Otter Assign refactor + Rmd format v1

[chrispyles]

Refactor the otter assign internals as they've become quite messy.

An (incomplete) list of ideas for this refactor:

• use factories for tasks like generating cells rather than always passing args/and Assignment instance around
• use jupytext to convert Rmd files to notebooks, removing the need for the Rmd adapter
• do Convert user-specified configuration management to fica #485

[chrispyles]

The new v1 format for Rmd should be similar to the example below. With this format, it should be simple to parse the notebook generated by jupytext and convert the HTML comments into raw cells to match the v1 assign format for notebooks.

---
title: "Otter Assign for Rmd Sample"
author: "Chris Pyles"
output: pdf_document
---

<!-- 
# ASSIGNMENT CONFIG
requirements: requirements.R
generate: true
solutions_pdf: true
seed:
    variable: rng_seed
    autograder_value: 42
    student_value: 90
-->

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

# Otter Assign for Rmd Sample

```{r}
library(ggplot2)
library(latex2exp)
rng_seed = 42
```

<!--
# BEGIN QUESTION
name: q1
points: 2
-->

**Question 1.** Write a function called `sieve` that takes in a positive integer `n` and returns a
`set` of the prime numbers less than or equal to `n`. Use the Sieve of Eratosthenes to find the primes.

<!-- # BEGIN SOLUTION -->

```{r}
sieve = function(n) {
  # BEGIN SOLUTION
  is_prime = rep(TRUE, n)
  p = 2
  while (p^2 <= n) {
    if (is_prime[p]) {
      is_prime[seq(p^2, n, p)] = FALSE
    }
    p = p + 1
  }
  is_prime[1] = FALSE
  return(seq(n)[is_prime])
  # END SOLUTION
}
```

<!-- # END SOLUTION -->

<!-- # BEGIN TESTS -->

```{r}
testthat::expect_equal(length(sieve(1)), 0)
```

```{r}
testthat::expect_equal(sieve(2), c(2))
```

```{r}
testthat::expect_equal(sieve(3), c(2, 3))
```

```{r}
testthat::expect_equal(sieve(20), c(2, 3, 5, 7, 11, 13, 17, 19))
```

```{r}
. = " # BEGIN TEST CONFIG
points: 1
hidden: true
" # END TEST CONFIG
testthat::expect_equal(sieve(100), c(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97))
```

<!-- # END TESTS -->

<!-- # END QUESTION -->

<!--
# BEGIN QUESTION
name: q2
manual: true
-->

**Question 2.** Evaluate the integral:

$$
\int x e^x \, \mathrm dx
$$

<!-- # BEGIN PROMPT -->

$$
\begin{aligned}
\int x e^x \, \mathrm dx &= \texttt{YOUR MATH HERE} \\
\end{aligned}
$$

<!-- # END PROMPT -->

<!-- # BEGIN SOLUTION -->

$$
\begin{aligned}
\int x e^x \, \mathrm dx &= x e^x - \int e^x \, \mathrm dx \\
&= x e^x - e^x + C \\
&= e^x (x - 1) + C \\
\end{aligned}
$$

<!-- # END SOLUTION -->

<!-- # END QUESTION -->

<!--
# BEGIN QUESTION
name: q3
manual: true
-->

**Question 3.** Graph the function $f(x) = \arctan \left ( e^x \right )$.

<!-- # BEGIN SOLUTION -->

```{r}
# BEGIN SOLUTION
xs = seq(-10, 10, length.out=100)
ys = atan(exp(xs))

ggplot(data.frame(xs, ys), aes(x=xs, y=ys)) +
  geom_line() +
  xlab(TeX("$x$")) +
  ylab(TeX("$f(x)$")) +
  labs(title=TeX("f(x) = arctan (e^x)")) +
  theme(legend.position = "none")
# END SOLUTION
```

<!-- # END SOLUTION -->

<!-- # END QUESTION -->

<!--
# BEGIN QUESTION
name: q4
-->

**Question 4.** Write a function `hailstone` that returns the hailstone sequence for a positive integer ``n`` as a ``list``.

<!-- # BEGIN SOLUTION -->

```{r}
hailstone = function(n) {
  # BEGIN SOLUTION
  if (n == 1) return(c(n))
  else if (n %% 2 == 0) return(c(n, hailstone(n %/% 2)))
  else return(c(n, hailstone(3 * n + 1)))
  # END SOLUTION
}
```

<!-- # END SOLUTION -->

<!-- # BEGIN TESTS -->

```{r}
testthat::expect_equal(hailstone(1), c(1))
```

```{r}
testthat::expect_equal(hailstone(2), c(2, 1))
```

```{r}
testthat::expect_equal(hailstone(4), c(4, 2, 1))
```

```{r}
. = " # BEGIN TEST CONFIG
points: 1.5
hidden: true
" # END TEST CONFIG
testthat::expect_equal(hailstone(20), c(20, 10, 5, 16, 8, 4, 2, 1))
```

```{r}
. = " # BEGIN TEST CONFIG
points: 1.5
hidden: true
" # END TEST CONFIG
testthat::expect_equal(hailstone(9), c(9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1))
```

<!-- # END TESTS -->

<!-- # END QUESTION -->

<!--
# BEGIN QUESTION
name: q5
-->

**Question 5.** Write a function ``gcd`` that takes in two positive integers ``a`` and ``b`` and
returns their greatest common divisor.

<!-- # BEGIN SOLUTION -->

```{r}
gcd = function(a, b) {
    # BEGIN SOLUTION
    if (a == 0) return(b)
    return(gcd(b %% a, a))
    # END SOLUTION
}
```

<!-- # END SOLUTION -->

<!-- # BEGIN TESTS -->

```{r}
testthat::expect_equal(gcd(1, 1), 1)
```

```{r}
testthat::expect_equal(gcd(2, 1), 1)
```

```{r}
testthat::expect_equal(gcd(5, 5), 5)
```

```{r}
testthat::expect_equal(gcd(10, 4), 2)
```

```{r}
. = " # BEGIN TEST CONFIG
points: 0.5
hidden: true
" # END TEST CONFIG
testthat::expect_equal(gcd(121, 11), 11)
```

```{r}
. = " # BEGIN TEST CONFIG
points: 0.5
hidden: true
" # END TEST CONFIG
testthat::expect_equal(gcd(807306, 622896), 4098)
```

<!-- # END TESTS -->

<!-- # END QUESTION -->

<!--
# BEGIN QUESTION
name: q6
-->

**Question 6.** Write a function `box_muller` that takes in two $U(0,1)$ random variables and
returns two standard normal random variables using the Box-Muller algorithm. Use this function to
generate 1000 standard normal random variables. Make sure to use the provided RNG to generate your
$U(0,1)$ RVs.

<!-- # BEGIN SOLUTION -->

```{r}
set.seed(rng_seed)

box_muller = function(u1, u2) {
    # BEGIN SOLUTION
    r = sqrt(-2 * log(u1))
    theta = 2 * pi * u2
    return(c(r * cos(theta), r * sin(theta)))
    # END SOLUTION
}

rvs = as.vector(mapply(box_muller, runif(500), runif(500))) # SOLUTION
```

<!-- # END SOLUTION -->

<!-- # BEGIN TESTS -->

```{r}
testthat::expect_equal(length(rvs), c(1000))
testthat::expect_equal(dim(rvs), NULL)
```

```{r}
testthat::expect_equal(box_muller(0.5, 0.5), c(-1.1774100225154747, 1.4419114153575892e-16), tolerance=1e-5)
```

```{r}
. = " # BEGIN TEST CONFIG
points: 1
hidden: true
" # END TEST CONFIG
set.seed(1048)
smpl = sample(rvs, 10)
testthat::expect_equal(smpl, c(0.488075015538835, 0.194588102062245, 0.166299257619479, 0.835872276439119, 1.24125215479332,
                               -0.249328292625541, -0.823870507269021, 0.610713558462346, -0.212479727693556, -1.63148721078593))
```

<!-- # END TESTS -->

<!-- # END QUESTION -->