2 * Carry out arithmetic to explore conversion of CPU clock ticks to nsec
4 * When we use the CPU clock for timing, we do the following:
6 * 1) Calibrate the CPU clock to relate the frequency of CPU clock ticks
9 * Using gettimeofday() or clock_gettime(), count how many CPU clock
10 * ticks occur per usec
12 * 2) Calculate conversion factors so that we can ultimately convert
13 * from clocks ticks to nsec with
14 * nsec = (ticks * clock_mult) >> clock_shift
16 * This is equivalent to
17 * nsec = ticks * (MULTIPLIER / cycles_per_nsec) / MULTIPLIER
19 * clock_mult = MULTIPLIER / cycles_per_nsec
20 * MULTIPLIER = 2^clock_shift
22 * It would be simpler to just calculate nsec = ticks / cycles_per_nsec,
23 * but all of this is necessary because of rounding when calculating
24 * cycles_per_nsec. With a 3.0GHz CPU, cycles_per_nsec would simply
25 * be 3. But with a 3.33GHz CPU or a 4.5GHz CPU, the fractional
26 * portion is lost with integer arithmetic.
28 * This multiply and shift calculation also has a performance benefit
29 * as multiplication and bit shift operations are faster than integer
32 * 3) Dynamically determine clock_shift and clock_mult at run time based
33 * on MAX_CLOCK_SEC and cycles_per_usec. MAX_CLOCK_SEC is the maximum
34 * duration for which the conversion will be valid.
36 * The primary constraint is that (ticks * clock_mult) must not overflow
37 * when ticks is at its maximum value.
40 * max_ticks = MAX_CLOCK_SEC * 1000000000 * cycles_per_nsec
41 * max_ticks * clock_mult <= ULLONG_MAX
42 * max_ticks * MULTIPLIER / cycles_per_nsec <= ULLONG_MAX
43 * MULTIPLIER <= ULLONG_MAX * cycles_per_nsec / max_ticks
45 * Then choose the largest clock_shift that satisfies
46 * 2^clock_shift <= ULLONG_MAX * cycles_per_nsec / max_ticks
48 * Finally calculate the appropriate clock_mult associated with clock_shift
49 * clock_mult = 2^clock_shift / cycles_per_nsec
51 * 4) In the code below we have cycles_per_usec and use
52 * cycles_per_nsec = cycles_per_usec / 1000
55 * The code below implements 4 clock tick to nsec conversion strategies
57 * i) 64-bit arithmetic for the (ticks * clock_mult) product with the
58 * conversion valid for at most MAX_CLOCK_SEC
60 * ii) NOT IMPLEMENTED Use 64-bit integers to emulate 128-bit multiplication
61 * for the (ticks * clock_mult) product
63 * iii) 64-bit arithmetic with clock ticks to nsec conversion occurring in
64 * two stages. The first stage counts the number of discrete, large chunks
65 * of time that have elapsed. To this is added the time represented by
66 * the remaining clock ticks. The advantage of this strategy is better
67 * accuracy because the (ticks * clock_mult) product used for final
70 * iv) 64-bit arithmetic with the clock ticks to nsec conversion occuring in
71 * two stages. This is carried out using locks to update the number of
72 * large time chunks (MAX_CLOCK_SEC_2STAGE) that have elapsed.
74 * v) 128-bit arithmetic used for the clock ticks to nsec conversion.
83 #include "lib/seqlock.h"
86 #define MAX_CLOCK_SEC 365*24*60*60ULL
87 #define MAX_CLOCK_SEC_2STAGE 60*60ULL
88 #define dprintf(...) if (DEBUG) { printf(__VA_ARGS__); }
91 __CLOCK64_BIT = 1 << 0,
92 __CLOCK128_BIT = 1 << 1,
93 __CLOCK_MULT_SHIFT = 1 << 2,
94 __CLOCK_EMULATE_128 = 1 << 3,
95 __CLOCK_2STAGE = 1 << 4,
96 __CLOCK_LOCK = 1 << 5,
98 CLOCK64_MULT_SHIFT = __CLOCK64_BIT | __CLOCK_MULT_SHIFT,
99 CLOCK64_EMULATE_128 = __CLOCK64_BIT | __CLOCK_EMULATE_128,
100 CLOCK64_2STAGE = __CLOCK64_BIT | __CLOCK_2STAGE,
101 CLOCK64_LOCK = __CLOCK64_BIT | __CLOCK_LOCK,
102 CLOCK128_MULT_SHIFT = __CLOCK128_BIT | __CLOCK_MULT_SHIFT,
105 struct seqlock clock_seqlock;
106 unsigned long long cycles_start;
107 unsigned long long elapsed_nsec;
109 unsigned int max_cycles_shift;
110 unsigned long long max_cycles_mask;
111 unsigned long long nsecs_for_max_cycles;
113 unsigned int clock_shift;
114 unsigned long long clock_mult;
116 unsigned long long *nsecs;
117 unsigned long long clock_mult64_128[2];
118 __uint128_t clock_mult128;
121 * Functions for carrying out 128-bit
122 * arithmetic using 64-bit integers
124 * 128-bit integers are stored as
125 * arrays of two 64-bit integers
127 * Ordering is little endian
129 * a[0] has the less significant bits
130 * a[1] has the more significant bits
132 * NOT FULLY IMPLEMENTED
134 void do_mult(unsigned long long a[2], unsigned long long b, unsigned long long product[2])
136 product[0] = product[1] = 0;
140 void do_div(unsigned long long a[2], unsigned long long b, unsigned long long c[2])
145 void do_shift64(unsigned long long a[2], unsigned int count)
147 a[0] = a[1] >> (count-64);
151 void do_shift(unsigned long long a[2], unsigned int count)
154 do_shift64(a, count);
163 void update_clock(unsigned long long t)
165 write_seqlock_begin(&clock_seqlock);
166 elapsed_nsec = (t >> max_cycles_shift) * nsecs_for_max_cycles;
167 cycles_start = t & ~max_cycles_mask;
168 write_seqlock_end(&clock_seqlock);
171 unsigned long long _get_nsec(int mode, unsigned long long t)
174 case CLOCK64_MULT_SHIFT: {
175 return (t * clock_mult) >> clock_shift;
177 case CLOCK64_EMULATE_128: {
178 unsigned long long product[2];
179 do_mult(clock_mult64_128, t, product);
180 do_shift(product, clock_shift);
183 case CLOCK64_2STAGE: {
184 unsigned long long multiples, nsec;
185 multiples = t >> max_cycles_shift;
186 dprintf("multiples=%llu\n", multiples);
187 nsec = multiples * nsecs_for_max_cycles;
188 nsec += ((t & max_cycles_mask) * clock_mult) >> clock_shift;
193 unsigned long long nsec;
195 seq = read_seqlock_begin(&clock_seqlock);
197 nsec += ((t - cycles_start) * clock_mult) >> clock_shift;
198 } while (read_seqlock_retry(&clock_seqlock, seq));
201 case CLOCK128_MULT_SHIFT: {
202 return (unsigned long long)((t * clock_mult128) >> clock_shift);
210 unsigned long long get_nsec(int mode, unsigned long long t)
212 if (mode == CLOCK64_LOCK) {
216 return _get_nsec(mode, t);
219 void calc_mult_shift(int mode, void *mult, unsigned int *shift, unsigned long long max_sec, unsigned long long cycles_per_usec)
221 unsigned long long max_ticks;
222 max_ticks = max_sec * cycles_per_usec * 1000000ULL;
225 case CLOCK64_MULT_SHIFT: {
226 unsigned long long max_mult, tmp;
227 unsigned int sft = 0;
230 * Calculate the largest multiplier that will not
231 * produce a 64-bit overflow in the multiplication
232 * step of the clock ticks to nsec conversion
234 max_mult = ULLONG_MAX / max_ticks;
235 dprintf("max_ticks=%llu, __builtin_clzll=%d, max_mult=%llu\n", max_ticks, __builtin_clzll(max_ticks), max_mult);
238 * Find the largest shift count that will produce
239 * a multiplier less than max_mult
241 tmp = max_mult * cycles_per_usec / 1000;
245 dprintf("tmp=%llu, sft=%u\n", tmp, sft);
249 *((unsigned long long *)mult) = (unsigned long long) ((1ULL << sft) * 1000 / cycles_per_usec);
252 case CLOCK64_EMULATE_128: {
253 unsigned long long max_mult[2], tmp[2];
254 unsigned int sft = 0;
257 * Calculate the largest multiplier that will not
258 * produce a 128-bit overflow in the multiplication
259 * step of the clock ticks to nsec conversion,
260 * but use only 64-bit integers in the process
262 max_mult[0] = max_mult[1] = ULLONG_MAX;
263 do_div(max_mult, max_ticks, max_mult);
264 dprintf("max_ticks=%llu, __builtin_clzll=%d, max_mult=0x%016llx%016llx\n",
265 max_ticks, __builtin_clzll(max_ticks), max_mult[1], max_mult[0]);
268 * Find the largest shift count that will produce
269 * a multiplier less than max_mult
271 do_div(max_mult, cycles_per_usec, tmp);
272 do_div(tmp, 1000ULL, tmp);
273 while (tmp[0] > 1 || tmp[1] > 1) {
276 dprintf("tmp=0x%016llx%016llx, sft=%u\n", tmp[1], tmp[0], sft);
280 // *((unsigned long long *)mult) = (__uint128_t) (((__uint128_t)1 << sft) * 1000 / cycles_per_usec);
283 case CLOCK64_2STAGE: {
284 unsigned long long tmp;
286 * This clock tick to nsec conversion requires two stages.
288 * Stage 1: Determine how many ~MAX_CLOCK_SEC_2STAGE periods worth of clock ticks
289 * have elapsed and set nsecs to the appropriate value for those
290 * ~MAX_CLOCK_SEC_2STAGE periods.
291 * Stage 2: Subtract the ticks for the elapsed ~MAX_CLOCK_SEC_2STAGE periods from
292 * Stage 1. Convert remaining clock ticks to nsecs and add to previously
295 * To optimize the arithmetic operations, use the greatest power of 2 ticks
296 * less than the number of ticks in MAX_CLOCK_SEC_2STAGE seconds.
299 // Use a period shorter than MAX_CLOCK_SEC here for better accuracy
300 calc_mult_shift(CLOCK64_MULT_SHIFT, mult, shift, MAX_CLOCK_SEC_2STAGE, cycles_per_usec);
302 // Find the greatest power of 2 clock ticks that is less than the ticks in MAX_CLOCK_SEC_2STAGE
303 max_cycles_shift = max_cycles_mask = 0;
304 tmp = MAX_CLOCK_SEC_2STAGE * 1000000ULL * cycles_per_usec;
305 dprintf("tmp=%llu, max_cycles_shift=%u\n", tmp, max_cycles_shift);
309 dprintf("tmp=%llu, max_cycles_shift=%u\n", tmp, max_cycles_shift);
311 // if use use (1ULL << max_cycles_shift) * 1000 / cycles_per_usec here we will
312 // have a discontinuity every (1ULL << max_cycles_shift) cycles
313 nsecs_for_max_cycles = (1ULL << max_cycles_shift) * *((unsigned long long *)mult) >> *shift;
315 // Use a bitmask to calculate ticks % (1ULL << max_cycles_shift)
316 for (tmp = 0; tmp < max_cycles_shift; tmp++)
317 max_cycles_mask |= 1ULL << tmp;
319 dprintf("max_cycles_shift=%u, 2^max_cycles_shift=%llu, nsecs_for_max_cycles=%llu, max_cycles_mask=%016llx\n",
320 max_cycles_shift, (1ULL << max_cycles_shift),
321 nsecs_for_max_cycles, max_cycles_mask);
328 * This clock tick to nsec conversion also requires two stages.
330 * Stage 1: Add to nsec the current running total of elapsed long periods
331 * Stage 2: Subtract from clock ticks the tick count corresponding to the
332 * most recently elapsed long period. Convert the remaining ticks to
333 * nsec and add to the previous nsec value.
335 * In practice the elapsed nsec from Stage 1 and the tick count subtracted
336 * in Stage 2 will be maintained in a separate thread.
339 calc_mult_shift(CLOCK64_2STAGE, mult, shift, MAX_CLOCK_SEC, cycles_per_usec);
343 case CLOCK128_MULT_SHIFT: {
344 __uint128_t max_mult, tmp;
345 unsigned int sft = 0;
348 * Calculate the largest multiplier that will not
349 * produce a 128-bit overflow in the multiplication
350 * step of the clock ticks to nsec conversion
352 max_mult = ((__uint128_t) ULLONG_MAX) << 64 | ULLONG_MAX;
353 max_mult /= max_ticks;
354 dprintf("max_ticks=%llu, __builtin_clzll=%d, max_mult=0x%016llx%016llx\n",
355 max_ticks, __builtin_clzll(max_ticks),
356 (unsigned long long) (max_mult >> 64),
357 (unsigned long long) max_mult);
360 * Find the largest shift count that will produce
361 * a multiplier less than max_mult
363 tmp = max_mult * cycles_per_usec / 1000;
367 dprintf("tmp=0x%016llx%016llx, sft=%u\n",
368 (unsigned long long) (tmp >> 64),
369 (unsigned long long) tmp, sft);
373 *((__uint128_t *)mult) = (__uint128_t) (((__uint128_t)1 << sft) * 1000 / cycles_per_usec);
379 int discontinuity(int mode, int delta_ticks, int delta_nsec, unsigned long long start, unsigned long len)
382 unsigned long mismatches = 0, bad_mismatches = 0;
383 unsigned long long delta, max_mismatch = 0;
384 unsigned long long *ns = nsecs;
386 for (i = 0; i < len; ns++, i++) {
387 *ns = get_nsec(mode, start + i);
388 if (i - delta_ticks >= 0) {
389 if (*ns > *(ns - delta_ticks))
390 delta = *ns - *(ns - delta_ticks);
392 delta = *(ns - delta_ticks) - *ns;
393 if (delta > delta_nsec)
396 delta = delta_nsec - delta;
401 if (delta > max_mismatch)
402 max_mismatch = delta;
406 assert(max_mismatch == 0 || max_mismatch == 1);
408 assert(max_mismatch == 0);
411 printf("%lu discontinuities (%lu%%) (%lu errors > 1ns, max delta = %lluns) for ticks = %llu...%llu\n",
412 mismatches, (mismatches * 100) / len, bad_mismatches, max_mismatch, start,
417 #define MIN_TICKS 1ULL
418 #define LEN 1000000000ULL
419 #define NSEC_ONE_SEC 1000000000ULL
421 long long test_clock(int mode, int cycles_per_usec, int fast_test, int quiet, int delta_ticks, int delta_nsec)
425 unsigned long long max_ticks;
426 unsigned long long nsecs;
428 unsigned long long test_ns[TESTLEN] =
429 {NSEC_ONE_SEC, NSEC_ONE_SEC,
430 NSEC_ONE_SEC, NSEC_ONE_SEC*60, NSEC_ONE_SEC*60*60,
431 NSEC_ONE_SEC*60*60*2, NSEC_ONE_SEC*60*60*4,
432 NSEC_ONE_SEC*60*60*8, NSEC_ONE_SEC*60*60*24};
433 unsigned long long test_ticks[TESTLEN];
435 max_ticks = MAX_CLOCK_SEC * (unsigned long long) cycles_per_usec * 1000000ULL;
438 case CLOCK64_MULT_SHIFT: {
442 case CLOCK64_EMULATE_128: {
443 mult = clock_mult64_128;
446 case CLOCK64_2STAGE: {
454 case CLOCK128_MULT_SHIFT: {
455 mult = &clock_mult128;
459 calc_mult_shift(mode, mult, &clock_shift, MAX_CLOCK_SEC, cycles_per_usec);
460 nsecs = get_nsec(mode, max_ticks);
461 delta = nsecs/1000000 - MAX_CLOCK_SEC*1000;
463 if (mode == CLOCK64_2STAGE) {
464 test_ns[0] = nsecs_for_max_cycles - 1;
465 test_ns[1] = nsecs_for_max_cycles;
466 test_ticks[0] = (1ULL << max_cycles_shift) - 1;
467 test_ticks[1] = (1ULL << max_cycles_shift);
469 for (i = 2; i < TESTLEN; i++)
470 test_ticks[i] = test_ns[i] / 1000 * cycles_per_usec;
473 for (i = 0; i < TESTLEN; i++)
474 test_ticks[i] = test_ns[i] / 1000 * cycles_per_usec;
478 printf("cycles_per_usec=%d, delta_ticks=%d, delta_nsec=%d, max_ticks=%llu, shift=%u, 2^shift=%llu\n",
479 cycles_per_usec, delta_ticks, delta_nsec, max_ticks, clock_shift, (1ULL << clock_shift));
483 case CLOCK64_MULT_SHIFT: {
484 printf("clock_mult=%llu, clock_mult / 2^clock_shift=%f\n",
485 clock_mult, (double) clock_mult / (1ULL << clock_shift));
488 case CLOCK64_EMULATE_128: {
489 printf("clock_mult=0x%016llx%016llx\n",
490 clock_mult64_128[1], clock_mult64_128[0]);
493 case CLOCK128_MULT_SHIFT: {
494 printf("clock_mult=0x%016llx%016llx\n",
495 (unsigned long long) (clock_mult128 >> 64),
496 (unsigned long long) clock_mult128);
500 printf("get_nsec(max_ticks) = %lluns, should be %lluns, error<=abs(%lld)ms\n",
501 nsecs, MAX_CLOCK_SEC*1000000000ULL, delta);
504 for (i = 0; i < TESTLEN; i++)
506 nsecs = get_nsec(mode, test_ticks[i]);
507 delta = nsecs > test_ns[i] ? nsecs - test_ns[i] : test_ns[i] - nsecs;
508 if (!quiet || delta > 0)
509 printf("get_nsec(%llu)=%llu, expected %llu, delta=%llu\n",
510 test_ticks[i], nsecs, test_ns[i], delta);
514 discontinuity(mode, delta_ticks, delta_nsec, max_ticks - LEN + 1, LEN);
515 discontinuity(mode, delta_ticks, delta_nsec, MIN_TICKS, LEN);
524 int main(int argc, char *argv[])
528 long long errors[10001];
531 nsecs = malloc(LEN * sizeof(unsigned long long));
532 assert(nsecs != NULL);
533 days = MAX_CLOCK_SEC / 60 / 60 / 24;
535 test_clock(CLOCK64_LOCK, 3333, 1, 0, 0, 0);
536 // test_clock(CLOCK64_MULT_SHIFT, 3333, 1, 0, 0, 0);
537 // test_clock(CLOCK128_MULT_SHIFT, 3333, 1, 0, 0, 0);
539 // Test 3 different clock types from 1000 to 10000 MHz
540 // and calculate average error
542 for (i = 1000, mean = 0.0; i <= 10000; i++) {
543 error = test_clock(CLOCK64_MULT_SHIFT, i, 1, 1, 0, 0);
544 errors[i] = error > 0 ? error : -1LL * error;
545 mean += (double) errors[i] / 9000;
547 printf(" 64-bit average error per %d days: %fms\n", days, mean);
549 for (i = 1000, mean = 0.0; i <= 10000; i++) {
550 error = test_clock(CLOCK64_2STAGE, i, 1, 1, 0, 0);
551 errors[i] = error > 0 ? error : -1LL * error;
552 mean += (double) errors[i] / 9000;
554 printf(" 64-bit two-stage average error per %d days: %fms\n", days, mean);
556 for (i = 1000, mean = 0.0; i <= 10000; i++) {
557 error = test_clock(CLOCK128_MULT_SHIFT, i, 1, 1, 0, 0);
558 errors[i] = error > 0 ? error : -1LL * error;
559 mean += (double) errors[i] / 9000;
561 printf(" 128-bit average error per %d days: %fms\n", days, mean);
563 test_clock(CLOCK64_LOCK, 1000, 1, 0, 1, 1);
564 test_clock(CLOCK64_LOCK, 1100, 1, 0, 11, 10);
565 test_clock(CLOCK64_LOCK, 3000, 1, 0, 3, 1);
566 test_clock(CLOCK64_LOCK, 3333, 1, 0, 3333, 1000);
567 test_clock(CLOCK64_LOCK, 3392, 1, 0, 424, 125);
568 test_clock(CLOCK64_LOCK, 4500, 1, 0, 9, 2);
569 test_clock(CLOCK64_LOCK, 5000, 1, 0, 5, 1);