commit bb91d78859dc446063287f7dee48f7771ce86499
parent b99a9cb3f540493b4ca64a86a05e4ab310476037
Author: NunoSempere <nuno.sempere@protonmail.com>
Date: Sat, 20 Jan 2024 14:02:59 +0100
add fermi paradox example
Diffstat:
2 files changed, 131 insertions(+), 8 deletions(-)
diff --git a/scratchpad/scratchpad b/scratchpad/scratchpad
Binary files differ.
diff --git a/scratchpad/scratchpad.c b/scratchpad/scratchpad.c
@@ -5,21 +5,144 @@
#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()
{
// 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* xs = malloc(sizeof(double) * n);
- for (int i = 0; i < n; i++) {
- xs[i] = sample_to(10, 100, seed);
+ double 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 fermi_paradox_naive(uint64_t* seed){
+ double n = fermi_naive(seed);
+ return (n > 1 ? 1 : 0);
+ }
+
+ double result;
+ for(int i=0; i<1000; i++){
+ result = fermi_naive(seed);
+ printf("result from fermi_naive: %lf\n", result);
+ printf("\n\n");
+ }
+ printf("result from naïve implementation: %lf\n", result);
+
+ // Thinking in log space
+ double 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.
+ When b ~ 0 (as it is), 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;
}
- 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 result2;
+
+ /*
+ for(int i=0; i<1000; i++){
+ result2 = fermi_logspace(seed);
+ printf("result from logspace implementation: %lf.2\n", result2);
+ printf("\n\n");
+ }
+ */
- printf("Size of uint64_t: %ld", sizeof(uint64_t*));
free(seed);
}
