1 //===-- Double-precision x^y 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/pow.h"
10 #include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2.
11 #include "hdr/errno_macros.h"
12 #include "hdr/fenv_macros.h"
13 #include "src/__support/CPP/bit.h"
14 #include "src/__support/FPUtil/FEnvImpl.h"
15 #include "src/__support/FPUtil/FPBits.h"
16 #include "src/__support/FPUtil/PolyEval.h"
17 #include "src/__support/FPUtil/double_double.h"
18 #include "src/__support/FPUtil/multiply_add.h"
19 #include "src/__support/FPUtil/nearest_integer.h"
20 #include "src/__support/FPUtil/sqrt.h" // Speedup for pow(x, 1/2) = sqrt(x)
21 #include "src/__support/common.h"
22 #include "src/__support/macros/config.h"
23 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
24
25 namespace LIBC_NAMESPACE_DECL {
26
27 using fputil::DoubleDouble;
28
29 namespace {
30
31 // Constants for log2(x) range reduction, generated by Sollya with:
32 // > for i from 0 to 127 do {
33 // r = 2^-8 * ceil( 2^8 * (1 - 2^(-8)) / (1 + i*2^-7) );
34 // b = nearestint(log2(r) * 2^41) * 2^-41;
35 // c = round(log2(r) - b, D, RN);
36 // print("{", -c, ",", -b, "},");
37 // };
38 // This is the same as -log2(RD[i]), with the least significant bits of the
39 // high part set to be 2^-41, so that the sum of high parts + e_x is exact in
40 // double precision.
41 // We also replace the first and the last ones to be 0.
42 constexpr DoubleDouble LOG2_R_DD[128] = {
43 {0.0, 0.0},
44 {-0x1.19b14945cf6bap-44, 0x1.72c7ba21p-7},
45 {-0x1.95539356f93dcp-43, 0x1.743ee862p-6},
46 {0x1.abe0a48f83604p-43, 0x1.184b8e4c5p-5},
47 {0x1.635577970e04p-43, 0x1.77394c9d9p-5},
48 {-0x1.401fbaaa67e3cp-45, 0x1.d6ebd1f2p-5},
49 {-0x1.5b1799ceaeb51p-43, 0x1.1bb32a6008p-4},
50 {0x1.7c407050799bfp-43, 0x1.4c560fe688p-4},
51 {0x1.da6339da288fcp-43, 0x1.7d60496cf8p-4},
52 {0x1.be4f6f22dbbadp-43, 0x1.960caf9ab8p-4},
53 {-0x1.c760bc9b188c4p-45, 0x1.c7b528b71p-4},
54 {0x1.164e932b2d51cp-44, 0x1.f9c95dc1dp-4},
55 {0x1.924ae921f7ecap-45, 0x1.097e38ce6p-3},
56 {-0x1.6d25a5b8a19b2p-44, 0x1.22dadc2ab4p-3},
57 {0x1.e50a1644ac794p-43, 0x1.3c6fb650ccp-3},
58 {0x1.f34baa74a7942p-43, 0x1.494f863b8cp-3},
59 {-0x1.8f7aac147fdc1p-46, 0x1.633a8bf438p-3},
60 {0x1.f84be19cb9578p-43, 0x1.7046031c78p-3},
61 {-0x1.66cccab240e9p-46, 0x1.8a8980abfcp-3},
62 {-0x1.3f7a55cd2af4cp-47, 0x1.97c1cb13c8p-3},
63 {0x1.3458cde69308cp-43, 0x1.b2602497d4p-3},
64 {-0x1.667f21fa8423fp-44, 0x1.bfc67a8p-3},
65 {0x1.d2fe4574e09b9p-47, 0x1.dac22d3e44p-3},
66 {0x1.367bde40c5e6dp-43, 0x1.e857d3d36p-3},
67 {0x1.d45da26510033p-46, 0x1.01d9bbcfa6p-2},
68 {-0x1.7204f55bbf90dp-44, 0x1.08bce0d96p-2},
69 {-0x1.d4f1b95e0ff45p-43, 0x1.169c05364p-2},
70 {0x1.c20d74c0211bfp-44, 0x1.1d982c9d52p-2},
71 {0x1.ad89a083e072ap-43, 0x1.249cd2b13cp-2},
72 {0x1.cd0cb4492f1bcp-43, 0x1.32bfee370ep-2},
73 {-0x1.2101a9685c779p-47, 0x1.39de8e155ap-2},
74 {0x1.9451cd394fe8dp-43, 0x1.4106017c3ep-2},
75 {0x1.661e393a16b95p-44, 0x1.4f6fbb2cecp-2},
76 {-0x1.c6d8d86531d56p-44, 0x1.56b22e6b58p-2},
77 {0x1.c1c885adb21d3p-43, 0x1.5dfdcf1eeap-2},
78 {0x1.3bb5921006679p-45, 0x1.6552b49986p-2},
79 {0x1.1d406db502403p-43, 0x1.6cb0f6865cp-2},
80 {0x1.55a63e278bad5p-43, 0x1.7b89f02cf2p-2},
81 {-0x1.66ae2a7ada553p-49, 0x1.8304d90c12p-2},
82 {-0x1.66cccab240e9p-45, 0x1.8a8980abfcp-2},
83 {-0x1.62404772a151dp-45, 0x1.921800924ep-2},
84 {0x1.ac9bca36fd02ep-44, 0x1.99b072a96cp-2},
85 {0x1.4bc302ffa76fbp-43, 0x1.a8ff97181p-2},
86 {0x1.01fea1ec47c71p-43, 0x1.b0b67f4f46p-2},
87 {-0x1.f20203b3186a6p-43, 0x1.b877c57b1cp-2},
88 {-0x1.2642415d47384p-45, 0x1.c043859e3p-2},
89 {-0x1.bc76a2753b99bp-50, 0x1.c819dc2d46p-2},
90 {-0x1.da93ae3a5f451p-43, 0x1.cffae611aep-2},
91 {-0x1.50e785694a8c6p-43, 0x1.d7e6c0abc4p-2},
92 {0x1.c56138c894641p-43, 0x1.dfdd89d586p-2},
93 {0x1.5669df6a2b592p-43, 0x1.e7df5fe538p-2},
94 {-0x1.ea92d9e0e8ac2p-48, 0x1.efec61b012p-2},
95 {0x1.a0331af2e6feap-43, 0x1.f804ae8d0cp-2},
96 {0x1.9518ce032f41dp-48, 0x1.0014332bep-1},
97 {-0x1.b3b3864c60011p-44, 0x1.042bd4b9a8p-1},
98 {-0x1.103e8f00d41c8p-45, 0x1.08494c66b9p-1},
99 {0x1.65be75cc3da17p-43, 0x1.0c6caaf0c5p-1},
100 {0x1.3676289cd3dd4p-43, 0x1.1096015deep-1},
101 {-0x1.41dfc7d7c3321p-43, 0x1.14c560fe69p-1},
102 {0x1.e0cda8bd74461p-44, 0x1.18fadb6e2dp-1},
103 {0x1.2a606046ad444p-44, 0x1.1d368296b5p-1},
104 {0x1.f9ea977a639cp-43, 0x1.217868b0c3p-1},
105 {-0x1.50520a377c7ecp-45, 0x1.25c0a0463cp-1},
106 {0x1.6e3cb71b554e7p-47, 0x1.2a0f3c3407p-1},
107 {-0x1.4275f1035e5e8p-48, 0x1.2e644fac05p-1},
108 {-0x1.4275f1035e5e8p-48, 0x1.2e644fac05p-1},
109 {-0x1.979a5db68721dp-45, 0x1.32bfee370fp-1},
110 {0x1.1ee969a95f529p-43, 0x1.37222bb707p-1},
111 {0x1.bb4b69336b66ep-43, 0x1.3b8b1c68fap-1},
112 {0x1.d5e6a8a4fb059p-45, 0x1.3ffad4e74fp-1},
113 {0x1.3106e404cabb7p-44, 0x1.44716a2c08p-1},
114 {0x1.3106e404cabb7p-44, 0x1.44716a2c08p-1},
115 {-0x1.9bcaf1aa4168ap-43, 0x1.48eef19318p-1},
116 {0x1.1646b761c48dep-44, 0x1.4d7380dcc4p-1},
117 {0x1.2f0c0bfe9dbecp-43, 0x1.51ff2e3021p-1},
118 {0x1.29904613e33cp-43, 0x1.5692101d9bp-1},
119 {0x1.1d406db502403p-44, 0x1.5b2c3da197p-1},
120 {0x1.1d406db502403p-44, 0x1.5b2c3da197p-1},
121 {-0x1.125d6cbcd1095p-44, 0x1.5fcdce2728p-1},
122 {-0x1.bd9b32266d92cp-43, 0x1.6476d98adap-1},
123 {0x1.54243b21709cep-44, 0x1.6927781d93p-1},
124 {0x1.54243b21709cep-44, 0x1.6927781d93p-1},
125 {-0x1.ce60916e52e91p-44, 0x1.6ddfc2a79p-1},
126 {0x1.f1f5ae718f241p-43, 0x1.729fd26b7p-1},
127 {-0x1.6eb9612e0b4f3p-43, 0x1.7767c12968p-1},
128 {-0x1.6eb9612e0b4f3p-43, 0x1.7767c12968p-1},
129 {0x1.fed21f9cb2cc5p-43, 0x1.7c37a9227ep-1},
130 {0x1.7f5dc57266758p-43, 0x1.810fa51bf6p-1},
131 {0x1.7f5dc57266758p-43, 0x1.810fa51bf6p-1},
132 {0x1.5b338360c2ae2p-43, 0x1.85efd062c6p-1},
133 {-0x1.96fc8f4b56502p-43, 0x1.8ad846cf37p-1},
134 {-0x1.96fc8f4b56502p-43, 0x1.8ad846cf37p-1},
135 {-0x1.bdc81c4db3134p-44, 0x1.8fc924c89bp-1},
136 {0x1.36c101ee1344p-43, 0x1.94c287492cp-1},
137 {0x1.36c101ee1344p-43, 0x1.94c287492cp-1},
138 {0x1.e41fa0a62e6aep-44, 0x1.99c48be206p-1},
139 {-0x1.d97ee9124773bp-46, 0x1.9ecf50bf44p-1},
140 {-0x1.d97ee9124773bp-46, 0x1.9ecf50bf44p-1},
141 {-0x1.3f94e00e7d6bcp-46, 0x1.a3e2f4ac44p-1},
142 {-0x1.6879fa00b120ap-43, 0x1.a8ff971811p-1},
143 {-0x1.6879fa00b120ap-43, 0x1.a8ff971811p-1},
144 {0x1.1659d8e2d7d38p-44, 0x1.ae255819fp-1},
145 {0x1.1e5e0ae0d3f8ap-43, 0x1.b35458761dp-1},
146 {0x1.1e5e0ae0d3f8ap-43, 0x1.b35458761dp-1},
147 {0x1.484a15babcf88p-43, 0x1.b88cb9a2abp-1},
148 {0x1.484a15babcf88p-43, 0x1.b88cb9a2abp-1},
149 {0x1.871a7610e40bdp-45, 0x1.bdce9dcc96p-1},
150 {-0x1.2d90e5edaeceep-43, 0x1.c31a27dd01p-1},
151 {-0x1.2d90e5edaeceep-43, 0x1.c31a27dd01p-1},
152 {-0x1.5dd31d962d373p-43, 0x1.c86f7b7ea5p-1},
153 {-0x1.5dd31d962d373p-43, 0x1.c86f7b7ea5p-1},
154 {-0x1.9ad57391924a7p-43, 0x1.cdcebd2374p-1},
155 {-0x1.3167ccc538261p-44, 0x1.d338120a6ep-1},
156 {-0x1.3167ccc538261p-44, 0x1.d338120a6ep-1},
157 {0x1.c7a4ff65ddbc9p-45, 0x1.d8aba045bp-1},
158 {0x1.c7a4ff65ddbc9p-45, 0x1.d8aba045bp-1},
159 {-0x1.f9ab3cf74babap-44, 0x1.de298ec0bbp-1},
160 {-0x1.f9ab3cf74babap-44, 0x1.de298ec0bbp-1},
161 {0x1.52842c1c1e586p-43, 0x1.e3b20546f5p-1},
162 {0x1.52842c1c1e586p-43, 0x1.e3b20546f5p-1},
163 {0x1.3c6764fc87b4ap-48, 0x1.e9452c8a71p-1},
164 {0x1.3c6764fc87b4ap-48, 0x1.e9452c8a71p-1},
165 {-0x1.a0976c0a2827dp-44, 0x1.eee32e2aedp-1},
166 {-0x1.a0976c0a2827dp-44, 0x1.eee32e2aedp-1},
167 {-0x1.a45314dc4fc42p-43, 0x1.f48c34bd1fp-1},
168 {-0x1.a45314dc4fc42p-43, 0x1.f48c34bd1fp-1},
169 {0x1.ef5d00e390ap-44, 0x1.fa406bd244p-1},
170 {0.0, 1.0},
171 };
172
is_odd_integer(double x)173 bool is_odd_integer(double x) {
174 using FPBits = fputil::FPBits<double>;
175 FPBits xbits(x);
176 uint64_t x_u = xbits.uintval();
177 unsigned x_e = static_cast<unsigned>(xbits.get_biased_exponent());
178 unsigned lsb =
179 static_cast<unsigned>(cpp::countr_zero(x_u | FPBits::EXP_MASK));
180 constexpr unsigned UNIT_EXPONENT =
181 static_cast<unsigned>(FPBits::EXP_BIAS + FPBits::FRACTION_LEN);
182 return (x_e + lsb == UNIT_EXPONENT);
183 }
184
is_integer(double x)185 bool is_integer(double x) {
186 using FPBits = fputil::FPBits<double>;
187 FPBits xbits(x);
188 uint64_t x_u = xbits.uintval();
189 unsigned x_e = static_cast<unsigned>(xbits.get_biased_exponent());
190 unsigned lsb =
191 static_cast<unsigned>(cpp::countr_zero(x_u | FPBits::EXP_MASK));
192 constexpr unsigned UNIT_EXPONENT =
193 static_cast<unsigned>(FPBits::EXP_BIAS + FPBits::FRACTION_LEN);
194 return (x_e + lsb >= UNIT_EXPONENT);
195 }
196
197 } // namespace
198
199 LLVM_LIBC_FUNCTION(double, pow, (double x, double y)) {
200 using FPBits = fputil::FPBits<double>;
201
202 FPBits xbits(x), ybits(y);
203
204 bool x_sign = xbits.sign() == Sign::NEG;
205 bool y_sign = ybits.sign() == Sign::NEG;
206
207 FPBits x_abs = xbits.abs();
208 FPBits y_abs = ybits.abs();
209
210 uint64_t x_mant = xbits.get_mantissa();
211 uint64_t y_mant = ybits.get_mantissa();
212 uint64_t x_u = xbits.uintval();
213 uint64_t x_a = x_abs.uintval();
214 uint64_t y_a = y_abs.uintval();
215
216 double e_x = static_cast<double>(xbits.get_exponent());
217 uint64_t sign = 0;
218
219 ///////// BEGIN - Check exceptional cases ////////////////////////////////////
220
221 // The double precision number that is closest to 1 is (1 - 2^-53), which has
222 // log2(1 - 2^-53) ~ -1.715...p-53.
223 // So if |y| > |1075 / log2(1 - 2^-53)|, and x is finite:
224 // |y * log2(x)| = 0 or > 1075.
225 // Hence x^y will either overflow or underflow if x is not zero.
226 if (LIBC_UNLIKELY(y_mant == 0 || y_a > 0x43d7'4910'd52d'3052 ||
227 x_u == FPBits::one().uintval() ||
228 x_u >= FPBits::inf().uintval() ||
229 x_u < FPBits::min_normal().uintval())) {
230 // Exceptional exponents.
231 if (y == 0.0)
232 return 1.0;
233
234 switch (y_a) {
235 case 0x3fe0'0000'0000'0000: { // y = +-0.5
236 // TODO: speed up x^(-1/2) with rsqrt(x) when available.
237 if (LIBC_UNLIKELY(
238 (x == 0.0 || x_u == FPBits::inf(Sign::NEG).uintval()))) {
239 // pow(-0, 1/2) = +0
240 // pow(-inf, 1/2) = +inf
241 // Make sure it works correctly for FTZ/DAZ.
242 return y_sign ? 1.0 / (x * x) : (x * x);
243 }
244 return y_sign ? (1.0 / fputil::sqrt<double>(x)) : fputil::sqrt<double>(x);
245 }
246 case 0x3ff0'0000'0000'0000: // y = +-1.0
247 return y_sign ? (1.0 / x) : x;
248 case 0x4000'0000'0000'0000: // y = +-2.0;
249 return y_sign ? (1.0 / (x * x)) : (x * x);
250 }
251
252 // |y| > |1075 / log2(1 - 2^-53)|.
253 if (y_a > 0x43d7'4910'd52d'3052) {
254 if (y_a >= 0x7ff0'0000'0000'0000) {
255 // y is inf or nan
256 if (y_mant != 0) {
257 // y is NaN
258 // pow(1, NaN) = 1
259 // pow(x, NaN) = NaN
260 return (x_u == FPBits::one().uintval()) ? 1.0 : y;
261 }
262
263 // Now y is +-Inf
264 if (x_abs.is_nan()) {
265 // pow(NaN, +-Inf) = NaN
266 return x;
267 }
268
269 if (x_a == 0x3ff0'0000'0000'0000) {
270 // pow(+-1, +-Inf) = 1.0
271 return 1.0;
272 }
273
274 if (x == 0.0 && y_sign) {
275 // pow(+-0, -Inf) = +inf and raise FE_DIVBYZERO
276 fputil::set_errno_if_required(EDOM);
277 fputil::raise_except_if_required(FE_DIVBYZERO);
278 return FPBits::inf().get_val();
279 }
280 // pow (|x| < 1, -inf) = +inf
281 // pow (|x| < 1, +inf) = 0.0
282 // pow (|x| > 1, -inf) = 0.0
283 // pow (|x| > 1, +inf) = +inf
284 return ((x_a < FPBits::one().uintval()) == y_sign)
285 ? FPBits::inf().get_val()
286 : 0.0;
287 }
288 // x^y will overflow / underflow in double precision. Set y to a
289 // large enough exponent but not too large, so that the computations
290 // won't overflow in double precision.
291 y = y_sign ? -0x1.0p100 : 0x1.0p100;
292 }
293
294 // y is finite and non-zero.
295
296 if (x_u == FPBits::one().uintval()) {
297 // pow(1, y) = 1
298 return 1.0;
299 }
300
301 // TODO: Speed things up with pow(2, y) = exp2(y) and pow(10, y) = exp10(y).
302
303 if (x == 0.0) {
304 bool out_is_neg = x_sign && is_odd_integer(y);
305 if (y_sign) {
306 // pow(0, negative number) = inf
307 fputil::set_errno_if_required(EDOM);
308 fputil::raise_except_if_required(FE_DIVBYZERO);
309 return FPBits::inf(out_is_neg ? Sign::NEG : Sign::POS).get_val();
310 }
311 // pow(0, positive number) = 0
312 return out_is_neg ? -0.0 : 0.0;
313 }
314
315 if (x_a == FPBits::inf().uintval()) {
316 bool out_is_neg = x_sign && is_odd_integer(y);
317 if (y_sign)
318 return out_is_neg ? -0.0 : 0.0;
319 return FPBits::inf(out_is_neg ? Sign::NEG : Sign::POS).get_val();
320 }
321
322 if (x_a > FPBits::inf().uintval()) {
323 // x is NaN.
324 // pow (aNaN, 0) is already taken care above.
325 return x;
326 }
327
328 // Normalize denormal inputs.
329 if (x_a < FPBits::min_normal().uintval()) {
330 e_x -= 64.0;
331 x_mant = FPBits(x * 0x1.0p64).get_mantissa();
332 }
333
334 // x is finite and negative, and y is a finite integer.
335 if (x_sign) {
336 if (is_integer(y)) {
337 x = -x;
338 if (is_odd_integer(y))
339 // sign = -1.0;
340 sign = 0x8000'0000'0000'0000;
341 } else {
342 // pow( negative, non-integer ) = NaN
343 fputil::set_errno_if_required(EDOM);
344 fputil::raise_except_if_required(FE_INVALID);
345 return FPBits::quiet_nan().get_val();
346 }
347 }
348 }
349
350 ///////// END - Check exceptional cases //////////////////////////////////////
351
352 // x^y = 2^( y * log2(x) )
353 // = 2^( y * ( e_x + log2(m_x) ) )
354 // First we compute log2(x) = e_x + log2(m_x)
355
356 // Extract exponent field of x.
357
358 // Use the highest 7 fractional bits of m_x as the index for look up tables.
359 unsigned idx_x = static_cast<unsigned>(x_mant >> (FPBits::FRACTION_LEN - 7));
360 // Add the hidden bit to the mantissa.
361 // 1 <= m_x < 2
362 FPBits m_x = FPBits(x_mant | 0x3ff0'0000'0000'0000);
363
364 // Reduced argument for log2(m_x):
365 // dx = r * m_x - 1.
366 // The computation is exact, and -2^-8 <= dx < 2^-7.
367 // Then m_x = (1 + dx) / r, and
368 // log2(m_x) = log2( (1 + dx) / r )
369 // = log2(1 + dx) - log2(r).
370
371 // In order for the overall computations x^y = 2^(y * log2(x)) to have the
372 // relative errors < 2^-52 (1ULP), we will need to evaluate the exponent part
373 // y * log2(x) with absolute errors < 2^-52 (or better, 2^-53). Since the
374 // whole exponent range for double precision is bounded by
375 // |y * log2(x)| < 1076 ~ 2^10, we need to evaluate log2(x) with absolute
376 // errors < 2^-53 * 2^-10 = 2^-63.
377
378 // With that requirement, we use the following degree-6 polynomial
379 // approximation:
380 // P(dx) ~ log2(1 + dx) / dx
381 // Generated by Sollya with:
382 // > P = fpminimax(log2(1 + x)/x, 6, [|D...|], [-2^-8, 2^-7]); P;
383 // > dirtyinfnorm(log2(1 + x) - x*P, [-2^-8, 2^-7]);
384 // 0x1.d03cc...p-66
385 constexpr double COEFFS[] = {0x1.71547652b82fep0, -0x1.71547652b82e7p-1,
386 0x1.ec709dc3b1fd5p-2, -0x1.7154766124215p-2,
387 0x1.2776bd90259d8p-2, -0x1.ec586c6f3d311p-3,
388 0x1.9c4775eccf524p-3};
389 // Error: ulp(dx^2) <= (2^-7)^2 * 2^-52 = 2^-66
390 // Extra errors from various computations and rounding directions, the overall
391 // errors we can be bounded by 2^-65.
392
393 double dx;
394 DoubleDouble dx_c0;
395
396 // Perform exact range reduction and exact product dx * c0.
397 #ifdef LIBC_TARGET_CPU_HAS_FMA
398 dx = fputil::multiply_add(RD[idx_x], m_x.get_val(), -1.0); // Exact
399 dx_c0 = fputil::exact_mult(COEFFS[0], dx);
400 #else
401 double c = FPBits(m_x.uintval() & 0x3fff'e000'0000'0000).get_val();
402 dx = fputil::multiply_add(RD[idx_x], m_x.get_val() - c, CD[idx_x]); // Exact
403 dx_c0 = fputil::exact_mult<28>(dx, COEFFS[0]); // Exact
404 #endif // LIBC_TARGET_CPU_HAS_FMA
405
406 double dx2 = dx * dx;
407 double c0 = fputil::multiply_add(dx, COEFFS[2], COEFFS[1]);
408 double c1 = fputil::multiply_add(dx, COEFFS[4], COEFFS[3]);
409 double c2 = fputil::multiply_add(dx, COEFFS[6], COEFFS[5]);
410
411 double p = fputil::polyeval(dx2, c0, c1, c2);
412
413 // s = e_x - log2(r) + dx * P(dx)
414 // Absolute error bound:
415 // |log2(x) - log2_x.hi - log2_x.lo| < 2^-65.
416
417 // Notice that e_x - log2(r).hi is exact, so we perform an exact sum of
418 // e_x - log2(r).hi and the high part of the product dx * c0:
419 // log2_x_hi.hi + log2_x_hi.lo = e_x - log2(r).hi + (dx * c0).hi
420 DoubleDouble log2_x_hi =
421 fputil::exact_add(e_x + LOG2_R_DD[idx_x].hi, dx_c0.hi);
422 // The low part is dx^2 * p + low part of (dx * c0) + low part of -log2(r).
423 double log2_x_lo =
424 fputil::multiply_add(dx2, p, dx_c0.lo + LOG2_R_DD[idx_x].lo);
425 // Perform accurate sums.
426 DoubleDouble log2_x = fputil::exact_add(log2_x_hi.hi, log2_x_lo);
427 log2_x.lo += log2_x_hi.lo;
428
429 // To compute 2^(y * log2(x)), we break the exponent into 3 parts:
430 // y * log(2) = hi + mid + lo, where
431 // hi is an integer
432 // mid * 2^6 is an integer
433 // |lo| <= 2^-7
434 // Then:
435 // x^y = 2^(y * log2(x)) = 2^hi * 2^mid * 2^lo,
436 // In which 2^mid is obtained from a look-up table of size 2^6 = 64 elements,
437 // and 2^lo ~ 1 + lo * P(lo).
438 // Thus, we have:
439 // hi + mid = 2^-6 * round( 2^6 * y * log2(x) )
440 // If we restrict the output such that |hi| < 150, (hi + mid) uses (8 + 6)
441 // bits, hence, if we use double precision to perform
442 // round( 2^6 * y * log2(x))
443 // the lo part is bounded by 2^-7 + 2^(-(52 - 14)) = 2^-7 + 2^-38
444
445 // In the following computations:
446 // y6 = 2^6 * y
447 // hm = 2^6 * (hi + mid) = round(2^6 * y * log2(x)) ~ round(y6 * s)
448 // lo6 = 2^6 * lo = 2^6 * (y - (hi + mid)) = y6 * log2(x) - hm.
449 double y6 = y * 0x1.0p6; // Exact.
450
451 DoubleDouble y6_log2_x = fputil::exact_mult(y6, log2_x.hi);
452 y6_log2_x.lo = fputil::multiply_add(y6, log2_x.lo, y6_log2_x.lo);
453
454 // Check overflow/underflow.
455 double scale = 1.0;
456
457 // |2^(hi + mid) - exp2_hi_mid| <= ulp(exp2_hi_mid) / 2
458 // Clamp the exponent part into smaller range that fits double precision.
459 // For those exponents that are out of range, the final conversion will round
460 // them correctly to inf/max float or 0/min float accordingly.
461 constexpr double UPPER_EXP_BOUND = 512.0 * 0x1.0p6;
462 if (LIBC_UNLIKELY(FPBits(y6_log2_x.hi).abs().get_val() >= UPPER_EXP_BOUND)) {
463 if (FPBits(y6_log2_x.hi).sign() == Sign::POS) {
464 scale = 0x1.0p512;
465 y6_log2_x.hi -= 512.0 * 64.0;
466 if (y6_log2_x.hi > 513.0 * 64.0)
467 y6_log2_x.hi = 513.0 * 64.0;
468 } else {
469 scale = 0x1.0p-512;
470 y6_log2_x.hi += 512.0 * 64.0;
471 if (y6_log2_x.hi < (-1076.0 + 512.0) * 64.0)
472 y6_log2_x.hi = -564.0 * 64.0;
473 }
474 }
475
476 double hm = fputil::nearest_integer(y6_log2_x.hi);
477
478 // lo6 = 2^6 * lo.
479 double lo6_hi = y6_log2_x.hi - hm;
480 double lo6 = lo6_hi + y6_log2_x.lo;
481
482 int hm_i = static_cast<int>(hm);
483 unsigned idx_y = static_cast<unsigned>(hm_i) & 0x3f;
484
485 // 2^hi
486 int64_t exp2_hi_i = static_cast<int64_t>(
487 static_cast<uint64_t>(static_cast<int64_t>(hm_i >> 6))
488 << FPBits::FRACTION_LEN);
489 // 2^mid
490 int64_t exp2_mid_hi_i =
491 static_cast<int64_t>(FPBits(EXP2_MID1[idx_y].hi).uintval());
492 int64_t exp2_mid_lo_i =
493 static_cast<int64_t>(FPBits(EXP2_MID1[idx_y].mid).uintval());
494 // (-1)^sign * 2^hi * 2^mid
495 // Error <= 2^hi * 2^-53
496 uint64_t exp2_hm_hi_i =
497 static_cast<uint64_t>(exp2_hi_i + exp2_mid_hi_i) + sign;
498 // The low part could be 0.
499 uint64_t exp2_hm_lo_i =
500 idx_y != 0 ? static_cast<uint64_t>(exp2_hi_i + exp2_mid_lo_i) + sign
501 : sign;
502 double exp2_hm_hi = FPBits(exp2_hm_hi_i).get_val();
503 double exp2_hm_lo = FPBits(exp2_hm_lo_i).get_val();
504
505 // Degree-5 polynomial approximation P(lo6) ~ 2^(lo6 / 2^6) = 2^(lo).
506 // Generated by Sollya with:
507 // > P = fpminimax(2^(x/64), 5, [|1, D...|], [-2^-1, 2^-1]);
508 // > dirtyinfnorm(2^(x/64) - P, [-0.5, 0.5]);
509 // 0x1.a2b77e618f5c4c176fd11b7659016cde5de83cb72p-60
510 constexpr double EXP2_COEFFS[] = {0x1p0,
511 0x1.62e42fefa39efp-7,
512 0x1.ebfbdff82a23ap-15,
513 0x1.c6b08d7076268p-23,
514 0x1.3b2ad33f8b48bp-31,
515 0x1.5d870c4d84445p-40};
516
517 double lo6_sqr = lo6 * lo6;
518
519 double d0 = fputil::multiply_add(lo6, EXP2_COEFFS[2], EXP2_COEFFS[1]);
520 double d1 = fputil::multiply_add(lo6, EXP2_COEFFS[4], EXP2_COEFFS[3]);
521 double pp = fputil::polyeval(lo6_sqr, d0, d1, EXP2_COEFFS[5]);
522
523 double r = fputil::multiply_add(exp2_hm_hi * lo6, pp, exp2_hm_lo);
524 r += exp2_hm_hi;
525
526 return r * scale;
527 }
528
529 } // namespace LIBC_NAMESPACE_DECL
530