vf = 0.0f;
80 }
81
82 *output++ = vf;
83 }
84 }
85