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/optimization.h" // LIBC_UNLIKELY
26
27#include <errno.h>
28
29namespace LIBC_NAMESPACE {
30
31using fputil::DoubleDouble;
32using fputil::TripleDouble;
33using Float128 = typename fputil::DyadicFloat<128>;
34
35using LIBC_NAMESPACE::operator""_u128;
36
37// log2(e)
38constexpr double LOG2_E = 0x1.71547652b82fep+0;
39
40// Error bounds:
41// Errors when using double precision.
42constexpr double ERR_D = 0x1.8p-63;
43// Errors when using double-double precision.
44constexpr double ERR_DD = 0x1.0p-99;
45
46// -2^-12 * log(2)
47// > a = -2^-12 * log(2);
48// > b = round(a, 30, RN);
49// > c = round(a - b, 30, RN);
50// > d = round(a - b - c, D, RN);
51// Errors < 1.5 * 2^-133
52constexpr double MLOG_2_EXP2_M12_HI = -0x1.62e42ffp-13;
53constexpr double MLOG_2_EXP2_M12_MID = 0x1.718432a1b0e26p-47;
54constexpr double MLOG_2_EXP2_M12_MID_30 = 0x1.718432ap-47;
55constexpr double MLOG_2_EXP2_M12_LO = 0x1.b0e2633fe0685p-79;
56
57namespace {
58
59// Polynomial approximations with double precision:
60// Return expm1(dx) / x ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
61// For |dx| < 2^-13 + 2^-30:
62// | output - expm1(dx) / dx | < 2^-51.
63LIBC_INLINE double poly_approx_d(double dx) {
64 // dx^2
65 double dx2 = dx * dx;
66 // c0 = 1 + dx / 2
67 double c0 = fputil::multiply_add(x: dx, y: 0.5, z: 1.0);
68 // c1 = 1/6 + dx / 24
69 double c1 =
70 fputil::multiply_add(x: dx, y: 0x1.5555555555555p-5, z: 0x1.5555555555555p-3);
71 // p = dx^2 * c1 + c0 = 1 + dx / 2 + dx^2 / 6 + dx^3 / 24
72 double p = fputil::multiply_add(x: dx2, y: c1, z: c0);
73 return p;
74}
75
76// Polynomial approximation with double-double precision:
77// Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^6 / 720
78// For |dx| < 2^-13 + 2^-30:
79// | output - exp(dx) | < 2^-101
80DoubleDouble poly_approx_dd(const DoubleDouble &dx) {
81 // Taylor polynomial.
82 constexpr DoubleDouble COEFFS[] = {
83 {.lo: 0, .hi: 0x1p0}, // 1
84 {.lo: 0, .hi: 0x1p0}, // 1
85 {.lo: 0, .hi: 0x1p-1}, // 1/2
86 {.lo: 0x1.5555555555555p-57, .hi: 0x1.5555555555555p-3}, // 1/6
87 {.lo: 0x1.5555555555555p-59, .hi: 0x1.5555555555555p-5}, // 1/24
88 {.lo: 0x1.1111111111111p-63, .hi: 0x1.1111111111111p-7}, // 1/120
89 {.lo: -0x1.f49f49f49f49fp-65, .hi: 0x1.6c16c16c16c17p-10}, // 1/720
90 };
91
92 DoubleDouble p = fputil::polyeval(x: dx, a0: COEFFS[0], a: COEFFS[1], a: COEFFS[2],
93 a: COEFFS[3], a: COEFFS[4], a: COEFFS[5], a: COEFFS[6]);
94 return p;
95}
96
97// Polynomial approximation with 128-bit precision:
98// Return exp(dx) ~ 1 + dx + dx^2 / 2 + ... + dx^7 / 5040
99// For |dx| < 2^-13 + 2^-30:
100// | output - exp(dx) | < 2^-126.
101Float128 poly_approx_f128(const Float128 &dx) {
102 constexpr Float128 COEFFS_128[]{
103 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
104 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
105 {Sign::POS, -128, 0x80000000'00000000'00000000'00000000_u128}, // 0.5
106 {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/6
107 {Sign::POS, -132, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/24
108 {Sign::POS, -134, 0x88888888'88888888'88888888'88888889_u128}, // 1/120
109 {Sign::POS, -137, 0xb60b60b6'0b60b60b'60b60b60'b60b60b6_u128}, // 1/720
110 {Sign::POS, -140, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // 1/5040
111 };
112
113 Float128 p = fputil::polyeval(x: dx, a0: COEFFS_128[0], a: COEFFS_128[1], a: COEFFS_128[2],
114 a: COEFFS_128[3], a: COEFFS_128[4], a: COEFFS_128[5],
115 a: COEFFS_128[6], a: COEFFS_128[7]);
116 return p;
117}
118
119// Compute exp(x) using 128-bit precision.
120// TODO(lntue): investigate triple-double precision implementation for this
121// step.
122Float128 exp_f128(double x, double kd, int idx1, int idx2) {
123 // Recalculate dx:
124
125 double t1 = fputil::multiply_add(x: kd, y: MLOG_2_EXP2_M12_HI, z: x); // exact
126 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact
127 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-133
128
129 Float128 dx = fputil::quick_add(
130 a: Float128(t1), b: fputil::quick_add(a: Float128(t2), b: Float128(t3)));
131
132 // TODO: Skip recalculating exp_mid1 and exp_mid2.
133 Float128 exp_mid1 =
134 fputil::quick_add(a: Float128(EXP2_MID1[idx1].hi),
135 b: fputil::quick_add(a: Float128(EXP2_MID1[idx1].mid),
136 b: Float128(EXP2_MID1[idx1].lo)));
137
138 Float128 exp_mid2 =
139 fputil::quick_add(a: Float128(EXP2_MID2[idx2].hi),
140 b: fputil::quick_add(a: Float128(EXP2_MID2[idx2].mid),
141 b: Float128(EXP2_MID2[idx2].lo)));
142
143 Float128 exp_mid = fputil::quick_mul(a: exp_mid1, b: exp_mid2);
144
145 Float128 p = poly_approx_f128(dx);
146
147 Float128 r = fputil::quick_mul(a: exp_mid, b: p);
148
149 r.exponent += static_cast<int>(kd) >> 12;
150
151 return r;
152}
153
154// Compute exp(x) with double-double precision.
155DoubleDouble exp_double_double(double x, double kd,
156 const DoubleDouble &exp_mid) {
157 // Recalculate dx:
158 // dx = x - k * 2^-12 * log(2)
159 double t1 = fputil::multiply_add(x: kd, y: MLOG_2_EXP2_M12_HI, z: x); // exact
160 double t2 = kd * MLOG_2_EXP2_M12_MID_30; // exact
161 double t3 = kd * MLOG_2_EXP2_M12_LO; // Error < 2^-130
162
163 DoubleDouble dx = fputil::exact_add(a: t1, b: t2);
164 dx.lo += t3;
165
166 // Degree-6 Taylor polynomial approximation in double-double precision.
167 // | p - exp(x) | < 2^-100.
168 DoubleDouble p = poly_approx_dd(dx);
169
170 // Error bounds: 2^-99.
171 DoubleDouble r = fputil::quick_mult(a: exp_mid, b: p);
172
173 return r;
174}
175
176// Check for exceptional cases when
177// |x| <= 2^-53 or x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9
178double set_exceptional(double x) {
179 using FPBits = typename fputil::FPBits<double>;
180 FPBits xbits(x);
181
182 uint64_t x_u = xbits.uintval();
183 uint64_t x_abs = xbits.abs().uintval();
184
185 // |x| <= 2^-53
186 if (x_abs <= 0x3ca0'0000'0000'0000ULL) {
187 // exp(x) ~ 1 + x
188 return 1 + x;
189 }
190
191 // x <= log(2^-1075) || x >= 0x1.6232bdd7abcd3p+9 or inf/nan.
192
193 // x <= log(2^-1075) or -inf/nan
194 if (x_u >= 0xc087'4910'd52d'3052ULL) {
195 // exp(-Inf) = 0
196 if (xbits.is_inf())
197 return 0.0;
198
199 // exp(nan) = nan
200 if (xbits.is_nan())
201 return x;
202
203 if (fputil::quick_get_round() == FE_UPWARD)
204 return FPBits::min_subnormal().get_val();
205 fputil::set_errno_if_required(ERANGE);
206 fputil::raise_except_if_required(FE_UNDERFLOW);
207 return 0.0;
208 }
209
210 // x >= round(log(MAX_NORMAL), D, RU) = 0x1.62e42fefa39fp+9 or +inf/nan
211 // x is finite
212 if (x_u < 0x7ff0'0000'0000'0000ULL) {
213 int rounding = fputil::quick_get_round();
214 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
215 return FPBits::max_normal().get_val();
216
217 fputil::set_errno_if_required(ERANGE);
218 fputil::raise_except_if_required(FE_OVERFLOW);
219 }
220 // x is +inf or nan
221 return x + FPBits::inf().get_val();
222}
223
224} // namespace
225
226LLVM_LIBC_FUNCTION(double, exp, (double x)) {
227 using FPBits = typename fputil::FPBits<double>;
228 FPBits xbits(x);
229
230 uint64_t x_u = xbits.uintval();
231
232 // Upper bound: max normal number = 2^1023 * (2 - 2^-52)
233 // > round(log (2^1023 ( 2 - 2^-52 )), D, RU) = 0x1.62e42fefa39fp+9
234 // > round(log (2^1023 ( 2 - 2^-52 )), D, RD) = 0x1.62e42fefa39efp+9
235 // > round(log (2^1023 ( 2 - 2^-52 )), D, RN) = 0x1.62e42fefa39efp+9
236 // > round(exp(0x1.62e42fefa39fp+9), D, RN) = infty
237
238 // Lower bound: min denormal number / 2 = 2^-1075
239 // > round(log(2^-1075), D, RN) = -0x1.74910d52d3052p9
240
241 // Another lower bound: min normal number = 2^-1022
242 // > round(log(2^-1022), D, RN) = -0x1.6232bdd7abcd2p9
243
244 // x < log(2^-1075) or x >= 0x1.6232bdd7abcd3p+9 or |x| < 2^-53.
245 if (LIBC_UNLIKELY(x_u >= 0xc0874910d52d3052 ||
246 (x_u < 0xbca0000000000000 && x_u >= 0x40862e42fefa39f0) ||
247 x_u < 0x3ca0000000000000)) {
248 return set_exceptional(x);
249 }
250
251 // Now log(2^-1075) <= x <= -2^-53 or 2^-53 <= x < log(2^1023 * (2 - 2^-52))
252
253 // Range reduction:
254 // Let x = log(2) * (hi + mid1 + mid2) + lo
255 // in which:
256 // hi is an integer
257 // mid1 * 2^6 is an integer
258 // mid2 * 2^12 is an integer
259 // then:
260 // exp(x) = 2^hi * 2^(mid1) * 2^(mid2) * exp(lo).
261 // With this formula:
262 // - multiplying by 2^hi is exact and cheap, simply by adding the exponent
263 // field.
264 // - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables.
265 // - exp(lo) ~ 1 + lo + a0 * lo^2 + ...
266 //
267 // They can be defined by:
268 // hi + mid1 + mid2 = 2^(-12) * round(2^12 * log_2(e) * x)
269 // If we store L2E = round(log2(e), D, RN), then:
270 // log2(e) - L2E ~ 1.5 * 2^(-56)
271 // So the errors when computing in double precision is:
272 // | x * 2^12 * log_2(e) - D(x * 2^12 * L2E) | <=
273 // <= | x * 2^12 * log_2(e) - x * 2^12 * L2E | +
274 // + | x * 2^12 * L2E - D(x * 2^12 * L2E) |
275 // <= 2^12 * ( |x| * 1.5 * 2^-56 + eps(x)) for RN
276 // 2^12 * ( |x| * 1.5 * 2^-56 + 2*eps(x)) for other rounding modes.
277 // So if:
278 // hi + mid1 + mid2 = 2^(-12) * round(x * 2^12 * L2E) is computed entirely
279 // in double precision, the reduced argument:
280 // lo = x - log(2) * (hi + mid1 + mid2) is bounded by:
281 // |lo| <= 2^-13 + (|x| * 1.5 * 2^-56 + 2*eps(x))
282 // < 2^-13 + (1.5 * 2^9 * 1.5 * 2^-56 + 2*2^(9 - 52))
283 // < 2^-13 + 2^-41
284 //
285
286 // The following trick computes the round(x * L2E) more efficiently
287 // than using the rounding instructions, with the tradeoff for less accuracy,
288 // and hence a slightly larger range for the reduced argument `lo`.
289 //
290 // To be precise, since |x| < |log(2^-1075)| < 1.5 * 2^9,
291 // |x * 2^12 * L2E| < 1.5 * 2^9 * 1.5 < 2^23,
292 // So we can fit the rounded result round(x * 2^12 * L2E) in int32_t.
293 // Thus, the goal is to be able to use an additional addition and fixed width
294 // shift to get an int32_t representing round(x * 2^12 * L2E).
295 //
296 // Assuming int32_t using 2-complement representation, since the mantissa part
297 // of a double precision is unsigned with the leading bit hidden, if we add an
298 // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^25 to the product, the
299 // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be
300 // considered as a proper 2-complement representations of x*2^12*L2E.
301 //
302 // One small problem with this approach is that the sum (x*2^12*L2E + C) in
303 // double precision is rounded to the least significant bit of the dorminant
304 // factor C. In order to minimize the rounding errors from this addition, we
305 // want to minimize e1. Another constraint that we want is that after
306 // shifting the mantissa so that the least significant bit of int32_t
307 // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without
308 // any adjustment. So combining these 2 requirements, we can choose
309 // C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence
310 // after right shifting the mantissa, the resulting int32_t has correct sign.
311 // With this choice of C, the number of mantissa bits we need to shift to the
312 // right is: 52 - 33 = 19.
313 //
314 // Moreover, since the integer right shifts are equivalent to rounding down,
315 // we can add an extra 0.5 so that it will become round-to-nearest, tie-to-
316 // +infinity. So in particular, we can compute:
317 // hmm = x * 2^12 * L2E + C,
318 // where C = 2^33 + 2^32 + 2^-1, then if
319 // k = int32_t(lower 51 bits of double(x * 2^12 * L2E + C) >> 19),
320 // the reduced argument:
321 // lo = x - log(2) * 2^-12 * k is bounded by:
322 // |lo| <= 2^-13 + 2^-41 + 2^-12*2^-19
323 // = 2^-13 + 2^-31 + 2^-41.
324 //
325 // Finally, notice that k only uses the mantissa of x * 2^12 * L2E, so the
326 // exponent 2^12 is not needed. So we can simply define
327 // C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and
328 // k = int32_t(lower 51 bits of double(x * L2E + C) >> 19).
329
330 // Rounding errors <= 2^-31 + 2^-41.
331 double tmp = fputil::multiply_add(x, y: LOG2_E, z: 0x1.8000'0000'4p21);
332 int k = static_cast<int>(cpp::bit_cast<uint64_t>(from: tmp) >> 19);
333 double kd = static_cast<double>(k);
334
335 uint32_t idx1 = (k >> 6) & 0x3f;
336 uint32_t idx2 = k & 0x3f;
337 int hi = k >> 12;
338
339 bool denorm = (hi <= -1022);
340
341 DoubleDouble exp_mid1{.lo: EXP2_MID1[idx1].mid, .hi: EXP2_MID1[idx1].hi};
342 DoubleDouble exp_mid2{.lo: EXP2_MID2[idx2].mid, .hi: EXP2_MID2[idx2].hi};
343
344 DoubleDouble exp_mid = fputil::quick_mult(a: exp_mid1, b: exp_mid2);
345
346 // |x - (hi + mid1 + mid2) * log(2) - dx| < 2^11 * eps(M_LOG_2_EXP2_M12.lo)
347 // = 2^11 * 2^-13 * 2^-52
348 // = 2^-54.
349 // |dx| < 2^-13 + 2^-30.
350 double lo_h = fputil::multiply_add(x: kd, y: MLOG_2_EXP2_M12_HI, z: x); // exact
351 double dx = fputil::multiply_add(x: kd, y: MLOG_2_EXP2_M12_MID, z: lo_h);
352
353 // We use the degree-4 Taylor polynomial to approximate exp(lo):
354 // exp(lo) ~ 1 + lo + lo^2 / 2 + lo^3 / 6 + lo^4 / 24 = 1 + lo * P(lo)
355 // So that the errors are bounded by:
356 // |P(lo) - expm1(lo)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58
357 // Let P_ be an evaluation of P where all intermediate computations are in
358 // double precision. Using either Horner's or Estrin's schemes, the evaluated
359 // errors can be bounded by:
360 // |P_(dx) - P(dx)| < 2^-51
361 // => |dx * P_(dx) - expm1(lo) | < 1.5 * 2^-64
362 // => 2^(mid1 + mid2) * |dx * P_(dx) - expm1(lo)| < 1.5 * 2^-63.
363 // Since we approximate
364 // 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo,
365 // We use the expression:
366 // (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~
367 // ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)
368 // with errors bounded by 1.5 * 2^-63.
369
370 double mid_lo = dx * exp_mid.hi;
371
372 // Approximate expm1(dx)/dx ~ 1 + dx / 2 + dx^2 / 6 + dx^3 / 24.
373 double p = poly_approx_d(dx);
374
375 double lo = fputil::multiply_add(x: p, y: mid_lo, z: exp_mid.lo);
376
377 if (LIBC_UNLIKELY(denorm)) {
378 if (auto r = ziv_test_denorm(hi, mid: exp_mid.hi, lo, err: ERR_D);
379 LIBC_LIKELY(r.has_value()))
380 return r.value();
381 } else {
382 double upper = exp_mid.hi + (lo + ERR_D);
383 double lower = exp_mid.hi + (lo - ERR_D);
384
385 if (LIBC_LIKELY(upper == lower)) {
386 // to multiply by 2^hi, a fast way is to simply add hi to the exponent
387 // field.
388 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
389 double r = cpp::bit_cast<double>(from: exp_hi + cpp::bit_cast<int64_t>(from: upper));
390 return r;
391 }
392 }
393
394 // Use double-double
395 DoubleDouble r_dd = exp_double_double(x, kd, exp_mid);
396
397 if (LIBC_UNLIKELY(denorm)) {
398 if (auto r = ziv_test_denorm(hi, mid: r_dd.hi, lo: r_dd.lo, err: ERR_DD);
399 LIBC_LIKELY(r.has_value()))
400 return r.value();
401 } else {
402 double upper_dd = r_dd.hi + (r_dd.lo + ERR_DD);
403 double lower_dd = r_dd.hi + (r_dd.lo - ERR_DD);
404
405 if (LIBC_LIKELY(upper_dd == lower_dd)) {
406 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
407 double r =
408 cpp::bit_cast<double>(from: exp_hi + cpp::bit_cast<int64_t>(from: upper_dd));
409 return r;
410 }
411 }
412
413 // Use 128-bit precision
414 Float128 r_f128 = exp_f128(x, kd, idx1, idx2);
415
416 return static_cast<double>(r_f128);
417}
418
419} // namespace LIBC_NAMESPACE
420

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