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_p5(size_t n,const float * input,float * output)15 void xnn_math_f32_expm1minus__scalar_rr2_p5(
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))))
32 // on [-log(2)/2, log(2)/2]
33 const float vc5 = 0x1.113780p-7f;
34 const float vc4 = 0x1.5704DCp-5f;
35 const float vc3 = 0x1.555634p-3f;
36 const float vc2 = 0x1.FFFE70p-2f;
37 const float vone = 1.0f;
38
39 for (; n != 0; n -= sizeof(float)) {
40 float vx = *input++;
41
42 // Compute reduced argument n := round(x / log(2)).
43 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
44 // the large number back. The trick with adding large number is valid only within certain bounds
45 // (|x / log(2)| <= 2**22, i.e. |x| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because inputs x are
46 // restricted to [-17.328680, 0].
47 // Note that addition-subtraction of the large number doesn't cause overflow for inputs in this range.
48 float vn = vx * vlog2e + vmagic_bias;
49
50 // Create a floating-point number s (scale) such that s == 2**n for valid inputs, i.e.
51 // -17.328680 <= x <= 0.0, and -25 <= n <= 0 accordingly.
52 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)).
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 // The function saturates at -1 for large negative inputs: expm1f(x) == -1.0f for x <= sat_cutoff ~= -17.328680.
63 // To guarantee this behaviour, we zero out s (scale) and t (reduced argument) for x <= sat_cutoff.
64 if XNN_UNPREDICTABLE(vx <= vsat_cutoff) {
65 vs = 0.0f;
66 vt = 0.0f;
67 }
68
69 // Compute degree-5 polynomial approximation for exp(t) - 1 on [-log(2)/2, log(2)/2].
70 // P(t) = t * (1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
71 // = t + t * (t * (c2 + t * (c3 + t * (c4 + t * c5)))) = t + t * p
72 float vp = vc5 * vt + vc4;
73 vp = vp * vt + vc3;
74 vp = vp * vt + vc2;
75 vp *= vt;
76
77 // Reconstruct the exp(x) - 1 value:
78 // exp(x) - 1 = s * (1 + t * (1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))) - 1
79 // = (s - 1) + s * (t + t * p)
80 // = ((t * s) + (t * s) * p) + (s - 1)
81 vt *= vs;
82 const float vsm1 = vs - vone;
83 vp = vp * vt + vt;
84 const float vf = vp + vsm1;
85
86 *output++ = vf;
87 }
88 }
89