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 "hdr/fenv_macros.h"
11#include "inv_trigf_utils.h"
12#include "src/__support/FPUtil/FEnvImpl.h"
13#include "src/__support/FPUtil/FPBits.h"
14#include "src/__support/FPUtil/PolyEval.h"
15#include "src/__support/FPUtil/double_double.h"
16#include "src/__support/FPUtil/multiply_add.h"
17#include "src/__support/FPUtil/nearest_integer.h"
18#include "src/__support/FPUtil/rounding_mode.h"
19#include "src/__support/macros/config.h"
20#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
21
22#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \
23 defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT)
24
25// We use float-float implementation to reduce size.
26#include "src/math/generic/atan2f_float.h"
27
28#else
29
30namespace LIBC_NAMESPACE_DECL {
31
32namespace {
33
34#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
35
36// Look up tables for accurate pass:
37
38// atan(i/16) with i = 0..16, generated by Sollya with:
39// > for i from 0 to 16 do {
40// a = round(atan(i/16), D, RN);
41// b = round(atan(i/16) - a, D, RN);
42// print("{", b, ",", a, "},");
43// };
44constexpr fputil::DoubleDouble ATAN_I[17] = {
45 {0.0, 0.0},
46 {-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
47 {-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
48 {0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
49 {0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
50 {-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
51 {-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
52 {-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
53 {0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
54 {-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
55 {-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
56 {0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
57 {0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
58 {0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
59 {-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
60 {-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
61 {0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
62};
63
64// Taylor polynomial, generated by Sollya with:
65// > for i from 0 to 8 do {
66// j = (-1)^(i + 1)/(2*i + 1);
67// a = round(j, D, RN);
68// b = round(j - a, D, RN);
69// print("{", b, ",", a, "},");
70// };
71constexpr fputil::DoubleDouble COEFFS[9] = {
72 {0.0, 1.0}, // 1
73 {-0x1.5555555555555p-56, -0x1.5555555555555p-2}, // -1/3
74 {-0x1.999999999999ap-57, 0x1.999999999999ap-3}, // 1/5
75 {-0x1.2492492492492p-57, -0x1.2492492492492p-3}, // -1/7
76 {0x1.c71c71c71c71cp-58, 0x1.c71c71c71c71cp-4}, // 1/9
77 {0x1.745d1745d1746p-59, -0x1.745d1745d1746p-4}, // -1/11
78 {-0x1.3b13b13b13b14p-58, 0x1.3b13b13b13b14p-4}, // 1/13
79 {-0x1.1111111111111p-60, -0x1.1111111111111p-4}, // -1/15
80 {0x1.e1e1e1e1e1e1ep-61, 0x1.e1e1e1e1e1e1ep-5}, // 1/17
81};
82
83// Veltkamp's splitting of a double precision into hi + lo, where the hi part is
84// slightly smaller than an even split, so that the product of
85// hi * (s1 * k + s2) is exact,
86// where:
87// s1, s2 are single precsion,
88// 1/16 <= s1/s2 <= 1
89// 1/16 <= k <= 1 is an integer.
90// So the maximal precision of (s1 * k + s2) is:
91// prec(s1 * k + s2) = 2 + log2(msb(s2)) - log2(lsb(k_d * s1))
92// = 2 + log2(msb(s1)) + 4 - log2(lsb(k_d)) - log2(lsb(s1))
93// = 2 + log2(lsb(s1)) + 23 + 4 - (-4) - log2(lsb(s1))
94// = 33.
95// Thus, the Veltkamp splitting constant is C = 2^33 + 1.
96// This is used when FMA instruction is not available.
97[[maybe_unused]] constexpr fputil::DoubleDouble split_d(double a) {
98 fputil::DoubleDouble r{0.0, 0.0};
99 constexpr double C = 0x1.0p33 + 1.0;
100 double t1 = C * a;
101 double t2 = a - t1;
102 r.hi = t1 + t2;
103 r.lo = a - r.hi;
104 return r;
105}
106
107// Compute atan( num_d / den_d ) in double-double precision.
108// num_d = min(|x|, |y|)
109// den_d = max(|x|, |y|)
110// q_d = num_d / den_d
111// idx, k_d = round( 2^4 * num_d / den_d )
112// final_sign = sign of the final result
113// const_term = the constant term in the final expression.
114float atan2f_double_double(double num_d, double den_d, double q_d, int idx,
115 double k_d, double final_sign,
116 const fputil::DoubleDouble &const_term) {
117 fputil::DoubleDouble q;
118 double num_r, den_r;
119
120 if (idx != 0) {
121 // The following range reduction is accurate even without fma for
122 // 1/16 <= n/d <= 1.
123 // atan(n/d) - atan(idx/16) = atan((n/d - idx/16) / (1 + (n/d) * (idx/16)))
124 // = atan((n - d*(idx/16)) / (d + n*idx/16))
125 k_d *= 0x1.0p-4;
126 num_r = fputil::multiply_add(k_d, -den_d, num_d); // Exact
127 den_r = fputil::multiply_add(k_d, num_d, den_d); // Exact
128 q.hi = num_r / den_r;
129 } else {
130 // For 0 < n/d < 1/16, we just need to calculate the lower part of their
131 // quotient.
132 q.hi = q_d;
133 num_r = num_d;
134 den_r = den_d;
135 }
136#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
137 q.lo = fputil::multiply_add(q.hi, -den_r, num_r) / den_r;
138#else
139 // Compute `(num_r - q.hi * den_r) / den_r` accurately without FMA
140 // instructions.
141 fputil::DoubleDouble q_hi_dd = split_d(q.hi);
142 double t1 = fputil::multiply_add(q_hi_dd.hi, -den_r, num_r); // Exact
143 double t2 = fputil::multiply_add(q_hi_dd.lo, -den_r, t1);
144 q.lo = t2 / den_r;
145#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
146
147 // Taylor polynomial, evaluating using Horner's scheme:
148 // P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
149 // + x^17/17
150 // = x*(1 + x^2*(-1/3 + x^2*(1/5 + x^2*(-1/7 + x^2*(1/9 + x^2*
151 // *(-1/11 + x^2*(1/13 + x^2*(-1/15 + x^2 * 1/17))))))))
152 fputil::DoubleDouble q2 = fputil::quick_mult(q, q);
153 fputil::DoubleDouble p_dd =
154 fputil::polyeval(q2, COEFFS[0], COEFFS[1], COEFFS[2], COEFFS[3],
155 COEFFS[4], COEFFS[5], COEFFS[6], COEFFS[7], COEFFS[8]);
156 fputil::DoubleDouble r_dd =
157 fputil::add(const_term, fputil::multiply_add(q, p_dd, ATAN_I[idx]));
158 r_dd.hi *= final_sign;
159 r_dd.lo *= final_sign;
160
161 // Make sure the sum is normalized:
162 fputil::DoubleDouble rr = fputil::exact_add(r_dd.hi, r_dd.lo);
163 // Round to odd.
164 uint64_t rr_bits = cpp::bit_cast<uint64_t>(rr.hi);
165 if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) {
166 Sign hi_sign = fputil::FPBits<double>(rr.hi).sign();
167 Sign lo_sign = fputil::FPBits<double>(rr.lo).sign();
168 if (hi_sign == lo_sign) {
169 ++rr_bits;
170 } else if ((rr_bits & fputil::FPBits<double>::FRACTION_MASK) > 0) {
171 --rr_bits;
172 }
173 }
174
175 return static_cast<float>(cpp::bit_cast<double>(rr_bits));
176}
177
178#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
179
180} // anonymous namespace
181
182// There are several range reduction steps we can take for atan2(y, x) as
183// follow:
184
185// * Range reduction 1: signness
186// atan2(y, x) will return a number between -PI and PI representing the angle
187// forming by the 0x axis and the vector (x, y) on the 0xy-plane.
188// In particular, we have that:
189// atan2(y, x) = atan( y/x ) if x >= 0 and y >= 0 (I-quadrant)
190// = pi + atan( y/x ) if x < 0 and y >= 0 (II-quadrant)
191// = -pi + atan( y/x ) if x < 0 and y < 0 (III-quadrant)
192// = atan( y/x ) if x >= 0 and y < 0 (IV-quadrant)
193// Since atan function is odd, we can use the formula:
194// atan(-u) = -atan(u)
195// to adjust the above conditions a bit further:
196// atan2(y, x) = atan( |y|/|x| ) if x >= 0 and y >= 0 (I-quadrant)
197// = pi - atan( |y|/|x| ) if x < 0 and y >= 0 (II-quadrant)
198// = -pi + atan( |y|/|x| ) if x < 0 and y < 0 (III-quadrant)
199// = -atan( |y|/|x| ) if x >= 0 and y < 0 (IV-quadrant)
200// Which can be simplified to:
201// atan2(y, x) = sign(y) * atan( |y|/|x| ) if x >= 0
202// = sign(y) * (pi - atan( |y|/|x| )) if x < 0
203
204// * Range reduction 2: reciprocal
205// Now that the argument inside atan is positive, we can use the formula:
206// atan(1/x) = pi/2 - atan(x)
207// to make the argument inside atan <= 1 as follow:
208// atan2(y, x) = sign(y) * atan( |y|/|x|) if 0 <= |y| <= x
209// = sign(y) * (pi/2 - atan( |x|/|y| ) if 0 <= x < |y|
210// = sign(y) * (pi - atan( |y|/|x| )) if 0 <= |y| <= -x
211// = sign(y) * (pi/2 + atan( |x|/|y| )) if 0 <= -x < |y|
212
213// * Range reduction 3: look up table.
214// After the previous two range reduction steps, we reduce the problem to
215// compute atan(u) with 0 <= u <= 1, or to be precise:
216// atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
217// An accurate polynomial approximation for the whole [0, 1] input range will
218// require a very large degree. To make it more efficient, we reduce the input
219// range further by finding an integer idx such that:
220// | n/d - idx/16 | <= 1/32.
221// In particular,
222// idx := 2^-4 * round(2^4 * n/d)
223// Then for the fast pass, we find a polynomial approximation for:
224// atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16)
225// For the accurate pass, we use the addition formula:
226// atan( n/d ) - atan( idx/16 ) = atan( (n/d - idx/16)/(1 + (n*idx)/(16*d)) )
227// = atan( (n - d * idx/16)/(d + n * idx/16) )
228// And finally we use Taylor polynomial to compute the RHS in the accurate pass:
229// atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 - u^11/11 + u^13/13 -
230// - u^15/15 + u^17/17
231// It's error in double-double precision is estimated in Sollya to be:
232// > P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
233// + x^17/17;
234// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
235// 0x1.aec6f...p-100
236// which is about rounding errors of double-double (2^-104).
237
238LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
239 using FPBits = typename fputil::FPBits<float>;
240 constexpr double IS_NEG[2] = {1.0, -1.0};
241 constexpr double PI = 0x1.921fb54442d18p1;
242 constexpr double PI_LO = 0x1.1a62633145c07p-53;
243 constexpr double PI_OVER_4 = 0x1.921fb54442d18p-1;
244 constexpr double PI_OVER_2 = 0x1.921fb54442d18p0;
245 constexpr double THREE_PI_OVER_4 = 0x1.2d97c7f3321d2p+1;
246 // Adjustment for constant term:
247 // CONST_ADJ[x_sign][y_sign][recip]
248 constexpr fputil::DoubleDouble CONST_ADJ[2][2][2] = {
249 {{{0.0, 0.0}, {-PI_LO / 2, -PI_OVER_2}},
250 {{-0.0, -0.0}, {-PI_LO / 2, -PI_OVER_2}}},
251 {{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}},
252 {{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}}}};
253
254 FPBits x_bits(x), y_bits(y);
255 bool x_sign = x_bits.sign().is_neg();
256 bool y_sign = y_bits.sign().is_neg();
257 x_bits.set_sign(Sign::POS);
258 y_bits.set_sign(Sign::POS);
259 uint32_t x_abs = x_bits.uintval();
260 uint32_t y_abs = y_bits.uintval();
261 uint32_t max_abs = x_abs > y_abs ? x_abs : y_abs;
262 uint32_t min_abs = x_abs <= y_abs ? x_abs : y_abs;
263 float num_f = FPBits(min_abs).get_val();
264 float den_f = FPBits(max_abs).get_val();
265 double num_d = static_cast<double>(num_f);
266 double den_d = static_cast<double>(den_f);
267
268 if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || num_d == 0.0)) {
269 if (x_bits.is_nan() || y_bits.is_nan()) {
270 if (x_bits.is_signaling_nan() || y_bits.is_signaling_nan())
271 fputil::raise_except_if_required(FE_INVALID);
272 return FPBits::quiet_nan().get_val();
273 }
274 double x_d = static_cast<double>(x);
275 double y_d = static_cast<double>(y);
276 size_t x_except = (x_d == 0.0) ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1);
277 size_t y_except = (y_d == 0.0) ? 0 : (y_abs == 0x7f80'0000 ? 2 : 1);
278
279 // Exceptional cases:
280 // EXCEPT[y_except][x_except][x_is_neg]
281 // with x_except & y_except:
282 // 0: zero
283 // 1: finite, non-zero
284 // 2: infinity
285 constexpr double EXCEPTS[3][3][2] = {
286 {{0.0, PI}, {0.0, PI}, {0.0, PI}},
287 {{PI_OVER_2, PI_OVER_2}, {0.0, 0.0}, {0.0, PI}},
288 {{PI_OVER_2, PI_OVER_2},
289 {PI_OVER_2, PI_OVER_2},
290 {PI_OVER_4, THREE_PI_OVER_4}},
291 };
292
293 double r = IS_NEG[y_sign] * EXCEPTS[y_except][x_except][x_sign];
294
295 return static_cast<float>(r);
296 }
297
298 bool recip = x_abs < y_abs;
299 double final_sign = IS_NEG[(x_sign != y_sign) != recip];
300 fputil::DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
301 double q_d = num_d / den_d;
302
303 double k_d = fputil::nearest_integer(q_d * 0x1.0p4);
304 int idx = static_cast<int>(k_d);
305 double r;
306
307#ifdef LIBC_MATH_HAS_SMALL_TABLES
308 double p = atan_eval_no_table(num_d, den_d, k_d * 0x1.0p-4);
309 r = final_sign * (p + (const_term.hi + ATAN_K_OVER_16[idx]));
310#else
311 q_d = fputil::multiply_add(k_d, -0x1.0p-4, q_d);
312
313 double p = atan_eval(q_d, idx);
314 r = final_sign *
315 fputil::multiply_add(q_d, p, const_term.hi + ATAN_COEFFS[idx][0]);
316#endif // LIBC_MATH_HAS_SMALL_TABLES
317
318#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
319 return static_cast<float>(r);
320#else
321 constexpr uint32_t LOWER_ERR = 4;
322 // Mask sticky bits in double precision before rounding to single precision.
323 constexpr uint32_t MASK =
324 mask_trailing_ones<uint32_t, fputil::FPBits<double>::SIG_LEN -
325 FPBits::SIG_LEN - 1>();
326 constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR;
327
328 uint32_t r_bits = static_cast<uint32_t>(cpp::bit_cast<uint64_t>(r)) & MASK;
329
330 // Ziv's rounding test.
331 if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR))
332 return static_cast<float>(r);
333
334 return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign,
335 const_term);
336#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
337}
338
339} // namespace LIBC_NAMESPACE_DECL
340
341#endif
342

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