1 // Copyright 2020 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_expm1minus__scalar_rr2_p6(size_t n,const float * input,float * output)15 void xnn_math_f32_expm1minus__scalar_rr2_p6(
16 size_t n,
17 const float* input,
18 float* output)
19 {
20 assert(n % (4 * 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 // The largest x for which expm1f(x) is saturated at -1.0f.
26 const float vsat_cutoff = -0x1.154246p+4f;
27 // Last 5 bits are zeroes
28 const float vminus_ln2_hi = -0x1.62E440p-1f;
29 const float vminus_ln2_lo = 0x1.0105C6p-21f;
30 // Coefficient of polynomial approximation
31 // exp(t) - 1 ~ t * (1 + t * (c2 + t * (c3 + t * (c4 + t * (c5 + t * c6)))))
32 // on [-log(2)/2, log(2)/2]
33 const float vc6 = 0x1.6b7338p-10f;
34 const float vc5 = 0x1.12278Ep-7f;
35 const float vc4 = 0x1.555716p-5f;
36 const float vc3 = 0x1.5554B0p-3f;
37 const float vc2 = 0x1.FFFFFEp-2f;
38 const float vone = 1.0f;
39
40 for (; n != 0; n -= sizeof(float)) {
41 float vx = *input++;
42
43 // Compute reduced argument n := round(x / log(2)).
44 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
45 // the large number back. The trick with adding large number is valid only within certain bounds
46 // (|x / log(2)| <= 2**22, i.e. |x| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because inputs x are
47 // restricted to [-17.328680, 0].
48 // Note that addition-subtraction of the large number doesn't cause overflow for inputs in this range.
49 float vn = vx * vlog2e + vmagic_bias;
50
51 // Create a floating-point number s (scale) such that s == 2**n for valid inputs, i.e.
52 // -17.328680 <= x <= 0.0, and -25 <= n <= 0 accordingly.
53 float vs = fp32_from_bits(fp32_to_bits(vn) << 23);
54
55 // Subtract the large number back to get final n := round(x / log(2)).
56 vn -= vmagic_bias;
57
58 // Compute reduced argument t := x - n * log(2).
59 // Use Cody-Waite range reduction method (note two constants to represent log(2)) to improve accuracy.
60 float vt = vn * vminus_ln2_hi + vx;
61 vt = vn * vminus_ln2_lo + vt;
62
63 // The function saturates at -1 for large negative inputs: expm1f(x) == -1.0f for x <= sat_cutoff ~= -17.328680.
64 // To guarantee this behaviour, we zero out s (scale) and t (reduced argument) for x <= sat_cutoff.
65 if XNN_UNPREDICTABLE(vx <= vsat_cutoff) {
66 vs = 0.0f;
67 vt = 0.0f;
68 }
69
70 // Compute degree-6 polynomial approximation for exp(t) - 1 on [-log(2)/2, log(2)/2].
71 // P(t) = t * (1 + t * (c2 + t * (c3 + t * (c4 + t * (c5 + t * c6)))))
72 // = t + t * (t * (c2 + t * (c3 + t * (c4 + t * (c5 + t * c6))))) = t + t * p
73 float vp = vc6 * vt + vc5;
74 vp = vp * vt + vc4;
75 vp = vp * vt + vc3;
76 vp = vp * vt + vc2;
77 vp *= vt;
78
79 // Reconstruct the exp(x) - 1 value:
80 // exp(x) - 1 = s * (1 + t * (1 + t * (c2 + t * (c3 + t * (c4 + t * (c5 + t * c6)))))) - 1
81 // = (s - 1) + s * (t + t * p)
82 // = ((t * s) + (t * s) * p) + (s - 1)
83 vt *= vs;
84 const float vsm1 = vs - vone;
85 vp = vp * vt + vt;
86 const float vf = vp + vsm1;
87
88 *output++ = vf;
89 }
90 }
91