1 //===-- Double-precision e^x 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/exp.h"
10 #include "common_constants.h" // Lookup tables EXP_M1 and EXP_M2.
11 #include "explogxf.h" // ziv_test_denorm.
12 #include "src/__support/CPP/bit.h"
13 #include "src/__support/CPP/optional.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/dyadic_float.h"
19 #include "src/__support/FPUtil/multiply_add.h"
20 #include "src/__support/FPUtil/nearest_integer.h"
21 #include "src/__support/FPUtil/rounding_mode.h"
22 #include "src/__support/FPUtil/triple_double.h"
23 #include "src/__support/common.h"
24 #include "src/__support/integer_literals.h"
25 #include "src/__support/macros/config.h"
26 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
27
28 namespace LIBC_NAMESPACE_DECL {
29
30 using fputil::DoubleDouble;
31 using fputil::TripleDouble;
32 using Float128 = typename fputil::DyadicFloat<128>;
33
34 using LIBC_NAMESPACE::operator""_u128;
35
36 // log2(e)
37 constexpr double LOG2_E = 0x1.71547652b82fep+0;
38
39 // Error bounds:
40 // Errors when using double precision.
41 constexpr double ERR_D = 0x1.8p-63;
42 // Errors when using double-double precision.
43 constexpr double ERR_DD = 0x1.0p-99;
44
45 // -2^-12 * log(2)
46 // > a = -2^-12 * log(2);
47 // > b = round(a, 30, RN);
48 // > c = round(a - b, 30, RN);
49 // > d = round(a - b - c, D, RN);
50 // Errors < 1.5 * 2^-133
51 constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13;
52 constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47;
53 constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47;
54 constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79;
55
56 namespace {
57
58 // Polynomial approximations with double precision:
59 // Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
60 // For |dx| < 2^-13 + 2^-30:
61 // | output - expm1(dx) / dx | < 2^-51.
poly_approx_d(double dx)62 LIBC_INLINE double poly_approx_d(double dx) {
63 // dx^2
64 double dx2 = dx * dx;
65 // c0 = 1 + dx / 2
66 double c0 = fputil::multiply_add(dx, 0.5, 1.0);
67 // c1 = 1/6 + dx / 24
68 double c1 =
69 fputil::multiply_add(dx, 0x1.5555555555555p-5, 0x1.5555555555555p-3);
70 // p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24
71 double p = fputil::multiply_add(dx2, c1, c0);
72 return p;
73 }
74
75 // Polynomial approximation with double-double precision:
76 // Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^6 / 720
77 // For |dx| < 2^-13 + 2^-30:
78 // | output - exp(dx) | < 2^-101
poly_approx_dd(const DoubleDouble & dx)79 DoubleDouble poly_approx_dd(const DoubleDouble &dx) {
80 // Taylor polynomial.
81 constexpr DoubleDouble COEFFS[] = {
82 {0, 0x1p0}, // 1
83 {0, 0x1p0}, // 1
84 {0, 0x1p-1}, // 1/2
85 {0x1.5555555555555p-57, 0x1.5555555555555p-3}, // 1/6
86 {0x1.5555555555555p-59, 0x1.5555555555555p-5}, // 1/24
87 {0x1.1111111111111p-63, 0x1.1111111111111p-7}, // 1/120
88 {-0x1.f49f49f49f49fp-65, 0x1.6c16c16c16c17p-10}, // 1/720
89 };
90
91 DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2],
92 COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]);
93 return p;
94 }
95
96 // Polynomial approximation with 128-bit precision:
97 // Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^7 / 5040
98 // For |dx| < 2^-13 + 2^-30:
99 // | output - exp(dx) | < 2^-126.
poly_approx_f128(const Float128 & dx)100 Float128 poly_approx_f128(const Float128 &dx) {
101 constexpr Float128 COEFFS_128[]{
102 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
103 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
104 {Sign::POS, -128, 0x80000000'00000000'00000000'00000000_u128}, // 0.5
105 {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/6
106 {Sign::POS, -132, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/24
107 {Sign::POS, -134, 0x88888888'88888888'88888888'88888889_u128}, // 1/120
108 {Sign::POS, -137, 0xb60b60b6'0b60b60b'60b60b60'b60b60b6_u128}, // 1/720
109 {Sign::POS, -140, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // 1/5040
110 };
111
112 Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2],
113 COEFFS_128[3], COEFFS_128[4], COEFFS_128[5],
114 COEFFS_128[6], COEFFS_128[7]);
115 return p;
116 }
117
118 // Compute exp(x) using 128-bit precision.
119 // TODO(lntue): investigate triple-double precision implementation for this
120 // step.
exp_f128(double x,double kd,int idx1,int idx2)121 Float128 exp_f128(double x, double kd, int idx1, int idx2) {
122 // Recalculate dx:
123
124 double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
125 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact
126 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-133
127
128 Float128 dx = fputil::quick_add(
129 Float128(t1), fputil::quick_add(Float128(t2), Float128(t3)));
130
131 // TODO: Skip recalculating exp_mid1 and exp_mid2.
132 Float128 exp_mid1 =
133 fputil::quick_add(Float128(EXP2_MID1[idx1].hi),
134 fputil::quick_add(Float128(EXP2_MID1[idx1].mid),
135 Float128(EXP2_MID1[idx1].lo)));
136
137 Float128 exp_mid2 =
138 fputil::quick_add(Float128(EXP2_MID2[idx2].hi),
139 fputil::quick_add(Float128(EXP2_MID2[idx2].mid),
140 Float128(EXP2_MID2[idx2].lo)));
141
142 Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2);
143
144 Float128 p = poly_approx_f128(dx);
145
146 Float128 r = fputil::quick_mul(exp_mid, p);
147
148 r.exponent += static_cast<int>(kd) >> 12;
149
150 return r;
151 }
152
153 // Compute exp(x) with double-double precision.
exp_double_double(double x,double kd,const DoubleDouble & exp_mid)154 DoubleDouble exp_double_double(double x, double kd,
155 const DoubleDouble &exp_mid) {
156 // Recalculate dx:
157 // dx = x - k * 2^-12 * log(2)
158 double t1 = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
159 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact
160 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-130
161
162 DoubleDouble dx = fputil::exact_add(t1, t2);
163 dx.lo += t3;
164
165 // Degree-6 Taylor polynomial approximation in double-double precision.
166 // | p - exp(x) | < 2^-100.
167 DoubleDouble p = poly_approx_dd(dx);
168
169 // Error bounds: 2^-99.
170 DoubleDouble r = fputil::quick_mult(exp_mid, p);
171
172 return r;
173 }
174
175 // Check for exceptional cases when
176 // |x| <= 2^-53 or x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9
set_exceptional(double x)177 double set_exceptional(double x) {
178 using FPBits = typename fputil::FPBits<double>;
179 FPBits xbits(x);
180
181 uint64_t x_u = xbits.uintval();
182 uint64_t x_abs = xbits.abs().uintval();
183
184 // |x| <= 2^-53
185 if (x_abs <= 0x3ca0'0000'0000'0000ULL) {
186 // exp(x) ~ 1 + x
187 return 1 + x;
188 }
189
190 // x <= log(2^-1075) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan.
191
192 // x <= log(2^-1075) or -inf/nan
193 if (x_u >= 0xc087'4910'd52d'3052ULL) {
194 // exp(-Inf) = 0
195 if (xbits.is_inf())
196 return 0.0;
197
198 // exp(nan) = nan
199 if (xbits.is_nan())
200 return x;
201
202 if (fputil::quick_get_round() == FE_UPWARD)
203 return FPBits::min_subnormal().get_val();
204 fputil::set_errno_if_required(ERANGE);
205 fputil::raise_except_if_required(FE_UNDERFLOW);
206 return 0.0;
207 }
208
209 // x >= round(log(MAX_NORMAL), D, RU) = 0x1.62e42fefa39fp+9 or +inf/nan
210 // x is finite
211 if (x_u < 0x7ff0'0000'0000'0000ULL) {
212 int rounding = fputil::quick_get_round();
213 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
214 return FPBits::max_normal().get_val();
215
216 fputil::set_errno_if_required(ERANGE);
217 fputil::raise_except_if_required(FE_OVERFLOW);
218 }
219 // x is +inf or nan
220 return x + FPBits::inf().get_val();
221 }
222
223 } // namespace
224
225 LLVM_LIBC_FUNCTION(double, exp, (double x)) {
226 using FPBits = typename fputil::FPBits<double>;
227 FPBits xbits(x);
228
229 uint64_t x_u = xbits.uintval();
230
231 // Upper bound: max normal number = 2^1023 * (2 - 2^-52)
232 // > round(log (2^1023 ( 2 - 2^-52 )), D, RU) = 0x1.62e42fefa39fp+9
233 // > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9
234 // > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9
235 // > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty
236
237 // Lower bound: min denormal number / 2 = 2^-1075
238 // > round(log(2^-1075), D, RN) = -0x1.74910d52d3052p9
239
240 // Another lower bound: min normal number = 2^-1022
241 // > round(log(2^-1022), D, RN) = -0x1.6232bdd7abcd2p9
242
243 // x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 or |x| < 2^-53.
244 if (LIBC_UNLIKELY(x_u >= 0xc0874910d52d3052 ||
245 (x_u < 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) ||
246 x_u < 0x3ca0000000000000)) {
247 return set_exceptional(x);
248 }
249
250 // Now log(2^-1075) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52))
251
252 // Range reduction:
253 // Let x = log(2) * (hi + mid1 + mid2) + lo
254 // in which:
255 // hi is an integer
256 // mid1 * 2^6 is an integer
257 // mid2 * 2^12 is an integer
258 // then:
259 // exp(x) = 2^hi * 2^(mid1) * 2^(mid2) * exp(lo).
260 // With this formula:
261 // - multiplying by 2^hi is exact and cheap, simply by adding the exponent
262 // field.
263 // - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables.
264 // - exp(lo) ~ 1 + lo + a0 * lo^2 + ...
265 //
266 // They can be defined by:
267 // hi + mid1 + mid2 = 2^(-12) * round(2^12 * log_2(e) * x)
268 // If we store L2E = round(log2(e), D, RN), then:
269 // log2(e) - L2E ~ 1.5 * 2^(-56)
270 // So the errors when computing in double precision is:
271 // | x * 2^12 * log_2(e) - D(x * 2^12 * L2E) | <=
272 // <= | x * 2^12 * log_2(e) - x * 2^12 * L2E | +
273 // + | x * 2^12 * L2E - D(x * 2^12 * L2E) |
274 // <= 2^12 * ( |x| * 1.5 * 2^-56 + eps(x)) for RN
275 // 2^12 * ( |x| * 1.5 * 2^-56 + 2*eps(x)) for other rounding modes.
276 // So if:
277 // hi + mid1 + mid2 = 2^(-12) * round(x * 2^12 * L2E) is computed entirely
278 // in double precision, the reduced argument:
279 // lo = x - log(2) * (hi + mid1 + mid2) is bounded by:
280 // |lo| <= 2^-13 + (|x| * 1.5 * 2^-56 + 2*eps(x))
281 // < 2^-13 + (1.5 * 2^9 * 1.5 * 2^-56 + 2*2^(9 - 52))
282 // < 2^-13 + 2^-41
283 //
284
285 // The following trick computes the round(x * L2E) more efficiently
286 // than using the rounding instructions, with the tradeoff for less accuracy,
287 // and hence a slightly larger range for the reduced argument `lo`.
288 //
289 // To be precise, since |x| < |log(2^-1075)| < 1.5 * 2^9,
290 // |x * 2^12 * L2E| < 1.5 * 2^9 * 1.5 < 2^23,
291 // So we can fit the rounded result round(x * 2^12 * L2E) in int32_t.
292 // Thus, the goal is to be able to use an additional addition and fixed width
293 // shift to get an int32_t representing round(x * 2^12 * L2E).
294 //
295 // Assuming int32_t using 2-complement representation, since the mantissa part
296 // of a double precision is unsigned with the leading bit hidden, if we add an
297 // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the
298 // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be
299 // considered as a proper 2-complement representations of x*2^12*L2E.
300 //
301 // One small problem with this approach is that the sum (x*2^12*L2E + C) in
302 // double precision is rounded to the least significant bit of the dorminant
303 // factor C. In order to minimize the rounding errors from this addition, we
304 // want to minimize e1. Another constraint that we want is that after
305 // shifting the mantissa so that the least significant bit of int32_t
306 // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without
307 // any adjustment. So combining these 2 requirements, we can choose
308 // C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence
309 // after right shifting the mantissa, the resulting int32_t has correct sign.
310 // With this choice of C, the number of mantissa bits we need to shift to the
311 // right is: 52 - 33 = 19.
312 //
313 // Moreover, since the integer right shifts are equivalent to rounding down,
314 // we can add an extra 0.5 so that it will become round-to-nearest, tie-to-
315 // +infinity. So in particular, we can compute:
316 // hmm = x * 2^12 * L2E + C,
317 // where C = 2^33 + 2^32 + 2^-1, then if
318 // k = int32_t(lower 51 bits of double(x * 2^12 * L2E + C) >> 19),
319 // the reduced argument:
320 // lo = x - log(2) * 2^-12 * k is bounded by:
321 // |lo| <= 2^-13 + 2^-41 + 2^-12*2^-19
322 // = 2^-13 + 2^-31 + 2^-41.
323 //
324 // Finally, notice that k only uses the mantissa of x * 2^12 * L2E, so the
325 // exponent 2^12 is not needed. So we can simply define
326 // C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and
327 // k = int32_t(lower 51 bits of double(x * L2E + C) >> 19).
328
329 // Rounding errors <= 2^-31 + 2^-41.
330 double tmp = fputil::multiply_add(x, LOG2_E, 0x1.8000'0000'4p21);
331 int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19);
332 double kd = static_cast<double>(k);
333
334 uint32_t idx1 = (k >> 6) & 0x3f;
335 uint32_t idx2 = k & 0x3f;
336 int hi = k >> 12;
337
338 bool denorm = (hi <= -1022);
339
340 DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};
341 DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};
342
343 DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);
344
345 // |x - (hi + mid1 + mid2) * log(2) - dx| < 2^11 * eps(M_LOG_2_EXP2_M12.lo)
346 // = 2^11 * 2^-13 * 2^-52
347 // = 2^-54.
348 // |dx| < 2^-13 + 2^-30.
349 double lo_h = fputil::multiply_add(kd, MLOG_2_EXP2_M12_HI, x); // exact
350 double dx = fputil::multiply_add(kd, MLOG_2_EXP2_M12_MID, lo_h);
351
352 // We use the degree-4 Taylor polynomial to approximate exp(lo):
353 // exp(lo) ~ 1 + lo + lo^2 / 2 + lo^3 / 6 + lo^4 / 24 = 1 + lo * P(lo)
354 // So that the errors are bounded by:
355 // |P(lo) - expm1(lo)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58
356 // Let P_ be an evaluation of P where all intermediate computations are in
357 // double precision. Using either Horner's or Estrin's schemes, the evaluated
358 // errors can be bounded by:
359 // |P_(dx) - P(dx)| < 2^-51
360 // => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64
361 // => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 2^-63.
362 // Since we approximate
363 // 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo,
364 // We use the expression:
365 // (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~
366 // ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)
367 // with errors bounded by 1.5 * 2^-63.
368
369 double mid_lo = dx * exp_mid.hi;
370
371 // Approximate expm1(dx)/dx ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
372 double p = poly_approx_d(dx);
373
374 double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);
375
376 if (LIBC_UNLIKELY(denorm)) {
377 if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D);
378 LIBC_LIKELY(r.has_value()))
379 return r.value();
380 } else {
381 double upper = exp_mid.hi + (lo + ERR_D);
382 double lower = exp_mid.hi + (lo - ERR_D);
383
384 if (LIBC_LIKELY(upper == lower)) {
385 // to multiply by 2^hi, a fast way is to simply add hi to the exponent
386 // field.
387 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
388 double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper));
389 return r;
390 }
391 }
392
393 // Use double-double
394 DoubleDouble r_dd = exp_double_double(x, kd, exp_mid);
395
396 if (LIBC_UNLIKELY(denorm)) {
397 if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD);
398 LIBC_LIKELY(r.has_value()))
399 return r.value();
400 } else {
401 double upper_dd = r_dd.hi + (r_dd.lo + ERR_DD);
402 double lower_dd = r_dd.hi + (r_dd.lo - ERR_DD);
403
404 if (LIBC_LIKELY(upper_dd == lower_dd)) {
405 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
406 double r =
407 cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd));
408 return r;
409 }
410 }
411
412 // Use 128-bit precision
413 Float128 r_f128 = exp_f128(x, kd, idx1, idx2);
414
415 return static_cast<double>(r_f128);
416 }
417
418 } // namespace LIBC_NAMESPACE_DECL
419