formatting pass, upkeep

README.md

@@ -334,6 +334,8 @@ But for more complicated use cases, my recommendation would be to not use parall
 - If you run the `sampler_parallel` function on two different inputs, their outputs will be correlated. E.g., if you run two lognormals, indices which have higher samples in one will tend to have higher samples in the other one. Why?
 - For a small amount of samples, if you run the `sampler_parallel` function, you will get better spread out random numbers than if you run things serially. Why?
 
+That said, I found adding parallelism to be an interesting an engaging task. Most recently, I even optimized the code to ensure that two threads weren't accessing the same cache line at the same time, and it was very satisfying to see a 30% improvement as a result.
+
 #### Extra: Algebraic manipulations
 
 `squiggle_more.c` has some functions to do some simple algebra manipulations: sums of normals and products of lognormals. You can see some example usage [here](examples/more/07_algebra/example.c) and [here](examples/more/08_algebra_and_conversion/example.c).

examples/core/03_gcc_nested_function/example.c

@@ -15,8 +15,16 @@ int main()
     int n_dists = 4;
 
     // These are nested functions. They will not compile without gcc.
-    double sample_0(uint64_t * seed) { UNUSED(seed); return 0; }
+    double sample_0(uint64_t * seed)
-    double sample_1(uint64_t * seed) { UNUSED(seed); return 1; }
+    {
+        UNUSED(seed);
+        return 0;
+    }
+    double sample_1(uint64_t * seed)
+    {
+        UNUSED(seed);
+        return 1;
+    }
     double sample_few(uint64_t * seed) { return sample_to(1, 3, seed); }
     double sample_many(uint64_t * seed) { return sample_to(2, 10, seed); }
 

examples/core/06_dissolving_fermi_paradox/example.c

@@ -0,0 +1,147 @@
+#include "../../../squiggle.h"
+#include <math.h>
+#include <stdint.h>
+#include <stdio.h>
+#include <stdlib.h>
+
+double sample_loguniform(double a, double b, uint64_t* seed)
+{
+    return exp(sample_uniform(log(a), log(b), seed));
+}
+
+int main()
+{
+    // Replicate <https://arxiv.org/pdf/1806.02404.pdf>, and in particular the red line in page 11.
+    // Could also be interesting to just produce and save many samples.
+
+    // set randomness seed
+    uint64_t* seed = malloc(sizeof(uint64_t));
+    *seed = UINT64_MAX / 64; // xorshift can't start with a seed of 0
+
+    double sample_fermi_naive(uint64_t * seed)
+    {
+        double rate_of_star_formation = sample_loguniform(1, 100, seed);
+        double fraction_of_stars_with_planets = sample_loguniform(0.1, 1, seed);
+        double number_of_habitable_planets_per_star_system = sample_loguniform(0.1, 1, seed);
+        double rate_of_life_formation_in_habitable_planets = sample_lognormal(1, 50, seed);
+        double fraction_of_habitable_planets_in_which_any_life_appears = -expm1(-rate_of_life_formation_in_habitable_planets);
+        // double fraction_of_habitable_planets_in_which_any_life_appears = 1-exp(-rate_of_life_formation_in_habitable_planets);
+        // but with more precision
+        double fraction_of_planets_with_life_in_which_intelligent_life_appears = sample_loguniform(0.001, 1, seed);
+        double fraction_of_intelligent_planets_which_are_detectable_as_such = sample_loguniform(0.01, 1, seed);
+        double longevity_of_detectable_civilizations = sample_loguniform(100, 10000000000, seed);
+
+        // printf(" rate_of_star_formation = %lf\n", rate_of_star_formation);
+        // printf(" fraction_of_stars_with_planets = %lf\n", fraction_of_stars_with_planets);
+        // printf(" number_of_habitable_planets_per_star_system = %lf\n", number_of_habitable_planets_per_star_system);
+        // printf(" rate_of_life_formation_in_habitable_planets = %.16lf\n", rate_of_life_formation_in_habitable_planets);
+        // printf(" fraction_of_habitable_planets_in_which_any_life_appears = %lf\n", fraction_of_habitable_planets_in_which_any_life_appears);
+        // printf(" fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", fraction_of_planets_with_life_in_which_intelligent_life_appears);
+        // printf(" fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", fraction_of_intelligent_planets_which_are_detectable_as_such);
+        // printf(" longevity_of_detectable_civilizations = %lf\n", longevity_of_detectable_civilizations);
+
+        // Expected number of civilizations in the Milky way;
+        // see footnote 3 (p. 5)
+        double n = rate_of_star_formation * fraction_of_stars_with_planets * number_of_habitable_planets_per_star_system * fraction_of_habitable_planets_in_which_any_life_appears * fraction_of_planets_with_life_in_which_intelligent_life_appears * fraction_of_intelligent_planets_which_are_detectable_as_such * longevity_of_detectable_civilizations;
+
+        return n;
+    }
+
+    double sample_fermi_paradox_naive(uint64_t * seed)
+    {
+        double n = sample_fermi_naive(seed);
+        return ((n > 1) ? 1 : 0);
+    }
+
+    double n = 1000000;
+    double naive_fermi_proportion = 0;
+    for (int i = 0; i < n; i++) {
+        double result = sample_fermi_paradox_naive(seed);
+        // printf("result: %lf\n", result);
+        naive_fermi_proportion += result;
+    }
+    printf("Naïve %% that we are not alone: %lf\n", naive_fermi_proportion / n);
+
+    // Thinking in log space
+    double sample_fermi_logspace(uint64_t * seed)
+    {
+        double log_rate_of_star_formation = sample_uniform(log(1), log(100), seed);
+        double log_fraction_of_stars_with_planets = sample_uniform(log(0.1), log(1), seed);
+        double log_number_of_habitable_planets_per_star_system = sample_uniform(log(0.1), log(1), seed);
+        double log_fraction_of_planets_with_life_in_which_intelligent_life_appears = sample_uniform(log(0.001), log(1), seed);
+        double log_fraction_of_intelligent_planets_which_are_detectable_as_such = sample_uniform(log(0.01), log(1), seed);
+        double log_longevity_of_detectable_civilizations = sample_uniform(log(100), log(10000000000), seed);
+
+        // printf(" log_rate_of_star_formation = %lf\n", log_rate_of_star_formation);
+        // printf(" log_fraction_of_stars_with_planets = %lf\n", log_fraction_of_stars_with_planets);
+        // printf(" log_number_of_habitable_planets_per_star_system = %lf\n", log_number_of_habitable_planets_per_star_system);
+        // printf(" log_fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", log_fraction_of_planets_with_life_in_which_intelligent_life_appears);
+        // printf(" log_fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", log_fraction_of_intelligent_planets_which_are_detectable_as_such);
+        // printf(" log_longevity_of_detectable_civilizations = %lf\n", log_longevity_of_detectable_civilizations);
+
+        double log_n1 = log_rate_of_star_formation + log_fraction_of_stars_with_planets + log_number_of_habitable_planets_per_star_system + log_fraction_of_planets_with_life_in_which_intelligent_life_appears + log_fraction_of_intelligent_planets_which_are_detectable_as_such + log_longevity_of_detectable_civilizations;
+        // printf("first part of calculation: %lf\n", log_n1);
+
+        /* Consider fraction_of_habitable_planets_in_which_any_life_appears separately.
+        Imprecisely, we could do:
+
+        double rate_of_life_formation_in_habitable_planets = sample_lognormal(1, 50, seed);
+        double fraction_of_habitable_planets_in_which_any_life_appears = 1- exp(-rate_of_life_formation_in_habitable_planets);
+        double log_fraction_of_habitable_planets_in_which_any_life_appears = log(1-fraction_of_habitable_planets_in_which_any_life_appears);
+        double n = exp(log_n1) * fraction_of_habitable_planets_in_which_any_life_appears;
+        // or: 
+        double n2 = exp(log_n1 + log(fraction_of_habitable_planets_in_which_any_life_appears))
+
+        However, we lose all precision here.
+
+        Now, say
+        a = underlying normal 
+        b = rate_of_life_formation_in_habitable_planets = exp(underlying normal)
+        c = 1 - exp(-b) = fraction_of_habitable_planets_in_which_any_life_appears
+        d = log(c)
+
+        Now, is there some way we can d more efficiently/precisely?
+        Turns out there is!
+
+        Looking at the Taylor expansion for c = 1 - exp(-b), it's b - b^2/2 + b^3/6 - x^b/24, etc.
+        // https://www.wolframalpha.com/input?i=1-exp%28-x%29
+        When b ~ 0 (as is often the case), this is close to b.
+
+        But now, if b ~ 0
+        c ~ b
+        and d = log(c) ~ log(b) = log(exp(a)) = a
+        */
+        double log_rate_of_life_formation_in_habitable_planets = sample_normal(1, 50, seed);
+        // printf("log_rate_of_life_formation_in_habitable_planets: %lf\n", log_rate_of_life_formation_in_habitable_planets);
+
+        double log_fraction_of_habitable_planets_in_which_any_life_appears;
+        if (log_rate_of_life_formation_in_habitable_planets < -32) {
+            log_fraction_of_habitable_planets_in_which_any_life_appears = log_rate_of_life_formation_in_habitable_planets;
+        } else {
+            double rate_of_life_formation_in_habitable_planets = exp(log_rate_of_life_formation_in_habitable_planets);
+            double fraction_of_habitable_planets_in_which_any_life_appears = -expm1(-rate_of_life_formation_in_habitable_planets);
+            log_fraction_of_habitable_planets_in_which_any_life_appears = log(fraction_of_habitable_planets_in_which_any_life_appears);
+        }
+        // printf(" log_fraction_of_habitable_planets_in_which_any_life_appears: %lf\n", log_fraction_of_habitable_planets_in_which_any_life_appears);
+
+        double log_n = log_n1 + log_fraction_of_habitable_planets_in_which_any_life_appears;
+
+        return log_n;
+    }
+
+    double sample_fermi_paradox_logspace(uint64_t * seed)
+    {
+        double n = sample_fermi_logspace(seed);
+        return ((n > 0) ? 1 : 0);
+    }
+
+    double logspace_fermi_proportion = 0;
+    for (int i = 0; i < n; i++) {
+        double result = sample_fermi_paradox_logspace(seed);
+        // printf("result: %lf\n", result);
+        logspace_fermi_proportion += result;
+    }
+    printf("Using more accurate logspace computations, %% that we are not alone: %lf\n", logspace_fermi_proportion / n);
+
+    free(seed);
+}

examples/core/06_dissolving_fermi_paradox/makefile

@@ -0,0 +1,56 @@
+# Interface: 
+#   make
+#   make build
+#   make format
+#   make run
+
+# Compiler
+CC=gcc
+# CC=tcc # <= faster compilation
+
+# Main file
+SRC=scratchpad.c ../squiggle.c ../squiggle_more.c
+OUTPUT=scratchpad
+
+## Dependencies
+MATH=-lm
+
+## Flags
+DEBUG= #'-g'
+STANDARD=-std=c99
+WARNINGS=-Wall
+OPTIMIZED=-O3  #-Ofast
+# OPENMP=-fopenmp
+
+## Formatter
+STYLE_BLUEPRINT=webkit
+FORMATTER=clang-format -i -style=$(STYLE_BLUEPRINT)
+
+## make build
+build: $(SRC)
+	$(CC) $(OPTIMIZED) $(DEBUG) $(SRC) $(MATH) -o $(OUTPUT)
+
+format: $(SRC)
+	$(FORMATTER) $(SRC)
+
+run: $(SRC) $(OUTPUT)
+	./$(OUTPUT)
+
+verify: $(SRC) $(OUTPUT)
+	./$(OUTPUT) | grep "NOT passed" -A 2 --group-separator='' || true
+
+time-linux: 
+	@echo "Requires /bin/time, found on GNU/Linux systems" && echo
+	
+	@echo "Running 100x and taking avg time $(OUTPUT)"
+	@t=$$(/usr/bin/time -f "%e" -p bash -c 'for i in {1..100}; do $(OUTPUT); done' 2>&1 >/dev/null | grep real | awk '{print $$2}' ); echo "scale=2; 1000 * $$t / 100" | bc | sed "s|^|Time using 1 thread: |" | sed 's|$$|ms|' && echo
+
+## Profiling
+
+profile-linux: 
+	echo "Requires perf, which depends on the kernel version, and might be in linux-tools package or similar"
+	echo "Must be run as sudo"
+	$(CC) $(SRC) $(MATH) -o $(OUTPUT)
+	sudo perf record ./$(OUTPUT)
+	sudo perf report
+	rm perf.data

examples/core/makefile

@@ -41,6 +41,8 @@ all:
 	$(CC) $(OPTIMIZED) $(DEBUG) $(WARN)  03_gcc_nested_function/$(SRC) $(DEPS) -o 03_gcc_nested_function/$(OUTPUT)
 	$(CC) $(OPTIMIZED) $(DEBUG) $(WARN) 04_gamma_beta/$(SRC)          $(DEPS) -o 04_gamma_beta/$(OUTPUT)
 	$(CC) $(OPTIMIZED) $(DEBUG) $(WARN) 05_hundred_lognormals/$(SRC)  $(DEPS) -o 05_hundred_lognormals/$(OUTPUT)
+	$(CC) $(OPTIMIZED) $(DEBUG) $(WARN) 06_dissolving_fermi_paradox/$(SRC)  $(DEPS) -o 06_dissolving_fermi_paradox/$(OUTPUT)
+
 
 format-all:
 	$(FORMATTER) 00_example_template/$(SRC)
@@ -49,6 +51,7 @@ format-all:
 	$(FORMATTER) 03_gcc_nested_function/$(SRC)
 	$(FORMATTER) 04_gamma_beta/$(SRC)
 	$(FORMATTER) 05_hundred_lognormals/$(SRC)
+	$(FORMATTER) 06_dissolving_fermi_paradox/$(SRC)
 
 run-all:
 	00_example_template/$(OUTPUT)
@@ -57,6 +60,7 @@ run-all:
 	03_gcc_nested_function/$(OUTPUT)
 	04_gamma_beta/$(OUTPUT)
 	05_hundred_lognormals/$(OUTPUT)
+	06_dissolving_fermi_paradox/$(OUTPUT)
 	
 ## make one DIR=01_one_sample
 one: $(DIR)/$(SRC)

scratchpad/scratchpad.c

@@ -1,25 +1,157 @@
 #include "../squiggle.h"
-#include "../squiggle_more.h"
+// #include "../squiggle_more.h"
 #include <math.h>
 #include <stdint.h>
 #include <stdio.h>
 #include <stdlib.h>
 
+double sample_loguniform(double a, double b, uint64_t* seed){
+    return exp(sample_uniform(log(a), log(b), seed));
+}
+
 int main()
 {
+    // Replicate <https://arxiv.org/pdf/1806.02404.pdf>, and in particular the red line in page 11.
+    // Could also be interesting to just produce and save many samples.
+
     // set randomness seed
     uint64_t* seed = malloc(sizeof(uint64_t));
-    *seed = 1000; // xorshift can't start with a seed of 0
+    *seed = UINT64_MAX/64; // xorshift can't start with a seed of 0
     
-    int n = 1000000;
+    double sample_fermi_naive(uint64_t* seed){
-    double* xs = malloc(sizeof(double) * n);
+        double rate_of_star_formation = sample_loguniform(1,100, seed);
-    for (int i = 0; i < n; i++) {
+        double fraction_of_stars_with_planets = sample_loguniform(0.1, 1, seed);
-        xs[i] = sample_to(10, 100, seed);
+        double number_of_habitable_planets_per_star_system = sample_loguniform(0.1, 1, seed);
+        double rate_of_life_formation_in_habitable_planets = sample_lognormal(1, 50, seed);
+        double fraction_of_habitable_planets_in_which_any_life_appears = -expm1(-rate_of_life_formation_in_habitable_planets);
+        // double fraction_of_habitable_planets_in_which_any_life_appears = 1-exp(-rate_of_life_formation_in_habitable_planets);
+        // but with more precision
+        double fraction_of_planets_with_life_in_which_intelligent_life_appears = sample_loguniform(0.001, 1, seed);
+        double fraction_of_intelligent_planets_which_are_detectable_as_such = sample_loguniform(0.01, 1, seed);
+        double longevity_of_detectable_civilizations = sample_loguniform(100, 10000000000, seed);
+
+        // printf(" rate_of_star_formation = %lf\n", rate_of_star_formation);
+        // printf(" fraction_of_stars_with_planets = %lf\n", fraction_of_stars_with_planets);
+        // printf(" number_of_habitable_planets_per_star_system = %lf\n", number_of_habitable_planets_per_star_system);
+        // printf(" rate_of_life_formation_in_habitable_planets = %.16lf\n", rate_of_life_formation_in_habitable_planets);
+        // printf(" fraction_of_habitable_planets_in_which_any_life_appears = %lf\n", fraction_of_habitable_planets_in_which_any_life_appears);
+        // printf(" fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", fraction_of_planets_with_life_in_which_intelligent_life_appears);
+        // printf(" fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", fraction_of_intelligent_planets_which_are_detectable_as_such);
+        // printf(" longevity_of_detectable_civilizations = %lf\n", longevity_of_detectable_civilizations);
+
+        // Expected number of civilizations in the Milky way;
+        // see footnote 3 (p. 5)
+        double n = rate_of_star_formation *
+          fraction_of_stars_with_planets *
+          number_of_habitable_planets_per_star_system * 
+          fraction_of_habitable_planets_in_which_any_life_appears * 
+          fraction_of_planets_with_life_in_which_intelligent_life_appears * 
+          fraction_of_intelligent_planets_which_are_detectable_as_such *
+          longevity_of_detectable_civilizations;
+
+        return n;
     }
-    ci ci_90 = array_get_90_ci(xs, n);
+
-    printf("Recovering confidence interval of sample_to(10, 100):\n  low: %f, high: %f\n", ci_90.low, ci_90.high);
+    double sample_fermi_paradox_naive(uint64_t* seed){
+        double n = sample_fermi_naive(seed);
+        return ((n > 1) ? 1 : 0);
+    }
+
+    double n = 1000000;
+    double naive_fermi_proportion = 0;
+    for(int i=0; i<n; i++){
+        double result = sample_fermi_paradox_naive(seed);
+        // printf("result: %lf\n", result);
+        naive_fermi_proportion+=result;
+    }
+    printf("Naïve %% that we are not alone: %lf\n", naive_fermi_proportion/n);
 
 
-    printf("Size of uint64_t: %ld", sizeof(uint64_t*));
+    // Thinking in log space
+    double sample_fermi_logspace(uint64_t* seed){
+        double log_rate_of_star_formation = sample_uniform(log(1), log(100), seed);
+        double log_fraction_of_stars_with_planets = sample_uniform(log(0.1), log(1), seed);
+        double log_number_of_habitable_planets_per_star_system = sample_uniform(log(0.1), log(1), seed);
+        double log_fraction_of_planets_with_life_in_which_intelligent_life_appears = sample_uniform(log(0.001), log(1), seed);
+        double log_fraction_of_intelligent_planets_which_are_detectable_as_such = sample_uniform(log(0.01), log(1), seed);
+        double log_longevity_of_detectable_civilizations = sample_uniform(log(100), log(10000000000), seed);
+
+        // printf(" log_rate_of_star_formation = %lf\n", log_rate_of_star_formation);
+        // printf(" log_fraction_of_stars_with_planets = %lf\n", log_fraction_of_stars_with_planets);
+        // printf(" log_number_of_habitable_planets_per_star_system = %lf\n", log_number_of_habitable_planets_per_star_system);
+        // printf(" log_fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", log_fraction_of_planets_with_life_in_which_intelligent_life_appears);
+        // printf(" log_fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", log_fraction_of_intelligent_planets_which_are_detectable_as_such);
+        // printf(" log_longevity_of_detectable_civilizations = %lf\n", log_longevity_of_detectable_civilizations);
+
+        double log_n1 = 
+            log_rate_of_star_formation +
+            log_fraction_of_stars_with_planets +
+            log_number_of_habitable_planets_per_star_system + 
+            log_fraction_of_planets_with_life_in_which_intelligent_life_appears + 
+            log_fraction_of_intelligent_planets_which_are_detectable_as_such +
+            log_longevity_of_detectable_civilizations;
+        // printf("first part of calculation: %lf\n", log_n1);
+
+        /* Consider fraction_of_habitable_planets_in_which_any_life_appears separately.
+        Imprecisely, we could do:
+
+        double rate_of_life_formation_in_habitable_planets = sample_lognormal(1, 50, seed);
+        double fraction_of_habitable_planets_in_which_any_life_appears = 1- exp(-rate_of_life_formation_in_habitable_planets);
+        double log_fraction_of_habitable_planets_in_which_any_life_appears = log(1-fraction_of_habitable_planets_in_which_any_life_appears);
+        double n = exp(log_n1) * fraction_of_habitable_planets_in_which_any_life_appears;
+        // or: 
+        double n2 = exp(log_n1 + log(fraction_of_habitable_planets_in_which_any_life_appears))
+
+        However, we lose all precision here.
+
+        Now, say
+        a = underlying normal 
+        b = rate_of_life_formation_in_habitable_planets = exp(underlying normal)
+        c = 1 - exp(-b) = fraction_of_habitable_planets_in_which_any_life_appears
+        d = log(c)
+
+        Now, is there some way we can d more efficiently/precisely?
+        Turns out there is!
+
+        Looking at the Taylor expansion for c = 1 - exp(-b), it's b - b^2/2 + b^3/6 - x^b/24, etc.
+        // https://www.wolframalpha.com/input?i=1-exp%28-x%29
+        When b ~ 0 (as is often the case), this is close to b.
+
+        But now, if b ~ 0
+        c ~ b
+        and d = log(c) ~ log(b) = log(exp(a)) = a
+        */
+        double log_rate_of_life_formation_in_habitable_planets = sample_normal(1, 50, seed);
+        // printf("log_rate_of_life_formation_in_habitable_planets: %lf\n", log_rate_of_life_formation_in_habitable_planets);
+
+        double log_fraction_of_habitable_planets_in_which_any_life_appears;
+        if(log_rate_of_life_formation_in_habitable_planets < -32){
+            log_fraction_of_habitable_planets_in_which_any_life_appears = log_rate_of_life_formation_in_habitable_planets;
+        } else{
+            double rate_of_life_formation_in_habitable_planets = exp(log_rate_of_life_formation_in_habitable_planets);
+            double fraction_of_habitable_planets_in_which_any_life_appears = -expm1(-rate_of_life_formation_in_habitable_planets);
+            log_fraction_of_habitable_planets_in_which_any_life_appears = log(fraction_of_habitable_planets_in_which_any_life_appears);
+        }
+        // printf(" log_fraction_of_habitable_planets_in_which_any_life_appears: %lf\n", log_fraction_of_habitable_planets_in_which_any_life_appears);
+
+        double log_n = log_n1 + log_fraction_of_habitable_planets_in_which_any_life_appears;
+
+        return log_n;
+    }
+
+    double sample_fermi_paradox_logspace(uint64_t* seed){
+        double n = sample_fermi_logspace(seed);
+        return ((n > 0) ? 1 : 0);
+    }
+
+    double logspace_fermi_proportion = 0;
+    for(int i=0; i<n; i++){
+        double result = sample_fermi_paradox_logspace(seed);
+        // printf("result: %lf\n", result);
+        logspace_fermi_proportion+=result;
+    }
+    printf("Using more accurate logspace computations, %% that we are not alone: %lf\n", logspace_fermi_proportion/n);
+    double result2;
+
     free(seed);
 }

squiggle_more.c

@@ -51,7 +51,6 @@ void sampler_parallel(double (*sampler)(uint64_t* seed), double* results, int n_
         // - 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);
-        // printf("#%ld: %lu\n",i, *seeds[i]);
 
         // Other initializations tried:
         // *seeds[i] = 1 + i;
@@ -69,17 +68,40 @@ void sampler_parallel(double (*sampler)(uint64_t* seed), double* results, int n_
 
             for (int j = lower_bound_inclusive; j < upper_bound_not_inclusive; j++) {
                 results[j] = sampler(&(cache_box[i].seed));
-                // In principle, these results[j] could also result in two threads competing for the same cache line.
+                /* 
-                // In practice, though,
+                t starts at 0 and ends at T
-                // a) this would happen infrequently
+                at t=0, 
-                // b) trying to unroll loops actually makes the code slower
+                  thread i accesses: results[i*quotient +0], 
-                // c) 8 results[j] are 8 doubles, which fit a cache line. If n_samples/n_threads
+                  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 are not doing multithreading
+        // we can just reuse a seed,
+        // this isn't problematic because we;ve now stopped doing multithreading
     }
 
     free(cache_box);