big README refactor

2024-02-01 20:24:44 +01:00 · 2024-02-01 20:24:44 +01:00 · b4c50996cd
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 # Folk Wisdom
 The README.md file was getting too long and messy, so I moved some of the grumpier comments here.
 ### Nested functions and compilation with tcc.
 GCC has an extension which allows a program to define a function inside another function. This makes squiggle.c code more linear and nicer to read, at the cost of becoming dependent on GCC and hence sacrificing portability and increasing compilation times. Conversely, compiling with tcc (tiny c compiler) is almost instantaneous, but leads to longer execution times and doesn't allow for nested functions.
 | GCC | tcc |
 | --- | --- | 
 | slower compilation | faster compilation | 
 | allows nested functions | doesn't allow nested functions |
 | faster execution | slower execution | 
 ~~My recommendation would be to use tcc while drawing a small number of samples for fast iteration, and then using gcc for the final version with lots of samples, and possibly with nested functions for ease of reading by others.~~
 My previous recommendation was to use tcc for marginally faster iteration, but nested functions are just really nice. So my current recommendation is to use gcc throughout, though keep in mind that modifying code to not use nested functions is relatively easy, so keep in mind that you can do that if you run in other environments.
 ### Correlated samples
 In the original [squiggle](https://www.squiggle-language.com/) language, there is some ambiguity about what this code means:
 ```js
 a = 1 to 10
 b = 2 * a
 c = b/a
 c
 ```
 Likewise in [squigglepy](https://github.com/rethinkpriorities/squigglepy):
 ```python
 import squigglepy as sq
 import numpy as np
 a = sq.to(1, 3)
 b = 2 * a  
 c = b / a 
 c_samples = sq.sample(c, 10)
 print(c_samples)
 ```
 Should `c` be equal to `2`? or should it be equal to 2 times the expected distribution of the ratio of two independent draws from a (`2 * a/a`, as it were)? You don't know, because you are not operating on samples, you are operating on magical objects whose internals are hidden from you.
 In squiggle.c, this ambiguity doesn't exist, at the cost of much greater overhead & verbosity:
 ```c
 // correlated samples
 // gcc -O3  correlated.c squiggle.c -lm -o correlated
 #include "squiggle.h"
 #include <stdint.h>
 #include <stdlib.h>
 #include <stdio.h>
 int main(){
    // set randomness seed
    uint64_t* seed = malloc(sizeof(uint64_t));
    *seed = 1000; // xorshift can't start with a seed of 0
    double a = sample_to(1, 10, seed);
    double b = 2 * a;
    double c = b / a;
    printf("a: %f, b: %f, c: %f\n", a, b, c);
    // a: 0.607162, b: 1.214325, c: 0.500000
    free(seed);
 }
 ```
 vs
 ```c
 // uncorrelated samples
 // gcc -O3    uncorrelated.c ../../squiggle.c -lm -o uncorrelated
 #include "squiggle.h"
 #include <stdint.h>
 #include <stdlib.h>
 #include <stdio.h>
 double draw_xyz(uint64_t* seed){
    // function could also be placed inside main with gcc nested functions extension.
    return sample_to(1, 20, seed);
 }
 int main(){
    // set randomness seed
    uint64_t* seed = malloc(sizeof(uint64_t));
    *seed = 1000; // xorshift can't start with a seed of 0
    double a = draw_xyz(seed);
    double b = 2 * draw_xyz(seed);
    double c = b / a;
    printf("a: %f, b: %f, c: %f\n", a, b, c);
    // a: 0.522484, b: 10.283501, c: 19.681936
    free(seed)
 }
 ```
 Exercise for the reader: What possible meanings could the following represent in [squiggle](https://www.squiggle-language.com/playground?v=0.8.6#code=eNqrVkpJTUsszSlxzk9JVbJSys3M08jLL8pNzNEw0FEw1NRUUKoFAOYsC1c%3D)? How would you implement each of those meanings in squiggle.c?
 ```
 min(normal(0, 1))
 ```
 Hint: See examples/more/13_parallelize_min
 ### Note on sampling strategies
 Right now, I am drawing samples using a random number generator. It requires some finesse, particularly when using parallelism. But it works fine.
 But..., what if we could do something more elegant, more ingenious? In particular, what if instead of drawing samples, we had a mesh of equally spaced points in the range of floats? Then we could, for a given number of samples, better estimate the, say, mean of the distribution we are trying to model...
 The problem with that is that if we have some code like:
 ```C
 double model(...){
  double a = sample_to(1, 10, i_mesh++);
  double b = sample_to(1, 2, i_mesh);
  return a * b;
 }
 ```
 Then this doesn't work, because the values of a and b will be correlated: when a is high, b will also be high. What might work would be something like this:
 ```C
 double* model(int n_samples){
  double* xs = malloc((size_t)n_samples * sizeof(double));
  for(int i_mesh=0; i_mesh < sqrt(n_samples); i_mesh++){
      for(int j_mesh=0; j_mesh < sqrt(n_samples); j_mesh++){
          double a = sample_to(1, 10, i_mesh);
          double b = sample_to(1, 2, j_mesh);
      }
  }
  return xs;
 }
 ```
 But that requires us to encode the shape of the model into the sampling function. It leads to an ugly nesting of for loops. It is a more complex approach. It is not [grug-brained](https://grugbrain.dev/). So every now and then I have to remember that this is not the way.
 ### Tests and the long tail of the lognormal
 Distribution functions can be tested with:
 ```sh
 cd tests
 make && make run
 ```
 `make verify` is an alias that runs all the tests and just displays the ones that are failing. 
 These tests are somewhat rudimentary: they get between 1M and 10M samples from a given sampling function, and check that their mean and standard deviations correspond to what they should theoretically should be.
 If you run `make run` (or `make verify`), you will see errors such as these:
 ```
 [-] Mean test for normal(47211.047473, 682197.019012) NOT passed.
 Mean of normal(47211.047473, 682197.019012): 46933.673278, vs expected mean: 47211.047473
 delta: -277.374195, relative delta: -0.005910
 [-] Std test for lognormal(4.584666, 2.180816) NOT passed.
 Std of lognormal(4.584666, 2.180816): 11443.588861, vs expected std: 11342.434900
 delta: 101.153961, relative delta: 0.008839
 [-] Std test for to(13839.861856, 897828.354318) NOT passed.
 Std of to(13839.861856, 897828.354318): 495123.630575, vs expected std: 498075.002499
 delta: -2951.371925, relative delta: -0.005961
 ```
 These tests I wouldn't worry about. Due to luck of the draw, their relative error is a bit over 0.005, or 0.5%, and so the test fails. But it would surprise me if that had some meaningful practical implication.
 The errors that should raise some worry are:
 ```
 [-] Mean test for lognormal(1.210013, 4.766882) NOT passed.
 Mean of lognormal(1.210013, 4.766882): 342337.257677, vs expected mean: 288253.061628
 delta: 54084.196049, relative delta: 0.157985
 [-] Std test for lognormal(1.210013, 4.766882) NOT passed.
 Std of lognormal(1.210013, 4.766882): 208107782.972184, vs expected std: 24776840217.604111
 delta: -24568732434.631927, relative delta: -118.057730
 [-] Mean test for lognormal(-0.195240, 4.883106) NOT passed.
 Mean of lognormal(-0.195240, 4.883106): 87151.733198, vs expected mean: 123886.818303
 delta: -36735.085104, relative delta: -0.421507
 [-] Std test for lognormal(-0.195240, 4.883106) NOT passed.
 Std of lognormal(-0.195240, 4.883106): 33837426.331671, vs expected std: 18657000192.914921
 delta: -18623162766.583248, relative delta: -550.371727
 [-] Mean test for lognormal(0.644931, 4.795860) NOT passed.
 Mean of lognormal(0.644931, 4.795860): 125053.904456, vs expected mean: 188163.894101
 delta: -63109.989645, relative delta: -0.504662
 [-] Std test for lognormal(0.644931, 4.795860) NOT passed.
 Std of lognormal(0.644931, 4.795860): 39976300.711166, vs expected std: 18577298706.170452
 delta: -18537322405.459286, relative delta: -463.707799
 ```
 What is happening in this case is that you are taking a normal, like `normal(-0.195240, 4.883106)`, and you are exponentiating it to arrive at a lognormal. But `normal(-0.195240, 4.883106)` is going to have some non-insignificant weight on, say, 18. But `exp(18) = 39976300`, and points like it are going to end up a nontrivial amount to the analytical mean and standard deviation, even though they have little probability mass.
 The reader can also check that for more plausible real-world values, like those fitting a lognormal to a really wide 90% confidence interval from 10 to 10k, errors aren't egregious:
 ```
 [x] Mean test for to(10.000000, 10000.000000) PASSED
 [-] Std test for to(10.000000, 10000.000000) NOT passed.
 Std of to(10.000000, 10000.000000): 23578.091775, vs expected std: 25836.381819
 delta: -2258.290043, relative delta: -0.095779
 ```
 Overall, I would caution that if you really care about the very far tails of distributions, you might want to instead use tools which can do some of the analytical manipulations for you, like the original Squiggle, Simple Squiggle (both linked below), or even doing lognormal multiplication by hand, relying on the fact that two lognormals multiplied together result in another lognormal with known shape.
 In fact, squiggle.c does have a few functions for algebraic manipulations of simple distributions at the end of squiggle.c. But these are pretty rudimentary, and I don't know whether I'll end up expanding or deleting them.
 ### Compiler warnings
 #### Harsh compilation
 By default, I've enabled -Wall -Wextra -Wdouble-promotion -Wconversion. However, these produce some false positive warnings, which I've dealt with through:
 - For conversion: Explicit casts, particularly from int to size_t when calling malloc.
 - For dealing with unused variables: Using an UNUSED macro. If you don't like that approach, you could add -Wno-unused-parameter to your makefile and remove the macro and its usage.
 Some resources on compiler flags: [1](https://nullprogram.com/blog/2023/04/29/), [2](https://news.ycombinator.com/item?id=7371806)
 #### Results of running clang-tidy
 clang-tidy is a utility to detect common errors in C/C++. You can run it with:
 ```
 make tidy
 ```
 So far in the history of this program it has emitted:
 - One false-positive warning about an issue I'd already taken care of (so I've suppressed the warning)
 - a warning about an unused variable
 I think this is good news in terms of making me more confident that this simple library is correct :).
 ### Boundaries between sampling functions and arrays of samples
 In squiggle.c, the boundary between working with sampler functions and arrays of samples is clear. Not so in the original squiggle, which hides this distinction from the user in the interest of accessibility. 
 ### Other gotchas
 - Even though the C standard is ambiguous about this, this code assumes that doubles are 64 bit precision (otherwise the xorshift code should be different).
 ## Related projects
 - [Squiggle](https://www.squiggle-language.com/)
 - [SquigglePy](https://github.com/rethinkpriorities/squigglepy)
 - [Simple Squiggle](https://nunosempere.com/blog/2022/04/17/simple-squiggle/)
 - [time to BOTEC](https://github.com/NunoSempere/time-to-botec)
 - [Find a beta distribution that fits your desired confidence interval](https://nunosempere.com/blog/2023/03/15/fit-beta/) 
--- a/README.md
+++ b/README.md
@ -1,56 +1,188 @@
 # squiggle.c 
-squiggle.c is a [grug-brained](https://grugbrain.dev/) self-contained C99 library that provides functions for simple Monte Carlo estimation, inspired by [Squiggle](https://www.squiggle-language.com/).
+squiggle.c is a self-contained C99 library that provides functions for simple Monte Carlo estimation, inspired by [Squiggle](https://www.squiggle-language.com/).
-## Why C?
+## Motivation
 ### What am I trying to do here? 
 - I am trying to build a reliable alternative to the original squiggle, that works for me and addresses my frustrations with it.
 - Some adjectives: [grug brain](https://grugbrain.dev/), [lindy](https://en.wikipedia.org/wiki/Lindy_effect), [suckless](https://suckless.org/)
 - I am trying to make something that is simple enough that I and others can fully understand. squiggle.c is less than 700 lines of C, with a core of <230 lines. You, a somewhat technically sophisticated reader, could just read it and grasp its contents, and is encouraged to do so.
 ### Why C? 
 - Because it is fast
 - Because I enjoy it
 - Because C is honest
- Because it will last long (because C is more [lindy](https://en.wikipedia.org/wiki/Lindy_effect))
+- Because it will last long 
 - Because it can fit in my head
 - Because if you can implement something in C, you can implement it anywhere else
 - Because it can be made faster if need be, e.g., with a multi-threading library like OpenMP, by implementing faster but more complex algorithms, or more simply, by inlining the sampling functions (adding an `inline` directive before their function declaration)
 - **Because there are few abstractions between it and machine code** (C => assembly => machine code with gcc, or C => machine code, with tcc), leading to fewer errors beyond the programmer's control.
 - Because it can fit in my head
 - Because if you can implement something in C, you can implement it anywhere else
 ### Design choices
 This code should aim to be correct, then simple, then fast.
 - It should be correct. The user should be able to rely on it and not think about whether errors come from the library.
  - Nonetheless, the user should understand the limitations of sampling-based methods. See the section on [Tests and the long tail of the lognormal](https://git.nunosempere.com/personal/squiggle.c/FOLK_WISDOM.md#tests-and-the-long-tail-of-the-lognormal) for a discussion of how sampling is bad at capturing some aspects of distributions with long tails.
 - It should be clear, conceptually simple. Simple for me to implement, simple for others to understand.
 - It should be fast. But when speed conflicts with simplicity, choose simplicity. For example, there might be several possible algorithms to sample a distribution, each of which is faster over part of the domain. In that case, it's conceptually simpler to just pick one algorithm, and pay the—normally small—performance penalty. 
  - In any case, though, the code should still be *way faster* than, say, Python.
 Note that being terse, or avoiding verbosity, is a non-goal. This is in part because of the constraints that C imposes. But it also aids with clarity and conceptual simplicity, as the issue of correlated samples illustrates in the next section.
 Caveats: Parallelism might hide monsters. The histogram function is pretty but was created with the aid of GPT, and so might have errors.
 ## Getting started 
-You can follow some example usage in the [examples/](examples]) folder. In [examples/core](examples/core/), we build up some functionality, starting from drawing one sample and finishing with a replication of [Dissolving the Fermi paradox](https://arxiv.org/abs/1806.02404). In [examples/more](examples/more), we present a few more complicated examples, like finding confidence intervals, a model of nuclear war, an estimate of how much exercise to do to lose 10kg, or an example using parallelism.
+### Example upfront
-## Commentary
+Download squiggle.c, for instance: 
-### squiggle.c is short
+```sh
 $ rm -r -f squiggle_c
 $ wget https://git.nunosempere.com/personal/squiggle.c/raw/branch/master/squiggle.c
 $ wget https://git.nunosempere.com/personal/squiggle.c/raw/branch/master/squiggle.h
 $ wget https://git.nunosempere.com/personal/squiggle.c/raw/branch/master/squiggle_more.c
 $ wget https://git.nunosempere.com/personal/squiggle.c/raw/branch/master/squiggle_more.h
 $ mkdir temp
 $ mv squiggle* temp
 $ mv temp squiggle_c
 ```
-[squiggle.c](squiggle.c) is less than 700 lines of C, with a core of <230 lines. The reader could just read it and grasp its contents, and is encouraged to do so.
+Write your model. The below is an overcomplicated example, because it has a long digression in the middle, which I think is funny. May change:
 ```C
 #include "squiggle_c/squiggle.h"
 #include <math.h>
 #include <stdint.h>
 #include <stdio.h>
 #include <stdlib.h>
 double sample_fermi_logspace(uint64_t * seed)
 {
    // Replicate <https://arxiv.org/pdf/1806.02404.pdf>, and in particular the red line in page 11.
    // 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_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 
        b = rate_of_life_formation_in_habitable_planets = exp(underlying normal) = exp(a)
        c = 1 - exp(-b) = fraction_of_habitable_planets_in_which_any_life_appears
        d = log(c)
        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
        Now, we could play around with estimating errors,
        and indeed if we want b^2/2 = exp(a)^2/2 < 10^(-n), i.e., to have n decimal digits of precision,
        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
        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 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 {
        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);
    }
    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 log_n = sample_fermi_logspace(seed);
    return ((log_n > 0) ? 1 : 0);
    // log_n > 0 => n > 1
 }
 int main()
 {
    // set randomness seed
    uint64_t* seed = malloc(sizeof(uint64_t));
    *seed = 1010101; // xorshift can't start with a seed of 0
    // Note: come back to choice of seed.
    double logspace_fermi_proportion = 0;
    int n_samples = 1000 * 1000;
    for (int i = 0; i < n_samples; i++) {
        double result = sample_are_we_alone_logspace(seed);
        logspace_fermi_proportion += result;
    }
    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);
 }
 ```
 Compile and run:
 ```
 $ gcc -O3 samples.c ./squiggle_c/squiggle.c  ./squiggle_c/squiggle_more.c -lm -fopenmp -o samples
 $ ./samples
 ```
 ### Core strategy
-This library provides some basic building blocks. The recommended strategy is to:
+The recommended strategy is to:
 1. Define sampler functions, which take a seed, and return 1 sample
 2. Compose those sampler functions to define your estimation model
 3. Produce an array of samples from a sampler function
 4. Get summary statistics for that array of samples.
-### Nested functions and compilation with tcc.
+### Examples
-GCC has an extension which allows a program to define a function inside another function. This makes squiggle.c code more linear and nicer to read, at the cost of becoming dependent on GCC and hence sacrificing portability and increasing compilation times. Conversely, compiling with tcc (tiny c compiler) is almost instantaneous, but leads to longer execution times and doesn't allow for nested functions.
+You can follow some example usage in the [examples/](examples]) folder. In [examples/core](examples/core/), we build up some functionality, starting from drawing one sample and finishing with a replication of [Dissolving the Fermi paradox](https://arxiv.org/abs/1806.02404). In [examples/more](examples/more), we present a few more complicated examples, like finding confidence intervals, a model of nuclear war, an estimate of how much exercise to do to lose 10kg, or an example using parallelism.
-| GCC | tcc |
+## Guarantees
 | --- | --- | 
 | slower compilation | faster compilation | 
 | allows nested functions | doesn't allow nested functions |
 | faster execution | slower execution | 
 ~~My recommendation would be to use tcc while drawing a small number of samples for fast iteration, and then using gcc for the final version with lots of samples, and possibly with nested functions for ease of reading by others.~~
 My previous recommendation was to use tcc for marginally faster iteration, but nested functions are just really nice. So my current recommendation is to use gcc throughout, though keep in mind that modifying code to not use nested functions is relatively easy, so keep in mind that you can do that if you run in other environments.
 ### Guarantees and licensing
 The motte:
- I offer no guarantees about stability, correctness, performance, etc. I might, for instance, abandon the version in C and rewrite it in Zig, Nim or Rust.
+- I offer no guarantees about stability, correctness, performance, etc. I might, for instance, abandon the version in C and rewrite it in Zig, Nim, Rust, Go.
 - This project mostly exists for my own usage & for my own amusement.
 - Caution! Think carefully before using this project for anything important.
 - If you wanted to pay me to provide some stability or correctness, guarantees, or to tweak this library for your own usage, or to teach you how to use it, you could do so [here](https://nunosempere.com/consulting).
@ -58,254 +190,16 @@ The motte:
 The bailey:
 - You can vendor the code, i.e., save it as a dependency together with your other files. This way, this renders you immune to any changes I may make. 
 - I've been hacking at this project for a while now, and I think I have a good grasp of its correctness and limitations. I've tried Nim and Zig, and I prefer C so far.
 - I think the core interface is not likely to change much, though I've recently changed the interface for parallelism and for getting confidence intervals.
 - I am using this code for a few important consulting projects, and I trust myself to operate it correctly.
 ## Licensing
 This project is released under the MIT license, a permissive open-source license. You can see it in the LICENSE.txt file. 
-### Design choices
+## squiggle extra functions
 This code should aim to be correct, then simple, then fast.
 - It should be correct. The user should be able to rely on it and not think about whether errors come from the library.
  - Nonetheless, the user should understand the limitations of sampling-based methods. See the section on [Tests and the long tail of the lognormal](https://git.nunosempere.com/personal/squiggle.c#tests-and-the-long-tail-of-the-lognormal) for a discussion of how sampling is bad at capturing some aspects of distributions with long tails.
 - It should be clear, conceptually simple. Simple for me to implement, simple for others to understand.
 - It should be fast. But when speed conflicts with simplicity, choose simplicity. For example, there might be several possible algorithms to sample a distribution, each of which is faster over part of the domain. In that case, it's conceptually simpler to just pick one algorithm, and pay the—normally small—performance penalty. In any case, though, the code should still be *way faster* than, say, Python.
 Note that being terse, or avoiding verbosity, is a non-goal. This is in part because of the constraints that C imposes. But it also aids with clarity and conceptual simplicity, as the issue of correlated samples illustrates in the next section.
 ### Correlated samples
 In the original [squiggle](https://www.squiggle-language.com/) language, there is some ambiguity about what this code means:
 ```js
 a = 1 to 10
 b = 2 * a
 c = b/a
 c
 ```
 Likewise in [squigglepy](https://github.com/rethinkpriorities/squigglepy):
 ```python
 import squigglepy as sq
 import numpy as np
 a = sq.to(1, 3)
 b = 2 * a  
 c = b / a 
 c_samples = sq.sample(c, 10)
 print(c_samples)
 ```
 Should `c` be equal to `2`? or should it be equal to 2 times the expected distribution of the ratio of two independent draws from a (`2 * a/a`, as it were)? You don't know, because you are not operating on samples, you are operating on magical objects whose internals are hidden from you.
 In squiggle.c, this ambiguity doesn't exist, at the cost of much greater overhead & verbosity:
 ```c
 // correlated samples
 // gcc -O3  correlated.c squiggle.c -lm -o correlated
 #include "squiggle.h"
 #include <stdint.h>
 #include <stdlib.h>
 #include <stdio.h>
 int main(){
    // set randomness seed
    uint64_t* seed = malloc(sizeof(uint64_t));
    *seed = 1000; // xorshift can't start with a seed of 0
    double a = sample_to(1, 10, seed);
    double b = 2 * a;
    double c = b / a;
    printf("a: %f, b: %f, c: %f\n", a, b, c);
    // a: 0.607162, b: 1.214325, c: 0.500000
    free(seed);
 }
 ```
 vs
 ```c
 // uncorrelated samples
 // gcc -O3    uncorrelated.c ../../squiggle.c -lm -o uncorrelated
 #include "squiggle.h"
 #include <stdint.h>
 #include <stdlib.h>
 #include <stdio.h>
 double draw_xyz(uint64_t* seed){
    // function could also be placed inside main with gcc nested functions extension.
    return sample_to(1, 20, seed);
 }
 int main(){
    // set randomness seed
    uint64_t* seed = malloc(sizeof(uint64_t));
    *seed = 1000; // xorshift can't start with a seed of 0
    double a = draw_xyz(seed);
    double b = 2 * draw_xyz(seed);
    double c = b / a;
    printf("a: %f, b: %f, c: %f\n", a, b, c);
    // a: 0.522484, b: 10.283501, c: 19.681936
    free(seed)
 }
 ```
 Exercise for the reader: What possible meanings could the following represent in [squiggle](https://www.squiggle-language.com/playground?v=0.8.6#code=eNqrVkpJTUsszSlxzk9JVbJSys3M08jLL8pNzNEw0FEw1NRUUKoFAOYsC1c%3D)? How would you implement each of those meanings in squiggle.c?
 ```
 min(normal(0, 1))
 ```
 Hint: See examples/more/13_parallelize_min
 ### Note on sampling strategies
 Right now, I am drawing samples using a random number generator. It requires some finesse, particularly when using parallelism. But it works fine.
 But..., what if we could do something more elegant, more ingenious? In particular, what if instead of drawing samples, we had a mesh of equally spaced points in the range of floats? Then we could, for a given number of samples, better estimate the, say, mean of the distribution we are trying to model...
 The problem with that is that if we have some code like:
 ```C
 double model(...){
  double a = sample_to(1, 10, i_mesh++);
  double b = sample_to(1, 2, i_mesh);
  return a * b;
 }
 ```
 Then this doesn't work, because the values of a and b will be correlated: when a is high, b will also be high. What might work would be something like this:
 ```C
 double* model(int n_samples){
  double* xs = malloc((size_t)n_samples * sizeof(double));
  for(int i_mesh=0; i_mesh < sqrt(n_samples); i_mesh++){
      for(int j_mesh=0; j_mesh < sqrt(n_samples); j_mesh++){
          double a = sample_to(1, 10, i_mesh);
          double b = sample_to(1, 2, j_mesh);
      }
  }
  return xs;
 }
 ```
 But that requires us to encode the shape of the model into the sampling function. It leads to an ugly nesting of for loops. It is a more complex approach. It is not [grug-brained](https://grugbrain.dev/). So every now and then I have to remember that this is not the way.
 ### Boundaries between sampling functions and arrays of samples
 In squiggle.c, the boundary between working with sampler functions and arrays of samples is clear. Not so in the original squiggle, which hides this distinction from the user in the interest of accessibility. 
 ### Tests and the long tail of the lognormal
 Distribution functions can be tested with:
 ```sh
 cd tests
 make && make run
 ```
 `make verify` is an alias that runs all the tests and just displays the ones that are failing. 
 These tests are somewhat rudimentary: they get between 1M and 10M samples from a given sampling function, and check that their mean and standard deviations correspond to what they should theoretically should be.
 If you run `make run` (or `make verify`), you will see errors such as these:
 ```
 [-] Mean test for normal(47211.047473, 682197.019012) NOT passed.
 Mean of normal(47211.047473, 682197.019012): 46933.673278, vs expected mean: 47211.047473
 delta: -277.374195, relative delta: -0.005910
 [-] Std test for lognormal(4.584666, 2.180816) NOT passed.
 Std of lognormal(4.584666, 2.180816): 11443.588861, vs expected std: 11342.434900
 delta: 101.153961, relative delta: 0.008839
 [-] Std test for to(13839.861856, 897828.354318) NOT passed.
 Std of to(13839.861856, 897828.354318): 495123.630575, vs expected std: 498075.002499
 delta: -2951.371925, relative delta: -0.005961
 ```
 These tests I wouldn't worry about. Due to luck of the draw, their relative error is a bit over 0.005, or 0.5%, and so the test fails. But it would surprise me if that had some meaningful practical implication.
 The errors that should raise some worry are:
 ```
 [-] Mean test for lognormal(1.210013, 4.766882) NOT passed.
 Mean of lognormal(1.210013, 4.766882): 342337.257677, vs expected mean: 288253.061628
 delta: 54084.196049, relative delta: 0.157985
 [-] Std test for lognormal(1.210013, 4.766882) NOT passed.
 Std of lognormal(1.210013, 4.766882): 208107782.972184, vs expected std: 24776840217.604111
 delta: -24568732434.631927, relative delta: -118.057730
 [-] Mean test for lognormal(-0.195240, 4.883106) NOT passed.
 Mean of lognormal(-0.195240, 4.883106): 87151.733198, vs expected mean: 123886.818303
 delta: -36735.085104, relative delta: -0.421507
 [-] Std test for lognormal(-0.195240, 4.883106) NOT passed.
 Std of lognormal(-0.195240, 4.883106): 33837426.331671, vs expected std: 18657000192.914921
 delta: -18623162766.583248, relative delta: -550.371727
 [-] Mean test for lognormal(0.644931, 4.795860) NOT passed.
 Mean of lognormal(0.644931, 4.795860): 125053.904456, vs expected mean: 188163.894101
 delta: -63109.989645, relative delta: -0.504662
 [-] Std test for lognormal(0.644931, 4.795860) NOT passed.
 Std of lognormal(0.644931, 4.795860): 39976300.711166, vs expected std: 18577298706.170452
 delta: -18537322405.459286, relative delta: -463.707799
 ```
 What is happening in this case is that you are taking a normal, like `normal(-0.195240, 4.883106)`, and you are exponentiating it to arrive at a lognormal. But `normal(-0.195240, 4.883106)` is going to have some non-insignificant weight on, say, 18. But `exp(18) = 39976300`, and points like it are going to end up a nontrivial amount to the analytical mean and standard deviation, even though they have little probability mass.
 The reader can also check that for more plausible real-world values, like those fitting a lognormal to a really wide 90% confidence interval from 10 to 10k, errors aren't egregious:
 ```
 [x] Mean test for to(10.000000, 10000.000000) PASSED
 [-] Std test for to(10.000000, 10000.000000) NOT passed.
 Std of to(10.000000, 10000.000000): 23578.091775, vs expected std: 25836.381819
 delta: -2258.290043, relative delta: -0.095779
 ```
 Overall, I would caution that if you really care about the very far tails of distributions, you might want to instead use tools which can do some of the analytical manipulations for you, like the original Squiggle, Simple Squiggle (both linked below), or even doing lognormal multiplication by hand, relying on the fact that two lognormals multiplied together result in another lognormal with known shape.
 In fact, squiggle.c does have a few functions for algebraic manipulations of simple distributions at the end of squiggle.c. But these are pretty rudimentary, and I don't know whether I'll end up expanding or deleting them.
 ### Compiler warnings
 #### Harsh compilation
 By default, I've enabled -Wall -Wextra -Wdouble-promotion -Wconversion. However, these produce some false positive warnings, which I've dealt with through:
 - For conversion: Explicit casts, particularly from int to size_t when calling malloc.
 - For dealing with unused variables: Using an UNUSED macro. If you don't like that approach, you could add -Wno-unused-parameter to your makefile and remove the macro and its usage.
 Some resources on compiler flags: [1](https://nullprogram.com/blog/2023/04/29/), [2](https://news.ycombinator.com/item?id=7371806)
 #### Results of running clang-tidy
 clang-tidy is a utility to detect common errors in C/C++. You can run it with:
 ```
 make tidy
 ```
 So far in the history of this program it has emitted:
 - One false-positive warning about an issue I'd already taken care of (so I've suppressed the warning)
 - a warning about an unused variable
 I think this is good news in terms of making me more confident that this simple library is correct :).
 ### Division between core functions and squiggle_more expansions
@ -385,109 +279,3 @@ Behaviour on error can be toggled by the `EXIT_ON_ERROR` variable. This library
 Overall, I'd describe the error handling capabilities of this library as pretty rudimentary. For example, this program might fail in surprising ways if you ask for a lognormal with negative standard deviation, because I haven't added error checking for that case yet.
 ### Other gotchas
 - Even though the C standard is ambiguous about this, this code assumes that doubles are 64 bit precision (otherwise the xorshift code should be different).
 ## Related projects
 - [Squiggle](https://www.squiggle-language.com/)
 - [SquigglePy](https://github.com/rethinkpriorities/squigglepy)
 - [Simple Squiggle](https://nunosempere.com/blog/2022/04/17/simple-squiggle/)
 - [time to BOTEC](https://github.com/NunoSempere/time-to-botec)
 - [Find a beta distribution that fits your desired confidence interval](https://nunosempere.com/blog/2023/03/15/fit-beta/) 
 ## Roadmap
 ### To do
 - [ ] Come up with a better headline example; fermi paradox paper is too complicated
 - [ ] Post on suckless subreddit
 - [ ] Drive in a few more real-life applications
  - [ ] US election modelling?
 - [ ] Look into using size_t instead of int for sample numbers
 - [ ] Reorganize code a little bit to reduce usage of gcc's nested functions
 ### Done
 - [x] Document print stats
 - [x] Document rudimentary algebra manipulations for normal/lognormal
 - [x] Think through whether to delete cdf => samples function => not for now
 - [x] Think through whether to:
  - simplify and just abort on error
  - complexify and use boxes for everything
  - leave as is
  - [x] Offer both options
 - [x] Add more functions to do algebra and get the 90% c.i. of normals, lognormals, betas, etc.
  - Think through which of these make sense.
 - [x] Systematize references
 - [x] Think through seed initialization
 - [x] Document parallelism
 - [x] Document confidence intervals
 - [x] Add example for only one sample
 - [x] Add example for many samples
 - [x] Use gcc extension to define functions nested inside main.
 - [x] Chain various `sample_mixture` functions
 - [x] Add beta distribution
  - See <https://stats.stackexchange.com/questions/502146/how-does-numpy-generate-samples-from-a-beta-distribution> for a faster method.
 - [x] Use OpenMP for acceleration
 - [x] Add function to get sample when given a cdf
 - [x] Don't have a single header file.
 - [x] Structure project a bit better
 - [x] Simplify `PROCESS_ERROR` macro
 - [x] Add README
  - [x] Schema: a function which takes a sample and manipulates it,
  - [x] and at the end, an array of samples.
  - [x] Explain boxes
  - [x] Explain nested functions
  - [x] Explain exit on error
  - [x] Explain individual examples
 - [x] Rename functions to something more self-explanatory, e.g,. `sample_unit_normal`.
 - [x] Add summarization functions: mean, std
 - [x] Add sampling from a gamma distribution
  - https://dl.acm.org/doi/pdf/10.1145/358407.358414
 - [x] Explain correlated samples
 - [x] Test summary statistics for each of the distributions.
  - [x] For uniform
  - [x] For normal
  - [x] For lognormal
  - [x] For lognormal (to syntax)
  - [x] For beta distribution
 - [x] Clarify gamma/standard gamma
 - [x] Add efficient sampling from a beta distribution
  - https://dl.acm.org/doi/10.1145/358407.358414
  - https://link.springer.com/article/10.1007/bf02293108
  - https://stats.stackexchange.com/questions/502146/how-does-numpy-generate-samples-from-a-beta-distribution
  - https://github.com/numpy/numpy/blob/5cae51e794d69dd553104099305e9f92db237c53/numpy/random/src/distributions/distributions.c
 - [x] Pontificate about lognormal tests
 - [x] Give warning about sampling-based methods.
 - [x] Have some more complicated & realistic example
 - [x] Add summarization functions: 90% ci (or all c.i.?) 
 - [x] Link to the examples in the examples section.
 - [x] Add a few functions for doing simple algebra on normals, and lognormals
  - [x] Add prototypes
  - [x] Use named structs
  - [x] Add to header file
  - [x] Provide example algebra
  - [x] Add conversion between 90% ci and parameters.
  - [x] Use that conversion in conjunction with small algebra.
  - [x] Consider ergonomics of using ci instead of c_i
    - [x] use named struct instead
    - [x] demonstrate and document feeding a struct directly to a function; my_function((struct c_i){.low = 1, .high = 2});
  - [x] Move to own file? Or signpost in file? => signposted in file.
 - [x] Write twitter thread: now [here](https://twitter.com/NunoSempere/status/1707041153210564959); retweets appreciated.
 - [x] Write better confidence interval code that:
  - Gets number of samples as an input
  - Gets either a sampler function or a list of samples
  - is O(n), not O(nlog(n))
  - Parallelizes stuff
 ### Discarded
 - [ ] ~~Disambiguate sample_laplace--successes vs failures || successes vs total trials as two distinct and differently named functions~~
 - [ ] ~~Support all distribution functions in <https://www.squiggle-language.com/docs/Api/Dist>~~
 - [ ] ~~Add a custom preprocessor to allow simple nested functions that don't rely on local scope?~~
 - [ ] ~~Add tests in Stan?~~
 - [ ] ~~Test results for lognormal manipulations~~
 - [ ] ~~Consider desirability of defining shortcuts for algebra functions. Adds a level of magic, though.~~
 - [ ] ~~Think about whether to write a simple version of this for [uxn](https://100r.co/site/uxn.html), a minimalist portable programming stack which, sadly, doesn't have doubles (64 bit floats)~~
--- a/ROADMAP.md
+++ b/ROADMAP.md
@ -0,0 +1,97 @@
 # Roadmap
 ## To do
 - [ ] Big refactor
  - [ ] Come up with a better headline example; fermi paradox paper is too complicated
  - [ ] Make README.md less messy
  - [ ] Give examples of new functions
 - [ ] Post on suckless subreddit
 - [ ] Drive in a few more real-life applications
  - [ ] US election modelling?
 - [ ] Look into using size_t instead of int for sample numbers
 - [ ] Reorganize code a little bit to reduce usage of gcc's nested functions
 ## Done
 - [x] Document print stats
 - [x] Document rudimentary algebra manipulations for normal/lognormal
 - [x] Think through whether to delete cdf => samples function => not for now
 - [x] Think through whether to:
  - simplify and just abort on error
  - complexify and use boxes for everything
  - leave as is
  - [x] Offer both options
 - [x] Add more functions to do algebra and get the 90% c.i. of normals, lognormals, betas, etc.
  - Think through which of these make sense.
 - [x] Systematize references
 - [x] Think through seed initialization
 - [x] Document parallelism
 - [x] Document confidence intervals
 - [x] Add example for only one sample
 - [x] Add example for many samples
 - [x] Use gcc extension to define functions nested inside main.
 - [x] Chain various `sample_mixture` functions
 - [x] Add beta distribution
  - See <https://stats.stackexchange.com/questions/502146/how-does-numpy-generate-samples-from-a-beta-distribution> for a faster method.
 - [x] Use OpenMP for acceleration
 - [x] Add function to get sample when given a cdf
 - [x] Don't have a single header file.
 - [x] Structure project a bit better
 - [x] Simplify `PROCESS_ERROR` macro
 - [x] Add README
  - [x] Schema: a function which takes a sample and manipulates it,
  - [x] and at the end, an array of samples.
  - [x] Explain boxes
  - [x] Explain nested functions
  - [x] Explain exit on error
  - [x] Explain individual examples
 - [x] Rename functions to something more self-explanatory, e.g,. `sample_unit_normal`.
 - [x] Add summarization functions: mean, std
 - [x] Add sampling from a gamma distribution
  - https://dl.acm.org/doi/pdf/10.1145/358407.358414
 - [x] Explain correlated samples
 - [x] Test summary statistics for each of the distributions.
  - [x] For uniform
  - [x] For normal
  - [x] For lognormal
  - [x] For lognormal (to syntax)
  - [x] For beta distribution
 - [x] Clarify gamma/standard gamma
 - [x] Add efficient sampling from a beta distribution
  - https://dl.acm.org/doi/10.1145/358407.358414
  - https://link.springer.com/article/10.1007/bf02293108
  - https://stats.stackexchange.com/questions/502146/how-does-numpy-generate-samples-from-a-beta-distribution
  - https://github.com/numpy/numpy/blob/5cae51e794d69dd553104099305e9f92db237c53/numpy/random/src/distributions/distributions.c
 - [x] Pontificate about lognormal tests
 - [x] Give warning about sampling-based methods.
 - [x] Have some more complicated & realistic example
 - [x] Add summarization functions: 90% ci (or all c.i.?) 
 - [x] Link to the examples in the examples section.
 - [x] Add a few functions for doing simple algebra on normals, and lognormals
  - [x] Add prototypes
  - [x] Use named structs
  - [x] Add to header file
  - [x] Provide example algebra
  - [x] Add conversion between 90% ci and parameters.
  - [x] Use that conversion in conjunction with small algebra.
  - [x] Consider ergonomics of using ci instead of c_i
    - [x] use named struct instead
    - [x] demonstrate and document feeding a struct directly to a function; my_function((struct c_i){.low = 1, .high = 2});
  - [x] Move to own file? Or signpost in file? => signposted in file.
 - [x] Write twitter thread: now [here](https://twitter.com/NunoSempere/status/1707041153210564959); retweets appreciated.
 - [x] Write better confidence interval code that:
  - Gets number of samples as an input
  - Gets either a sampler function or a list of samples
  - is O(n), not O(nlog(n))
  - Parallelizes stuff
 ## Discarded
 - [ ] ~~Disambiguate sample_laplace--successes vs failures || successes vs total trials as two distinct and differently named functions~~
 - [ ] ~~Support all distribution functions in <https://www.squiggle-language.com/docs/Api/Dist>~~
 - [ ] ~~Add a custom preprocessor to allow simple nested functions that don't rely on local scope?~~
 - [ ] ~~Add tests in Stan?~~
 - [ ] ~~Test results for lognormal manipulations~~
 - [ ] ~~Consider desirability of defining shortcuts for algebra functions. Adds a level of magic, though.~~
 - [ ] ~~Think about whether to write a simple version of this for [uxn](https://100r.co/site/uxn.html), a minimalist portable programming stack which, sadly, doesn't have doubles (64 bit floats)~~