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squiggle.c/squiggle_more.c

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#include "squiggle.h"
#include <float.h>
#include <limits.h>
#include <math.h>
#include <omp.h>
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h> // memcpy
/* Cache optimizations */
#define CACHE_LINE_SIZE 64
// getconf LEVEL1_DCACHE_LINESIZE
// <https://stackoverflow.com/questions/794632/programmatically-get-the-cache-line-size>
typedef struct seed_cache_box_t {
uint64_t seed;
char padding[CACHE_LINE_SIZE - sizeof(uint64_t)];
// Cache line size is 64 *bytes*, uint64_t is 64 *bits* (8 bytes). Different units!
} seed_cache_box;
// This avoids "false sharing", i.e., different threads competing for the same cache line
// Dealing with this shaves 4ms from a 12ms process, or a third of runtime
// <http://www.nic.uoregon.edu/~khuck/ts/acumem-report/manual_html/ch06s07.html>
/* Parallel sampler */
void sampler_parallel(double (*sampler)(uint64_t* seed), double* results, int n_threads, int n_samples)
{
// Terms of the division:
// a = b * quotient + reminder
// a = b * (a/b) + (a%b)
// dividend: a
// divisor: b
// quotient = a/b
// reminder = a%b
// "divisor's multiple" := b*(a/b)
// now, we have n_samples and n_threads
// to make our life easy, each thread will have a number of samples of: a/b (quotient)
// and we'll compute the remainder of samples separately
// to possibly do by Jorge: improve so that the remainder is included in the threads
int quotient = n_samples / n_threads;
int divisor_multiple = quotient * n_threads;
// uint64_t** seeds = malloc((size_t)n_threads * sizeof(uint64_t*));
seed_cache_box* cache_box = (seed_cache_box*)malloc(sizeof(seed_cache_box) * (size_t)n_threads);
// seed_cache_box cache_box[n_threads]; // we could use the C stack. On normal linux machines, it's 8MB ($ ulimit -s). However, it doesn't quite feel right.
srand(1);
for (int i = 0; i < n_threads; i++) {
// Constraints:
// - xorshift can't start with 0
// - the seeds should be reasonably separated and not correlated
cache_box[i].seed = (uint64_t)rand() * (UINT64_MAX / RAND_MAX);
// Other initializations tried:
// *seeds[i] = 1 + i;
// *seeds[i] = (i + 0.5)*(UINT64_MAX/n_threads);
// *seeds[i] = (i + 0.5)*(UINT64_MAX/n_threads) + constant * i;
}
int i;
#pragma omp parallel private(i)
{
#pragma omp for
for (i = 0; i < n_threads; i++) {
// It's possible I don't need the for, and could instead call omp
// in some different way and get the thread number with omp_get_thread_num()
int lower_bound_inclusive = i * quotient;
int upper_bound_not_inclusive = ((i + 1) * quotient); // note the < in the for loop below,
for (int j = lower_bound_inclusive; j < upper_bound_not_inclusive; j++) {
results[j] = sampler(&(cache_box[i].seed));
/*
t starts at 0 and ends at T
at t=0,
thread i accesses: results[i*quotient +0],
thread i+1 acccesses: results[(i+1)*quotient +0]
at t=T
thread i accesses: results[(i+1)*quotient -1]
thread i+1 acccesses: results[(i+2)*quotient -1]
The results[j] that are directly adjacent are
results[(i+1)*quotient -1] (accessed by thread i at time T)
results[(i+1)*quotient +0] (accessed by thread i+1 at time 0)
and these are themselves adjacent to
results[(i+1)*quotient -2] (accessed by thread i at time T-1)
results[(i+1)*quotient +1] (accessed by thread i+1 at time 2)
If T is large enough, which it is, two threads won't access the same
cache line at the same time.
Pictorially:
at t=0 ....i.........I.........
at t=T .............i.........I
and the two never overlap
Note that results[j] is a double, a double has 8 bytes (64 bits)
8 doubles fill a cache line of 64 bytes.
So we specifically won't get problems as long as n_samples/n_threads > 8
n_threads is normally 16, so n_samples > 128
Note also that this is only a problem in terms of speed, if n_samples<128
the results are still computed, it'll just be slower
*/
}
}
}
for (int j = divisor_multiple; j < n_samples; j++) {
results[j] = sampler(&(cache_box[0].seed));
// we can just reuse a seed,
// this isn't problematic because we;ve now stopped doing multithreading
}
free(cache_box);
}
/* Get confidence intervals, given a sampler */
typedef struct ci_t {
double low;
double high;
} ci;
inline static void swp(int i, int j, double xs[])
{
double tmp = xs[i];
xs[i] = xs[j];
xs[j] = tmp;
}
static int partition(int low, int high, double xs[], int length)
{
if (low > high || high >= length) {
printf("Invariant violated for function partition in %s (%d)", __FILE__, __LINE__);
exit(1);
}
// Note: the scratchpad/ folder in commit 578bfa27 has printfs sprinkled throughout
int pivot = low + (int)floor((high - low) / 2);
double pivot_value = xs[pivot];
swp(pivot, high, xs);
int gt = low; /* This pointer will iterate until finding an element which is greater than the pivot. Then it will move elements that are smaller before it--more specifically, it will move elements to its position and then increment. As a result all elements between gt and i will be greater than the pivot. */
for (int i = low; i < high; i++) {
if (xs[i] < pivot_value) {
swp(gt, i, xs);
gt++;
}
}
swp(high, gt, xs);
return gt;
}
static double quickselect(int k, double xs[], int n)
{
// https://en.wikipedia.org/wiki/Quickselect
double* ys = malloc((size_t)n * sizeof(double));
memcpy(ys, xs, (size_t)n * sizeof(double));
// ^: don't rearrange item order in the original array
int low = 0;
int high = n - 1;
for (;;) {
if (low == high) {
double result = ys[low];
free(ys);
return result;
}
int pivot = partition(low, high, ys, n);
if (pivot == k) {
double result = ys[pivot];
free(ys);
return result;
} else if (k < pivot) {
high = pivot - 1;
} else {
low = pivot + 1;
}
}
}
ci array_get_ci(ci interval, double* xs, int n)
{
int low_k = (int)floor(interval.low * n);
int high_k = (int)ceil(interval.high * n);
ci result = {
.low = quickselect(low_k, xs, n),
.high = quickselect(high_k, xs, n),
};
return result;
}
ci array_get_90_ci(double xs[], int n)
{
return array_get_ci((ci) { .low = 0.05, .high = 0.95 }, xs, n);
}
double array_get_median(double xs[], int n){
int median_k = (int)floor(0.5 * n);
return quickselect(median_k, xs, n)
}
void array_print_stats(double xs[], int n){
ci ci_90 = array_get_ci((ci) { .low = 0.05, .high = 0.95 }, xs, n);
ci ci_80 = array_get_ci((ci) { .low = 0.1, .high = 0.9 }, xs, n);
ci ci_50 = array_get_ci((ci) { .low = 0.25, .high = 0.75 }, xs, n);
double median = array_get_median(xs, n);
double mean = array_mean(xs, n);
double std = array_std(xs, n);
printf("Mean: %lf\n"
" Std: %lf\n"
" 5%%: %lf\n"
" 10%%: %lf\n"
" 25%%: %lf\n"
" 50%%: %lf\n"
" 75%%: %lf\n"
" 90%%: %lf\n"
" 95%%: %lf\n",
mean, std, ci_90.low, ci_80.low, ci_50.low, median, ci_50.high, ci_80.high, ci_90.high);
}
// Replicate some of the above functions over samplers
// However, in the future I'll delete this
// There should be a clear boundary between working with samplers and working with an array of samples
ci sampler_get_ci(ci interval, double (*sampler)(uint64_t*), int n, uint64_t* seed)
{
UNUSED(seed); // don't want to use it right now, but want to preserve ability to do so (e.g., remove parallelism from internals). Also nicer for consistency.
double* xs = malloc((size_t)n * sizeof(double));
sampler_parallel(sampler, xs, 16, n);
ci result = array_get_ci(interval, xs, n);
free(xs);
return result;
}
ci sampler_get_90_ci(double (*sampler)(uint64_t*), int n, uint64_t* seed)
{
return sampler_get_ci((ci) { .low = 0.05, .high = 0.95 }, sampler, n, seed);
}
/* Algebra manipulations */
#define NORMAL90CONFIDENCE 1.6448536269514727
typedef struct normal_params_t {
double mean;
double std;
} normal_params;
normal_params algebra_sum_normals(normal_params a, normal_params b)
{
normal_params result = {
.mean = a.mean + b.mean,
.std = sqrt((a.std * a.std) + (b.std * b.std)),
};
return result;
}
typedef struct lognormal_params_t {
double logmean;
double logstd;
} lognormal_params;
lognormal_params algebra_product_lognormals(lognormal_params a, lognormal_params b)
{
lognormal_params result = {
.logmean = a.logmean + b.logmean,
.logstd = sqrt((a.logstd * a.logstd) + (b.logstd * b.logstd)),
};
return result;
}
lognormal_params convert_ci_to_lognormal_params(ci x)
{
double loghigh = log(x.high);
double loglow = log(x.low);
double logmean = (loghigh + loglow) / 2.0;
double logstd = (loghigh - loglow) / (2.0 * NORMAL90CONFIDENCE);
lognormal_params result = { .logmean = logmean, .logstd = logstd };
return result;
}
ci convert_lognormal_params_to_ci(lognormal_params y)
{
double h = y.logstd * NORMAL90CONFIDENCE;
double loghigh = y.logmean + h;
double loglow = y.logmean - h;
ci result = { .low = exp(loglow), .high = exp(loghigh) };
return result;
}
/* Scaffolding to handle errors */
// We will sample from an arbitrary cdf
// and that operation might fail
// so we build some scaffolding here
#define MAX_ERROR_LENGTH 500
#define EXIT_ON_ERROR 0
#define PROCESS_ERROR(error_msg) process_error(error_msg, EXIT_ON_ERROR, __FILE__, __LINE__)
typedef struct box_t {
int empty;
double content;
char* error_msg;
} box;
box process_error(const char* error_msg, int should_exit, char* file, int line)
{
if (should_exit) {
printf("%s, @, in %s (%d)", error_msg, file, line);
exit(1);
} else {
char error_msg[MAX_ERROR_LENGTH];
snprintf(error_msg, MAX_ERROR_LENGTH, "@, in %s (%d)", file, line); // NOLINT: We are being carefull here by considering MAX_ERROR_LENGTH explicitly.
box error = { .empty = 1, .error_msg = error_msg };
return error;
}
}
/* Invert an arbitrary cdf at a point */
// Version #1:
// - input: (cdf: double => double, p)
// - output: Box(number|error)
box inverse_cdf_double(double cdf(double), double p)
{
// given a cdf: [-Inf, Inf] => [0,1]
// returns a box with either
// x such that cdf(x) = p
// or an error
// if EXIT_ON_ERROR is set to 1, it exits instead of providing an error
double low = -1.0;
double high = 1.0;
// 1. Make sure that cdf(low) < p < cdf(high)
int interval_found = 0;
while ((!interval_found) && (low > -DBL_MAX / 4) && (high < DBL_MAX / 4)) {
// for floats, use FLT_MAX instead
// Note that this approach is overkill
// but it's also the *correct* thing to do.
int low_condition = (cdf(low) < p);
int high_condition = (p < cdf(high));
if (low_condition && high_condition) {
interval_found = 1;
} else if (!low_condition) {
low = low * 2;
} else if (!high_condition) {
high = high * 2;
}
}
if (!interval_found) {
return PROCESS_ERROR("Interval containing the target value not found, in function inverse_cdf");
} else {
int convergence_condition = 0;
int count = 0;
while (!convergence_condition && (count < (INT_MAX / 2))) {
double mid = (high + low) / 2;
int mid_not_new = (mid == low) || (mid == high);
// double width = high - low;
// if ((width < 1e-8) || mid_not_new){
if (mid_not_new) {
convergence_condition = 1;
} else {
double mid_sign = cdf(mid) - p;
if (mid_sign < 0) {
low = mid;
} else if (mid_sign > 0) {
high = mid;
} else if (mid_sign == 0) {
low = mid;
high = mid;
}
}
}
if (convergence_condition) {
box result = { .empty = 0, .content = low };
return result;
} else {
return PROCESS_ERROR("Search process did not converge, in function inverse_cdf");
}
}
}
// Version #2:
// - input: (cdf: double => Box(number|error), p)
// - output: Box(number|error)
box inverse_cdf_box(box cdf_box(double), double p)
{
// given a cdf: [-Inf, Inf] => Box([0,1])
// returns a box with either
// x such that cdf(x) = p
// or an error
// if EXIT_ON_ERROR is set to 1, it exits instead of providing an error
double low = -1.0;
double high = 1.0;
// 1. Make sure that cdf(low) < p < cdf(high)
int interval_found = 0;
while ((!interval_found) && (low > -DBL_MAX / 4) && (high < DBL_MAX / 4)) {
// for floats, use FLT_MAX instead
// Note that this approach is overkill
// but it's also the *correct* thing to do.
box cdf_low = cdf_box(low);
if (cdf_low.empty) {
return PROCESS_ERROR(cdf_low.error_msg);
}
box cdf_high = cdf_box(high);
if (cdf_high.empty) {
return PROCESS_ERROR(cdf_low.error_msg);
}
int low_condition = (cdf_low.content < p);
int high_condition = (p < cdf_high.content);
if (low_condition && high_condition) {
interval_found = 1;
} else if (!low_condition) {
low = low * 2;
} else if (!high_condition) {
high = high * 2;
}
}
if (!interval_found) {
return PROCESS_ERROR("Interval containing the target value not found, in function inverse_cdf");
} else {
int convergence_condition = 0;
int count = 0;
while (!convergence_condition && (count < (INT_MAX / 2))) {
double mid = (high + low) / 2;
int mid_not_new = (mid == low) || (mid == high);
// double width = high - low;
if (mid_not_new) {
// if ((width < 1e-8) || mid_not_new){
convergence_condition = 1;
} else {
box cdf_mid = cdf_box(mid);
if (cdf_mid.empty) {
return PROCESS_ERROR(cdf_mid.error_msg);
}
double mid_sign = cdf_mid.content - p;
if (mid_sign < 0) {
low = mid;
} else if (mid_sign > 0) {
high = mid;
} else if (mid_sign == 0) {
low = mid;
high = mid;
}
}
}
if (convergence_condition) {
box result = { .empty = 0, .content = low };
return result;
} else {
return PROCESS_ERROR("Search process did not converge, in function inverse_cdf");
}
}
}
/* Sample from an arbitrary cdf */
// Before: invert an arbitrary cdf at a point
// Now: from an arbitrary cdf, get a sample
box sampler_cdf_box(box cdf(double), uint64_t* seed)
{
double p = sample_unit_uniform(seed);
box result = inverse_cdf_box(cdf, p);
return result;
}
box sampler_cdf_double(double cdf(double), uint64_t* seed)
{
double p = sample_unit_uniform(seed);
box result = inverse_cdf_double(cdf, p);
return result;
}
double sampler_cdf_danger(box cdf(double), uint64_t* seed)
{
double p = sample_unit_uniform(seed);
box result = inverse_cdf_box(cdf, p);
if (result.empty) {
exit(1);
} else {
return result.content;
}
}
/* array print: potentially useful for debugging */
void array_print(double xs[], int n)
{
printf("[");
for (int i = 0; i < n - 1; i++) {
printf("%f, ", xs[i]);
}
printf("%f", xs[n - 1]);
printf("]\n");
}
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