1 //===-- Double-precision tan 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/tan.h"
10 #include "hdr/errno_macros.h"
11 #include "src/__support/FPUtil/FEnvImpl.h"
12 #include "src/__support/FPUtil/FPBits.h"
13 #include "src/__support/FPUtil/PolyEval.h"
14 #include "src/__support/FPUtil/double_double.h"
15 #include "src/__support/FPUtil/dyadic_float.h"
16 #include "src/__support/FPUtil/except_value_utils.h"
17 #include "src/__support/FPUtil/multiply_add.h"
18 #include "src/__support/FPUtil/rounding_mode.h"
19 #include "src/__support/common.h"
20 #include "src/__support/macros/config.h"
21 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
22 #include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
23 #include "src/math/generic/range_reduction_double_common.h"
24
25 #ifdef LIBC_TARGET_CPU_HAS_FMA
26 #include "range_reduction_double_fma.h"
27 #else
28 #include "range_reduction_double_nofma.h"
29 #endif // LIBC_TARGET_CPU_HAS_FMA
30
31 namespace LIBC_NAMESPACE_DECL {
32
33 using DoubleDouble = fputil::DoubleDouble;
34 using Float128 = typename fputil::DyadicFloat<128>;
35
36 namespace {
37
tan_eval(const DoubleDouble & u,DoubleDouble & result)38 LIBC_INLINE double tan_eval(const DoubleDouble &u, DoubleDouble &result) {
39 // Evaluate tan(y) = tan(x - k * (pi/128))
40 // We use the degree-9 Taylor approximation:
41 // tan(y) ~ P(y) = y + y^3/3 + 2*y^5/15 + 17*y^7/315 + 62*y^9/2835
42 // Then the error is bounded by:
43 // |tan(y) - P(y)| < 2^-6 * |y|^11 < 2^-6 * 2^-66 = 2^-72.
44 // For y ~ u_hi + u_lo, fully expanding the polynomial and drop any terms
45 // < ulp(u_hi^3) gives us:
46 // P(y) = y + y^3/3 + 2*y^5/15 + 17*y^7/315 + 62*y^9/2835 = ...
47 // ~ u_hi + u_hi^3 * (1/3 + u_hi^2 * (2/15 + u_hi^2 * (17/315 +
48 // + u_hi^2 * 62/2835))) +
49 // + u_lo (1 + u_hi^2 * (1 + u_hi^2 * 2/3))
50 double u_hi_sq = u.hi * u.hi; // Error < ulp(u_hi^2) < 2^(-6 - 52) = 2^-58.
51 // p1 ~ 17/315 + u_hi^2 62 / 2835.
52 double p1 =
53 fputil::multiply_add(u_hi_sq, 0x1.664f4882c10fap-6, 0x1.ba1ba1ba1ba1cp-5);
54 // p2 ~ 1/3 + u_hi^2 2 / 15.
55 double p2 =
56 fputil::multiply_add(u_hi_sq, 0x1.1111111111111p-3, 0x1.5555555555555p-2);
57 // q1 ~ 1 + u_hi^2 * 2/3.
58 double q1 = fputil::multiply_add(u_hi_sq, 0x1.5555555555555p-1, 1.0);
59 double u_hi_3 = u_hi_sq * u.hi;
60 double u_hi_4 = u_hi_sq * u_hi_sq;
61 // p3 ~ 1/3 + u_hi^2 * (2/15 + u_hi^2 * (17/315 + u_hi^2 * 62/2835))
62 double p3 = fputil::multiply_add(u_hi_4, p1, p2);
63 // q2 ~ 1 + u_hi^2 * (1 + u_hi^2 * 2/3)
64 double q2 = fputil::multiply_add(u_hi_sq, q1, 1.0);
65 double tan_lo = fputil::multiply_add(u_hi_3, p3, u.lo * q2);
66 // Overall, |tan(y) - (u_hi + tan_lo)| < ulp(u_hi^3) <= 2^-71.
67 // And the relative errors is:
68 // |(tan(y) - (u_hi + tan_lo)) / tan(y) | <= 2*ulp(u_hi^2) < 2^-64
69 result = fputil::exact_add(u.hi, tan_lo);
70 return fputil::multiply_add(fputil::FPBits<double>(u_hi_3).abs().get_val(),
71 0x1.0p-51, 0x1.0p-102);
72 }
73
74 #ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
75 // Accurate evaluation of tan for small u.
tan_eval(const Float128 & u)76 [[maybe_unused]] Float128 tan_eval(const Float128 &u) {
77 Float128 u_sq = fputil::quick_mul(u, u);
78
79 // tan(x) ~ x + x^3/3 + x^5 * 2/15 + x^7 * 17/315 + x^9 * 62/2835 +
80 // + x^11 * 1382/155925 + x^13 * 21844/6081075 +
81 // + x^15 * 929569/638512875 + x^17 * 6404582/10854718875
82 // Relative errors < 2^-127 for |u| < pi/256.
83 constexpr Float128 TAN_COEFFS[] = {
84 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1
85 {Sign::POS, -129, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1
86 {Sign::POS, -130, 0x88888888'88888888'88888888'88888889_u128}, // 2/15
87 {Sign::POS, -132, 0xdd0dd0dd'0dd0dd0d'd0dd0dd0'dd0dd0dd_u128}, // 17/315
88 {Sign::POS, -133, 0xb327a441'6087cf99'6b5dd24e'ec0b327a_u128}, // 62/2835
89 {Sign::POS, -134,
90 0x91371aaf'3611e47a'da8e1cba'7d900eca_u128}, // 1382/155925
91 {Sign::POS, -136,
92 0xeb69e870'abeefdaf'e606d2e4'd1e65fbc_u128}, // 21844/6081075
93 {Sign::POS, -137,
94 0xbed1b229'5baf15b5'0ec9af45'a2619971_u128}, // 929569/638512875
95 {Sign::POS, -138,
96 0x9aac1240'1b3a2291'1b2ac7e3'e4627d0a_u128}, // 6404582/10854718875
97 };
98
99 return fputil::quick_mul(
100 u, fputil::polyeval(u_sq, TAN_COEFFS[0], TAN_COEFFS[1], TAN_COEFFS[2],
101 TAN_COEFFS[3], TAN_COEFFS[4], TAN_COEFFS[5],
102 TAN_COEFFS[6], TAN_COEFFS[7], TAN_COEFFS[8]));
103 }
104
105 // Calculation a / b = a * (1/b) for Float128.
106 // Using the initial approximation of q ~ (1/b), then apply 2 Newton-Raphson
107 // iterations, before multiplying by a.
newton_raphson_div(const Float128 & a,Float128 b,double q)108 [[maybe_unused]] Float128 newton_raphson_div(const Float128 &a, Float128 b,
109 double q) {
110 Float128 q0(q);
111 constexpr Float128 TWO(2.0);
112 b.sign = (b.sign == Sign::POS) ? Sign::NEG : Sign::POS;
113 Float128 q1 =
114 fputil::quick_mul(q0, fputil::quick_add(TWO, fputil::quick_mul(b, q0)));
115 Float128 q2 =
116 fputil::quick_mul(q1, fputil::quick_add(TWO, fputil::quick_mul(b, q1)));
117 return fputil::quick_mul(a, q2);
118 }
119 #endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
120
121 } // anonymous namespace
122
123 LLVM_LIBC_FUNCTION(double, tan, (double x)) {
124 using FPBits = typename fputil::FPBits<double>;
125 FPBits xbits(x);
126
127 uint16_t x_e = xbits.get_biased_exponent();
128
129 DoubleDouble y;
130 unsigned k;
131 LargeRangeReduction range_reduction_large{};
132
133 // |x| < 2^16
134 if (LIBC_LIKELY(x_e < FPBits::EXP_BIAS + FAST_PASS_EXPONENT)) {
135 // |x| < 2^-7
136 if (LIBC_UNLIKELY(x_e < FPBits::EXP_BIAS - 7)) {
137 // |x| < 2^-27, |tan(x) - x| < ulp(x)/2.
138 if (LIBC_UNLIKELY(x_e < FPBits::EXP_BIAS - 27)) {
139 // Signed zeros.
140 if (LIBC_UNLIKELY(x == 0.0))
141 return x + x; // Make sure it works with FTZ/DAZ.
142
143 #ifdef LIBC_TARGET_CPU_HAS_FMA
144 return fputil::multiply_add(x, 0x1.0p-54, x);
145 #else
146 if (LIBC_UNLIKELY(x_e < 4)) {
147 int rounding_mode = fputil::quick_get_round();
148 if ((xbits.sign() == Sign::POS && rounding_mode == FE_UPWARD) ||
149 (xbits.sign() == Sign::NEG && rounding_mode == FE_DOWNWARD))
150 return FPBits(xbits.uintval() + 1).get_val();
151 }
152 return fputil::multiply_add(x, 0x1.0p-54, x);
153 #endif // LIBC_TARGET_CPU_HAS_FMA
154 }
155 // No range reduction needed.
156 k = 0;
157 y.lo = 0.0;
158 y.hi = x;
159 } else {
160 // Small range reduction.
161 k = range_reduction_small(x, y);
162 }
163 } else {
164 // Inf or NaN
165 if (LIBC_UNLIKELY(x_e > 2 * FPBits::EXP_BIAS)) {
166 // tan(+-Inf) = NaN
167 if (xbits.get_mantissa() == 0) {
168 fputil::set_errno_if_required(EDOM);
169 fputil::raise_except_if_required(FE_INVALID);
170 }
171 return x + FPBits::quiet_nan().get_val();
172 }
173
174 // Large range reduction.
175 k = range_reduction_large.fast(x, y);
176 }
177
178 DoubleDouble tan_y;
179 [[maybe_unused]] double err = tan_eval(y, tan_y);
180
181 // Look up sin(k * pi/128) and cos(k * pi/128)
182 #ifdef LIBC_MATH_HAS_SMALL_TABLES
183 // Memory saving versions. Use 65-entry table:
__anon2c32d4080202(unsigned kk) 184 auto get_idx_dd = [](unsigned kk) -> DoubleDouble {
185 unsigned idx = (kk & 64) ? 64 - (kk & 63) : (kk & 63);
186 DoubleDouble ans = SIN_K_PI_OVER_128[idx];
187 if (kk & 128) {
188 ans.hi = -ans.hi;
189 ans.lo = -ans.lo;
190 }
191 return ans;
192 };
193 DoubleDouble msin_k = get_idx_dd(k + 128);
194 DoubleDouble cos_k = get_idx_dd(k + 64);
195 #else
196 // Fast look up version, but needs 256-entry table.
197 // cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).
198 DoubleDouble msin_k = SIN_K_PI_OVER_128[(k + 128) & 255];
199 DoubleDouble cos_k = SIN_K_PI_OVER_128[(k + 64) & 255];
200 #endif // LIBC_MATH_HAS_SMALL_TABLES
201
202 // After range reduction, k = round(x * 128 / pi) and y = x - k * (pi / 128).
203 // So k is an integer and -pi / 256 <= y <= pi / 256.
204 // Then tan(x) = sin(x) / cos(x)
205 // = sin((k * pi/128 + y) / cos((k * pi/128 + y)
206 // = (cos(y) * sin(k*pi/128) + sin(y) * cos(k*pi/128)) /
207 // / (cos(y) * cos(k*pi/128) - sin(y) * sin(k*pi/128))
208 // = (sin(k*pi/128) + tan(y) * cos(k*pi/128)) /
209 // / (cos(k*pi/128) - tan(y) * sin(k*pi/128))
210 DoubleDouble cos_k_tan_y = fputil::quick_mult(tan_y, cos_k);
211 DoubleDouble msin_k_tan_y = fputil::quick_mult(tan_y, msin_k);
212
213 // num_dd = sin(k*pi/128) + tan(y) * cos(k*pi/128)
214 DoubleDouble num_dd = fputil::exact_add<false>(cos_k_tan_y.hi, -msin_k.hi);
215 // den_dd = cos(k*pi/128) - tan(y) * sin(k*pi/128)
216 DoubleDouble den_dd = fputil::exact_add<false>(msin_k_tan_y.hi, cos_k.hi);
217 num_dd.lo += cos_k_tan_y.lo - msin_k.lo;
218 den_dd.lo += msin_k_tan_y.lo + cos_k.lo;
219
220 #ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
221 double tan_x = (num_dd.hi + num_dd.lo) / (den_dd.hi + den_dd.lo);
222 return tan_x;
223 #else
224 // Accurate test and pass for correctly rounded implementation.
225
226 // Accurate double-double division
227 DoubleDouble tan_x = fputil::div(num_dd, den_dd);
228
229 // Simple error bound: |1 / den_dd| < 2^(1 + floor(-log2(den_dd)))).
230 uint64_t den_inv = (static_cast<uint64_t>(FPBits::EXP_BIAS + 1)
231 << (FPBits::FRACTION_LEN + 1)) -
232 (FPBits(den_dd.hi).uintval() & FPBits::EXP_MASK);
233
234 // For tan_x = (num_dd + err) / (den_dd + err), the error is bounded by:
235 // | tan_x - num_dd / den_dd | <= err * ( 1 + | tan_x * den_dd | ).
236 double tan_err =
237 err * fputil::multiply_add(FPBits(den_inv).get_val(),
238 FPBits(tan_x.hi).abs().get_val(), 1.0);
239
240 double err_higher = tan_x.lo + tan_err;
241 double err_lower = tan_x.lo - tan_err;
242
243 double tan_upper = tan_x.hi + err_higher;
244 double tan_lower = tan_x.hi + err_lower;
245
246 // Ziv's rounding test.
247 if (LIBC_LIKELY(tan_upper == tan_lower))
248 return tan_upper;
249
250 Float128 u_f128;
251 if (LIBC_LIKELY(x_e < FPBits::EXP_BIAS + FAST_PASS_EXPONENT))
252 u_f128 = range_reduction_small_f128(x);
253 else
254 u_f128 = range_reduction_large.accurate();
255
256 Float128 tan_u = tan_eval(u_f128);
257
__anon2c32d4080302(unsigned kk) 258 auto get_sin_k = [](unsigned kk) -> Float128 {
259 unsigned idx = (kk & 64) ? 64 - (kk & 63) : (kk & 63);
260 Float128 ans = SIN_K_PI_OVER_128_F128[idx];
261 if (kk & 128)
262 ans.sign = Sign::NEG;
263 return ans;
264 };
265
266 // cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).
267 Float128 sin_k_f128 = get_sin_k(k);
268 Float128 cos_k_f128 = get_sin_k(k + 64);
269 Float128 msin_k_f128 = get_sin_k(k + 128);
270
271 // num_f128 = sin(k*pi/128) + tan(y) * cos(k*pi/128)
272 Float128 num_f128 =
273 fputil::quick_add(sin_k_f128, fputil::quick_mul(cos_k_f128, tan_u));
274 // den_f128 = cos(k*pi/128) - tan(y) * sin(k*pi/128)
275 Float128 den_f128 =
276 fputil::quick_add(cos_k_f128, fputil::quick_mul(msin_k_f128, tan_u));
277
278 // tan(x) = (sin(k*pi/128) + tan(y) * cos(k*pi/128)) /
279 // / (cos(k*pi/128) - tan(y) * sin(k*pi/128))
280 // TODO: The initial seed 1.0/den_dd.hi for Newton-Raphson reciprocal can be
281 // reused from DoubleDouble fputil::div in the fast pass.
282 Float128 result = newton_raphson_div(num_f128, den_f128, 1.0 / den_dd.hi);
283
284 // TODO: Add assertion if Ziv's accuracy tests fail in debug mode.
285 // https://github.com/llvm/llvm-project/issues/96452.
286 return static_cast<double>(result);
287
288 #endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
289 }
290
291 } // namespace LIBC_NAMESPACE_DECL
292