1//===-- Single-precision atan2f 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/atan2f.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/optimization.h" // LIBC_UNLIKELY
18
19namespace LIBC_NAMESPACE {
20
21namespace {
22
23// Look up tables for accurate pass:
24
25// atan(i/16) with i = 0..16, generated by Sollya with:
26// > for i from 0 to 16 do {
27// a = round(atan(i/16), D, RN);
28// b = round(atan(i/16) - a, D, RN);
29// print("{", b, ",", a, "},");
30// };
31constexpr fputil::DoubleDouble ATAN_I[17] = {
32 {.lo: 0.0, .hi: 0.0},
33 {.lo: -0x1.c934d86d23f1dp-60, .hi: 0x1.ff55bb72cfdeap-5},
34 {.lo: -0x1.cd37686760c17p-59, .hi: 0x1.fd5ba9aac2f6ep-4},
35 {.lo: 0x1.347b0b4f881cap-58, .hi: 0x1.7b97b4bce5b02p-3},
36 {.lo: 0x1.8ab6e3cf7afbdp-57, .hi: 0x1.f5b75f92c80ddp-3},
37 {.lo: -0x1.963a544b672d8p-57, .hi: 0x1.362773707ebccp-2},
38 {.lo: -0x1.c63aae6f6e918p-56, .hi: 0x1.6f61941e4def1p-2},
39 {.lo: -0x1.24dec1b50b7ffp-56, .hi: 0x1.a64eec3cc23fdp-2},
40 {.lo: 0x1.a2b7f222f65e2p-56, .hi: 0x1.dac670561bb4fp-2},
41 {.lo: -0x1.d5b495f6349e6p-56, .hi: 0x1.0657e94db30dp-1},
42 {.lo: -0x1.928df287a668fp-58, .hi: 0x1.1e00babdefeb4p-1},
43 {.lo: 0x1.1021137c71102p-55, .hi: 0x1.345f01cce37bbp-1},
44 {.lo: 0x1.2419a87f2a458p-56, .hi: 0x1.4978fa3269ee1p-1},
45 {.lo: 0x1.0028e4bc5e7cap-57, .hi: 0x1.5d58987169b18p-1},
46 {.lo: -0x1.8c34d25aadef6p-56, .hi: 0x1.700a7c5784634p-1},
47 {.lo: -0x1.bf76229d3b917p-56, .hi: 0x1.819d0b7158a4dp-1},
48 {.lo: 0x1.1a62633145c07p-55, .hi: 0x1.921fb54442d18p-1},
49};
50
51// Taylor polynomial, generated by Sollya with:
52// > for i from 0 to 8 do {
53// j = (-1)^(i + 1)/(2*i + 1);
54// a = round(j, D, RN);
55// b = round(j - a, D, RN);
56// print("{", b, ",", a, "},");
57// };
58constexpr fputil::DoubleDouble COEFFS[9] = {
59 {.lo: 0.0, .hi: 1.0}, // 1
60 {.lo: -0x1.5555555555555p-56, .hi: -0x1.5555555555555p-2}, // -1/3
61 {.lo: -0x1.999999999999ap-57, .hi: 0x1.999999999999ap-3}, // 1/5
62 {.lo: -0x1.2492492492492p-57, .hi: -0x1.2492492492492p-3}, // -1/7
63 {.lo: 0x1.c71c71c71c71cp-58, .hi: 0x1.c71c71c71c71cp-4}, // 1/9
64 {.lo: 0x1.745d1745d1746p-59, .hi: -0x1.745d1745d1746p-4}, // -1/11
65 {.lo: -0x1.3b13b13b13b14p-58, .hi: 0x1.3b13b13b13b14p-4}, // 1/13
66 {.lo: -0x1.1111111111111p-60, .hi: -0x1.1111111111111p-4}, // -1/15
67 {.lo: 0x1.e1e1e1e1e1e1ep-61, .hi: 0x1.e1e1e1e1e1e1ep-5}, // 1/17
68};
69
70// Veltkamp's splitting of a double precision into hi + lo, where the hi part is
71// slightly smaller than an even split, so that the product of
72// hi * (s1 * k + s2) is exact,
73// where:
74// s1, s2 are single precsion,
75// 1/16 <= s1/s2 <= 1
76// 1/16 <= k <= 1 is an integer.
77// So the maximal precision of (s1 * k + s2) is:
78// prec(s1 * k + s2) = 2 + log2(msb(s2)) - log2(lsb(k_d * s1))
79// = 2 + log2(msb(s1)) + 4 - log2(lsb(k_d)) - log2(lsb(s1))
80// = 2 + log2(lsb(s1)) + 23 + 4 - (-4) - log2(lsb(s1))
81// = 33.
82// Thus, the Veltkamp splitting constant is C = 2^33 + 1.
83// This is used when FMA instruction is not available.
84[[maybe_unused]] constexpr fputil::DoubleDouble split_d(double a) {
85 fputil::DoubleDouble r{.lo: 0.0, .hi: 0.0};
86 constexpr double C = 0x1.0p33 + 1.0;
87 double t1 = C * a;
88 double t2 = a - t1;
89 r.hi = t1 + t2;
90 r.lo = a - r.hi;
91 return r;
92}
93
94// Compute atan( num_d / den_d ) in double-double precision.
95// num_d = min(|x|, |y|)
96// den_d = max(|x|, |y|)
97// q_d = num_d / den_d
98// idx, k_d = round( 2^4 * num_d / den_d )
99// final_sign = sign of the final result
100// const_term = the constant term in the final expression.
101float atan2f_double_double(double num_d, double den_d, double q_d, int idx,
102 double k_d, double final_sign,
103 const fputil::DoubleDouble &const_term) {
104 fputil::DoubleDouble q;
105 double num_r, den_r;
106
107 if (idx != 0) {
108 // The following range reduction is accurate even without fma for
109 // 1/16 <= n/d <= 1.
110 // atan(n/d) - atan(idx/16) = atan((n/d - idx/16) / (1 + (n/d) * (idx/16)))
111 // = atan((n - d*(idx/16)) / (d + n*idx/16))
112 k_d *= 0x1.0p-4;
113 num_r = fputil::multiply_add(x: k_d, y: -den_d, z: num_d); // Exact
114 den_r = fputil::multiply_add(x: k_d, y: num_d, z: den_d); // Exact
115 q.hi = num_r / den_r;
116 } else {
117 // For 0 < n/d < 1/16, we just need to calculate the lower part of their
118 // quotient.
119 q.hi = q_d;
120 num_r = num_d;
121 den_r = den_d;
122 }
123#ifdef LIBC_TARGET_CPU_HAS_FMA
124 q.lo = fputil::multiply_add(x: q.hi, y: -den_r, z: num_r) / den_r;
125#else
126 // Compute `(num_r - q.hi * den_r) / den_r` accurately without FMA
127 // instructions.
128 fputil::DoubleDouble q_hi_dd = split_d(q.hi);
129 double t1 = fputil::multiply_add(q_hi_dd.hi, -den_r, num_r); // Exact
130 double t2 = fputil::multiply_add(q_hi_dd.lo, -den_r, t1);
131 q.lo = t2 / den_r;
132#endif // LIBC_TARGET_CPU_HAS_FMA
133
134 // Taylor polynomial, evaluating using Horner's scheme:
135 // P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
136 // + x^17/17
137 // = x*(1 + x^2*(-1/3 + x^2*(1/5 + x^2*(-1/7 + x^2*(1/9 + x^2*
138 // *(-1/11 + x^2*(1/13 + x^2*(-1/15 + x^2 * 1/17))))))))
139 fputil::DoubleDouble q2 = fputil::quick_mult(a: q, b: q);
140 fputil::DoubleDouble p_dd =
141 fputil::polyeval(x: q2, a0: COEFFS[0], a: COEFFS[1], a: COEFFS[2], a: COEFFS[3],
142 a: COEFFS[4], a: COEFFS[5], a: COEFFS[6], a: COEFFS[7], a: COEFFS[8]);
143 fputil::DoubleDouble r_dd =
144 fputil::add(a: const_term, b: fputil::multiply_add(a: q, b: p_dd, c: ATAN_I[idx]));
145 r_dd.hi *= final_sign;
146 r_dd.lo *= final_sign;
147
148 // Make sure the sum is normalized:
149 fputil::DoubleDouble rr = fputil::exact_add(a: r_dd.hi, b: r_dd.lo);
150 // Round to odd.
151 uint64_t rr_bits = cpp::bit_cast<uint64_t>(from: rr.hi);
152 if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) {
153 Sign hi_sign = fputil::FPBits<double>(rr.hi).sign();
154 Sign lo_sign = fputil::FPBits<double>(rr.lo).sign();
155 if (hi_sign == lo_sign) {
156 ++rr_bits;
157 } else if ((rr_bits & fputil::FPBits<double>::FRACTION_MASK) > 0) {
158 --rr_bits;
159 }
160 }
161
162 return static_cast<float>(cpp::bit_cast<double>(from: rr_bits));
163}
164
165} // anonymous namespace
166
167// There are several range reduction steps we can take for atan2(y, x) as
168// follow:
169
170// * Range reduction 1: signness
171// atan2(y, x) will return a number between -PI and PI representing the angle
172// forming by the 0x axis and the vector (x, y) on the 0xy-plane.
173// In particular, we have that:
174// atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)
175// = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)
176// = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)
177// = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)
178// Since atan function is odd, we can use the formula:
179// atan(-u) = -atan(u)
180// to adjust the above conditions a bit further:
181// atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)
182// = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)
183// = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)
184// = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)
185// Which can be simplified to:
186// atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0
187// = sign(y) * (pi - atan( |y|/|x| )) if x < 0
188
189// * Range reduction 2: reciprocal
190// Now that the argument inside atan is positive, we can use the formula:
191// atan(1/x) = pi/2 - atan(x)
192// to make the argument inside atan <= 1 as follow:
193// atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x
194// = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|
195// = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x
196// = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|
197
198// * Range reduction 3: look up table.
199// After the previous two range reduction steps, we reduce the problem to
200// compute atan(u) with 0 <= u <= 1, or to be precise:
201// atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
202// An accurate polynomial approximation for the whole [0, 1] input range will
203// require a very large degree. To make it more efficient, we reduce the input
204// range further by finding an integer idx such that:
205// | n/d - idx/16 | <= 1/32.
206// In particular,
207// idx := 2^-4 * round(2^4 * n/d)
208// Then for the fast pass, we find a polynomial approximation for:
209// atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16)
210// For the accurate pass, we use the addition formula:
211// atan( n/d ) - atan( idx/16 ) = atan( (n/d - idx/16)/(1 + (n*idx)/(16*d)) )
212// = atan( (n - d * idx/16)/(d + n * idx/16) )
213// And finally we use Taylor polynomial to compute the RHS in the accurate pass:
214// atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 - u^11/11 + u^13/13 -
215// - u^15/15 + u^17/17
216// It's error in double-double precision is estimated in Sollya to be:
217// > P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
218// + x^17/17;
219// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
220// 0x1.aec6f...p-100
221// which is about rounding errors of double-double (2^-104).
222
223LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
224 using FPBits = typename fputil::FPBits<float>;
225 constexpr double IS_NEG[2] = {1.0, -1.0};
226 constexpr double PI = 0x1.921fb54442d18p1;
227 constexpr double PI_LO = 0x1.1a62633145c07p-53;
228 constexpr double PI_OVER_4 = 0x1.921fb54442d18p-1;
229 constexpr double PI_OVER_2 = 0x1.921fb54442d18p0;
230 constexpr double THREE_PI_OVER_4 = 0x1.2d97c7f3321d2p+1;
231 // Adjustment for constant term:
232 // CONST_ADJ[x_sign][y_sign][recip]
233 constexpr fputil::DoubleDouble CONST_ADJ[2][2][2] = {
234 {{{.lo: 0.0, .hi: 0.0}, {.lo: -PI_LO / 2, .hi: -PI_OVER_2}},
235 {{.lo: -0.0, .hi: -0.0}, {.lo: -PI_LO / 2, .hi: -PI_OVER_2}}},
236 {{{.lo: -PI_LO, .hi: -PI}, {.lo: PI_LO / 2, .hi: PI_OVER_2}},
237 {{.lo: -PI_LO, .hi: -PI}, {.lo: PI_LO / 2, .hi: PI_OVER_2}}}};
238
239 FPBits x_bits(x), y_bits(y);
240 bool x_sign = x_bits.sign().is_neg();
241 bool y_sign = y_bits.sign().is_neg();
242 x_bits.set_sign(Sign::POS);
243 y_bits.set_sign(Sign::POS);
244 uint32_t x_abs = x_bits.uintval();
245 uint32_t y_abs = y_bits.uintval();
246 uint32_t max_abs = x_abs > y_abs ? x_abs : y_abs;
247 uint32_t min_abs = x_abs <= y_abs ? x_abs : y_abs;
248
249 if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || min_abs == 0U)) {
250 if (x_bits.is_nan() || y_bits.is_nan())
251 return FPBits::quiet_nan().get_val();
252 size_t x_except = x_abs == 0 ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1);
253 size_t y_except = y_abs == 0 ? 0 : (y_abs == 0x7f80'0000 ? 2 : 1);
254
255 // Exceptional cases:
256 // EXCEPT[y_except][x_except][x_is_neg]
257 // with x_except & y_except:
258 // 0: zero
259 // 1: finite, non-zero
260 // 2: infinity
261 constexpr double EXCEPTS[3][3][2] = {
262 {{0.0, PI}, {0.0, PI}, {0.0, PI}},
263 {{PI_OVER_2, PI_OVER_2}, {0.0, 0.0}, {0.0, PI}},
264 {{PI_OVER_2, PI_OVER_2},
265 {PI_OVER_2, PI_OVER_2},
266 {PI_OVER_4, THREE_PI_OVER_4}},
267 };
268
269 double r = IS_NEG[y_sign] * EXCEPTS[y_except][x_except][x_sign];
270
271 return static_cast<float>(r);
272 }
273
274 bool recip = x_abs < y_abs;
275 double final_sign = IS_NEG[(x_sign != y_sign) != recip];
276 fputil::DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
277 double num_d = static_cast<double>(FPBits(min_abs).get_val());
278 double den_d = static_cast<double>(FPBits(max_abs).get_val());
279 double q_d = num_d / den_d;
280
281 double k_d = fputil::nearest_integer(x: q_d * 0x1.0p4f);
282 int idx = static_cast<int>(k_d);
283 q_d = fputil::multiply_add(x: k_d, y: -0x1.0p-4, z: q_d);
284
285 double p = atan_eval(x: q_d, i: idx);
286 double r = final_sign *
287 fputil::multiply_add(x: q_d, y: p, z: const_term.hi + ATAN_COEFFS[idx][0]);
288
289 constexpr uint32_t LOWER_ERR = 4;
290 // Mask sticky bits in double precision before rounding to single precision.
291 constexpr uint32_t MASK =
292 mask_trailing_ones<uint32_t, fputil::FPBits<double>::SIG_LEN -
293 FPBits::SIG_LEN - 1>();
294 constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR;
295
296 uint32_t r_bits = static_cast<uint32_t>(cpp::bit_cast<uint64_t>(from: r)) & MASK;
297
298 // Ziv's rounding test.
299 if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR))
300 return static_cast<float>(r);
301
302 return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign,
303 const_term);
304}
305
306} // namespace LIBC_NAMESPACE
307

source code of libc/src/math/generic/atan2f.cpp