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
41VPCS_ATTR
42static v_f64_t
43specialcase (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
57VPCS_ATTR
58v_f64_t
59V_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}
93VPCS_ALIAS
94#endif
95