commit 1e786170793775025dcbe7c7757e21475bdd89cf
parent 9e5d2db23b806e4003fe3b7b22e3bfeaf517f61b
Author: NunoSempere <nuno.sempere@protonmail.com>
Date: Sun, 21 Jan 2024 11:44:28 +0100
streamline fermi estimation a little
Diffstat:
3 files changed, 111 insertions(+), 98 deletions(-)
diff --git a/examples/core/06_dissolving_fermi_paradox/example b/examples/core/06_dissolving_fermi_paradox/example
Binary files differ.
diff --git a/examples/core/06_dissolving_fermi_paradox/example.c b/examples/core/06_dissolving_fermi_paradox/example.c
@@ -4,88 +4,23 @@
#include <stdio.h>
#include <stdlib.h>
-#define VERBOSE 0
-
-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
-
- // Do this naïvely, without worrying that much about numerical precision
- 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);
-
- if(VERBOSE) printf(" rate_of_star_formation = %lf\n", rate_of_star_formation);
- if(VERBOSE) printf(" fraction_of_stars_with_planets = %lf\n", fraction_of_stars_with_planets);
- if(VERBOSE) printf(" number_of_habitable_planets_per_star_system = %lf\n", number_of_habitable_planets_per_star_system);
- if(VERBOSE) printf(" rate_of_life_formation_in_habitable_planets = %.16lf\n", rate_of_life_formation_in_habitable_planets);
- if(VERBOSE) printf(" fraction_of_habitable_planets_in_which_any_life_appears = %lf\n", fraction_of_habitable_planets_in_which_any_life_appears);
- if(VERBOSE) printf(" fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", fraction_of_planets_with_life_in_which_intelligent_life_appears);
- if(VERBOSE) printf(" fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", fraction_of_intelligent_planets_which_are_detectable_as_such);
- if(VERBOSE) 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_are_we_alone_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_are_we_alone_naive(seed);
- if(VERBOSE) printf("result: %lf\n", result);
- naive_fermi_proportion += result;
- }
- printf("Naïve %% that we are not alone: %lf\n", naive_fermi_proportion / n);
+ *seed = 1001; // xorshift can't start with a seed of 0
- // Thinking in log space
- // Taking care of numerical precision
double sample_fermi_logspace(uint64_t * seed)
{
+ // You can see a simple version of this function in naive.c in this same folder
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);
-
- if(VERBOSE) printf(" log_rate_of_star_formation = %lf\n", log_rate_of_star_formation);
- if(VERBOSE) printf(" log_fraction_of_stars_with_planets = %lf\n", log_fraction_of_stars_with_planets);
- if(VERBOSE) printf(" log_number_of_habitable_planets_per_star_system = %lf\n", log_number_of_habitable_planets_per_star_system);
- if(VERBOSE) 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);
- if(VERBOSE) printf(" log_fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", log_fraction_of_intelligent_planets_which_are_detectable_as_such);
- if(VERBOSE) 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;
- if(VERBOSE) printf("first part of calculation: %lf\n", log_n1);
+ double log_rate_of_life_formation_in_habitable_planets = sample_normal(1, 50, seed);
+ double log_fraction_of_habitable_planets_in_which_any_life_appears;
/*
Consider:
a = underlying normal
@@ -93,15 +28,12 @@ int main()
c = 1 - exp(-b) = fraction_of_habitable_planets_in_which_any_life_appears
d = log(c)
- Now, is there some way we can get 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.
+ 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
+ But now, if b ~ 0, c ~ b
and d = log(c) ~ log(b) = log(exp(a)) = a
Now, we could play around with estimating errors,
@@ -109,15 +41,20 @@ int main()
we could compute this as e.g., a < (nlog(10) + log(2))/2
so for example if we want ten digits of precision, that's a < -11
- But more empirically, the two numbers do become really close around 11 or so, and at 38 that calculation results in a -inf (so probably an overflow.)
+ Empirically, the two numbers as calculated in C do become really close around 11 or so,
+ and at 38 that calculation results in a -inf (so probably a floating point error or similar.)
So we should be using that formula for somewhere between -38 << a < -11
- I chose -16 for the sake of it after playing with:
- <https://www.wolframalpha.com/input?i=log%281-exp%28-exp%28-16%29%29%29>
- */
- double log_rate_of_life_formation_in_habitable_planets = sample_normal(1, 50, seed);
- if(VERBOSE) 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;
+ I chose -16 as a happy medium after playing around with
+ double invert(double x){
+ return log(1-exp(-exp(-x)));
+ }
+ for(int i=0; i<64; i++){
+ double j = i;
+ printf("for %lf, log(1-exp(-exp(-x))) is calculated as... %lf\n", j, invert(j));
+ }
+ and <https://www.wolframalpha.com/input?i=log%281-exp%28-exp%28-16%29%29%29>
+ */
if (log_rate_of_life_formation_in_habitable_planets < -16) {
log_fraction_of_habitable_planets_in_which_any_life_appears = log_rate_of_life_formation_in_habitable_planets;
} else {
@@ -125,36 +62,33 @@ int main()
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);
}
- if(VERBOSE) 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;
+ 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);
+
+ double log_n = log_rate_of_star_formation + log_fraction_of_stars_with_planets + log_number_of_habitable_planets_per_star_system + log_fraction_of_habitable_planets_in_which_any_life_appears + 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;
return log_n;
}
double sample_are_we_alone_logspace(uint64_t * seed)
{
- double n = sample_fermi_logspace(seed);
- return ((n > 0) ? 1 : 0);
+ double log_n = sample_fermi_logspace(seed);
+ return ((log_n > 0) ? 1 : 0);
+ // log_n > 0 => n > 1
}
double logspace_fermi_proportion = 0;
- for (int i = 0; i < n; i++) {
+ int n_samples = 1000 * 1000;
+ for (int i = 0; i < n_samples; i++) {
double result = sample_are_we_alone_logspace(seed);
- if(VERBOSE) 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 p_not_alone = logspace_fermi_proportion / n_samples;
+ printf("Probability that we are not alone: %lf (%.lf%%)\n", p_not_alone, p_not_alone * 100);
free(seed);
- /*
- double invert(double x){
- return log(1-exp(-exp(-x)));
- }
- for(int i=0; i<64; i++){
- double j = i;
- printf("for %lf, log(1-exp(-exp(-x))) is calculated as... %lf\n", j, invert(j));
- }
- */
}
+
diff --git a/examples/core/06_dissolving_fermi_paradox/naive.c b/examples/core/06_dissolving_fermi_paradox/naive.c
@@ -0,0 +1,79 @@
+#include "../../../squiggle.h"
+#include <math.h>
+#include <stdint.h>
+#include <stdio.h>
+#include <stdlib.h>
+
+#define VERBOSE 0
+
+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
+
+ // Do this naïvely, without worrying that much about numerical precision
+ 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);
+
+ if(VERBOSE) printf(" rate_of_star_formation = %lf\n", rate_of_star_formation);
+ if(VERBOSE) printf(" fraction_of_stars_with_planets = %lf\n", fraction_of_stars_with_planets);
+ if(VERBOSE) printf(" number_of_habitable_planets_per_star_system = %lf\n", number_of_habitable_planets_per_star_system);
+ if(VERBOSE) printf(" rate_of_life_formation_in_habitable_planets = %.16lf\n", rate_of_life_formation_in_habitable_planets);
+ if(VERBOSE) printf(" fraction_of_habitable_planets_in_which_any_life_appears = %lf\n", fraction_of_habitable_planets_in_which_any_life_appears);
+ if(VERBOSE) printf(" fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", fraction_of_planets_with_life_in_which_intelligent_life_appears);
+ if(VERBOSE) printf(" fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", fraction_of_intelligent_planets_which_are_detectable_as_such);
+ if(VERBOSE) 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_are_we_alone_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_are_we_alone_naive(seed);
+ if(VERBOSE) printf("result: %lf\n", result);
+ naive_fermi_proportion += result;
+ }
+ printf("Naïve %% that we are not alone: %lf\n", naive_fermi_proportion / n);
+
+ free(seed);
+
+ /*
+ double invert(double x){
+ return log(1-exp(-exp(-x)));
+ }
+ for(int i=0; i<64; i++){
+ double j = i;
+ printf("for %lf, log(1-exp(-exp(-x))) is calculated as... %lf\n", j, invert(j));
+ }
+ */
+}
