1/*
2 * Double-precision SVE log(1+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
13static const struct data
14{
15  double poly[19];
16  double ln2_hi, ln2_lo;
17  uint64_t hfrt2_top, onemhfrt2_top, inf, mone;
18} data = {
19  /* Generated using Remez in [ sqrt(2)/2 - 1, sqrt(2) - 1]. Order 20
20     polynomial, however first 2 coefficients are 0 and 1 so are not stored.  */
21  .poly = { -0x1.ffffffffffffbp-2, 0x1.55555555551a9p-2, -0x1.00000000008e3p-2,
22	    0x1.9999999a32797p-3, -0x1.555555552fecfp-3, 0x1.249248e071e5ap-3,
23	    -0x1.ffffff8bf8482p-4, 0x1.c71c8f07da57ap-4, -0x1.9999ca4ccb617p-4,
24	    0x1.7459ad2e1dfa3p-4, -0x1.554d2680a3ff2p-4, 0x1.3b4c54d487455p-4,
25	    -0x1.2548a9ffe80e6p-4, 0x1.0f389a24b2e07p-4, -0x1.eee4db15db335p-5,
26	    0x1.e95b494d4a5ddp-5, -0x1.15fdf07cb7c73p-4, 0x1.0310b70800fcfp-4,
27	    -0x1.cfa7385bdb37ep-6, },
28  .ln2_hi = 0x1.62e42fefa3800p-1,
29  .ln2_lo = 0x1.ef35793c76730p-45,
30  /* top32(asuint64(sqrt(2)/2)) << 32.  */
31  .hfrt2_top = 0x3fe6a09e00000000,
32  /* (top32(asuint64(1)) - top32(asuint64(sqrt(2)/2))) << 32.  */
33  .onemhfrt2_top = 0x00095f6200000000,
34  .inf = 0x7ff0000000000000,
35  .mone = 0xbff0000000000000,
36};
37
38#define AbsMask 0x7fffffffffffffff
39#define BottomMask 0xffffffff
40
41static svfloat64_t NOINLINE
42special_case (svbool_t special, svfloat64_t x, svfloat64_t y)
43{
44  return sv_call_f64 (log1p, x, y, special);
45}
46
47/* Vector approximation for log1p using polynomial on reduced interval. Maximum
48   observed error is 2.46 ULP:
49   _ZGVsMxv_log1p(0x1.654a1307242a4p+11) got 0x1.fd5565fb590f4p+2
50					want 0x1.fd5565fb590f6p+2.  */
51svfloat64_t SV_NAME_D1 (log1p) (svfloat64_t x, svbool_t pg)
52{
53  const struct data *d = ptr_barrier (&data);
54  svuint64_t ix = svreinterpret_u64 (x);
55  svuint64_t ax = svand_x (pg, ix, AbsMask);
56  svbool_t special
57      = svorr_z (pg, svcmpge (pg, ax, d->inf), svcmpge (pg, ix, d->mone));
58
59  /* With x + 1 = t * 2^k (where t = f + 1 and k is chosen such that f
60			   is in [sqrt(2)/2, sqrt(2)]):
61     log1p(x) = k*log(2) + log1p(f).
62
63     f may not be representable exactly, so we need a correction term:
64     let m = round(1 + x), c = (1 + x) - m.
65     c << m: at very small x, log1p(x) ~ x, hence:
66     log(1+x) - log(m) ~ c/m.
67
68     We therefore calculate log1p(x) by k*log2 + log1p(f) + c/m.  */
69
70  /* Obtain correctly scaled k by manipulation in the exponent.
71     The scalar algorithm casts down to 32-bit at this point to calculate k and
72     u_red. We stay in double-width to obtain f and k, using the same constants
73     as the scalar algorithm but shifted left by 32.  */
74  svfloat64_t m = svadd_x (pg, x, 1);
75  svuint64_t mi = svreinterpret_u64 (m);
76  svuint64_t u = svadd_x (pg, mi, d->onemhfrt2_top);
77
78  svint64_t ki = svsub_x (pg, svreinterpret_s64 (svlsr_x (pg, u, 52)), 0x3ff);
79  svfloat64_t k = svcvt_f64_x (pg, ki);
80
81  /* Reduce x to f in [sqrt(2)/2, sqrt(2)].  */
82  svuint64_t utop
83      = svadd_x (pg, svand_x (pg, u, 0x000fffff00000000), d->hfrt2_top);
84  svuint64_t u_red = svorr_x (pg, utop, svand_x (pg, mi, BottomMask));
85  svfloat64_t f = svsub_x (pg, svreinterpret_f64 (u_red), 1);
86
87  /* Correction term c/m.  */
88  svfloat64_t cm = svdiv_x (pg, svsub_x (pg, x, svsub_x (pg, m, 1)), m);
89
90  /* Approximate log1p(x) on the reduced input using a polynomial. Because
91     log1p(0)=0 we choose an approximation of the form:
92	x + C0*x^2 + C1*x^3 + C2x^4 + ...
93     Hence approximation has the form f + f^2 * P(f)
94     where P(x) = C0 + C1*x + C2x^2 + ...
95     Assembling this all correctly is dealt with at the final step.  */
96  svfloat64_t f2 = svmul_x (pg, f, f), f4 = svmul_x (pg, f2, f2),
97	      f8 = svmul_x (pg, f4, f4), f16 = svmul_x (pg, f8, f8);
98  svfloat64_t p = sv_estrin_18_f64_x (pg, f, f2, f4, f8, f16, d->poly);
99
100  svfloat64_t ylo = svmla_x (pg, cm, k, d->ln2_lo);
101  svfloat64_t yhi = svmla_x (pg, f, k, d->ln2_hi);
102  svfloat64_t y = svmla_x (pg, svadd_x (pg, ylo, yhi), f2, p);
103
104  if (unlikely (svptest_any (pg, special)))
105    return special_case (special, x, y);
106
107  return y;
108}
109
110PL_SIG (SV, D, 1, log1p, -0.9, 10.0)
111PL_TEST_ULP (SV_NAME_D1 (log1p), 1.97)
112PL_TEST_SYM_INTERVAL (SV_NAME_D1 (log1p), 0.0, 0x1p-23, 50000)
113PL_TEST_SYM_INTERVAL (SV_NAME_D1 (log1p), 0x1p-23, 0.001, 50000)
114PL_TEST_SYM_INTERVAL (SV_NAME_D1 (log1p), 0.001, 1.0, 50000)
115PL_TEST_INTERVAL (SV_NAME_D1 (log1p), 1, inf, 10000)
116PL_TEST_INTERVAL (SV_NAME_D1 (log1p), -1, -inf, 10)
117