1 //===-- Double-precision atan2 function -----------------------------------===//
2 //
3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4 // See https://llvm.org/LICENSE.txt for license information.
5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6 //
7 //===----------------------------------------------------------------------===//
8
9 #include "src/math/atan2.h"
10 #include "inv_trigf_utils.h"
11 #include "src/__support/FPUtil/FPBits.h"
12 #include "src/__support/FPUtil/PolyEval.h"
13 #include "src/__support/FPUtil/double_double.h"
14 #include "src/__support/FPUtil/multiply_add.h"
15 #include "src/__support/FPUtil/nearest_integer.h"
16 #include "src/__support/FPUtil/rounding_mode.h"
17 #include "src/__support/macros/config.h"
18 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
19
20 namespace LIBC_NAMESPACE_DECL {
21
22 namespace {
23
24 using DoubleDouble = fputil::DoubleDouble;
25
26 // atan(i/64) with i = 0..64, generated by Sollya with:
27 // > for i from 0 to 64 do {
28 // a = round(atan(i/64), D, RN);
29 // b = round(atan(i/64) - a, D, RN);
30 // print("{", b, ",", a, "},");
31 // };
32 constexpr fputil::DoubleDouble ATAN_I[65] = {
33 {0.0, 0.0},
34 {-0x1.220c39d4dff5p-61, 0x1.fff555bbb729bp-7},
35 {-0x1.5ec431444912cp-60, 0x1.ffd55bba97625p-6},
36 {-0x1.86ef8f794f105p-63, 0x1.7fb818430da2ap-5},
37 {-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
38 {0x1.ac4ce285df847p-58, 0x1.3f59f0e7c559dp-4},
39 {-0x1.cfb654c0c3d98p-58, 0x1.7ee182602f10fp-4},
40 {0x1.f7b8f29a05987p-58, 0x1.be39ebe6f07c3p-4},
41 {-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
42 {-0x1.b485914dacf8cp-59, 0x1.1e1fafb043727p-3},
43 {0x1.61a3b0ce9281bp-57, 0x1.3d6eee8c6626cp-3},
44 {-0x1.054ab2c010f3dp-58, 0x1.5c9811e3ec26ap-3},
45 {0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
46 {0x1.cf601e7b4348ep-59, 0x1.9a6a8e96c8626p-3},
47 {0x1.17b10d2e0e5abp-61, 0x1.b90d7529260a2p-3},
48 {0x1.c648d1534597ep-57, 0x1.d77d5df205736p-3},
49 {0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
50 {0x1.62e47390cb865p-56, 0x1.09dc597d86362p-2},
51 {0x1.30ca4748b1bf9p-57, 0x1.18bf5a30bf178p-2},
52 {-0x1.077cdd36dfc81p-56, 0x1.278372057ef46p-2},
53 {-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
54 {-0x1.5d5e43c55b3bap-56, 0x1.44aa436c2af0ap-2},
55 {-0x1.2566480884082p-57, 0x1.530ad9951cd4ap-2},
56 {-0x1.a725715711fp-56, 0x1.614840309cfe2p-2},
57 {-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
58 {0x1.69c885c2b249ap-56, 0x1.7d5604b63b3f7p-2},
59 {0x1.b6d0ba3748fa8p-56, 0x1.8b24d394a1b25p-2},
60 {0x1.9e6c988fd0a77p-56, 0x1.98cd5454d6b18p-2},
61 {-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
62 {0x1.ae187b1ca504p-56, 0x1.b3a911da65c6cp-2},
63 {-0x1.cc1ce70934c34p-56, 0x1.c0db4c94ec9fp-2},
64 {-0x1.a2cfa4418f1adp-56, 0x1.cde53432c1351p-2},
65 {0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
66 {0x1.0e53dc1bf3435p-56, 0x1.e77eb7f175a34p-2},
67 {-0x1.a3992dc382a23p-57, 0x1.f40dd0b541418p-2},
68 {-0x1.b32c949c9d593p-55, 0x1.0039c73c1a40cp-1},
69 {-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
70 {0x1.974fa13b5404fp-58, 0x1.0c6145b5b43dap-1},
71 {-0x1.2bdaee1c0ee35p-58, 0x1.1255d9bfbd2a9p-1},
72 {0x1.c621cec00c301p-55, 0x1.1835a88be7c13p-1},
73 {-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
74 {0x1.c421c9f38224ep-57, 0x1.23b71e2cc9e6ap-1},
75 {-0x1.09e73b0c6c087p-56, 0x1.2958e59308e31p-1},
76 {0x1.c5d5e9ff0cf8dp-55, 0x1.2ee628406cbcap-1},
77 {0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
78 {-0x1.2304331d8bf46p-55, 0x1.39c391cd4171ap-1},
79 {0x1.ecf8b492644fp-56, 0x1.3f13fb89e96f4p-1},
80 {-0x1.f76d0163f79c8p-56, 0x1.445065b795b56p-1},
81 {0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
82 {0x1.4a33dbeb3796cp-55, 0x1.4e8de5bb6ec04p-1},
83 {-0x1.1bb74abda520cp-55, 0x1.538f57b89061fp-1},
84 {-0x1.5e5c9d8c5a95p-56, 0x1.587d81f732fbbp-1},
85 {0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
86 {-0x1.2b785350ee8c1p-57, 0x1.6220d115d7b8ep-1},
87 {-0x1.6ea6febe8bbbap-56, 0x1.66d663923e087p-1},
88 {-0x1.a80386188c50ep-55, 0x1.6b798920b3d99p-1},
89 {-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
90 {0x1.7b2a6165884a1p-59, 0x1.748978fba8e0fp-1},
91 {0x1.406a08980374p-55, 0x1.78f6bbd5d315ep-1},
92 {0x1.560821e2f3aa9p-55, 0x1.7d528289fa093p-1},
93 {-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
94 {0x1.6b66e7fc8b8c3p-57, 0x1.85d69576cc2c5p-1},
95 {-0x1.55b9a5e177a1bp-55, 0x1.89ff5ff57f1f8p-1},
96 {-0x1.ec182ab042f61p-56, 0x1.8e17aa99cc05ep-1},
97 {0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
98 };
99
100 // Approximate atan(x) for |x| <= 2^-7.
101 // Using degree-9 Taylor polynomial:
102 // P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9;
103 // Then the absolute error is bounded by:
104 // |atan(x) - P(x)| < |x|^11/11 < 2^(-7*11) / 11 < 2^-80.
105 // And the relative error is bounded by:
106 // |(atan(x) - P(x))/atan(x)| < |x|^10 / 10 < 2^-73.
107 // For x = x_hi + x_lo, fully expand the polynomial and drop any terms less than
108 // ulp(x_hi^3 / 3) gives us:
109 // P(x) ~ x_hi - x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 +
110 // + x_lo * (1 - x_hi^2 + x_hi^4)
atan_eval(const DoubleDouble & x)111 DoubleDouble atan_eval(const DoubleDouble &x) {
112 DoubleDouble p;
113 p.hi = x.hi;
114 double x_hi_sq = x.hi * x.hi;
115 // c0 ~ x_hi^2 * 1/5 - 1/3
116 double c0 = fputil::multiply_add(x_hi_sq, 0x1.999999999999ap-3,
117 -0x1.5555555555555p-2);
118 // c1 ~ x_hi^2 * 1/9 - 1/7
119 double c1 = fputil::multiply_add(x_hi_sq, 0x1.c71c71c71c71cp-4,
120 -0x1.2492492492492p-3);
121 // x_hi^3
122 double x_hi_3 = x_hi_sq * x.hi;
123 // x_hi^4
124 double x_hi_4 = x_hi_sq * x_hi_sq;
125 // d0 ~ 1/3 - x_hi^2 / 5 + x_hi^4 / 7 - x_hi^6 / 9
126 double d0 = fputil::multiply_add(x_hi_4, c1, c0);
127 // x_lo - x_lo * x_hi^2 + x_lo * x_hi^4
128 double d1 = fputil::multiply_add(x_hi_4 - x_hi_sq, x.lo, x.lo);
129 // p.lo ~ -x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 +
130 // + x_lo * (1 - x_hi^2 + x_hi^4)
131 p.lo = fputil::multiply_add(x_hi_3, d0, d1);
132 return p;
133 }
134
135 } // anonymous namespace
136
137 // There are several range reduction steps we can take for atan2(y, x) as
138 // follow:
139
140 // * Range reduction 1: signness
141 // atan2(y, x) will return a number between -PI and PI representing the angle
142 // forming by the 0x axis and the vector (x, y) on the 0xy-plane.
143 // In particular, we have that:
144 // atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)
145 // = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)
146 // = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)
147 // = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)
148 // Since atan function is odd, we can use the formula:
149 // atan(-u) = -atan(u)
150 // to adjust the above conditions a bit further:
151 // atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)
152 // = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)
153 // = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)
154 // = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)
155 // Which can be simplified to:
156 // atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0
157 // = sign(y) * (pi - atan( |y|/|x| )) if x < 0
158
159 // * Range reduction 2: reciprocal
160 // Now that the argument inside atan is positive, we can use the formula:
161 // atan(1/x) = pi/2 - atan(x)
162 // to make the argument inside atan <= 1 as follow:
163 // atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x
164 // = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|
165 // = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x
166 // = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|
167
168 // * Range reduction 3: look up table.
169 // After the previous two range reduction steps, we reduce the problem to
170 // compute atan(u) with 0 <= u <= 1, or to be precise:
171 // atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
172 // An accurate polynomial approximation for the whole [0, 1] input range will
173 // require a very large degree. To make it more efficient, we reduce the input
174 // range further by finding an integer idx such that:
175 // | n/d - idx/64 | <= 1/128.
176 // In particular,
177 // idx := round(2^6 * n/d)
178 // Then for the fast pass, we find a polynomial approximation for:
179 // atan( n/d ) ~ atan( idx/64 ) + (n/d - idx/64) * Q(n/d - idx/64)
180 // For the accurate pass, we use the addition formula:
181 // atan( n/d ) - atan( idx/64 ) = atan( (n/d - idx/64)/(1 + (n*idx)/(64*d)) )
182 // = atan( (n - d*(idx/64))/(d + n*(idx/64)) )
183 // And for the fast pass, we use degree-9 Taylor polynomial to compute the RHS:
184 // atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9
185 // with absolute errors bounded by:
186 // |atan(u) - P(u)| < |u|^11 / 11 < 2^-80
187 // and relative errors bounded by:
188 // |(atan(u) - P(u)) / P(u)| < u^10 / 11 < 2^-73.
189
190 LLVM_LIBC_FUNCTION(double, atan2, (double y, double x)) {
191 using FPBits = fputil::FPBits<double>;
192
193 constexpr double IS_NEG[2] = {1.0, -1.0};
194 constexpr DoubleDouble ZERO = {0.0, 0.0};
195 constexpr DoubleDouble MZERO = {-0.0, -0.0};
196 constexpr DoubleDouble PI = {0x1.1a62633145c07p-53, 0x1.921fb54442d18p+1};
197 constexpr DoubleDouble MPI = {-0x1.1a62633145c07p-53, -0x1.921fb54442d18p+1};
198 constexpr DoubleDouble PI_OVER_2 = {0x1.1a62633145c07p-54,
199 0x1.921fb54442d18p0};
200 constexpr DoubleDouble MPI_OVER_2 = {-0x1.1a62633145c07p-54,
201 -0x1.921fb54442d18p0};
202 constexpr DoubleDouble PI_OVER_4 = {0x1.1a62633145c07p-55,
203 0x1.921fb54442d18p-1};
204 constexpr DoubleDouble THREE_PI_OVER_4 = {0x1.a79394c9e8a0ap-54,
205 0x1.2d97c7f3321d2p+1};
206 // Adjustment for constant term:
207 // CONST_ADJ[x_sign][y_sign][recip]
208 constexpr DoubleDouble CONST_ADJ[2][2][2] = {
209 {{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}},
210 {{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}};
211
212 FPBits x_bits(x), y_bits(y);
213 bool x_sign = x_bits.sign().is_neg();
214 bool y_sign = y_bits.sign().is_neg();
215 x_bits = x_bits.abs();
216 y_bits = y_bits.abs();
217 uint64_t x_abs = x_bits.uintval();
218 uint64_t y_abs = y_bits.uintval();
219 bool recip = x_abs < y_abs;
220 uint64_t min_abs = recip ? x_abs : y_abs;
221 uint64_t max_abs = !recip ? x_abs : y_abs;
222 unsigned min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
223 unsigned max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
224
225 double num = FPBits(min_abs).get_val();
226 double den = FPBits(max_abs).get_val();
227
228 // Check for exceptional cases, whether inputs are 0, inf, nan, or close to
229 // overflow, or close to underflow.
230 if (LIBC_UNLIKELY(max_exp > 0x7ffU - 128U || min_exp < 128U)) {
231 if (x_bits.is_nan() || y_bits.is_nan())
232 return FPBits::quiet_nan().get_val();
233 unsigned x_except = x == 0.0 ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1);
234 unsigned y_except = y == 0.0 ? 0 : (FPBits(y_abs).is_inf() ? 2 : 1);
235
236 // Exceptional cases:
237 // EXCEPT[y_except][x_except][x_is_neg]
238 // with x_except & y_except:
239 // 0: zero
240 // 1: finite, non-zero
241 // 2: infinity
242 constexpr DoubleDouble EXCEPTS[3][3][2] = {
243 {{ZERO, PI}, {ZERO, PI}, {ZERO, PI}},
244 {{PI_OVER_2, PI_OVER_2}, {ZERO, ZERO}, {ZERO, PI}},
245 {{PI_OVER_2, PI_OVER_2},
246 {PI_OVER_2, PI_OVER_2},
247 {PI_OVER_4, THREE_PI_OVER_4}},
248 };
249
250 if ((x_except != 1) || (y_except != 1)) {
251 DoubleDouble r = EXCEPTS[y_except][x_except][x_sign];
252 return fputil::multiply_add(IS_NEG[y_sign], r.hi, IS_NEG[y_sign] * r.lo);
253 }
254 bool scale_up = min_exp < 128U;
255 bool scale_down = max_exp > 0x7ffU - 128U;
256 // At least one input is denormal, multiply both numerator and denominator
257 // by some large enough power of 2 to normalize denormal inputs.
258 if (scale_up) {
259 num *= 0x1.0p64;
260 if (!scale_down)
261 den *= 0x1.0p64;
262 } else if (scale_down) {
263 den *= 0x1.0p-64;
264 if (!scale_up)
265 num *= 0x1.0p-64;
266 }
267
268 min_abs = FPBits(num).uintval();
269 max_abs = FPBits(den).uintval();
270 min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
271 max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
272 }
273
274 double final_sign = IS_NEG[(x_sign != y_sign) != recip];
275 DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
276 unsigned exp_diff = max_exp - min_exp;
277 // We have the following bound for normalized n and d:
278 // 2^(-exp_diff - 1) < n/d < 2^(-exp_diff + 1).
279 if (LIBC_UNLIKELY(exp_diff > 54)) {
280 return fputil::multiply_add(final_sign, const_term.hi,
281 final_sign * (const_term.lo + num / den));
282 }
283
284 double k = fputil::nearest_integer(64.0 * num / den);
285 unsigned idx = static_cast<unsigned>(k);
286 // k = idx / 64
287 k *= 0x1.0p-6;
288
289 // Range reduction:
290 // atan(n/d) - atan(k/64) = atan((n/d - k/64) / (1 + (n/d) * (k/64)))
291 // = atan((n - d * k/64)) / (d + n * k/64))
292 DoubleDouble num_k = fputil::exact_mult(num, k);
293 DoubleDouble den_k = fputil::exact_mult(den, k);
294
295 // num_dd = n - d * k
296 DoubleDouble num_dd = fputil::exact_add(num - den_k.hi, -den_k.lo);
297 // den_dd = d + n * k
298 DoubleDouble den_dd = fputil::exact_add(den, num_k.hi);
299 den_dd.lo += num_k.lo;
300
301 // q = (n - d * k) / (d + n * k)
302 DoubleDouble q = fputil::div(num_dd, den_dd);
303 // p ~ atan(q)
304 DoubleDouble p = atan_eval(q);
305
306 DoubleDouble r = fputil::add(const_term, fputil::add(ATAN_I[idx], p));
307 r.hi *= final_sign;
308 r.lo *= final_sign;
309
310 return r.hi + r.lo;
311 }
312
313 } // namespace LIBC_NAMESPACE_DECL
314