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.h>
11 #include <xnnpack/math-stubs.h>
12
13
xnn_math_f32_expm1minus__scalar_rr2_p5(size_t n,const float * input,float * output)14 void xnn_math_f32_expm1minus__scalar_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 float vmagic_bias = 0x1.8000FEp23f;
23 const float vlog2e = 0x1.715476p+0f;
24 // The largest x for which expm1f(x) is saturated at -1.0f.
25 const float vsat_cutoff = -0x1.154246p+4f;
26 // Last 5 bits are zeroes
27 const float vminus_ln2_hi = -0x1.62E440p-1f;
28 const float vminus_ln2_lo = 0x1.0105C6p-21f;
29 // Coefficient of polynomial approximation
30 // exp(t) - 1 ~ t * (1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
31 // on [-log(2)/2, log(2)/2]
32 const float vc5 = 0x1.113780p-7f;
33 const float vc4 = 0x1.5704DCp-5f;
34 const float vc3 = 0x1.555634p-3f;
35 const float vc2 = 0x1.FFFE70p-2f;
36 const float vone = 1.0f;
37
38 for (; n != 0; n -= sizeof(float)) {
39 float vx = *input++;
40
41 // Compute reduced argument n := round(x / log(2)).
42 // We do it by adding a large number (magic bias), which cause rounding of the result to integer, then subtracing
43 // the large number back. The trick with adding large number is valid only within certain bounds
44 // (|x / log(2)| <= 2**22, i.e. |x| <= 0x1.62E43p+21 = 2907270.0), but that is acceptable, because inputs x are
45 // restricted to [-17.328680, 0].
46 // Note that addition-subtraction of the large number doesn't cause overflow for inputs in this range.
47 float vn = vx * vlog2e + vmagic_bias;
48
49 // Create a floating-point number s (scale) such that s == 2**n for valid inputs, i.e.
50 // -17.328680 <= x <= 0.0, and -25 <= n <= 0 accordingly.
51 float vs = uint32_as_float(float_as_uint32(vn) << 23);
52
53 // Subtract the large number back to get final n := round(x / log(2)).
54 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 float vt = vn * vminus_ln2_hi + vx;
59 vt = vn * vminus_ln2_lo + vt;
60
61 // The function saturates at -1 for large negative inputs: expm1f(x) == -1.0f for x <= sat_cutoff ~= -17.328680.
62 // To guarantee this behaviour, we zero out s (scale) and t (reduced argument) for x <= sat_cutoff.
63 if XNN_UNPREDICTABLE(vx <= vsat_cutoff) {
64 vs = 0.0f;
65 vt = 0.0f;
66 }
67
68 // Compute degree-5 polynomial approximation for exp(t) - 1 on [-log(2)/2, log(2)/2].
69 // P(t) = t * (1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))
70 // = t + t * (t * (c2 + t * (c3 + t * (c4 + t * c5)))) = t + t * p
71 float vp = vc5 * vt + vc4;
72 vp = vp * vt + vc3;
73 vp = vp * vt + vc2;
74 vp *= vt;
75
76 // Reconstruct the exp(x) - 1 value:
77 // exp(x) - 1 = s * (1 + t * (1 + t * (c2 + t * (c3 + t * (c4 + t * c5))))) - 1
78 // = (s - 1) + s * (t + t * p)
79 // = ((t * s) + (t * s) * p) + (s - 1)
80 vt *= vs;
81 const float vsm1 = vs - vone;
82 vp = vp * vt + vt;
83 const float vf = vp + vsm1;
84
85 *output++ = vf;
86 }
87 }
88