Line data Source code
1 : /*-------------------------------------------------------------------------
2 : *
3 : * int128.h
4 : * Roll-our-own 128-bit integer arithmetic.
5 : *
6 : * We make use of the native int128 type if there is one, otherwise
7 : * implement things the hard way based on two int64 halves.
8 : *
9 : * See src/test/modules/test_int128 for a simple test harness for this file.
10 : *
11 : * Copyright (c) 2017-2025, PostgreSQL Global Development Group
12 : *
13 : * src/include/common/int128.h
14 : *
15 : *-------------------------------------------------------------------------
16 : */
17 : #ifndef INT128_H
18 : #define INT128_H
19 :
20 : /*
21 : * For testing purposes, use of native int128 can be switched on/off by
22 : * predefining USE_NATIVE_INT128.
23 : */
24 : #ifndef USE_NATIVE_INT128
25 : #ifdef HAVE_INT128
26 : #define USE_NATIVE_INT128 1
27 : #else
28 : #define USE_NATIVE_INT128 0
29 : #endif
30 : #endif
31 :
32 : /*
33 : * If native int128 support is enabled, INT128 is just int128. Otherwise, it
34 : * is a structure with separate 64-bit high and low parts.
35 : *
36 : * We lay out the INT128 structure with the same content and byte ordering
37 : * that a native int128 type would (probably) have. This makes no difference
38 : * for ordinary use of INT128, but allows union'ing INT128 with int128 for
39 : * testing purposes.
40 : *
41 : * PG_INT128_HI_INT64 and PG_INT128_LO_UINT64 allow the (signed) high and
42 : * (unsigned) low 64-bit integer parts to be extracted portably on all
43 : * platforms.
44 : */
45 : #if USE_NATIVE_INT128
46 :
47 : typedef int128 INT128;
48 :
49 : #define PG_INT128_HI_INT64(i128) ((int64) ((i128) >> 64))
50 : #define PG_INT128_LO_UINT64(i128) ((uint64) (i128))
51 :
52 : #else
53 :
54 : typedef struct
55 : {
56 : #ifdef WORDS_BIGENDIAN
57 : int64 hi; /* most significant 64 bits, including sign */
58 : uint64 lo; /* least significant 64 bits, without sign */
59 : #else
60 : uint64 lo; /* least significant 64 bits, without sign */
61 : int64 hi; /* most significant 64 bits, including sign */
62 : #endif
63 : } INT128;
64 :
65 : #define PG_INT128_HI_INT64(i128) ((i128).hi)
66 : #define PG_INT128_LO_UINT64(i128) ((i128).lo)
67 :
68 : #endif
69 :
70 : /*
71 : * Construct an INT128 from (signed) high and (unsigned) low 64-bit integer
72 : * parts.
73 : */
74 : static inline INT128
75 72 : make_int128(int64 hi, uint64 lo)
76 : {
77 : #if USE_NATIVE_INT128
78 72 : return (((int128) hi) << 64) + lo;
79 : #else
80 : INT128 val;
81 :
82 : val.hi = hi;
83 : val.lo = lo;
84 : return val;
85 : #endif
86 : }
87 :
88 : /*
89 : * Add an unsigned int64 value into an INT128 variable.
90 : */
91 : static inline void
92 10000000 : int128_add_uint64(INT128 *i128, uint64 v)
93 : {
94 : #if USE_NATIVE_INT128
95 : *i128 += v;
96 : #else
97 : /*
98 : * First add the value to the .lo part, then check to see if a carry needs
99 : * to be propagated into the .hi part. Since this is unsigned integer
100 : * arithmetic, which is just modular arithmetic, a carry is needed if the
101 : * new .lo part is less than the old .lo part (i.e., if modular
102 : * wrap-around occurred). Writing this in the form below, rather than
103 : * using an "if" statement causes modern compilers to produce branchless
104 : * machine code identical to the native code.
105 : */
106 10000000 : uint64 oldlo = i128->lo;
107 :
108 10000000 : i128->lo += v;
109 10000000 : i128->hi += (i128->lo < oldlo);
110 : #endif
111 10000000 : }
112 :
113 : /*
114 : * Add a signed int64 value into an INT128 variable.
115 : */
116 : static inline void
117 2557482 : int128_add_int64(INT128 *i128, int64 v)
118 : {
119 : #if USE_NATIVE_INT128
120 557482 : *i128 += v;
121 : #else
122 : /*
123 : * This is much like the above except that the carry logic differs for
124 : * negative v -- we need to subtract 1 from the .hi part if the new .lo
125 : * value is greater than the old .lo value. That can be achieved without
126 : * any branching by adding the sign bit from v (v >> 63 = 0 or -1) to the
127 : * previous result (for negative v, if the new .lo value is less than the
128 : * old .lo value, the two terms cancel and we leave the .hi part
129 : * unchanged, otherwise we subtract 1 from the .hi part). With modern
130 : * compilers this often produces machine code identical to the native
131 : * code.
132 : */
133 2000000 : uint64 oldlo = i128->lo;
134 :
135 2000000 : i128->lo += v;
136 2000000 : i128->hi += (i128->lo < oldlo) + (v >> 63);
137 : #endif
138 2557482 : }
139 :
140 : /*
141 : * Add an INT128 value into an INT128 variable.
142 : */
143 : static inline void
144 2000048 : int128_add_int128(INT128 *i128, INT128 v)
145 : {
146 : #if USE_NATIVE_INT128
147 48 : *i128 += v;
148 : #else
149 2000000 : int128_add_uint64(i128, v.lo);
150 2000000 : i128->hi += v.hi;
151 : #endif
152 2000048 : }
153 :
154 : /*
155 : * Subtract an unsigned int64 value from an INT128 variable.
156 : */
157 : static inline void
158 8000000 : int128_sub_uint64(INT128 *i128, uint64 v)
159 : {
160 : #if USE_NATIVE_INT128
161 : *i128 -= v;
162 : #else
163 : /*
164 : * This is like int128_add_uint64(), except we must propagate a borrow to
165 : * (subtract 1 from) the .hi part if the new .lo part is greater than the
166 : * old .lo part.
167 : */
168 8000000 : uint64 oldlo = i128->lo;
169 :
170 8000000 : i128->lo -= v;
171 8000000 : i128->hi -= (i128->lo > oldlo);
172 : #endif
173 8000000 : }
174 :
175 : /*
176 : * Subtract a signed int64 value from an INT128 variable.
177 : */
178 : static inline void
179 2000312 : int128_sub_int64(INT128 *i128, int64 v)
180 : {
181 : #if USE_NATIVE_INT128
182 312 : *i128 -= v;
183 : #else
184 : /* Like int128_add_int64() with the sign of v inverted */
185 2000000 : uint64 oldlo = i128->lo;
186 :
187 2000000 : i128->lo -= v;
188 2000000 : i128->hi -= (i128->lo > oldlo) + (v >> 63);
189 : #endif
190 2000312 : }
191 :
192 : /*
193 : * INT64_HI_INT32 extracts the most significant 32 bits of int64 as int32.
194 : * INT64_LO_UINT32 extracts the least significant 32 bits as uint32.
195 : */
196 : #define INT64_HI_INT32(i64) ((int32) ((i64) >> 32))
197 : #define INT64_LO_UINT32(i64) ((uint32) (i64))
198 :
199 : /*
200 : * Add the 128-bit product of two int64 values into an INT128 variable.
201 : */
202 : static inline void
203 2517332 : int128_add_int64_mul_int64(INT128 *i128, int64 x, int64 y)
204 : {
205 : #if USE_NATIVE_INT128
206 : /*
207 : * XXX with a stupid compiler, this could actually be less efficient than
208 : * the non-native implementation; maybe we should do it by hand always?
209 : */
210 517332 : *i128 += (int128) x * (int128) y;
211 : #else
212 : /* INT64_HI_INT32 must use arithmetic right shift */
213 : StaticAssertDecl(((int64) -1 >> 1) == (int64) -1,
214 : "arithmetic right shift is needed");
215 :
216 : /*----------
217 : * Form the 128-bit product x * y using 64-bit arithmetic.
218 : * Considering each 64-bit input as having 32-bit high and low parts,
219 : * we can compute
220 : *
221 : * x * y = ((x.hi << 32) + x.lo) * (((y.hi << 32) + y.lo)
222 : * = (x.hi * y.hi) << 64 +
223 : * (x.hi * y.lo) << 32 +
224 : * (x.lo * y.hi) << 32 +
225 : * x.lo * y.lo
226 : *
227 : * Each individual product is of 32-bit terms so it won't overflow when
228 : * computed in 64-bit arithmetic. Then we just have to shift it to the
229 : * correct position while adding into the 128-bit result. We must also
230 : * keep in mind that the "lo" parts must be treated as unsigned.
231 : *----------
232 : */
233 :
234 : /* No need to work hard if product must be zero */
235 2000000 : if (x != 0 && y != 0)
236 : {
237 2000000 : int32 x_hi = INT64_HI_INT32(x);
238 2000000 : uint32 x_lo = INT64_LO_UINT32(x);
239 2000000 : int32 y_hi = INT64_HI_INT32(y);
240 2000000 : uint32 y_lo = INT64_LO_UINT32(y);
241 : int64 tmp;
242 :
243 : /* the first term */
244 2000000 : i128->hi += (int64) x_hi * (int64) y_hi;
245 :
246 : /* the second term: sign-extended with the sign of x */
247 2000000 : tmp = (int64) x_hi * (int64) y_lo;
248 2000000 : i128->hi += INT64_HI_INT32(tmp);
249 2000000 : int128_add_uint64(i128, ((uint64) INT64_LO_UINT32(tmp)) << 32);
250 :
251 : /* the third term: sign-extended with the sign of y */
252 2000000 : tmp = (int64) x_lo * (int64) y_hi;
253 2000000 : i128->hi += INT64_HI_INT32(tmp);
254 2000000 : int128_add_uint64(i128, ((uint64) INT64_LO_UINT32(tmp)) << 32);
255 :
256 : /* the fourth term: always unsigned */
257 2000000 : int128_add_uint64(i128, (uint64) x_lo * (uint64) y_lo);
258 : }
259 : #endif
260 2517332 : }
261 :
262 : /*
263 : * Subtract the 128-bit product of two int64 values from an INT128 variable.
264 : */
265 : static inline void
266 2000288 : int128_sub_int64_mul_int64(INT128 *i128, int64 x, int64 y)
267 : {
268 : #if USE_NATIVE_INT128
269 288 : *i128 -= (int128) x * (int128) y;
270 : #else
271 : /* As above, except subtract the 128-bit product */
272 2000000 : if (x != 0 && y != 0)
273 : {
274 2000000 : int32 x_hi = INT64_HI_INT32(x);
275 2000000 : uint32 x_lo = INT64_LO_UINT32(x);
276 2000000 : int32 y_hi = INT64_HI_INT32(y);
277 2000000 : uint32 y_lo = INT64_LO_UINT32(y);
278 : int64 tmp;
279 :
280 : /* the first term */
281 2000000 : i128->hi -= (int64) x_hi * (int64) y_hi;
282 :
283 : /* the second term: sign-extended with the sign of x */
284 2000000 : tmp = (int64) x_hi * (int64) y_lo;
285 2000000 : i128->hi -= INT64_HI_INT32(tmp);
286 2000000 : int128_sub_uint64(i128, ((uint64) INT64_LO_UINT32(tmp)) << 32);
287 :
288 : /* the third term: sign-extended with the sign of y */
289 2000000 : tmp = (int64) x_lo * (int64) y_hi;
290 2000000 : i128->hi -= INT64_HI_INT32(tmp);
291 2000000 : int128_sub_uint64(i128, ((uint64) INT64_LO_UINT32(tmp)) << 32);
292 :
293 : /* the fourth term: always unsigned */
294 2000000 : int128_sub_uint64(i128, (uint64) x_lo * (uint64) y_lo);
295 : }
296 : #endif
297 2000288 : }
298 :
299 : /*
300 : * Divide an INT128 variable by a signed int32 value, returning the quotient
301 : * and remainder. The remainder will have the same sign as *i128.
302 : *
303 : * Note: This provides no protection against dividing by 0, or dividing
304 : * INT128_MIN by -1, which overflows. It is the caller's responsibility to
305 : * guard against those.
306 : */
307 : static inline void
308 2045240 : int128_div_mod_int32(INT128 *i128, int32 v, int32 *remainder)
309 : {
310 : #if USE_NATIVE_INT128
311 45240 : int128 old_i128 = *i128;
312 :
313 45240 : *i128 /= v;
314 45240 : *remainder = (int32) (old_i128 - *i128 * v);
315 : #else
316 : /*
317 : * To avoid any intermediate values overflowing (as happens if INT64_MIN
318 : * is divided by -1), we first compute the quotient abs(*i128) / abs(v)
319 : * using unsigned 64-bit arithmetic, and then fix the signs up at the end.
320 : *
321 : * The quotient is computed using the short division algorithm described
322 : * in Knuth volume 2, section 4.3.1 exercise 16 (cf. div_var_int() in
323 : * numeric.c). Since the absolute value of the divisor is known to be at
324 : * most 2^31, the remainder carried from one digit to the next is at most
325 : * 2^31 - 1, and so there is no danger of overflow when this is combined
326 : * with the next digit (a 32-bit unsigned integer).
327 : */
328 : uint64 n_hi;
329 : uint64 n_lo;
330 : uint32 d;
331 : uint64 q;
332 : uint64 r;
333 : uint64 tmp;
334 :
335 : /* numerator: absolute value of *i128 */
336 2000000 : if (i128->hi < 0)
337 : {
338 998936 : n_hi = 0 - ((uint64) i128->hi);
339 998936 : n_lo = 0 - i128->lo;
340 998936 : if (n_lo != 0)
341 998936 : n_hi--;
342 : }
343 : else
344 : {
345 1001064 : n_hi = i128->hi;
346 1001064 : n_lo = i128->lo;
347 : }
348 :
349 : /* denomimator: absolute value of v */
350 2000000 : d = abs(v);
351 :
352 : /* quotient and remainder of high 64 bits */
353 2000000 : q = n_hi / d;
354 2000000 : r = n_hi % d;
355 2000000 : n_hi = q;
356 :
357 : /* quotient and remainder of next 32 bits (upper half of n_lo) */
358 2000000 : tmp = (r << 32) + (n_lo >> 32);
359 2000000 : q = tmp / d;
360 2000000 : r = tmp % d;
361 :
362 : /* quotient and remainder of last 32 bits (lower half of n_lo) */
363 2000000 : tmp = (r << 32) + (uint32) n_lo;
364 2000000 : n_lo = q << 32;
365 2000000 : q = tmp / d;
366 2000000 : r = tmp % d;
367 2000000 : n_lo += q;
368 :
369 : /* final remainder should have the same sign as *i128 */
370 2000000 : *remainder = i128->hi < 0 ? (int32) (0 - r) : (int32) r;
371 :
372 : /* store the quotient in *i128, negating it if necessary */
373 2000000 : if ((i128->hi < 0) != (v < 0))
374 : {
375 1000784 : n_hi = 0 - n_hi;
376 1000784 : n_lo = 0 - n_lo;
377 1000784 : if (n_lo != 0)
378 1000784 : n_hi--;
379 : }
380 2000000 : i128->hi = (int64) n_hi;
381 2000000 : i128->lo = n_lo;
382 : #endif
383 2045240 : }
384 :
385 : /*
386 : * Test if an INT128 value is zero.
387 : */
388 : static inline bool
389 45240 : int128_is_zero(INT128 x)
390 : {
391 : #if USE_NATIVE_INT128
392 45240 : return x == 0;
393 : #else
394 : return x.hi == 0 && x.lo == 0;
395 : #endif
396 : }
397 :
398 : /*
399 : * Return the sign of an INT128 value (returns -1, 0, or +1).
400 : */
401 : static inline int
402 8796 : int128_sign(INT128 x)
403 : {
404 : #if USE_NATIVE_INT128
405 8796 : if (x < 0)
406 12 : return -1;
407 8784 : if (x > 0)
408 8572 : return 1;
409 212 : return 0;
410 : #else
411 : if (x.hi < 0)
412 : return -1;
413 : if (x.hi > 0)
414 : return 1;
415 : if (x.lo > 0)
416 : return 1;
417 : return 0;
418 : #endif
419 : }
420 :
421 : /*
422 : * Compare two INT128 values, return -1, 0, or +1.
423 : */
424 : static inline int
425 4138754 : int128_compare(INT128 x, INT128 y)
426 : {
427 : #if USE_NATIVE_INT128
428 138754 : if (x < y)
429 64782 : return -1;
430 73972 : if (x > y)
431 54272 : return 1;
432 19700 : return 0;
433 : #else
434 4000000 : if (x.hi < y.hi)
435 998880 : return -1;
436 3001120 : if (x.hi > y.hi)
437 1001120 : return 1;
438 2000000 : if (x.lo < y.lo)
439 999082 : return -1;
440 1000918 : if (x.lo > y.lo)
441 1000918 : return 1;
442 0 : return 0;
443 : #endif
444 : }
445 :
446 : /*
447 : * Widen int64 to INT128.
448 : */
449 : static inline INT128
450 279858 : int64_to_int128(int64 v)
451 : {
452 : #if USE_NATIVE_INT128
453 279858 : return (INT128) v;
454 : #else
455 : INT128 val;
456 :
457 : val.lo = (uint64) v;
458 : val.hi = (v < 0) ? -INT64CONST(1) : INT64CONST(0);
459 : return val;
460 : #endif
461 : }
462 :
463 : /*
464 : * Convert INT128 to int64 (losing any high-order bits).
465 : * This also works fine for casting down to uint64.
466 : */
467 : static inline int64
468 2338 : int128_to_int64(INT128 val)
469 : {
470 : #if USE_NATIVE_INT128
471 2338 : return (int64) val;
472 : #else
473 : return (int64) val.lo;
474 : #endif
475 : }
476 :
477 : #endif /* INT128_H */
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