1 /*
2 * Double-precision SVE 2^x function.
3 *
4 * Copyright (c) 2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "sv_math.h"
9 #include "poly_sve_f64.h"
10 #include "pl_sig.h"
11 #include "pl_test.h"
12
13 #define N (1 << V_EXP_TABLE_BITS)
14
15 #define BigBound 1022
16 #define UOFlowBound 1280
17
18 static const struct data
19 {
20 double poly[4];
21 double shift, big_bound, uoflow_bound;
22 } data = {
23 /* Coefficients are computed using Remez algorithm with
24 minimisation of the absolute error. */
25 .poly = { 0x1.62e42fefa3686p-1, 0x1.ebfbdff82c241p-3, 0x1.c6b09b16de99ap-5,
26 0x1.3b2abf5571ad8p-7 },
27 .shift = 0x1.8p52 / N,
28 .uoflow_bound = UOFlowBound,
29 .big_bound = BigBound,
30 };
31
32 #define SpecialOffset 0x6000000000000000 /* 0x1p513. */
33 /* SpecialBias1 + SpecialBias1 = asuint(1.0). */
34 #define SpecialBias1 0x7000000000000000 /* 0x1p769. */
35 #define SpecialBias2 0x3010000000000000 /* 0x1p-254. */
36
37 /* Update of both special and non-special cases, if any special case is
38 detected. */
39 static inline svfloat64_t
special_case(svbool_t pg,svfloat64_t s,svfloat64_t y,svfloat64_t n,const struct data * d)40 special_case (svbool_t pg, svfloat64_t s, svfloat64_t y, svfloat64_t n,
41 const struct data *d)
42 {
43 /* s=2^n may overflow, break it up into s=s1*s2,
44 such that exp = s + s*y can be computed as s1*(s2+s2*y)
45 and s1*s1 overflows only if n>0. */
46
47 /* If n<=0 then set b to 0x6, 0 otherwise. */
48 svbool_t p_sign = svcmple (pg, n, 0.0); /* n <= 0. */
49 svuint64_t b = svdup_u64_z (p_sign, SpecialOffset);
50
51 /* Set s1 to generate overflow depending on sign of exponent n. */
52 svfloat64_t s1 = svreinterpret_f64 (svsubr_x (pg, b, SpecialBias1));
53 /* Offset s to avoid overflow in final result if n is below threshold. */
54 svfloat64_t s2 = svreinterpret_f64 (
55 svadd_x (pg, svsub_x (pg, svreinterpret_u64 (s), SpecialBias2), b));
56
57 /* |n| > 1280 => 2^(n) overflows. */
58 svbool_t p_cmp = svacgt (pg, n, d->uoflow_bound);
59
60 svfloat64_t r1 = svmul_x (pg, s1, s1);
61 svfloat64_t r2 = svmla_x (pg, s2, s2, y);
62 svfloat64_t r0 = svmul_x (pg, r2, s1);
63
64 return svsel (p_cmp, r1, r0);
65 }
66
67 /* Fast vector implementation of exp2.
68 Maximum measured error is 1.65 ulp.
69 _ZGVsMxv_exp2(-0x1.4c264ab5b559bp-6) got 0x1.f8db0d4df721fp-1
70 want 0x1.f8db0d4df721dp-1. */
SV_NAME_D1(exp2)71 svfloat64_t SV_NAME_D1 (exp2) (svfloat64_t x, svbool_t pg)
72 {
73 const struct data *d = ptr_barrier (&data);
74 svbool_t no_big_scale = svacle (pg, x, d->big_bound);
75 svbool_t special = svnot_z (pg, no_big_scale);
76
77 /* Reduce x to k/N + r, where k is integer and r in [-1/2N, 1/2N]. */
78 svfloat64_t shift = sv_f64 (d->shift);
79 svfloat64_t kd = svadd_x (pg, x, shift);
80 svuint64_t ki = svreinterpret_u64 (kd);
81 /* kd = k/N. */
82 kd = svsub_x (pg, kd, shift);
83 svfloat64_t r = svsub_x (pg, x, kd);
84
85 /* scale ~= 2^(k/N). */
86 svuint64_t idx = svand_x (pg, ki, N - 1);
87 svuint64_t sbits = svld1_gather_index (pg, __v_exp_data, idx);
88 /* This is only a valid scale when -1023*N < k < 1024*N. */
89 svuint64_t top = svlsl_x (pg, ki, 52 - V_EXP_TABLE_BITS);
90 svfloat64_t scale = svreinterpret_f64 (svadd_x (pg, sbits, top));
91
92 /* Approximate exp2(r) using polynomial. */
93 svfloat64_t r2 = svmul_x (pg, r, r);
94 svfloat64_t p = sv_pairwise_poly_3_f64_x (pg, r, r2, d->poly);
95 svfloat64_t y = svmul_x (pg, r, p);
96
97 /* Assemble exp2(x) = exp2(r) * scale. */
98 if (unlikely (svptest_any (pg, special)))
99 return special_case (pg, scale, y, kd, d);
100 return svmla_x (pg, scale, scale, y);
101 }
102
103 PL_SIG (SV, D, 1, exp2, -9.9, 9.9)
104 PL_TEST_ULP (SV_NAME_D1 (exp2), 1.15)
105 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (exp2), 0, BigBound, 1000)
106 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (exp2), BigBound, UOFlowBound, 100000)
107 PL_TEST_SYM_INTERVAL (SV_NAME_D1 (exp2), UOFlowBound, inf, 1000)
108