squiggle.c/scratchpad/scratchpad.c

#include "../squiggle.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 = 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);
    double result2;

    free(seed);
}