Gaussian Integral Approximation

• Write a function gaussian_integral (or gaussianIntegral depending on the language's case convention) that takes:

  • a 64-bit floating point number $a$ (you may assume $a > 0$),
  • a 64-bit integer $n$ (you may assume $n \ge 1$),

  and returns an approximation of

$$ \int_{-a}^{a} e^{-x^2} \, dx $$

using $n$ equal-width subintervals and the left-endpoint Riemann sum.

• Let

$$ \Delta x = \frac{2a}{n} $$

and approximate the integral by

$$ \sum_{i=0}^{n-1} e^{-x_i^2} \, \Delta x, \quad x_i = -a + i \Delta x. $$

• Use:
  • 64-bit floating point numbers for $a$ and the return value,
  • an integer for $n$.

Note

As $a$ becomes large, this value approaches $\sqrt{\pi}$.

• Boilerplate source files {go,jl,ml,rs}/gaussian_integral.{go,jl,ml,rs} containing the test code is generated and shown below.

• Edit the source files either by opening them in a text editor (e.g., vscode), or editing the cells below and executing them.

1. AI tutor

1-1. Prepare

• Your personal AI tutor is provided for questions and feedback
• Execute the following cell before you use it

import heytutor

1-2. Examples

1-2-1. A general question

%%hey
How to write a function in Go?

1-2-2. A hint on this specific problem

%%hey
Give me a hint on this problem for Rust

1-2-3. NEW: A few builtin variables

• {file:FILENAME} is the content of FILE
• {bash[-1]} is the output of the last %%bash_ cell, {bash[-2]} that of the second last %%bash_ cell, etc.
• {problem} is the content of the file you specified by %%hey problem_file=foo.md
• {answer} is the content of the file you specified by %%hey answer_file=go/foo.go

1-2-4. Help when you struggle

%%hey answer_file=go/foo.go
I get this error when I compile it. What's wrong?"

My program:
{answer}

Error message:
{bash[-1]}

1-2-5. Ask feedback

• You are encouraged to ask a feedback once you think you are done with the problem, to know if there is a better answer. You can do so by something like:

%%hey problem_file=foo.md answer_file=go/foo.md
Give me a feedback to my answer.

Problem:
{problem}

My Answer:
{answer}

2. Go

2-1. Baseline code

import heytutor

%%writefile_ go/gaussian_integral.go
package main
import "math"
import "fmt"

/** begin my answer */

func gaussianIntegral(a float64, n int64) float64 {
	dx := 2.0 * a / float64(n)
	sum := 0.0
	for i := int64(0); i < n; i++ {
		x := -a + float64(i)*dx
		sum += math.Exp(-x*x) * dx
	}
	return sum
}
/** end my answer */

func main() {
	if !(math.Abs(gaussianIntegral(1, 1000) - 1.493648) < 1.0e-6) { panic("wrong") }
	if !(math.Abs(gaussianIntegral(2, 2000) - 1.764163) < 1.0e-6) { panic("wrong") }
	if !(math.Abs(gaussianIntegral(10, 10000) - math.Sqrt(math.Pi)) < 1.0e-6) { panic("wrong") }
	fmt.Println("OK")
}

2-2. Compile

%%bash_
export PATH=${PATH}:~/.local/go/bin:~/go/bin
go build -o go/gaussian_integral go/gaussian_integral.go

• Note: when you run go or other Go commands in a terminal (SSH or Jupyter terminal), you need to execute the first line (export PATH=${PATH}:~/go/bin)
• You may consider adding that line in your ~/.bash_profile

2-3. Run

%%bash_
go/gaussian_integral

2-4. Ask Questions or Get Feedback

%%hey problem_file=gaussian_integral.md answer_file=go/gaussian_integral.go

Problem:
{problem}
My Answer (between /** begin my answer */ and /** end my answer */):
{answer}

Give me a feedback to my answer.

3. Julia

3-1. Baseline code

import heytutor

%%writefile_ jl/gaussian_integral.jl
### begin my answer

function gaussian_integral(a, n)
    dx = 2a / n
    s = 0.0
    for i in 0:n-1
        x = -a + i * dx
        s += exp(-(x * x)) * dx
    end
    s
end
### end my answer

function main()
    @assert abs(gaussian_integral(1, 1000) - 1.493648) < 1.0e-6
    @assert abs(gaussian_integral(2, 2000) - 1.764163) < 1.0e-6
    @assert abs(gaussian_integral(10, 10000) - sqrt(pi)) < 1.0e-6
    println("OK")
end

main()

3-2. Compile

• Julia code is compiled "just in time" (compiled upon executed), so does not need a specific action for compilation before you run

3-3. Run

%%bash_
export PATH=${PATH}:~/.juliaup/bin
julia jl/gaussian_integral.jl

• Note: when you run julia or other Julia commands in a terminal (SSH or Jupyter terminal), you need to execute the first line (export PATH=${PATH}:~/.juliaup/bin)
• You may consider adding that line in your ~/.bash_profile

3-4. Interactive execution

• julia command also serves is an interactive command for Julia programs

• You can run a source code and continue interaction

$ julia -i jl/gaussian_integral.jl

• For trial and error, you may also consider creating a Julia notebook

3-5. Ask Questions or Get Feedback

%%hey problem_file=gaussian_integral.md answer_file=jl/gaussian_integral.jl

Problem:
{problem}

My Answer (between ### begin my answer and ### end my answer):
{answer}

Give me a feedback to my answer.

4. OCaml

4-1. Baseline code

import heytutor

%%writefile_ ml/gaussian_integral.ml
(** begin my answer *)

let gaussian_integral a n =
  let dx = 2.0 *. a /. float_of_int n in
  let s = ref 0.0 in
  for i = 0 to n - 1 do
    let x = -. a +. float_of_int i *. dx in
    s := !s +. exp (-. x *. x) *. dx
  done;
  !s;;
(** end my answer *)

let main () =
  assert (abs_float (gaussian_integral  1.0 1000)  -. 1.493648 < 1.0e-6);
  assert (abs_float (gaussian_integral  2.0 2000)  -. 1.764163 < 1.0e-6);
  assert (abs_float (gaussian_integral 10.0 10000) -. Float.sqrt Float.pi < 1.0e-6);
  Printf.printf "OK\n"
;;

main()

4-2. Compile

%%bash_
eval $(opam env)
ocamlc ml/gaussian_integral.ml -o ml/gaussian_integral

• Note: when you run ocamlc or other OCaml commands (see below) in a terminal (SSH or Jupyter terminal), you need to execute the first line (eval $(opam env))
• You may consider adding that line in your ~/.bash_profile

4-3. Run

%%bash_
ml/gaussian_integral

4-4. Interactive execution

• ocaml command is an interactive command for OCaml programs

• In terminal (Jupyter or SSH), you can directly run a source code

$ eval $(opam env)   # once in your session or put it in ~/.bash_profile
$ ocaml ml/gaussian_integral.ml

• You can run a source code and continue interaction

$ eval $(opam env)   # once in your session or put it in ~/.bash_profile
$ ocaml -init ml/gaussian_integral.ml

• For trial and error, you may also consider creating an OCaml notebook

4-5. Ask Questions or Get Feedback

%%hey problem_file=gaussian_integral.md answer_file=ml/gaussian_integral.ml

Problem:
{problem}

My Answer (between (** begin my answer *) and (** end my answer *)):
{answer}

Give me a feedback to my answer.

5. Rust

5-1. Baseline code

import heytutor

%%writefile_ rs/gaussian_integral.rs
/** begin my answer */

fn gaussian_integral(a: f64, n: i64) -> f64 {
    let dx = 2.0 * a / (n as f64);
    let mut sum = 0.0;
    for i in 0..n {
        let x = -a + (i as f64) * dx;
        sum += (-x * x).exp() * dx;
    }
    sum
}
/** end my answer */

fn main() {
    let pi = std::f64::consts::PI;
    assert!((gaussian_integral(1.0, 1000) - 1.493648).abs() < 1.0e-6);
    assert!((gaussian_integral(2.0, 2000) - 1.764163).abs() < 1.0e-6);
    assert!((gaussian_integral(10.0, 10000) - pi.sqrt()).abs() < 1.0e-6);
    println!("OK");
}

5-2. Compile

%%bash_
. ~/.cargo/env
rustc rs/gaussian_integral.rs -o rs/gaussian_integral

• Note: when you run rustc or other Rust commands in a terminal (SSH or Jupyter terminal), you need to execute the first line (. ~/.cargo/env)
• You may consider adding that line in your ~/.bash_profile

5-3. Run

%%bash_
rs/gaussian_integral

5-4. Ask Questions or Get Feedback

%%hey problem_file=gaussian_integral.md answer_file=rs/gaussian_integral.rs

Problem:
{problem}

My Answer (between /** begin my answer */ and /** end my answer */):
{answer}

Give me a feedback to my answer.