1 //===-- Double-precision 10^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/exp10.h"
10 #include "common_constants.h" // Lookup tables EXP2_MID1 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
29 namespace LIBC_NAMESPACE {
30
31 using fputil::DoubleDouble;
32 using fputil::TripleDouble;
33 using Float128 = typename fputil::DyadicFloat<128>;
34
35 using LIBC_NAMESPACE::operator""_u128;
36
37 // log2(10)
38 constexpr double LOG2_10 = 0x1.a934f0979a371p+1;
39
40 // -2^-12 * log10(2)
41 // > a = -2^-12 * log10(2);
42 // > b = round(a, 32, RN);
43 // > c = round(a - b, 32, RN);
44 // > d = round(a - b - c, D, RN);
45 // Errors < 1.5 * 2^-144
46 constexpr double MLOG10_2_EXP2_M12_HI = -0x1.3441350ap-14;
47 constexpr double MLOG10_2_EXP2_M12_MID = 0x1.0c0219dc1da99p-51;
48 constexpr double MLOG10_2_EXP2_M12_MID_32 = 0x1.0c0219dcp-51;
49 constexpr double MLOG10_2_EXP2_M12_LO = 0x1.da994fd20dba2p-87;
50
51 // Error bounds:
52 // Errors when using double precision.
53 constexpr double ERR_D = 0x1.8p-63;
54
55 // Errors when using double-double precision.
56 constexpr double ERR_DD = 0x1.8p-99;
57
58 namespace {
59
60 // Polynomial approximations with double precision. Generated by Sollya with:
61 // > P = fpminimax((10^x - 1)/x, 3, [|D...|], [-2^-14, 2^-14]);
62 // > P;
63 // Error bounds:
64 // | output - (10^dx - 1) / dx | < 2^-52.
poly_approx_d(double dx)65 LIBC_INLINE double poly_approx_d(double dx) {
66 // dx^2
67 double dx2 = dx * dx;
68 double c0 =
69 fputil::multiply_add(dx, 0x1.53524c73cea6ap+1, 0x1.26bb1bbb55516p+1);
70 double c1 =
71 fputil::multiply_add(dx, 0x1.2bd75cc6afc65p+0, 0x1.0470587aa264cp+1);
72 double p = fputil::multiply_add(dx2, c1, c0);
73 return p;
74 }
75
76 // Polynomial approximation with double-double precision. Generated by Solya
77 // with:
78 // > P = fpminimax((10^x - 1)/x, 5, [|DD...|], [-2^-14, 2^-14]);
79 // Error bounds:
80 // | output - 10^(dx) | < 2^-101
poly_approx_dd(const DoubleDouble & dx)81 DoubleDouble poly_approx_dd(const DoubleDouble &dx) {
82 // Taylor polynomial.
83 constexpr DoubleDouble COEFFS[] = {
84 {0, 0x1p0},
85 {-0x1.f48ad494e927bp-53, 0x1.26bb1bbb55516p1},
86 {-0x1.e2bfab3191cd2p-53, 0x1.53524c73cea69p1},
87 {0x1.80fb65ec3b503p-53, 0x1.0470591de2ca4p1},
88 {0x1.338fc05e21e55p-54, 0x1.2bd7609fd98c4p0},
89 {0x1.d4ea116818fbp-56, 0x1.1429ffd519865p-1},
90 {-0x1.872a8ff352077p-57, 0x1.a7ed70847c8b3p-3},
91
92 };
93
94 DoubleDouble p = fputil::polyeval(dx, COEFFS[0], COEFFS[1], COEFFS[2],
95 COEFFS[3], COEFFS[4], COEFFS[5], COEFFS[6]);
96 return p;
97 }
98
99 // Polynomial approximation with 128-bit precision:
100 // Return exp(dx) ~ 1 + a0 * dx + a1 * dx^2 + ... + a6 * dx^7
101 // For |dx| < 2^-14:
102 // | output - 10^dx | < 1.5 * 2^-124.
poly_approx_f128(const Float128 & dx)103 Float128 poly_approx_f128(const Float128 &dx) {
104 constexpr Float128 COEFFS_128[]{
105 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0
106 {Sign::POS, -126, 0x935d8ddd'aaa8ac16'ea56d62b'82d30a2d_u128},
107 {Sign::POS, -126, 0xa9a92639'e753443a'80a99ce7'5f4d5bdb_u128},
108 {Sign::POS, -126, 0x82382c8e'f1652304'6a4f9d7d'bf6c9635_u128},
109 {Sign::POS, -124, 0x12bd7609'fd98c44c'34578701'9216c7af_u128},
110 {Sign::POS, -127, 0x450a7ff4'7535d889'cc41ed7e'0d27aee5_u128},
111 {Sign::POS, -130, 0xd3f6b844'702d636b'8326bb91'a6e7601d_u128},
112 {Sign::POS, -130, 0x45b937f0'd05bb1cd'fa7b46df'314112a9_u128},
113 };
114
115 Float128 p = fputil::polyeval(dx, COEFFS_128[0], COEFFS_128[1], COEFFS_128[2],
116 COEFFS_128[3], COEFFS_128[4], COEFFS_128[5],
117 COEFFS_128[6], COEFFS_128[7]);
118 return p;
119 }
120
121 // Compute 10^(x) using 128-bit precision.
122 // TODO(lntue): investigate triple-double precision implementation for this
123 // step.
exp10_f128(double x,double kd,int idx1,int idx2)124 Float128 exp10_f128(double x, double kd, int idx1, int idx2) {
125 double t1 = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact
126 double t2 = kd * MLOG10_2_EXP2_M12_MID_32; // exact
127 double t3 = kd * MLOG10_2_EXP2_M12_LO; // Error < 2^-144
128
129 Float128 dx = fputil::quick_add(
130 Float128(t1), fputil::quick_add(Float128(t2), Float128(t3)));
131
132 // TODO: Skip recalculating exp_mid1 and exp_mid2.
133 Float128 exp_mid1 =
134 fputil::quick_add(Float128(EXP2_MID1[idx1].hi),
135 fputil::quick_add(Float128(EXP2_MID1[idx1].mid),
136 Float128(EXP2_MID1[idx1].lo)));
137
138 Float128 exp_mid2 =
139 fputil::quick_add(Float128(EXP2_MID2[idx2].hi),
140 fputil::quick_add(Float128(EXP2_MID2[idx2].mid),
141 Float128(EXP2_MID2[idx2].lo)));
142
143 Float128 exp_mid = fputil::quick_mul(exp_mid1, exp_mid2);
144
145 Float128 p = poly_approx_f128(dx);
146
147 Float128 r = fputil::quick_mul(exp_mid, p);
148
149 r.exponent += static_cast<int>(kd) >> 12;
150
151 return r;
152 }
153
154 // Compute 10^x with double-double precision.
exp10_double_double(double x,double kd,const DoubleDouble & exp_mid)155 DoubleDouble exp10_double_double(double x, double kd,
156 const DoubleDouble &exp_mid) {
157 // Recalculate dx:
158 // dx = x - k * 2^-12 * log10(2)
159 double t1 = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact
160 double t2 = kd * MLOG10_2_EXP2_M12_MID_32; // exact
161 double t3 = kd * MLOG10_2_EXP2_M12_LO; // Error < 2^-140
162
163 DoubleDouble dx = fputil::exact_add(t1, t2);
164 dx.lo += t3;
165
166 // Degree-6 polynomial approximation in double-double precision.
167 // | p - 10^x | < 2^-103.
168 DoubleDouble p = poly_approx_dd(dx);
169
170 // Error bounds: 2^-102.
171 DoubleDouble r = fputil::quick_mult(exp_mid, p);
172
173 return r;
174 }
175
176 // When output is denormal.
exp10_denorm(double x)177 double exp10_denorm(double x) {
178 // Range reduction.
179 double tmp = fputil::multiply_add(x, LOG2_10, 0x1.8000'0000'4p21);
180 int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19);
181 double kd = static_cast<double>(k);
182
183 uint32_t idx1 = (k >> 6) & 0x3f;
184 uint32_t idx2 = k & 0x3f;
185
186 int hi = k >> 12;
187
188 DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};
189 DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};
190 DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);
191
192 // |dx| < 1.5 * 2^-15 + 2^-31 < 2^-14
193 double lo_h = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact
194 double dx = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_MID, lo_h);
195
196 double mid_lo = dx * exp_mid.hi;
197
198 // Approximate (10^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4.
199 double p = poly_approx_d(dx);
200
201 double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);
202
203 if (auto r = ziv_test_denorm(hi, exp_mid.hi, lo, ERR_D);
204 LIBC_LIKELY(r.has_value()))
205 return r.value();
206
207 // Use double-double
208 DoubleDouble r_dd = exp10_double_double(x, kd, exp_mid);
209
210 if (auto r = ziv_test_denorm(hi, r_dd.hi, r_dd.lo, ERR_DD);
211 LIBC_LIKELY(r.has_value()))
212 return r.value();
213
214 // Use 128-bit precision
215 Float128 r_f128 = exp10_f128(x, kd, idx1, idx2);
216
217 return static_cast<double>(r_f128);
218 }
219
220 // Check for exceptional cases when:
221 // * log10(1 - 2^-54) < x < log10(1 + 2^-53)
222 // * x >= log10(2^1024)
223 // * x <= log10(2^-1022)
224 // * x is inf or nan
set_exceptional(double x)225 double set_exceptional(double x) {
226 using FPBits = typename fputil::FPBits<double>;
227 FPBits xbits(x);
228
229 uint64_t x_u = xbits.uintval();
230 uint64_t x_abs = xbits.abs().uintval();
231
232 // |x| < log10(1 + 2^-53)
233 if (x_abs <= 0x3c8bcb7b1526e50e) {
234 // 10^(x) ~ 1 + x/2
235 return fputil::multiply_add(x, 0.5, 1.0);
236 }
237
238 // x <= log10(2^-1022) || x >= log10(2^1024) or inf/nan.
239 if (x_u >= 0xc0733a7146f72a42) {
240 // x <= log10(2^-1075) or -inf/nan
241 if (x_u > 0xc07439b746e36b52) {
242 // exp(-Inf) = 0
243 if (xbits.is_inf())
244 return 0.0;
245
246 // exp(nan) = nan
247 if (xbits.is_nan())
248 return x;
249
250 if (fputil::quick_get_round() == FE_UPWARD)
251 return FPBits::min_subnormal().get_val();
252 fputil::set_errno_if_required(ERANGE);
253 fputil::raise_except_if_required(FE_UNDERFLOW);
254 return 0.0;
255 }
256
257 return exp10_denorm(x);
258 }
259
260 // x >= log10(2^1024) or +inf/nan
261 // x is finite
262 if (x_u < 0x7ff0'0000'0000'0000ULL) {
263 int rounding = fputil::quick_get_round();
264 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO)
265 return FPBits::max_normal().get_val();
266
267 fputil::set_errno_if_required(ERANGE);
268 fputil::raise_except_if_required(FE_OVERFLOW);
269 }
270 // x is +inf or nan
271 return x + FPBits::inf().get_val();
272 }
273
274 } // namespace
275
276 LLVM_LIBC_FUNCTION(double, exp10, (double x)) {
277 using FPBits = typename fputil::FPBits<double>;
278 FPBits xbits(x);
279
280 uint64_t x_u = xbits.uintval();
281
282 // x <= log10(2^-1022) or x >= log10(2^1024) or
283 // log10(1 - 2^-54) < x < log10(1 + 2^-53).
284 if (LIBC_UNLIKELY(x_u >= 0xc0733a7146f72a42 ||
285 (x_u <= 0xbc7bcb7b1526e50e && x_u >= 0x40734413509f79ff) ||
286 x_u < 0x3c8bcb7b1526e50e)) {
287 return set_exceptional(x);
288 }
289
290 // Now log10(2^-1075) < x <= log10(1 - 2^-54) or
291 // log10(1 + 2^-53) < x < log10(2^1024)
292
293 // Range reduction:
294 // Let x = log10(2) * (hi + mid1 + mid2) + lo
295 // in which:
296 // hi is an integer
297 // mid1 * 2^6 is an integer
298 // mid2 * 2^12 is an integer
299 // then:
300 // 10^(x) = 2^hi * 2^(mid1) * 2^(mid2) * 10^(lo).
301 // With this formula:
302 // - multiplying by 2^hi is exact and cheap, simply by adding the exponent
303 // field.
304 // - 2^(mid1) and 2^(mid2) are stored in 2 x 64-element tables.
305 // - 10^(lo) ~ 1 + a0*lo + a1 * lo^2 + ...
306 //
307 // We compute (hi + mid1 + mid2) together by perform the rounding on
308 // x * log2(10) * 2^12.
309 // Since |x| < |log10(2^-1075)| < 2^9,
310 // |x * 2^12| < 2^9 * 2^12 < 2^21,
311 // So we can fit the rounded result round(x * 2^12) in int32_t.
312 // Thus, the goal is to be able to use an additional addition and fixed width
313 // shift to get an int32_t representing round(x * 2^12).
314 //
315 // Assuming int32_t using 2-complement representation, since the mantissa part
316 // of a double precision is unsigned with the leading bit hidden, if we add an
317 // extra constant C = 2^e1 + 2^e2 with e1 > e2 >= 2^23 to the product, the
318 // part that are < 2^e2 in resulted mantissa of (x*2^12*L2E + C) can be
319 // considered as a proper 2-complement representations of x*2^12.
320 //
321 // One small problem with this approach is that the sum (x*2^12 + C) in
322 // double precision is rounded to the least significant bit of the dorminant
323 // factor C. In order to minimize the rounding errors from this addition, we
324 // want to minimize e1. Another constraint that we want is that after
325 // shifting the mantissa so that the least significant bit of int32_t
326 // corresponds to the unit bit of (x*2^12*L2E), the sign is correct without
327 // any adjustment. So combining these 2 requirements, we can choose
328 // C = 2^33 + 2^32, so that the sign bit corresponds to 2^31 bit, and hence
329 // after right shifting the mantissa, the resulting int32_t has correct sign.
330 // With this choice of C, the number of mantissa bits we need to shift to the
331 // right is: 52 - 33 = 19.
332 //
333 // Moreover, since the integer right shifts are equivalent to rounding down,
334 // we can add an extra 0.5 so that it will become round-to-nearest, tie-to-
335 // +infinity. So in particular, we can compute:
336 // hmm = x * 2^12 + C,
337 // where C = 2^33 + 2^32 + 2^-1, then if
338 // k = int32_t(lower 51 bits of double(x * 2^12 + C) >> 19),
339 // the reduced argument:
340 // lo = x - log10(2) * 2^-12 * k is bounded by:
341 // |lo| = |x - log10(2) * 2^-12 * k|
342 // = log10(2) * 2^-12 * | x * log2(10) * 2^12 - k |
343 // <= log10(2) * 2^-12 * (2^-1 + 2^-19)
344 // < 1.5 * 2^-2 * (2^-13 + 2^-31)
345 // = 1.5 * (2^-15 * 2^-31)
346 //
347 // Finally, notice that k only uses the mantissa of x * 2^12, so the
348 // exponent 2^12 is not needed. So we can simply define
349 // C = 2^(33 - 12) + 2^(32 - 12) + 2^(-13 - 12), and
350 // k = int32_t(lower 51 bits of double(x + C) >> 19).
351
352 // Rounding errors <= 2^-31.
353 double tmp = fputil::multiply_add(x, LOG2_10, 0x1.8000'0000'4p21);
354 int k = static_cast<int>(cpp::bit_cast<uint64_t>(tmp) >> 19);
355 double kd = static_cast<double>(k);
356
357 uint32_t idx1 = (k >> 6) & 0x3f;
358 uint32_t idx2 = k & 0x3f;
359
360 int hi = k >> 12;
361
362 DoubleDouble exp_mid1{EXP2_MID1[idx1].mid, EXP2_MID1[idx1].hi};
363 DoubleDouble exp_mid2{EXP2_MID2[idx2].mid, EXP2_MID2[idx2].hi};
364 DoubleDouble exp_mid = fputil::quick_mult(exp_mid1, exp_mid2);
365
366 // |dx| < 1.5 * 2^-15 + 2^-31 < 2^-14
367 double lo_h = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_HI, x); // exact
368 double dx = fputil::multiply_add(kd, MLOG10_2_EXP2_M12_MID, lo_h);
369
370 // We use the degree-4 polynomial to approximate 10^(lo):
371 // 10^(lo) ~ 1 + a0 * lo + a1 * lo^2 + a2 * lo^3 + a3 * lo^4
372 // = 1 + lo * P(lo)
373 // So that the errors are bounded by:
374 // |P(lo) - (10^lo - 1)/lo| < |lo|^4 / 64 < 2^(-13 * 4) / 64 = 2^-58
375 // Let P_ be an evaluation of P where all intermediate computations are in
376 // double precision. Using either Horner's or Estrin's schemes, the evaluated
377 // errors can be bounded by:
378 // |P_(lo) - P(lo)| < 2^-51
379 // => |lo * P_(lo) - (2^lo - 1) | < 2^-65
380 // => 2^(mid1 + mid2) * |lo * P_(lo) - expm1(lo)| < 2^-64.
381 // Since we approximate
382 // 2^(mid1 + mid2) ~ exp_mid.hi + exp_mid.lo,
383 // We use the expression:
384 // (exp_mid.hi + exp_mid.lo) * (1 + dx * P_(dx)) ~
385 // ~ exp_mid.hi + (exp_mid.hi * dx * P_(dx) + exp_mid.lo)
386 // with errors bounded by 2^-64.
387
388 double mid_lo = dx * exp_mid.hi;
389
390 // Approximate (10^dx - 1)/dx ~ 1 + a0*dx + a1*dx^2 + a2*dx^3 + a3*dx^4.
391 double p = poly_approx_d(dx);
392
393 double lo = fputil::multiply_add(p, mid_lo, exp_mid.lo);
394
395 double upper = exp_mid.hi + (lo + ERR_D);
396 double lower = exp_mid.hi + (lo - ERR_D);
397
398 if (LIBC_LIKELY(upper == lower)) {
399 // To multiply by 2^hi, a fast way is to simply add hi to the exponent
400 // field.
401 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
402 double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper));
403 return r;
404 }
405
406 // Exact outputs when x = 1, 2, ..., 22 + hard to round with x = 23.
407 // Quick check mask: 0x800f'ffffU = ~(bits of 1.0 | ... | bits of 23.0)
408 if (LIBC_UNLIKELY((x_u & 0x8000'ffff'ffff'ffffULL) == 0ULL)) {
409 switch (x_u) {
410 case 0x3ff0000000000000: // x = 1.0
411 return 10.0;
412 case 0x4000000000000000: // x = 2.0
413 return 100.0;
414 case 0x4008000000000000: // x = 3.0
415 return 1'000.0;
416 case 0x4010000000000000: // x = 4.0
417 return 10'000.0;
418 case 0x4014000000000000: // x = 5.0
419 return 100'000.0;
420 case 0x4018000000000000: // x = 6.0
421 return 1'000'000.0;
422 case 0x401c000000000000: // x = 7.0
423 return 10'000'000.0;
424 case 0x4020000000000000: // x = 8.0
425 return 100'000'000.0;
426 case 0x4022000000000000: // x = 9.0
427 return 1'000'000'000.0;
428 case 0x4024000000000000: // x = 10.0
429 return 10'000'000'000.0;
430 case 0x4026000000000000: // x = 11.0
431 return 100'000'000'000.0;
432 case 0x4028000000000000: // x = 12.0
433 return 1'000'000'000'000.0;
434 case 0x402a000000000000: // x = 13.0
435 return 10'000'000'000'000.0;
436 case 0x402c000000000000: // x = 14.0
437 return 100'000'000'000'000.0;
438 case 0x402e000000000000: // x = 15.0
439 return 1'000'000'000'000'000.0;
440 case 0x4030000000000000: // x = 16.0
441 return 10'000'000'000'000'000.0;
442 case 0x4031000000000000: // x = 17.0
443 return 100'000'000'000'000'000.0;
444 case 0x4032000000000000: // x = 18.0
445 return 1'000'000'000'000'000'000.0;
446 case 0x4033000000000000: // x = 19.0
447 return 10'000'000'000'000'000'000.0;
448 case 0x4034000000000000: // x = 20.0
449 return 100'000'000'000'000'000'000.0;
450 case 0x4035000000000000: // x = 21.0
451 return 1'000'000'000'000'000'000'000.0;
452 case 0x4036000000000000: // x = 22.0
453 return 10'000'000'000'000'000'000'000.0;
454 case 0x4037000000000000: // x = 23.0
455 return 0x1.52d02c7e14af6p76 + x;
456 }
457 }
458
459 // Use double-double
460 DoubleDouble r_dd = exp10_double_double(x, kd, exp_mid);
461
462 double upper_dd = r_dd.hi + (r_dd.lo + ERR_DD);
463 double lower_dd = r_dd.hi + (r_dd.lo - ERR_DD);
464
465 if (LIBC_LIKELY(upper_dd == lower_dd)) {
466 // To multiply by 2^hi, a fast way is to simply add hi to the exponent
467 // field.
468 int64_t exp_hi = static_cast<int64_t>(hi) << FPBits::FRACTION_LEN;
469 double r = cpp::bit_cast<double>(exp_hi + cpp::bit_cast<int64_t>(upper_dd));
470 return r;
471 }
472
473 // Use 128-bit precision
474 Float128 r_f128 = exp10_f128(x, kd, idx1, idx2);
475
476 return static_cast<double>(r_f128);
477 }
478
479 } // namespace LIBC_NAMESPACE
480