1 /*
2 * Double-precision vector e^x function.
3 *
4 * Copyright (c) 2019, Arm Limited.
5 * SPDX-License-Identifier: MIT
6 */
7
8 #include "mathlib.h"
9 #include "v_math.h"
10 #if V_SUPPORTED
11 #include "v_exp.h"
12
13 #if V_EXP_TABLE_BITS == 7
14 /* maxerr: 1.88 +0.5 ulp
15 rel error: 1.4337*2^-53
16 abs error: 1.4299*2^-53 in [ -ln2/256, ln2/256 ]. */
17 #define C1 v_f64 (0x1.ffffffffffd43p-2)
18 #define C2 v_f64 (0x1.55555c75adbb2p-3)
19 #define C3 v_f64 (0x1.55555da646206p-5)
20 #define InvLn2 v_f64 (0x1.71547652b82fep7) /* N/ln2. */
21 #define Ln2hi v_f64 (0x1.62e42fefa39efp-8) /* ln2/N. */
22 #define Ln2lo v_f64 (0x1.abc9e3b39803f3p-63)
23 #elif V_EXP_TABLE_BITS == 8
24 /* maxerr: 0.54 +0.5 ulp
25 rel error: 1.4318*2^-58
26 abs error: 1.4299*2^-58 in [ -ln2/512, ln2/512 ]. */
27 #define C1 v_f64 (0x1.fffffffffffd4p-2)
28 #define C2 v_f64 (0x1.5555571d6b68cp-3)
29 #define C3 v_f64 (0x1.5555576a59599p-5)
30 #define InvLn2 v_f64 (0x1.71547652b82fep8)
31 #define Ln2hi v_f64 (0x1.62e42fefa39efp-9)
32 #define Ln2lo v_f64 (0x1.abc9e3b39803f3p-64)
33 #endif
34
35 #define N (1 << V_EXP_TABLE_BITS)
36 #define Tab __v_exp_data
37 #define IndexMask v_u64 (N - 1)
38 #define Shift v_f64 (0x1.8p+52)
39 #define Thres v_f64 (704.0)
40
41 VPCS_ATTR
42 static v_f64_t
specialcase(v_f64_t s,v_f64_t y,v_f64_t n)43 specialcase (v_f64_t s, v_f64_t y, v_f64_t n)
44 {
45 v_f64_t absn = v_abs_f64 (n);
46
47 /* 2^(n/N) may overflow, break it up into s1*s2. */
48 v_u64_t b = v_cond_u64 (n <= v_f64 (0.0)) & v_u64 (0x6000000000000000);
49 v_f64_t s1 = v_as_f64_u64 (v_u64 (0x7000000000000000) - b);
50 v_f64_t s2 = v_as_f64_u64 (v_as_u64_f64 (s) - v_u64 (0x3010000000000000) + b);
51 v_u64_t cmp = v_cond_u64 (absn > v_f64 (1280.0 * N));
52 v_f64_t r1 = s1 * s1;
53 v_f64_t r0 = v_fma_f64 (y, s2, s2) * s1;
54 return v_as_f64_u64 ((cmp & v_as_u64_f64 (r1)) | (~cmp & v_as_u64_f64 (r0)));
55 }
56
57 VPCS_ATTR
58 v_f64_t
V_NAME(exp)59 V_NAME(exp) (v_f64_t x)
60 {
61 v_f64_t n, r, r2, s, y, z;
62 v_u64_t cmp, u, e, i;
63
64 cmp = v_cond_u64 (v_abs_f64 (x) > Thres);
65
66 /* n = round(x/(ln2/N)). */
67 z = v_fma_f64 (x, InvLn2, Shift);
68 u = v_as_u64_f64 (z);
69 n = z - Shift;
70
71 /* r = x - n*ln2/N. */
72 r = x;
73 r = v_fma_f64 (-Ln2hi, n, r);
74 r = v_fma_f64 (-Ln2lo, n, r);
75
76 e = u << (52 - V_EXP_TABLE_BITS);
77 i = u & IndexMask;
78
79 /* y = exp(r) - 1 ~= r + C1 r^2 + C2 r^3 + C3 r^4. */
80 r2 = r * r;
81 y = v_fma_f64 (C2, r, C1);
82 y = v_fma_f64 (C3, r2, y);
83 y = v_fma_f64 (y, r2, r);
84
85 /* s = 2^(n/N). */
86 u = v_lookup_u64 (Tab, i);
87 s = v_as_f64_u64 (u + e);
88
89 if (unlikely (v_any_u64 (cmp)))
90 return specialcase (s, y, n);
91 return v_fma_f64 (y, s, s);
92 }
93 VPCS_ALIAS
94 #endif
95