# `Calendrical.Islamic.Observational`

Implementation of the *observational* (sighting-based) Islamic
calendar.

Unlike the tabular calendars in `Calendrical.Islamic.Civil` and
`Calendrical.Islamic.Tbla`, the start of each lunar month in this
calendar is determined by **actual visibility of the new crescent
moon** at a sample location. The location used here is **Cairo,
Egypt** (30.1° N, 31.3° E, 200 m), the canonical "Islamic location"
used by Dershowitz & Reingold's *Calendrical Calculations*.

This is the calendar identified by the CLDR `:islamic` calendar
type. Days are assumed to begin at midnight.

## Visibility model

Crescent visibility is computed by `Astro.new_visible_crescent/3`
using the **Odeh (2006)** criterion by default — a modern
empirical model fitted against 737 historical sightings. The Odeh
criterion replaces Reingold's older Shaukat criterion and is the
basis for several national Islamic calendar committees.

A month begins on the day after the evening its crescent is first
predicted to be visible: that evening is the eve of the month's first
day, when the Islamic day begins. Months are 29 or 30 days long
depending on whether the crescent was visible on the eve of the 30th
day; years are 354 or 355 days long. Because the Umm al-Qura rule
declares a month once the moon sets after the sun, before a crescent
can be seen, each month begins on the same day as in
`Calendrical.Islamic.UmmAlQura` or a day later (so for every month of
1400–1470 AH).

## Performance

Each calendar conversion makes a small number of crescent
visibility calls (typically 1-3, plus one new-moon search). The
underlying astronomical computations are not cached, so users that
need to convert many dates may benefit from caching results
externally.

## Reference

* Dershowitz & Reingold, *Calendrical Calculations* (4th ed.),
  Chapter 14, "The Islamic and Saudi Arabian Calendars".

* Odeh, M. Sh. (2004), *New Criterion for Lunar Crescent
  Visibility*. Experimental Astronomy 18, 39–64.

* CLDR `:islamic` calendar type.

See also `Calendrical.Islamic.Rgsa` for the same algorithm applied
at Mecca, and `Calendrical.Islamic.UmmAlQura` for the official
Saudi tabular calendar.

ICU's `islamic` calendar, and its `islamic-rgsa` alias, begin a month
on the day after the conjunction rather than on a sighting, so they run
a day ahead of this calendar in most months. Against the dates Saudi
Arabia announces, this calendar is within a day and ICU is sometimes a
day early.

# `day`

```elixir
@type day() :: 1..30
```

# `month`

```elixir
@type month() :: 1..12
```

# `year`

```elixir
@type year() :: integer()
```

# `add`

Adds whole years, quarters or months to the fields of a naive
datetime, keeping its time of day: the `add/9` callback
[`CalendarInterval`](https://hexdocs.pm/calendar_interval) steps an
interval of a year or a month with. The date moves as `shift/6`
moves it, clamping the day into a shorter month.

### Arguments

* `year`, `month` and `day` are the date's fields in this calendar.

* `hour`, `minute`, `second` and `microsecond` are the time of day,
  returned unchanged.

* `precision` is `:year`, `:quarter` or `:month`.

* `step` is the number of years, quarters or months to add, which
  may be negative.

### Returns

* `{year, month, day, hour, minute, second, microsecond}`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {year, month, day, 12, 30, 0, {0, 0}} =
    ...>   Calendrical.Islamic.Observational.add(date.year, date.month, date.day, 12, 30, 0, {0, 0}, :month, 1)
    iex> {year, month, day} == Calendrical.Islamic.Observational.shift(date.year, date.month, date.day, :months, 1, [])
    true

# `calendar_base`

Returns what this calendar's year is built of, `:month` or `:week`,
as its `info/0` states.

### Returns

* `:week` for a calendar whose year is built of weeks.

* `:month` otherwise.

### Examples

    iex> Calendrical.Islamic.Observational.calendar_base() in [:month, :week]
    true

# `calendar_from_cldr_calendar_type`

Returns the calendar module whose CLDR calendar type is the one
given, whichever calendar is asked
(`Calendrical.calendar_from_cldr_calendar_type/1`).

### Arguments

* `calendar_type` is a CLDR calendar type as an atom or a string.

### Returns

* `{:ok, calendar_module}`, or

* `{:error, exception}` for a calendar type no calendar module has.

### Examples

    iex> Calendrical.Islamic.Observational.calendar_from_cldr_calendar_type(:gregorian)
    {:ok, Calendrical.Gregorian}

    iex> {:error, %Localize.UnknownCalendarError{}} =
    ...>   Calendrical.Islamic.Observational.calendar_from_cldr_calendar_type(:not_a_calendar)

# `cardinal_day`

Returns the day a date is written with: the day of its
`written_date/3`.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The day of the date's `written_date/3`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar
  does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {_year, _month, written_day} = Calendrical.Islamic.Observational.written_date(date.year, date.month, date.day)
    iex> Calendrical.Islamic.Observational.cardinal_day(date.year, date.month, date.day) == written_day
    true

# `cardinal_month`

Returns the CLDR month a month of the year names without a year
(`Calendrical.Month.cldr_month/2`).

### Arguments

* `month` is a month of this calendar.

### Returns

* The CLDR month the month is named in the year that holds
  1 January 2000, the month itself in the Hebrew calendar, whose
  names depend on the year, and the month itself where the
  reference year has no such month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2000-01-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.cardinal_month(1) == Calendrical.Islamic.Observational.cldr_month(date.year, 1)
    true

# `cardinal_year`

Returns the year a date is written with: the year of its
`written_date/3`.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The year of the date's `written_date/3`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar
  does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {written_year, _month, _day} = Calendrical.Islamic.Observational.written_date(date.year, date.month, date.day)
    iex> Calendrical.Islamic.Observational.cardinal_year(date.year, date.month, date.day) == written_year
    true

# `cldr_calendar_type`

Returns the CLDR calendar type whose data names this calendar's
months, days and quarters, as its `info/0` states.

### Returns

* A CLDR calendar type such as `:gregorian` or `:hebrew`.

### Examples

    iex> Calendrical.Islamic.Observational.cldr_calendar_type() == Calendrical.Islamic.Observational.info().cldr_calendar_type
    true

# `cldr_calendar_type`

Returns the CLDR calendar type whose data names a date's months and
days: the member in effect on the date in a composite calendar, and
this calendar's own otherwise (`Calendrical.Info.cldr_calendar_type/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* A CLDR calendar type.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> in_effect = Calendrical.Info.in_effect(Calendrical.Islamic.Observational, date.year, date.month, date.day)
    iex> Calendrical.Islamic.Observational.cldr_calendar_type(date.year, date.month, date.day) == in_effect.cldr_calendar_type()
    true

# `cldr_month`

Returns the CLDR month of a month of a year
(`Calendrical.Month.cldr_month/3`).

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* A CLDR month number, or `{month, :leap}` where the month's name
  has a leap form, or

* `{:error, exception}` for a month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> cldr_month = Calendrical.Islamic.Observational.cldr_month(date.year, date.month)
    iex> Calendrical.Islamic.Observational.written_month_from_cldr_month(date.year, cldr_month) ==
    ...>   Calendrical.Islamic.Observational.numeric_month(date.year, date.month, date.day)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Islamic.Observational.cldr_month(date.year, 99)

# `cyclic_year`

Returns the cyclic year of a date (`Calendrical.Era.cyclic_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The date's place in the sexagenary cycle, `1` to `60`, in an East
  Asian calendar, and its year in a calendar whose years run in no
  such cycle, or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> cyclic_year = Calendrical.Islamic.Observational.cyclic_year(date.year, date.month, date.day)
    iex> cyclic_year in 1..60 or cyclic_year == date.year
    true

    iex> {:error, %Calendrical.MissingFieldsError{}} = Calendrical.Islamic.Observational.cyclic_year(nil, 1, 1)

# `date_at`

```elixir
@spec date_at(DateTime.t(), Keyword.t()) :: {:ok, Date.t()} | {:error, Exception.t()}
```

Returns the observational Hijri date in effect at a given instant, under a
chosen day-start convention.

The Islamic day begins at sunset, so after sunset at the observation point
(Cairo) the date is already the following day. This maps an absolute instant
to the Hijri date, choosing the boundary with the `:day_start` option, always
evaluated **at Cairo** (the calendar's reference location).

For the plain civil-day mapping (the calendar's default), convert a date
directly with `Date.convert/2` instead.

### Arguments

* `datetime` is a `t:DateTime.t/0` — an absolute instant. Its own time zone
  fixes the instant; the day boundary is always taken at Cairo.

### Options

* `:day_start` selects the moment the day begins:

  * `:midnight` (the default) — 00:00 Cairo standard time, the ordinary
    civil-day mapping.

  * `:evening` — 18:00 Cairo standard time, the fixed-clock proxy for sunset.

  * `:sunset` — true sunset (Maghrib) at Cairo, computed via `Astro`.

### Returns

* `{:ok, date}` — an observational Hijri `t:Date.t/0`.

* `{:error, %Calendrical.InvalidOptionValueError{}}` for an invalid
  `:day_start` or `:nightfall_angle`.

* `{:error, %Calendrical.UnsupportedDateRangeError{}}` for a date outside
  the range the calendar or the installed JPL ephemeris covers.

* `{:error, %Calendrical.NoDuskError{}}` where the sun does not set, or
  nightfall does not occur, at the location on that day.

### Examples

    iex> {:ok, date} = Calendrical.Islamic.Observational.date_at(~U[2025-03-01 06:00:00Z])
    iex> date.calendar
    Calendrical.Islamic.Observational

# `date_from_cardinal_date`

Returns the date of this calendar that written figures name
(`date_from_written_date/3`), as fields.

### Arguments

* `year`, `month` and `day` are the figures a date is written with,
  as `written_date/3` returns them.

### Returns

* `{year, month, day}`, the date of this calendar, or

* `{:error, %Calendrical.InvalidDateError{}}` where the figures name no date.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {year, month, day} = Calendrical.Islamic.Observational.written_date(date.year, date.month, date.day)
    iex> Calendrical.Islamic.Observational.date_from_cardinal_date(year, month, day) == {date.year, date.month, date.day}
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.date_from_cardinal_date(date.year, 0, 1)

# `date_from_day_of_year`

Returns the date of a day of a year, counted from the year's first
day.

### Arguments

* `year` is a year of this calendar.

* `day_of_year` is the day of the year, counted from 1.

### Returns

* A `t:Date.t/0` of this calendar, or

* `{:error, %Calendrical.InvalidDateError{}}` for a day the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> day_of_year = Calendrical.Islamic.Observational.day_of_year(date.year, date.month, date.day)
    iex> Calendrical.Islamic.Observational.date_from_day_of_year(date.year, day_of_year) == date
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.date_from_day_of_year(date.year, 1000)

# `date_from_iso_days`

```elixir
@spec date_from_iso_days(integer()) :: {year(), month(), day()}
```

Returns an observational Islamic `{year, month, day}` for the
given ISO day number.

### Arguments

* `iso_days` is an integer count of days since the proleptic
  ISO epoch.

### Returns

* A three-tuple `{year, month, day}` in the observational Islamic
  calendar.

* Raises `Calendrical.UnsupportedDateRangeError` if `iso_days`
  falls outside the range of the underlying JPL ephemeris.

### Examples

    iex> Calendrical.Islamic.Observational.date_from_iso_days(739_252)
    {1445, 6, 19}

# `date_from_week_date`

Returns the date of a day of a week of a week-year
(`Calendrical.Week.date_from_week_date/4`).

### Arguments

* `week_year` is a week-year of this calendar.

* `week` is a week of the week-year.

* `day` is the day of the week, numbered from 1 for the day the
  calendar's weeks begin on.

### Returns

* A `t:Date.t/0` of this calendar, or

* `{:error, %Calendrical.InvalidDateError{}}` where the week has no such day.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {week_year, week, day} = Calendrical.Islamic.Observational.week_date_from_date(date.year, date.month, date.day)
    iex> Calendrical.Islamic.Observational.date_from_week_date(week_year, week, day) == date
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.date_from_week_date(date.year, 1, 8)

# `date_from_written_date`

Returns the date of this calendar that written figures name: the
inverse of `written_date/3`, the fields themselves where they are a
date.

### Arguments

* `year`, `month` and `day` are the figures a date is written with,
  as `written_date/3` returns them.

### Returns

* `{:ok, {year, month, day}}`, the date of this calendar, or

* `{:error, :invalid_date}` where the figures name no date.

The reason is an atom, not an exception: Localize's date parser reads
it, as `c:Calendrical.date_from_written_date/3` sets out.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {year, month, day} = Calendrical.Islamic.Observational.written_date(date.year, date.month, date.day)
    iex> Calendrical.Islamic.Observational.date_from_written_date(year, month, day) == {:ok, {date.year, date.month, date.day}}
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.date_from_written_date(date.year, 0, 1)
    {:error, :invalid_date}

# `date_to_iso_days`

```elixir
@spec date_to_iso_days(year(), month(), day()) :: integer()
```

Returns the number of ISO days for the given observational
Islamic `year`, `month`, and `day`.

### Arguments

* `year` is any Hijri year as an integer.

* `month` is a Hijri month in the range `1..12`.

* `day` is a Hijri day-of-month in the range `1..30`.

### Returns

* An integer count of days since the proleptic ISO epoch.

* Raises `Calendrical.UnsupportedDateRangeError` if the date falls
  outside the range of the underlying JPL ephemeris.

### Examples

    iex> Calendrical.Islamic.Observational.date_to_iso_days(1446, 1, 1)
    739439

# `dates_in_gregorian_year`

Returns the dates of this calendar, earliest first, whose month and
day are those given and that fall in a Gregorian year.

### Arguments

* `gregorian_year` is the Gregorian year to search within.

* `month` and `day` are a month and a day of this calendar.

### Returns

* A list of zero, one or two `t:Date.t/0` of this calendar, earliest
  first, since this calendar's years and the Gregorian years drift.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> date in Calendrical.Islamic.Observational.dates_in_gregorian_year(2024, date.month, date.day)
    true

# `day_numbers`

Returns the days a month of a year has, as runs of their numbers
(`Calendrical.Month.day_numbers/3`).

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* A list of ranges: one from 1 in every month of every calendar but
  a composite, whose month a change of calendar falls in has the
  days that are left, and `[]` for a month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.day_numbers(date.year, date.month) ==
    ...>   [1..Calendrical.Islamic.Observational.days_in_month(date.year, date.month)//1]
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.day_numbers(date.year, 99)
    []

# `day_of_era`

Returns the day of era and era of a date (`Calendrical.Era.day_of_era/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{day_of_era, era}`, where the day of the era counts from the
  era's first day, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for a
  missing field.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {day_of_era, era} = Calendrical.Islamic.Observational.day_of_era(date.year, date.month, date.day)
    iex> {_year_of_era, year_era} = Calendrical.Islamic.Observational.year_of_era(date.year, date.month, date.day)
    iex> day_of_era >= 1 and era == year_era
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.day_of_era(date.year, date.month, 0)

# `day_of_week`

Returns the day of the week of a date (`Calendrical.Week.day_of_week/5`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

* `starting_on` is `:default` for the day this calendar's weeks
  begin on, or the name of a weekday, `:monday` to `:sunday`.

### Returns

* `{day_of_week, 1, 7}`, the day numbered from the day the week
  begins on, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for a
  missing field.

### Examples

    # 1 June 2024 is a Saturday
    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.day_of_week(date.year, date.month, date.day, :monday)
    {6, 1, 7}

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.day_of_week(date.year, date.month, 0, :monday)

# `day_of_year`

Returns the day of the year of a date (`Calendrical.Period.day_of_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The day of the year, counted from 1 for its first day, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for a
  missing field.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> first_day = Calendrical.Islamic.Observational.year(date.year).first
    iex> Calendrical.Islamic.Observational.day_of_year(date.year, date.month, date.day) == Date.diff(date, first_day) + 1
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.day_of_year(date.year, date.month, 0)

# `days_in_month`

Returns the number of days in a month without a year
(`Calendrical.Month.days_in_month/2`).

### Arguments

* `month` is a month of this calendar.

### Returns

* The number of days where the month has as many in every year, or

* `{:ambiguous, lengths}` where its length depends on the year, the
  lengths a range or, where they are not consecutive, a list, or

* `{:error, %Calendrical.NotDefinedError{}}` for a month the calendar has in no year, or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, where
  the month is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {fewest, most} = Calendrical.Islamic.Observational.days_in_month_bounds(date.month)
    iex> case Calendrical.Islamic.Observational.days_in_month(date.month) do
    ...>   days when is_integer(days) -> fewest == days and most == days
    ...>   {:ambiguous, lengths} -> Enum.min(lengths) == fewest and Enum.max(lengths) == most
    ...> end
    true

    iex> {:error, %Calendrical.NotDefinedError{}} = Calendrical.Islamic.Observational.days_in_month(99)

# `days_in_month`

```elixir
@spec days_in_month(Calendar.year(), Calendar.month()) ::
  Calendar.day() | {:error, Exception.t()}
@spec days_in_month(year(), month()) :: 29..30 | {:error, Exception.t()}
```

Returns the number of days in the given Hijri `year` and `month`
(29 or 30, determined by actual crescent visibility).

### Arguments

* `year` is any Hijri year as an integer.

* `month` is a Hijri month in the range `1..12`.

### Returns

* The number of days in the month (`29` or `30`), or

* `{:error, %Calendrical.UnsupportedDateRangeError{}}` for a month the
  installed ephemeris does not cover, or

* `{:error, %Calendrical.InvalidMonthError{}}` for a month that is not
  `1..12`, or

* `{:error, %Calendrical.MissingFieldsError{}}` where the year or the
  month is `nil`.

### Examples

    iex> Calendrical.Islamic.Observational.days_in_month(1446, 1)
    30

    iex> Calendrical.Islamic.Observational.days_in_month(1446, 5)
    29

# `days_in_month_bounds`

Returns the fewest and the most days a month has in any year
(`Calendrical.Month.days_in_month_bounds/2`).

### Arguments

* `month` is a month of this calendar.

### Returns

* `{fewest, most}`, or

* `{:error, exception}` for a month the calendar has in no year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {fewest, most} = Calendrical.Islamic.Observational.days_in_month_bounds(date.month)
    iex> Calendrical.Islamic.Observational.days_in_month(date.year, date.month) in fewest..most
    true

    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Islamic.Observational.days_in_month_bounds(99)

# `days_in_week`

Returns the number of days in a week.

### Returns

* `7`.

### Examples

    iex> Calendrical.Islamic.Observational.days_in_week()
    7

# `days_in_year`

Returns the number of days in a year (`Calendrical.Period.days_in_year/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* The number of days, or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.days_in_year(date.year) == Enum.count(Calendrical.Islamic.Observational.year(date.year))
    true

    iex> {:error, %Calendrical.MissingFieldsError{}} = Calendrical.Islamic.Observational.days_in_year(nil)

# `diff`

Returns the number of whole units between two dates, counted
positionally.

### Arguments

* `from` and `to` are dates of this calendar as `{year, month, day}`
  tuples.

* `date_part` is one of `:years`, `:quarters`, `:months`, `:weeks`
  or `:days`.

### Returns

* The largest number of `date_part`s that can be added to `from`
  without passing `to`, negative where `to` is before `from`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a `from` or `to`
  the calendar does not have, `{:error, %Calendrical.MissingFieldsError{}}`
  for one missing a field, or `{:error, %Calendrical.InvalidPartError{}}`
  for a `date_part` other than the five.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> from = {date.year, date.month, date.day}
    iex> Calendrical.Islamic.Observational.diff(from, {date.year + 1, date.month, date.day}, :years)
    1

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> later = Date.add(date, 10)
    iex> Calendrical.Islamic.Observational.diff({later.year, later.month, later.day}, {date.year, date.month, date.day}, :days)
    -10

# `epoch`

Returns the epoch of this calendar: the ISO day number of
its first day.

### Returns

* The epoch as an integer number of days since the ISO epoch.

### Examples

    iex> Calendrical.Islamic.Observational.epoch() |> is_integer()
    true

# `era_calendar_type`

Returns the CLDR calendar type that names this calendar's eras, as
its `info/0` states.

### Returns

* A CLDR calendar type such as `:gregorian` or `:japanese`.

### Examples

    iex> Calendrical.Islamic.Observational.era_calendar_type() == Calendrical.Islamic.Observational.info().era_calendar_type
    true

# `extended_year`

Returns the extended year of a date (`Calendrical.Era.extended_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The year numbered without a gap at 0, so that in a calendar with
  no year 0 the year before 1 is 0, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have, or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.extended_year(date.year, date.month, date.day) == date.year
    true

    iex> {:error, %Calendrical.MissingFieldsError{}} = Calendrical.Islamic.Observational.extended_year(nil, 1, 1)

# `info`

Returns the static facts about this calendar, as
`t:Calendrical.Info.t/0`.

### Returns

* A `t:Calendrical.Info.t/0` describing this calendar.

### Examples

    iex> Calendrical.Islamic.Observational.info().calendar
    Calendrical.Islamic.Observational

# `iso_week_of_year`

Returns the ISO 8601 week-year and week of the day a date names
(`Calendrical.Week.iso_week_of_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{week_year, week}` of ISO 8601, or

* `{:error, exception}` for a date the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.iso_week_of_year(date.year, date.month, date.day)
    {2024, 22}

# `leap_month`

Returns the position of a year's leap month, or `nil` where the
year has none.

### Arguments

* `year` is a year of this calendar.

### Returns

* The leap month's position in the year, or

* `nil` where the year has no leap month, as for anything but an
  integer year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> leap_month = Calendrical.Islamic.Observational.leap_month(date.year)
    iex> is_nil(leap_month) or leap_month in 1..Calendrical.Islamic.Observational.months_in_year(date.year)
    true

# `leap_year?`

```elixir
@spec leap_year?(year()) :: boolean()
```

Returns whether the given Hijri `year` is a 355-day year (the
observational analogue of a "leap year").

Year length varies between 354 and 355 days depending on actual
crescent sightings, so this must be computed by comparing the
starts of two successive years.

### Arguments

* `year` is any Hijri year as an integer.

### Returns

* `true` if the year contains 355 days; otherwise `false`.

### Examples

    iex> Calendrical.Islamic.Observational.leap_year?(1447)
    false

    iex> Calendrical.Islamic.Observational.leap_year?(1446)
    true

# `location`

```elixir
@spec location() :: Geo.PointZ.t()
```

Returns the geographic location used to determine crescent
visibility for this calendar.

### Returns

* A `t:Geo.PointZ.t/0` for Cairo, Egypt (Reingold's canonical
  "Islamic location").

### Examples

    iex> Calendrical.Islamic.Observational.location().coordinates
    {31.3, 30.1, 200.0}

# `lunar_month_of_year`

Returns the traditional month at a position in a year
(`Calendrical.Month.traditional_month/3`), or an error for a place
the year does not have.

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* The month's traditional number, or `{month, :leap}` for a leap
  month, or

* `{:error, %Calendrical.InvalidMonthError{}}` for a month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.lunar_month_of_year(date.year, date.month) ==
    ...>   Calendrical.Islamic.Observational.traditional_month(date.year, date.month)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Islamic.Observational.lunar_month_of_year(date.year, 99)

# `month`

Returns the days of a month of a year (`Calendrical.Period.month/3`).

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* A `t:Date.Range.t/0`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a month the year does not have, or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for a
  missing year or month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> date in Calendrical.Islamic.Observational.month(date.year, date.month)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.month(date.year, 99)

# `month_from_traditional`

Returns the position of a traditional month in a year
(`Calendrical.Month.month_from_traditional/3`).

### Arguments

* `year` is a year of this calendar.

* `traditional_month` is a month's traditional number, or
  `{month, :leap}` for a leap month.

### Returns

* The month's position in the year, or

* `{:error, exception}` where the year has no such month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> traditional_month = Calendrical.Islamic.Observational.traditional_month(date.year, date.month)
    iex> Calendrical.Islamic.Observational.month_from_traditional(date.year, traditional_month) == date.month
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Islamic.Observational.month_from_traditional(date.year, 99)

# `month_numbers`

Returns the months a year has, as runs of their numbers
(`Calendrical.Month.month_numbers/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* A list of ranges: one from 1 in every year of every calendar but a
  composite, whose year a change of calendar falls in has the months
  that are left, and `[]` for a year the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.month_numbers(date.year) == [1..Calendrical.Islamic.Observational.months_in_year(date.year)//1]
    true

# `month_of_year`

Returns the CLDR month of a month of a year: the index of its name
in this calendar's CLDR data, with `{month, :leap}` where the name
has a leap form (`Calendrical.Month.cldr_month/3`).

### Arguments

* `year`, `month` and `day` are a date of this calendar. The year
  and the day may be `nil`: a month's name needs only the month.

### Returns

* A CLDR month number, or `{month, :leap}`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a day the month does not have, or

* `{:error, exception}` for a month the year does not have, or a
  `Calendrical.MissingFieldsError` where the month is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.month_of_year(date.year, date.month, date.day) == Calendrical.Islamic.Observational.cldr_month(date.year, date.month)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.month_of_year(date.year, date.month, 0)

# `months_in_leap_year`

Returns the number of months in a leap year: the most months a year
of this calendar has.

### Returns

* A positive integer.

### Examples

    iex> Calendrical.Islamic.Observational.months_in_leap_year() >= Calendrical.Islamic.Observational.months_in_ordinary_year()
    true

# `months_in_ordinary_year`

Returns the number of months in an ordinary year: the fewest months
a year of this calendar has.

### Returns

* A positive integer.

### Examples

    iex> Calendrical.Islamic.Observational.months_in_ordinary_year() <= Calendrical.Islamic.Observational.months_in_leap_year()
    true

# `months_in_year`

Returns the number of months in a year without a year: the count
where every year has as many, `{:ambiguous, range}` where a leap
year has more, and `{:error, %Calendrical.NotDefinedError{}}` for a composite calendar
(`Calendrical.Info.months_in_year/1`).

### Returns

* The number of months where every year has as many, or

* `{:ambiguous, fewest..most}` where a leap year has more, or

* `{:error, %Calendrical.NotDefinedError{}}` for a composite calendar, whose bounds
  `months_in_year_bounds/0` gives.

### Examples

    iex> case Calendrical.Islamic.Observational.months_in_year() do
    ...>   months when is_integer(months) -> months == Calendrical.Islamic.Observational.months_in_leap_year()
    ...>   {:ambiguous, fewest..most//1} -> {fewest, most} == Calendrical.Islamic.Observational.info().months_in_year
    ...>   {:error, %Calendrical.NotDefinedError{}} -> is_list(Calendrical.Islamic.Observational.info().members)
    ...> end
    true

# `months_in_year`

Returns the number of months in a
given `year`.

### Arguments

* `year` is the year in this calendar.

### Returns

* The number of months in the year, as a positive integer, or

* `{:error, %Calendrical.MissingFieldsError{}}` where `year` is `nil`, or

* `{:error, %Calendrical.InvalidDateError{}}` where `year` is no integer.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.months_in_year(date.year) >= date.month
    true

# `months_in_year_bounds`

Returns the fewest and the most months a year has
(`Calendrical.Month.months_in_year_bounds/1`).

### Returns

* `{fewest, most}`, which in a composite calendar counts the months
  left in a year a change of calendar falls in.

### Examples

    iex> {fewest, most} = Calendrical.Islamic.Observational.months_in_year_bounds()
    iex> fewest <= Calendrical.Islamic.Observational.months_in_ordinary_year() and
    ...>   most >= Calendrical.Islamic.Observational.months_in_leap_year()
    true

# `naive_datetime_from_iso_days`

Converts ISO days and a day fraction to a naive datetime of this
calendar, as `c:Calendar.naive_datetime_from_iso_days/1` asks.

### Arguments

* `iso_days` is `{days, {parts_in_day, parts_per_day}}`, a day and a
  fraction of it.

### Returns

* `{year, month, day, hour, minute, second, microsecond}` of this
  calendar.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {year, month, day, 12, 0, 0, {0, 6}} =
    ...>   Calendrical.Islamic.Observational.naive_datetime_from_iso_days({739403, {43200000000, 86400000000}})
    iex> {year, month, day} == {date.year, date.month, date.day}
    true

# `naive_datetime_to_iso_days`

Converts a naive datetime of this calendar to ISO days and a day
fraction, as `c:Calendar.naive_datetime_to_iso_days/7` asks.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

* `hour`, `minute`, `second` and `microsecond` are a time of day,
  the microsecond as `{microseconds, precision}`.

### Returns

* `{iso_days, {parts_in_day, parts_per_day}}`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.naive_datetime_to_iso_days(date.year, date.month, date.day, 12, 0, 0, {0, 6})
    {739403, {43200000000, 86400000000}}

# `named_month`

Returns the days of a named month in a year
(`Calendrical.Month.named_month/3`).

### Arguments

* `year` is a year of this calendar.

* `named_month` is the CLDR month that names the month.

### Returns

* A list of `t:Date.Range.t/0`, one for each counted month that
  carries the name, earliest first, and `[]` where none does.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> cldr_month = Calendrical.Islamic.Observational.cldr_month(date.year, date.month)
    iex> Calendrical.Islamic.Observational.named_month(date.year, cldr_month) |> Enum.any?(&(date in &1))
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.named_month(date.year, 99)
    []

# `numeric_month`

Returns the number a date's month is written with in figures: the
month of its `written_date/3`.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The month of the date's `written_date/3`: a month number, or
  `{month, :leap}` for a leap month of a lunisolar calendar, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar
  does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {_year, written_month, _day} = Calendrical.Islamic.Observational.written_date(date.year, date.month, date.day)
    iex> Calendrical.Islamic.Observational.numeric_month(date.year, date.month, date.day) == written_month
    true

# `ordinal_month_from_traditional`

Returns the position in a year of a traditional month
(`Calendrical.Month.month_from_traditional/3`), as `{:ok, month}`.

### Arguments

* `year` is a year of this calendar.

* `traditional_month` is a month's traditional number, or
  `{month, :leap}` for a leap month.

### Returns

* `{:ok, month}`, the month's position in the year, or

* `{:error, %Calendrical.InvalidMonthError{}}` for a month or a leap
  month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> traditional_month = Calendrical.Islamic.Observational.traditional_month(date.year, date.month)
    iex> Calendrical.Islamic.Observational.ordinal_month_from_traditional(date.year, traditional_month) == {:ok, date.month}
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Islamic.Observational.ordinal_month_from_traditional(date.year, 99)

# `parse_date`

Parses a date written in this calendar's own notation.

### Arguments

* `string` is a date written `YYYY-MM-DD`, with a leading `-` for a
  year before 1, or, in a calendar of weeks, `YYYY-Www-D`.

### Returns

* `{:ok, {year, month, day}}`, or

* `{:error, :invalid_format}` where the text is not in that form or
  its fields are not digits, or

* `{:error, :invalid_date}` where the text names no date of this
  calendar.

The reason is an atom, not an exception, because `c:Calendar.parse_date/1`
requires one: `Date.from_iso8601/2` reads it.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.parse_date(Date.to_string(date)) == {:ok, {date.year, date.month, date.day}}
    true

    iex> Calendrical.Islamic.Observational.parse_date("not a date")
    {:error, :invalid_format}

# `parse_naive_datetime`

Parses a naive datetime written in this calendar's own notation.

### Arguments

* `string` is a date as `parse_date/1` reads it, then `T` or a
  space and a time of day.

### Returns

* `{:ok, {year, month, day, hour, minute, second, microsecond}}`, or

* `{:error, reason}`, such as `:invalid_format` or `:invalid_date`,
  where the text is no such datetime.

The reason is an atom, not an exception, because
`c:Calendar.parse_naive_datetime/1` requires one:
`NaiveDateTime.from_iso8601/2` reads it.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.parse_naive_datetime(Date.to_string(date) <> "T12:00:00") ==
    ...>   {:ok, {date.year, date.month, date.day, 12, 0, 0, {0, 0}}}
    true

    iex> Calendrical.Islamic.Observational.parse_naive_datetime("2024-06-01")
    {:error, :invalid_format}

# `parse_utc_datetime`

Parses a UTC datetime written in this calendar's own notation.

### Arguments

* `string` is a date as `parse_date/1` reads it, then `T` or a
  space, a time of day and an offset from UTC, `Z` or `±HH:MM`.

### Returns

* `{:ok, {year, month, day, hour, minute, second, microsecond}, offset}`,
  the datetime in UTC and the offset in seconds it was written with,
  or

* `{:error, reason}`, such as `:invalid_format`, `:invalid_date` or
  `:missing_offset`, where the text is no such datetime.

The reason is an atom, not an exception, because
`c:Calendar.parse_utc_datetime/1` requires one:
`DateTime.from_iso8601/2` reads it.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.parse_utc_datetime(Date.to_string(date) <> "T12:00:00Z") ==
    ...>   {:ok, {date.year, date.month, date.day, 12, 0, 0, {0, 0}}, 0}
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.parse_utc_datetime(Date.to_string(date) <> "T12:00:00")
    {:error, :missing_offset}

# `parsing_calendar`

Returns the calendar a date written for this calendar is parsed in,
as its `info/0` states.

### Returns

* A calendar module.

### Examples

    iex> Calendrical.Islamic.Observational.parsing_calendar() == Calendrical.Islamic.Observational.info().parsing_calendar
    true

# `parsing_calendars`

Returns the calendars a date written for this calendar may be
written in: the members of a composite calendar, and this calendar
alone otherwise (`Calendrical.Info.parsing_calendars/1`).

### Returns

* A non-empty list of calendar modules, the members of a composite
  calendar in the order they take effect.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> in_effect = Calendrical.Info.in_effect(Calendrical.Islamic.Observational, date.year, date.month, date.day)
    iex> in_effect in Calendrical.Islamic.Observational.parsing_calendars()
    true

# `periods_in_year`

Returns the number of periods in a year: its months.

### Arguments

* `year` is a year of this calendar.

### Returns

* The number of months in the year, as `months_in_year/1` returns
  it.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.periods_in_year(date.year) == Calendrical.Islamic.Observational.months_in_year(date.year)
    true

# `quadrimester`

Returns the days of a quadrimester of a year
(`Calendrical.Period.quadrimester/3`).

### Arguments

* `year` is a year of this calendar.

* `quadrimester` is the quadrimester, `1` to `3`.

### Returns

* A `t:Date.Range.t/0`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a quadrimester or a year the
  calendar does not have, or

* `{:error, %Calendrical.MissingFieldsError{}}` where the year or the
  quadrimester is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.quadrimester(date.year, 1).first == Calendrical.Islamic.Observational.year(date.year).first
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.quadrimester(date.year, 4)

# `quarter`

Returns the days of a quarter of a year (`Calendrical.Period.quarter/3`).

### Arguments

* `year` is a year of this calendar.

* `quarter` is the quarter, `1` to `4`.

### Returns

* A `t:Date.Range.t/0`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a quarter or a year the calendar
  does not have, or

* `{:error, %Calendrical.MissingFieldsError{}}` where the year or the
  quarter is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> quarter = Calendrical.Islamic.Observational.quarter_of_year(date.year, date.month, date.day)
    iex> date in Calendrical.Islamic.Observational.quarter(date.year, quarter)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.quarter(date.year, 5)

# `quarter_of_year`

Returns the quarter of the year a date falls in
(`Calendrical.Period.quarter_of_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The quarter, `1` to `4`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for a
  missing year or month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.quarter_of_year(date.year, date.month, date.day) in 1..4
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.quarter_of_year(date.year, date.month, 0)

# `related_gregorian_year`

Returns the related Gregorian year of a date
(`Calendrical.Era.related_gregorian_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* The Gregorian year the date's calendar year begins in, or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:ok, first_day} = Date.convert(Calendrical.Islamic.Observational.year(date.year).first, Calendrical.Gregorian)
    iex> Calendrical.Islamic.Observational.related_gregorian_year(date.year, date.month, date.day) == first_day.year
    true

# `semester`

Returns the days of a semester of a year (`Calendrical.Period.semester/3`).

### Arguments

* `year` is a year of this calendar.

* `semester` is the semester, `1` or `2`.

### Returns

* A `t:Date.Range.t/0`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a semester or a year the calendar
  does not have, or

* `{:error, %Calendrical.MissingFieldsError{}}` where the year or the
  semester is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.semester(date.year, 1).first == Calendrical.Islamic.Observational.year(date.year).first
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.semester(date.year, 3)

# `shift`

Returns the date a number of years, quarters, months, weeks or days
on from a date of this calendar, by position, with the traditional
month kept across years and a day the resulting month lacks clamped
to its last day unless `clamp: false` (`Calendrical.Shift.shift/7`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

* `date_part` is one of `:years`, `:quarters`, `:months`, `:weeks`
  or `:days`.

* `increment` is the number of units, which may be negative.

* `options` is a keyword list of options.

### Options

* `:clamp` is whether a day the resulting month lacks becomes its
  last day. The default is `true`.

### Returns

* `{year, month, day}` of this calendar, or

* `{:error, %Calendrical.InvalidDateError{}}` where `year`, `month` and
  `day` are no date of this calendar, or

* `{:error, %Calendrical.MissingFieldsError{}}` where a field is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.shift(date.year, date.month, date.day, :days, 0)
    {date.year, date.month, date.day}

# `shift_date`

Shifts a date by a duration: its years and months together, as
`shift/6` adds months, and then its weeks and days.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

* `duration` is a `t:Duration.t/0`.

### Returns

* `{year, month, day}` of this calendar.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {year, month, day} = Calendrical.Islamic.Observational.shift_date(date.year, date.month, date.day, Duration.new!(day: 7))
    iex> Date.new!(year, month, day, Calendrical.Islamic.Observational) == Date.add(date, 7)
    true

# `shift_naive_datetime`

Shifts a naive datetime by a duration: its date as `shift_date/4`
shifts it, and its time as `Calendar.ISO` does.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

* `hour`, `minute`, `second` and `microsecond` are a time of day,
  the microsecond as `{microseconds, precision}`.

* `duration` is a `t:Duration.t/0`.

### Returns

* `{year, month, day, hour, minute, second, microsecond}` of this
  calendar.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> next_day = Date.add(date, 1)
    iex> Calendrical.Islamic.Observational.shift_naive_datetime(date.year, date.month, date.day, 23, 0, 0, {0, 0}, Duration.new!(hour: 2)) ==
    ...>   {next_day.year, next_day.month, next_day.day, 1, 0, 0, {0, 0}}
    true

# `shift_time`

Shifts a time by a duration, as `Calendar.ISO` does.

### Arguments

* `hour`, `minute`, `second` and `microsecond` are a time of day,
  the microsecond as `{microseconds, precision}`.

* `duration` is a `t:Duration.t/0` of time units.

### Returns

* `{hour, minute, second, microsecond}`, wrapped around midnight.

### Examples

    iex> Calendrical.Islamic.Observational.shift_time(23, 0, 0, {0, 0}, Duration.new!(hour: 2))
    {1, 0, 0, {0, 0}}

# `solar_term`

Returns the day of a solar term of a Gregorian year, reckoned at the
place this calendar's `info/0` names, or at the Chinese calendar's
meridian, the traditional reference, where it names none.

### Arguments

* `index` is the solar term's number, `1` (lichun, 315° of solar
  longitude) to `24`, every 15° on.

* `gregorian_year` is a Gregorian year.

### Returns

* `{:ok, date}`, a `t:Date.t/0` in `Calendrical.Gregorian`, or

* `{:error, %Calendrical.InvalidSolarTermError{}}` where `index` is not
  `1` to `24`.

### Examples

    # Qingming, the fifth solar term
    iex> Calendrical.Islamic.Observational.solar_term(5, 2025)
    {:ok, ~D[2025-04-04 Calendrical.Gregorian]}

    iex> Calendrical.Islamic.Observational.solar_term(25, 2025)
    {:error, %Calendrical.InvalidSolarTermError{index: 25}}

# `traditional_leap_month`

Returns the traditional number of the month a year's leap month
follows, or `nil` (`Calendrical.Month.traditional_leap_month/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* The traditional number of the month the leap month follows, or

* `nil` where the year has no leap month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> is_nil(Calendrical.Islamic.Observational.traditional_leap_month(date.year)) ==
    ...>   is_nil(Calendrical.Islamic.Observational.leap_month(date.year))
    true

# `traditional_month`

Returns a month of a year as this calendar names it
(`Calendrical.Month.traditional_month/3`).

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* The month's traditional number, or `{month, :leap}` for a leap
  month, or

* `{:error, exception}` for a month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.traditional_month(date.year, date.month) ==
    ...>   Enum.at(Calendrical.Islamic.Observational.traditional_months(date.year), date.month - 1)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Islamic.Observational.traditional_month(date.year, 99)

# `traditional_months`

Returns a year's months in order, named traditionally
(`Calendrical.Month.traditional_months/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* A list of traditional months, the nth naming the year's nth month,
  with `{month, :leap}` for a leap month, and `[]` for a year the
  calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> length(Calendrical.Islamic.Observational.traditional_months(date.year)) ==
    ...>   Calendrical.Islamic.Observational.months_in_year(date.year)
    true

# `valid_date?`

```elixir
@spec valid_date?(year(), month(), day()) :: boolean()
```

Determines if the given `year`, `month`, and `day` form a valid
observational Islamic date.

### Arguments

* `year` is any positive Hijri year as an integer.

* `month` is a Hijri month in the range `1..12`.

* `day` is a Hijri day-of-month in the range `1..30`.

### Returns

* `true` if the date is valid; otherwise `false`.

### Examples

    iex> Calendrical.Islamic.Observational.valid_date?(1446, 1, 30)
    true

    iex> Calendrical.Islamic.Observational.valid_date?(1446, 5, 30)
    false

# `week`

Returns the seven days of a week of a year (`Calendrical.Week.week/3`).

### Arguments

* `year` is a year of this calendar.

* `week` is a week of the year.

### Returns

* A `t:Date.Range.t/0` of the week's seven days, which may begin in
  the year before or end in the year after, or

* `{:error, %Calendrical.InvalidDateError{}}` for a week the year does not have, or

* `{:error, %Calendrical.MissingFieldsError{}}` where the year or the
  week is `nil`.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {week_year, week} = Calendrical.Islamic.Observational.week_of_year(date.year, date.month, date.day)
    iex> days = Calendrical.Islamic.Observational.week(week_year, week)
    iex> {Enum.count(days), date in days}
    {7, true}

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.week(date.year, 0)

# `week_date_from_date`

Returns the week-year, the week and the day of the week of a date
(`Calendrical.Week.week_date_from_date/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{week_year, week, day_of_week}`, the day numbered from the day
  the calendar's weeks begin on, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {week_year, week, day} = Calendrical.Islamic.Observational.week_date_from_date(date.year, date.month, date.day)
    iex> {week_year, week} == Calendrical.Islamic.Observational.week_of_year(date.year, date.month, date.day) and day in 1..7
    true

# `week_of_month`

Returns the month and the week of the month a date is in
(`Calendrical.Week.week_of_month/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{month, week}`, where the month is the one before or after where
  the date falls in that month's last or first week, or

* `{:error, exception}` for a date the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {month, week} = Calendrical.Islamic.Observational.week_of_month(date.year, date.month, date.day)
    iex> month in (date.month - 1)..(date.month + 1) and week >= 1
    true

# `week_of_year`

Returns the week-year and the week of the year a date is in
(`Calendrical.Week.week_of_year/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{week_year, week}`, where the week-year is the year before or
  after where the date falls in that year's last or first week, or

* `{:error, exception}` for a date the calendar does not have: a
  `Calendrical.MissingFieldsError` for a date missing a field, and a
  `Calendrical.InvalidDateError` otherwise.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {week_year, week} = Calendrical.Islamic.Observational.week_of_year(date.year, date.month, date.day)
    iex> date in Calendrical.Islamic.Observational.week(week_year, week)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.week_of_year(date.year, date.month, 0)

# `weeks_in_week_year`

Returns the number of weeks in a week-year
(`Calendrical.Week.weeks_in_week_year/2`).

### Arguments

* `week_year` is a week-year of this calendar.

### Returns

* The number of weeks, or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {week_year, week, _day} = Calendrical.Islamic.Observational.week_date_from_date(date.year, date.month, date.day)
    iex> week <= Calendrical.Islamic.Observational.weeks_in_week_year(week_year)
    true

# `weeks_in_year`

Returns the number of weeks in a year (`Calendrical.Week.weeks_in_year/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* The number of weeks, or

* `{:error, exception}` for a year the calendar does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {week_year, week} = Calendrical.Islamic.Observational.week_of_year(date.year, date.month, date.day)
    iex> week <= Calendrical.Islamic.Observational.weeks_in_year(week_year)
    true

# `weeks_of_month`

Returns the weeks of a month of a year in order
(`Calendrical.Week.weeks_of_month/3`).

### Arguments

* `year` and `month` are a month of this calendar.

### Returns

* A list of `t:Date.Range.t/0`, one for each week `week_of_month/3`
  names for the month, the first of which may begin in the month
  before and the last run into the month after, or

* `{:error, exception}` for a month the year does not have.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.weeks_of_month(date.year, date.month) |> Enum.all?(&(Enum.count(&1) == 7))
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidMonthError{}} = Calendrical.Islamic.Observational.weeks_of_month(date.year, 99)

# `written_date`

Returns the figures a date is written with: its own fields, in a
calendar whose dates are written as its own.

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{year, month, day}` as the date is written, where the month of a
  lunisolar calendar is its traditional number, `{month, :leap}`
  for a leap month.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {year, month, day} = Calendrical.Islamic.Observational.written_date(date.year, date.month, date.day)
    iex> Calendrical.Islamic.Observational.date_from_written_date(year, month, day) == {:ok, {date.year, date.month, date.day}}
    true

# `written_month_from_cldr_month`

Returns the month a CLDR month is written with in a year
(`Calendrical.Month.written_month_from_cldr_month/3`).

### Arguments

* `year` is a year of this calendar.

* `cldr_month` is a CLDR month number, or `{month, :leap}`.

### Returns

* The month as `written_date/3` writes it, or

* `{:error, :invalid_month}` where the year has no month of that
  name. The reason is an atom, not an exception: Localize's date parser
  reads it, where `Calendrical.Month.written_month_from_cldr_month/3`
  answers with an exception.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {_year, written_month, _day} = Calendrical.Islamic.Observational.written_date(date.year, date.month, date.day)
    iex> cldr_month = Calendrical.Islamic.Observational.cldr_month(date.year, date.month)
    iex> Calendrical.Islamic.Observational.written_month_from_cldr_month(date.year, cldr_month) == written_month
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.written_month_from_cldr_month(date.year, 99)
    {:error, :invalid_month}

# `year`

Returns the days of a year (`Calendrical.Period.year/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* A `t:Date.Range.t/0`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for
  no year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> date in Calendrical.Islamic.Observational.year(date.year)
    true

    iex> {:error, %Calendrical.MissingFieldsError{}} = Calendrical.Islamic.Observational.year(nil)

# `year_of_era`

Returns the year of era and era of a year (`Calendrical.Era.year_of_era/2`).

### Arguments

* `year` is a year of this calendar.

### Returns

* `{year_of_era, era}`, the era in effect on the year's first day,
  or

* `{:error, %Calendrical.InvalidDateError{}}` for a year the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> Calendrical.Islamic.Observational.year_of_era(date.year) ==
    ...>   Calendrical.Islamic.Observational.year_of_era(date.year, date.month, date.day)
    true

# `year_of_era`

Returns the year of era and era of a date (`Calendrical.Era.year_of_era/4`).

### Arguments

* `year`, `month` and `day` are a date of this calendar.

### Returns

* `{year_of_era, era}`, or

* `{:error, %Calendrical.InvalidDateError{}}` for a date the calendar does not have,
  or

* `{:error, exception}`, a `Calendrical.MissingFieldsError`, for no
  year.

### Examples

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {year_of_era, era} = Calendrical.Islamic.Observational.year_of_era(date.year, date.month, date.day)
    iex> year_of_era >= 1 and is_integer(era)
    true

    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> {:error, %Calendrical.InvalidDateError{}} = Calendrical.Islamic.Observational.year_of_era(date.year, 99, 1)

# `years_in_cycle`

Returns the number of years after which this calendar's years
repeat, as its `info/0` states, or `{:error, %Calendrical.NotDefinedError{}}`.

### Returns

* The number of years in the cycle, or

* `{:error, %Calendrical.NotDefinedError{}}` where the calendar states none.

### Examples

    # A year has as many days as the year one cycle later
    iex> {:ok, date} = Date.convert(~D[2024-06-01], Calendrical.Islamic.Observational)
    iex> case Calendrical.Islamic.Observational.years_in_cycle() do
    ...>   {:error, %Calendrical.NotDefinedError{}} -> true
    ...>   years -> Calendrical.Islamic.Observational.days_in_year(date.year) == Calendrical.Islamic.Observational.days_in_year(date.year + years)
    ...> end
    true

---

*Consult [api-reference.md](api-reference.md) for complete listing*