aboutsummaryrefslogtreecommitdiff
path: root/libc/src/math/generic/atan_utils.h
blob: 24c7271b7e4eca1f0b802ab655073bbd3807d587 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
//===-- Collection of utils for atan/atan2 ----------------------*- C++ -*-===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//

#ifndef LLVM_LIBC_SRC_MATH_GENERIC_ATAN_UTILS_H
#define LLVM_LIBC_SRC_MATH_GENERIC_ATAN_UTILS_H

#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/double_double.h"
#include "src/__support/FPUtil/dyadic_float.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/integer_literals.h"
#include "src/__support/macros/config.h"

namespace LIBC_NAMESPACE_DECL {

namespace {

using DoubleDouble = fputil::DoubleDouble;
using Float128 = fputil::DyadicFloat<128>;

// atan(i/64) with i = 0..64, generated by Sollya with:
// > for i from 0 to 64 do {
//     a = round(atan(i/64), D, RN);
//     b = round(atan(i/64) - a, D, RN);
//     print("{", b, ",", a, "},");
//   };
constexpr DoubleDouble ATAN_I[65] = {
    {0.0, 0.0},
    {-0x1.220c39d4dff5p-61, 0x1.fff555bbb729bp-7},
    {-0x1.5ec431444912cp-60, 0x1.ffd55bba97625p-6},
    {-0x1.86ef8f794f105p-63, 0x1.7fb818430da2ap-5},
    {-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
    {0x1.ac4ce285df847p-58, 0x1.3f59f0e7c559dp-4},
    {-0x1.cfb654c0c3d98p-58, 0x1.7ee182602f10fp-4},
    {0x1.f7b8f29a05987p-58, 0x1.be39ebe6f07c3p-4},
    {-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
    {-0x1.b485914dacf8cp-59, 0x1.1e1fafb043727p-3},
    {0x1.61a3b0ce9281bp-57, 0x1.3d6eee8c6626cp-3},
    {-0x1.054ab2c010f3dp-58, 0x1.5c9811e3ec26ap-3},
    {0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
    {0x1.cf601e7b4348ep-59, 0x1.9a6a8e96c8626p-3},
    {0x1.17b10d2e0e5abp-61, 0x1.b90d7529260a2p-3},
    {0x1.c648d1534597ep-57, 0x1.d77d5df205736p-3},
    {0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
    {0x1.62e47390cb865p-56, 0x1.09dc597d86362p-2},
    {0x1.30ca4748b1bf9p-57, 0x1.18bf5a30bf178p-2},
    {-0x1.077cdd36dfc81p-56, 0x1.278372057ef46p-2},
    {-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
    {-0x1.5d5e43c55b3bap-56, 0x1.44aa436c2af0ap-2},
    {-0x1.2566480884082p-57, 0x1.530ad9951cd4ap-2},
    {-0x1.a725715711fp-56, 0x1.614840309cfe2p-2},
    {-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
    {0x1.69c885c2b249ap-56, 0x1.7d5604b63b3f7p-2},
    {0x1.b6d0ba3748fa8p-56, 0x1.8b24d394a1b25p-2},
    {0x1.9e6c988fd0a77p-56, 0x1.98cd5454d6b18p-2},
    {-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
    {0x1.ae187b1ca504p-56, 0x1.b3a911da65c6cp-2},
    {-0x1.cc1ce70934c34p-56, 0x1.c0db4c94ec9fp-2},
    {-0x1.a2cfa4418f1adp-56, 0x1.cde53432c1351p-2},
    {0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
    {0x1.0e53dc1bf3435p-56, 0x1.e77eb7f175a34p-2},
    {-0x1.a3992dc382a23p-57, 0x1.f40dd0b541418p-2},
    {-0x1.b32c949c9d593p-55, 0x1.0039c73c1a40cp-1},
    {-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
    {0x1.974fa13b5404fp-58, 0x1.0c6145b5b43dap-1},
    {-0x1.2bdaee1c0ee35p-58, 0x1.1255d9bfbd2a9p-1},
    {0x1.c621cec00c301p-55, 0x1.1835a88be7c13p-1},
    {-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
    {0x1.c421c9f38224ep-57, 0x1.23b71e2cc9e6ap-1},
    {-0x1.09e73b0c6c087p-56, 0x1.2958e59308e31p-1},
    {0x1.c5d5e9ff0cf8dp-55, 0x1.2ee628406cbcap-1},
    {0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
    {-0x1.2304331d8bf46p-55, 0x1.39c391cd4171ap-1},
    {0x1.ecf8b492644fp-56, 0x1.3f13fb89e96f4p-1},
    {-0x1.f76d0163f79c8p-56, 0x1.445065b795b56p-1},
    {0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
    {0x1.4a33dbeb3796cp-55, 0x1.4e8de5bb6ec04p-1},
    {-0x1.1bb74abda520cp-55, 0x1.538f57b89061fp-1},
    {-0x1.5e5c9d8c5a95p-56, 0x1.587d81f732fbbp-1},
    {0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
    {-0x1.2b785350ee8c1p-57, 0x1.6220d115d7b8ep-1},
    {-0x1.6ea6febe8bbbap-56, 0x1.66d663923e087p-1},
    {-0x1.a80386188c50ep-55, 0x1.6b798920b3d99p-1},
    {-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
    {0x1.7b2a6165884a1p-59, 0x1.748978fba8e0fp-1},
    {0x1.406a08980374p-55, 0x1.78f6bbd5d315ep-1},
    {0x1.560821e2f3aa9p-55, 0x1.7d528289fa093p-1},
    {-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
    {0x1.6b66e7fc8b8c3p-57, 0x1.85d69576cc2c5p-1},
    {-0x1.55b9a5e177a1bp-55, 0x1.89ff5ff57f1f8p-1},
    {-0x1.ec182ab042f61p-56, 0x1.8e17aa99cc05ep-1},
    {0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
};

// Approximate atan(x) for |x| <= 2^-7.
// Using degree-9 Taylor polynomial:
//  P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9;
// Then the absolute error is bounded by:
//   |atan(x) - P(x)| < |x|^11/11 < 2^(-7*11) / 11 < 2^-80.
// And the relative error is bounded by:
//   |(atan(x) - P(x))/atan(x)| < |x|^10 / 10 < 2^-73.
// For x = x_hi + x_lo, fully expand the polynomial and drop any terms less than
//   ulp(x_hi^3 / 3) gives us:
// P(x) ~ x_hi - x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 +
//        + x_lo * (1 - x_hi^2 + x_hi^4)
// Since p.lo is ~ x^3/3, the relative error from rounding is bounded by:
//   |(atan(x) - P(x))/atan(x)| < ulp(x^2) <= 2^(-14-52) = 2^-66.
[[maybe_unused]] DoubleDouble atan_eval(const DoubleDouble &x) {
  DoubleDouble p;
  p.hi = x.hi;
  double x_hi_sq = x.hi * x.hi;
  // c0 ~ x_hi^2 * 1/5 - 1/3
  double c0 = fputil::multiply_add(x_hi_sq, 0x1.999999999999ap-3,
                                   -0x1.5555555555555p-2);
  // c1 ~ x_hi^2 * 1/9 - 1/7
  double c1 = fputil::multiply_add(x_hi_sq, 0x1.c71c71c71c71cp-4,
                                   -0x1.2492492492492p-3);
  // x_hi^3
  double x_hi_3 = x_hi_sq * x.hi;
  // x_hi^4
  double x_hi_4 = x_hi_sq * x_hi_sq;
  // d0 ~ 1/3 - x_hi^2 / 5 + x_hi^4 / 7 - x_hi^6 / 9
  double d0 = fputil::multiply_add(x_hi_4, c1, c0);
  // x_lo - x_lo * x_hi^2 + x_lo * x_hi^4
  double d1 = fputil::multiply_add(x_hi_4 - x_hi_sq, x.lo, x.lo);
  // p.lo ~ -x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 +
  //        + x_lo * (1 - x_hi^2 + x_hi^4)
  p.lo = fputil::multiply_add(x_hi_3, d0, d1);
  return p;
}

// Float128 versions.
// atan(i/64) with i = 0..64, generated by Sollya with:
// > for i from 1 to 64 do {
//     a = round(atan(i/64), 128, RN);
//     ll = ceil(log2(a));
//     b = 2^ll + a;
//     print("{Sign::POS, ", 2^(ll - 128), ",", b, "},");
// };
constexpr Float128 ATAN_I_F128[65] = {
    {Sign::POS, 0, 0_u128},
    {Sign::POS, -134, 0xfffaaadd'db94d5bb'e78c5640'15f76048_u128},
    {Sign::POS, -133, 0xffeaaddd'4bb12542'779d776d'da8c6214_u128},
    {Sign::POS, -132, 0xbfdc0c21'86d14fcf'220e10d6'1df56ec7_u128},
    {Sign::POS, -132, 0xffaaddb9'67ef4e36'cb2792dc'0e2e0d51_u128},
    {Sign::POS, -131, 0x9facf873'e2aceb58'99c50bbf'08e6cdf6_u128},
    {Sign::POS, -131, 0xbf70c130'17887460'93567e78'4cf83676_u128},
    {Sign::POS, -131, 0xdf1cf5f3'783e1bef'71e5340b'30e5d9ef_u128},
    {Sign::POS, -131, 0xfeadd4d5'617b6e32'c897989f'3e888ef8_u128},
    {Sign::POS, -130, 0x8f0fd7d8'21b93725'bd375929'83a0af9a_u128},
    {Sign::POS, -130, 0x9eb77746'331362c3'47619d25'0360fe85_u128},
    {Sign::POS, -130, 0xae4c08f1'f6134efa'b54d3fef'0c2de994_u128},
    {Sign::POS, -130, 0xbdcbda5e'72d81134'7b0b4f88'1c9c7488_u128},
    {Sign::POS, -130, 0xcd35474b'643130e7'b00f3da1'a46eeb3b_u128},
    {Sign::POS, -130, 0xdc86ba94'93051022'f621a5c1'cb552f03_u128},
    {Sign::POS, -130, 0xebbeaef9'02b9b38c'91a2a68b'2fbd78e8_u128},
    {Sign::POS, -130, 0xfadbafc9'6406eb15'6dc79ef5'f7a217e6_u128},
    {Sign::POS, -129, 0x84ee2cbe'c31b12c5'c8e72197'0cabd3a3_u128},
    {Sign::POS, -129, 0x8c5fad18'5f8bc130'ca4748b1'bf88298d_u128},
    {Sign::POS, -129, 0x93c1b902'bf7a2df1'06459240'6fe1447a_u128},
    {Sign::POS, -129, 0x9b13b9b8'3f5e5e69'c5abb498'd27af328_u128},
    {Sign::POS, -129, 0xa25521b6'15784d45'43787549'88b8d9e3_u128},
    {Sign::POS, -129, 0xa9856cca'8e6a4eda'99b7f77b'f7d9e8c1_u128},
    {Sign::POS, -129, 0xb0a42018'4e7f0cb1'b51d51dc'200a0fc3_u128},
    {Sign::POS, -129, 0xb7b0ca0f'26f78473'8aa32122'dcfe4483_u128},
    {Sign::POS, -129, 0xbeab025b'1d9fbad3'910b8564'93411026_u128},
    {Sign::POS, -129, 0xc59269ca'50d92b6d'a1746e91'f50a28de_u128},
    {Sign::POS, -129, 0xcc66aa2a'6b58c33c'd9311fa1'4ed9b7c4_u128},
    {Sign::POS, -129, 0xd327761e'611fe5b6'427c95e9'001e7136_u128},
    {Sign::POS, -129, 0xd9d488ed'32e3635c'30f6394a'0806345d_u128},
    {Sign::POS, -129, 0xe06da64a'764f7c67'c631ed96'798cb804_u128},
    {Sign::POS, -129, 0xe6f29a19'609a84ba'60b77ce1'ca6dc2c8_u128},
    {Sign::POS, -129, 0xed63382b'0dda7b45'6fe445ec'bc3a8d03_u128},
    {Sign::POS, -129, 0xf3bf5bf8'bad1a21c'a7b837e6'86adf3fa_u128},
    {Sign::POS, -129, 0xfa06e85a'a0a0be5c'66d23c7d'5dc8ecc2_u128},
    {Sign::POS, -128, 0x801ce39e'0d205c99'a6d6c6c5'4d938596_u128},
    {Sign::POS, -128, 0x832bf4a6'd9867e2a'4b6a09cb'61a515c1_u128},
    {Sign::POS, -128, 0x8630a2da'da1ed065'd3e84ed5'013ca37e_u128},
    {Sign::POS, -128, 0x892aecdf'de9547b5'094478fc'472b4afc_u128},
    {Sign::POS, -128, 0x8c1ad445'f3e09b8c'439d8018'60205921_u128},
    {Sign::POS, -128, 0x8f005d5e'f7f59f9b'5c835e16'65c43748_u128},
    {Sign::POS, -128, 0x91db8f16'64f350e2'10e4f9c1'126e0220_u128},
    {Sign::POS, -128, 0x94ac72c9'847186f6'18c4f393'f78a32f9_u128},
    {Sign::POS, -128, 0x97731420'365e538b'abd3fe19'f1aeb6b3_u128},
    {Sign::POS, -128, 0x9a2f80e6'71bdda20'4226f8e2'204ff3bd_u128},
    {Sign::POS, -128, 0x9ce1c8e6'a0b8cdb9'f799c4e8'174cf11c_u128},
    {Sign::POS, -128, 0x9f89fdc4'f4b7a1ec'f8b49264'4f0701e0_u128},
    {Sign::POS, -128, 0xa22832db'cadaae08'92fe9c08'637af0e6_u128},
    {Sign::POS, -128, 0xa4bc7d19'34f70924'19a87f2a'457dac9f_u128},
    {Sign::POS, -128, 0xa746f2dd'b7602294'67b7d66f'2d74e019_u128},
    {Sign::POS, -128, 0xa9c7abdc'4830f5c8'916a84b5'be7933f6_u128},
    {Sign::POS, -128, 0xac3ec0fb'997dd6a1'a36273a5'6afa8ef4_u128},
    {Sign::POS, -128, 0xaeac4c38'b4d8c080'14725e2f'3e52070a_u128},
    {Sign::POS, -128, 0xb110688a'ebdc6f6a'43d65788'b9f6a7b5_u128},
    {Sign::POS, -128, 0xb36b31c9'1f043691'59014174'4462f93a_u128},
    {Sign::POS, -128, 0xb5bcc490'59ecc4af'f8f3cee7'5e3907d5_u128},
    {Sign::POS, -128, 0xb8053e2b'c2319e73'cb2da552'10a4443d_u128},
    {Sign::POS, -128, 0xba44bc7d'd470782f'654c2cb1'0942e386_u128},
    {Sign::POS, -128, 0xbc7b5dea'e98af280'd4113006'e80fb290_u128},
    {Sign::POS, -128, 0xbea94144'fd049aac'1043c5e7'55282e7d_u128},
    {Sign::POS, -128, 0xc0ce85b8'ac526640'89dd62c4'6e92fa25_u128},
    {Sign::POS, -128, 0xc2eb4abb'661628b5'b373fe45'c61bb9fb_u128},
    {Sign::POS, -128, 0xc4ffaffa'bf8fbd54'8cb43d10'bc9e0221_u128},
    {Sign::POS, -128, 0xc70bd54c'e602ee13'e7d54fbd'09f2be38_u128},
    {Sign::POS, -128, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128},
};

// Degree-13 minimax polynomial generated by Sollya with:
// > P = fpminimax(atan(x), [|1, 3, 5, 7, 9, 11, 13|], [|1, 128...|],
//                 [0, 2^-7]);
// > dirtyinfnorm(atan(x) - P, [0, 2^-7]);
// 0x1.26016ad97f323875760f869684c0898d7b7bb8bep-122
constexpr Float128 ATAN_POLY_F128[] = {
    {Sign::NEG, -129, 0xaaaaaaaa'aaaaaaaa'aaaaaaa6'003c5d1d_u128},
    {Sign::POS, -130, 0xcccccccc'cccccccc'cca00232'8776b063_u128},
    {Sign::NEG, -130, 0x92492492'49249201'27f5268a'cb24aec0_u128},
    {Sign::POS, -131, 0xe38e38e3'8dce3d96'626a1643'f8eb68f3_u128},
    {Sign::NEG, -131, 0xba2e8b7a'ea4ad00f'005a35c7'6ef609b1_u128},
    {Sign::POS, -131, 0x9d82765e'd22a7d92'ac09c405'c0a69214_u128},
};

// Approximate atan for |x| <= 2^-7.
[[maybe_unused]] Float128 atan_eval(const Float128 &x) {
  Float128 x_sq = fputil::quick_mul(x, x);
  Float128 x3 = fputil::quick_mul(x, x_sq);
  Float128 p = fputil::polyeval(x_sq, ATAN_POLY_F128[0], ATAN_POLY_F128[1],
                                ATAN_POLY_F128[2], ATAN_POLY_F128[3],
                                ATAN_POLY_F128[4], ATAN_POLY_F128[5]);
  return fputil::multiply_add(x3, p, x);
}

} // anonymous namespace

} // namespace LIBC_NAMESPACE_DECL

#endif // LLVM_LIBC_SRC_MATH_GENERIC_ATAN_UTILS_H