Add complex-numbers exercise (#100)

config.json

@@ -95,6 +95,14 @@
         "prerequisites": [],
         "difficulty": 2
       },
+      {
+        "slug": "complex-numbers",
+        "name": "Complex Numbers",
+        "uuid": "b7934691-ac3e-4aee-acd2-5e01536c43e8",
+        "practices": [],
+        "prerequisites": [],
+        "difficulty": 2
+      },
       {
         "slug": "darts",
         "name": "Darts",

exercises/practice/complex-numbers/.docs/instructions.md

@@ -0,0 +1,29 @@
+# Instructions
+
+A complex number is a number in the form `a + b * i` where `a` and `b` are real and `i` satisfies `i^2 = -1`.
+
+`a` is called the real part and `b` is called the imaginary part of `z`.
+The conjugate of the number `a + b * i` is the number `a - b * i`.
+The absolute value of a complex number `z = a + b * i` is a real number `|z| = sqrt(a^2 + b^2)`. The square of the absolute value `|z|^2` is the result of multiplication of `z` by its complex conjugate.
+
+The sum/difference of two complex numbers involves adding/subtracting their real and imaginary parts separately:
+`(a + i * b) + (c + i * d) = (a + c) + (b + d) * i`,
+`(a + i * b) - (c + i * d) = (a - c) + (b - d) * i`.
+
+Multiplication result is by definition
+`(a + i * b) * (c + i * d) = (a * c - b * d) + (b * c + a * d) * i`.
+
+The reciprocal of a non-zero complex number is
+`1 / (a + i * b) = a/(a^2 + b^2) - b/(a^2 + b^2) * i`.
+
+Dividing a complex number `a + i * b` by another `c + i * d` gives:
+`(a + i * b) / (c + i * d) = (a * c + b * d)/(c^2 + d^2) + (b * c - a * d)/(c^2 + d^2) * i`.
+
+Raising e to a complex exponent can be expressed as `e^(a + i * b) = e^a * e^(i * b)`, the last term of which is given by Euler's formula `e^(i * b) = cos(b) + i * sin(b)`.
+
+Implement the following operations:
+
+- addition, subtraction, multiplication and division of two complex numbers,
+- conjugate, absolute value, exponent of a given complex number.
+
+Assume the programming language you are using does not have an implementation of complex numbers.

exercises/practice/complex-numbers/.meta/Example.roc

@@ -0,0 +1,53 @@
+module [real, imaginary, add, sub, mul, div, conjugate, abs, exp]
+
+Complex : { re : F64, im : F64 }
+
+real : Complex -> F64
+real = \z -> z.re
+
+imaginary : Complex -> F64
+imaginary = \z -> z.im
+
+add : Complex, Complex -> Complex
+add = \z1, z2 -> {
+    re: z1.re + z2.re,
+    im: z1.im + z2.im,
+}
+
+sub : Complex, Complex -> Complex
+sub = \z1, z2 -> {
+    re: z1.re - z2.re,
+    im: z1.im - z2.im,
+}
+
+mul : Complex, Complex -> Complex
+mul = \z1, z2 -> {
+    re: z1.re * z2.re - z1.im * z2.im,
+    im: z1.re * z2.im + z1.im * z2.re,
+}
+
+div : Complex, Complex -> Complex
+div = \z1, z2 ->
+    denominator = z2.re * z2.re + z2.im * z2.im
+    {
+        re: (z1.re * z2.re + z1.im * z2.im) / denominator,
+        im: (z1.im * z2.re - z1.re * z2.im) / denominator,
+    }
+
+conjugate : Complex -> Complex
+conjugate = \z -> {
+    re: z.re,
+    im: -z.im,
+}
+
+abs : Complex -> F64
+abs = \z ->
+    z.re * z.re + z.im * z.im |> Num.sqrt
+
+exp : Complex -> Complex
+exp = \z ->
+    factor = Num.e |> Num.pow z.re
+    {
+        re: factor * Num.cos z.im,
+        im: factor * Num.sin z.im,
+    }

exercises/practice/complex-numbers/.meta/config.json

@@ -0,0 +1,19 @@
+{
+  "authors": [
+    "ageron"
+  ],
+  "files": {
+    "solution": [
+      "ComplexNumbers.roc"
+    ],
+    "test": [
+      "complex-numbers-test.roc"
+    ],
+    "example": [
+      ".meta/Example.roc"
+    ]
+  },
+  "blurb": "Implement complex numbers.",
+  "source": "Wikipedia",
+  "source_url": "https://en.wikipedia.org/wiki/Complex_number"
+}

exercises/practice/complex-numbers/.meta/plugins.py

@@ -0,0 +1,25 @@
+float_map = {
+    "e": "Num.e",
+    "ln(2)": "Num.log 2f64",
+    "ln(2)/2": "Num.log 2f64 / 2",
+    "pi": "Num.pi",
+    "pi/4": "Num.pi / 4",
+}
+
+
+def to_roc(value):
+    if isinstance(value, str):
+        return float_map[value]
+    else:
+        return f"{value}"
+
+
+def to_complex_number(value):
+    if isinstance(value, list):
+        assert len(value) == 2
+        re = to_roc(value[0])
+        im = to_roc(value[1])
+    else:
+        re = to_roc(value)
+        im = 0
+    return f"{{ re: {re}, im: {im} }}"

exercises/practice/complex-numbers/.meta/template.j2

@@ -0,0 +1,51 @@
+{%- import "generator_macros.j2" as macros with context -%}
+{{ macros.canonical_ref() }}
+{{ macros.header() }}
+
+import {{ exercise | to_pascal }} exposing [real, imaginary, add, sub, mul, div, conjugate, abs, exp]
+
+isApproxEq = \z1, z2 ->
+    z1.re |> Num.isApproxEq z2.re {} && z1.im |> Num.isApproxEq z2.im {}
+
+{% for supercase in cases %}
+###
+### {{ supercase["description"] }}
+###
+
+{% for case in supercase["cases"] -%}
+{%- if case["cases"] %}
+## {{ case["description"] }}
+{%- set subcases = case["cases"] %}
+{%- else %}
+{%- set subcases = [case] %}
+{%- endif %}
+
+{% for subcase in subcases -%}
+# {{ subcase["description"] }}
+{%- if subcase["input"]["z"] %}
+expect
+    z = {{ plugins.to_complex_number(subcase["input"]["z"]) }}
+    result = {{ subcase["property"] }} z
+    {%- if subcase["expected"] is iterable and subcase["expected"] is not string %}
+    expected = {{ plugins.to_complex_number(subcase["expected"]) }}
+    result |> isApproxEq expected
+    {%- else %}
+    result |> Num.isApproxEq {{ subcase["expected"] | to_roc }} {}
+    {%- endif %}
+
+{%- elif subcase["input"]["z1"] %}
+expect
+    z1 = {{ plugins.to_complex_number(subcase["input"]["z1"]) }}
+    z2 = {{ plugins.to_complex_number(subcase["input"]["z2"]) }}
+    result = z1 |> {{ subcase["property"] }} z2
+    expected = {{ plugins.to_complex_number(subcase["expected"]) }}
+    result |> isApproxEq expected
+{%- else %}
+# This test case is not implemented: perhaps you can implement it?
+{%- endif %}
+
+{% endfor %}
+
+{% endfor %}
+{% endfor %}
+

exercises/practice/complex-numbers/.meta/tests.toml

@@ -0,0 +1,130 @@
+# This is an auto-generated file.
+#
+# Regenerating this file via `configlet sync` will:
+# - Recreate every `description` key/value pair
+# - Recreate every `reimplements` key/value pair, where they exist in problem-specifications
+# - Remove any `include = true` key/value pair (an omitted `include` key implies inclusion)
+# - Preserve any other key/value pair
+#
+# As user-added comments (using the # character) will be removed when this file
+# is regenerated, comments can be added via a `comment` key.
+
+[9f98e133-eb7f-45b0-9676-cce001cd6f7a]
+description = "Real part -> Real part of a purely real number"
+
+[07988e20-f287-4bb7-90cf-b32c4bffe0f3]
+description = "Real part -> Real part of a purely imaginary number"
+
+[4a370e86-939e-43de-a895-a00ca32da60a]
+description = "Real part -> Real part of a number with real and imaginary part"
+
+[9b3fddef-4c12-4a99-b8f8-e3a42c7ccef6]
+description = "Imaginary part -> Imaginary part of a purely real number"
+
+[a8dafedd-535a-4ed3-8a39-fda103a2b01e]
+description = "Imaginary part -> Imaginary part of a purely imaginary number"
+
+[0f998f19-69ee-4c64-80ef-01b086feab80]
+description = "Imaginary part -> Imaginary part of a number with real and imaginary part"
+
+[a39b7fd6-6527-492f-8c34-609d2c913879]
+description = "Imaginary unit"
+
+[9a2c8de9-f068-4f6f-b41c-82232cc6c33e]
+description = "Arithmetic -> Addition -> Add purely real numbers"
+
+[657c55e1-b14b-4ba7-bd5c-19db22b7d659]
+description = "Arithmetic -> Addition -> Add purely imaginary numbers"
+
+[4e1395f5-572b-4ce8-bfa9-9a63056888da]
+description = "Arithmetic -> Addition -> Add numbers with real and imaginary part"
+
+[1155dc45-e4f7-44b8-af34-a91aa431475d]
+description = "Arithmetic -> Subtraction -> Subtract purely real numbers"
+
+[f95e9da8-acd5-4da4-ac7c-c861b02f774b]
+description = "Arithmetic -> Subtraction -> Subtract purely imaginary numbers"
+
+[f876feb1-f9d1-4d34-b067-b599a8746400]
+description = "Arithmetic -> Subtraction -> Subtract numbers with real and imaginary part"
+
+[8a0366c0-9e16-431f-9fd7-40ac46ff4ec4]
+description = "Arithmetic -> Multiplication -> Multiply purely real numbers"
+
+[e560ed2b-0b80-4b4f-90f2-63cefc911aaf]
+description = "Arithmetic -> Multiplication -> Multiply purely imaginary numbers"
+
+[4d1d10f0-f8d4-48a0-b1d0-f284ada567e6]
+description = "Arithmetic -> Multiplication -> Multiply numbers with real and imaginary part"
+
+[b0571ddb-9045-412b-9c15-cd1d816d36c1]
+description = "Arithmetic -> Division -> Divide purely real numbers"
+
+[5bb4c7e4-9934-4237-93cc-5780764fdbdd]
+description = "Arithmetic -> Division -> Divide purely imaginary numbers"
+
+[c4e7fef5-64ac-4537-91c2-c6529707701f]
+description = "Arithmetic -> Division -> Divide numbers with real and imaginary part"
+
+[c56a7332-aad2-4437-83a0-b3580ecee843]
+description = "Absolute value -> Absolute value of a positive purely real number"
+
+[cf88d7d3-ee74-4f4e-8a88-a1b0090ecb0c]
+description = "Absolute value -> Absolute value of a negative purely real number"
+
+[bbe26568-86c1-4bb4-ba7a-da5697e2b994]
+description = "Absolute value -> Absolute value of a purely imaginary number with positive imaginary part"
+
+[3b48233d-468e-4276-9f59-70f4ca1f26f3]
+description = "Absolute value -> Absolute value of a purely imaginary number with negative imaginary part"
+
+[fe400a9f-aa22-4b49-af92-51e0f5a2a6d3]
+description = "Absolute value -> Absolute value of a number with real and imaginary part"
+
+[fb2d0792-e55a-4484-9443-df1eddfc84a2]
+description = "Complex conjugate -> Conjugate a purely real number"
+
+[e37fe7ac-a968-4694-a460-66cb605f8691]
+description = "Complex conjugate -> Conjugate a purely imaginary number"
+
+[f7704498-d0be-4192-aaf5-a1f3a7f43e68]
+description = "Complex conjugate -> Conjugate a number with real and imaginary part"
+
+[6d96d4c6-2edb-445b-94a2-7de6d4caaf60]
+description = "Complex exponential function -> Euler's identity/formula"
+
+[2d2c05a0-4038-4427-a24d-72f6624aa45f]
+description = "Complex exponential function -> Exponential of 0"
+
+[ed87f1bd-b187-45d6-8ece-7e331232c809]
+description = "Complex exponential function -> Exponential of a purely real number"
+
+[08eedacc-5a95-44fc-8789-1547b27a8702]
+description = "Complex exponential function -> Exponential of a number with real and imaginary part"
+
+[d2de4375-7537-479a-aa0e-d474f4f09859]
+description = "Complex exponential function -> Exponential resulting in a number with real and imaginary part"
+
+[06d793bf-73bd-4b02-b015-3030b2c952ec]
+description = "Operations between real numbers and complex numbers -> Add real number to complex number"
+
+[d77dbbdf-b8df-43f6-a58d-3acb96765328]
+description = "Operations between real numbers and complex numbers -> Add complex number to real number"
+
+[20432c8e-8960-4c40-ba83-c9d910ff0a0f]
+description = "Operations between real numbers and complex numbers -> Subtract real number from complex number"
+
+[b4b38c85-e1bf-437d-b04d-49bba6e55000]
+description = "Operations between real numbers and complex numbers -> Subtract complex number from real number"
+
+[dabe1c8c-b8f4-44dd-879d-37d77c4d06bd]
+description = "Operations between real numbers and complex numbers -> Multiply complex number by real number"
+
+[6c81b8c8-9851-46f0-9de5-d96d314c3a28]
+description = "Operations between real numbers and complex numbers -> Multiply real number by complex number"
+
+[8a400f75-710e-4d0c-bcb4-5e5a00c78aa0]
+description = "Operations between real numbers and complex numbers -> Divide complex number by real number"
+
+[9a867d1b-d736-4c41-a41e-90bd148e9d5e]
+description = "Operations between real numbers and complex numbers -> Divide real number by complex number"

exercises/practice/complex-numbers/ComplexNumbers.roc

@@ -0,0 +1,39 @@
+module [real, imaginary, add, sub, mul, div, conjugate, abs, exp]
+
+Complex : { re : F64, im : F64 }
+
+real : Complex -> F64
+real = \z ->
+    crash "Please implement the 'real' function"
+
+imaginary : Complex -> F64
+imaginary = \z ->
+    crash "Please implement the 'imaginary' function"
+
+add : Complex, Complex -> Complex
+add = \z1, z2 ->
+    crash "Please implement the 'add' function"
+
+sub : Complex, Complex -> Complex
+sub = \z1, z2 ->
+    crash "Please implement the 'sub' function"
+
+mul : Complex, Complex -> Complex
+mul = \z1, z2 ->
+    crash "Please implement the 'mul' function"
+
+div : Complex, Complex -> Complex
+div = \z1, z2 ->
+    crash "Please implement the 'div' function"
+
+conjugate : Complex -> Complex
+conjugate = \z ->
+    crash "Please implement the 'conjugate' function"
+
+abs : Complex -> F64
+abs = \z ->
+    crash "Please implement the 'abs' function"
+
+exp : Complex -> Complex
+exp = \z ->
+    crash "Please implement the 'exp' function"