---
title: "acos"
slug: "dev-database-reference-sql-functions-numeric-acos"
updated: 2026-10-30T18:18:51Z
published: 2026-10-30T18:18:51Z
canonical: "docs.vastdata.com/dev-database-reference-sql-functions-numeric-acos"
---

> ## Documentation Index
> Fetch the complete documentation index at: https://docs.vastdata.com/llms.txt
> Use this file to discover all available pages before exploring further.

# acos

*Applies to VAST Cluster 5.5.*

*[Functions](/develop/docs/dev-database-reference-sql-functions-index) › [Numeric and math functions](/develop/docs/dev-database-reference-sql-functions-numeric) › Inverse trigonometry.* Returns the arc cosine of a number: the angle, in radians from 0 to pi, whose cosine is that number. The argument must be from -1 to 1. A common use is turning a cosine similarity into an angle.

## Syntax

```
acos(x)
```

| Argument | Type | Description |
| --- | --- | --- |
| `x` | Any number (`DOUBLE`, `REAL`, an integer or a `DECIMAL`) | The cosine, from -1 to 1. |

**Returns:** `DOUBLE`, from 0 to pi.

## Set up the examples

The examples connect with the ADBC driver, as in [VAST Query Engine (ADBC)](/develop/docs/dev-database-integrations-adbc), and work in a schema called `docs_sql_fn_numeric_acos`. Set `QE_ENDPOINT`, `AWS_ACCESS_KEY_ID`, `AWS_SECRET_ACCESS_KEY` and `VASTDB_BUCKET` ([SQL reference](/develop/docs/dev-database-reference-sql-index#example-connect-and-use-a-transaction)), and run the blocks in order in one Python session. Below each block is the output you should see.

This block connects, defines the `run()` and `plan()` helpers, and creates a table of six cosine similarities between pairs of vectors. Row 3 has no value:

```
import os
import re

import adbc_driver_manager
import adbc_driver_manager.dbapi as dbapi
import adbc_driver_vastdb

BUCKET = os.environ["VASTDB_BUCKET"]
SCHEMA = f'"{BUCKET}/docs_sql_fn_numeric_acos"'

conn = dbapi.connect(
    driver=adbc_driver_vastdb.get_driver_path(),
    db_kwargs={
        "vast.db.endpoint": os.environ["QE_ENDPOINT"],
        "vast.db.access_key": os.environ["AWS_ACCESS_KEY_ID"],
        "vast.db.secret_key": os.environ["AWS_SECRET_ACCESS_KEY"],
    },
    autocommit=True,
)

def server_message(e):
    """The server's message in a driver error, without the request ID or the echoed statement."""
    text = re.sub(r"\[req=0x[0-9a-f]+\]: ", "", str(e))
    found = (re.search(r'message: "(.*?)", request_id', text, re.S)
             or re.search(r"description: (.*)\)", text, re.S))
    message = found.group(1) if found else text.strip().splitlines()[-1]
    message = re.sub(r"^Plan query: '.*?' failed with: ", "", message, flags=re.S)
    message = re.split(r"\\n|\n", message)[0].replace('\\"', '"')
    return message.replace(BUCKET, "<bucket>")

def run(sql):
    """Run one statement. Print the rows it returns, or the error."""
    try:
        with conn.cursor() as cur:
            cur.execute(sql)
            rows = cur.fetchall() if cur.description else []
    except (adbc_driver_manager.Error, OSError) as e:
        print(f"{type(e).__name__}: {server_message(e)}")
    else:
        print(rows)

def plan(sql):
    """Print the plan without its ~N Rows lines: the table's row count, which can still be 0 just after an insert."""
    with conn.cursor() as cur:
        cur.execute("EXPLAIN " + sql)
        steps = cur.fetchall()[0][1]
    print("\n".join(line for line in steps.splitlines() if "Rows" not in line))

TABLE = f"{SCHEMA}.similarities"
run(f"CREATE SCHEMA IF NOT EXISTS {SCHEMA}")
run(f"CREATE TABLE IF NOT EXISTS {TABLE} (id BIGINT, similarity DOUBLE)")
run(f"INSERT INTO {TABLE} VALUES "
    "(1, 0.5), (2, -0.5), (3, NULL), (4, 0.0), (5, 1.0), (6, -1.0)")
```

```
[]
[]
[(6,)]
```

## What it does

`acos(x)` is the inverse of [cos](/develop/docs/dev-database-reference-sql-functions-numeric-cos) on the range 0 to pi: it returns the angle `a`, in radians, for which `cos(a) = x`. A cosine of 1 is an angle of 0, a cosine of 0 is a right angle (pi/2), and a cosine of -1 is pi. To get degrees, wrap it in [degrees](/develop/docs/dev-database-reference-sql-functions-numeric-degrees).

| Call | Result |
| --- | --- |
| `acos(1)` | `0.0` |
| `acos(0.5)` | `1.0471975511965979` (pi/3) |
| `acos(0)` | `1.5707963267948966` (pi/2) |
| `acos(-1)` | `3.141592653589793` (pi) |

The result is a `DOUBLE` for every numeric argument type: an integer, a `REAL`, a `DECIMAL` or a `DOUBLE`.

```
run("SELECT acos(1), acos(0.5), acos(0), acos(-1)")
run("SELECT typeof(acos(1::INTEGER)), typeof(acos(0.5::FLOAT)), typeof(acos(0.5))")
```

```
[(0.0, 1.0471975511965979, 1.5707963267948966, 3.141592653589793)]
[('DOUBLE', 'DOUBLE', 'DOUBLE')]
```

## NULLs and edge cases

| Input | Result |
| --- | --- |
| `NULL` | `NULL` |
| `&apos;nan&apos;::DOUBLE` | `nan` |
| `-0.0::DOUBLE` | `1.5707963267948966`, the same as for `0` |
| The smallest positive `DOUBLE` (`5e-324`) | `1.5707963267948966` |
| `&apos;12&apos;` (a string literal holding a number) | Converted to `DOUBLE`, then fails: `ACOS is undefined outside [-1,1]` |
| A value above 1 or below -1, such as `1.0000000001` or `-2` | Error: `ACOS is undefined outside [-1,1]` |
| `&apos;inf&apos;::DOUBLE`, `&apos;-inf&apos;::DOUBLE` | Error: `input value inf is out of range for numeric function` (`-inf` for minus infinity) |
| `&apos;abc&apos;` (a string that isn't a number) | Error: `Could not convert string &apos;abc&apos; to DOUBLE` |
| A `BLOB`, date, boolean, list or interval | Error: `No function matches the given name and argument types`, for example `&apos;acos(DATE)&apos;` |

The same cases, as the cluster returns them. They use the helpers from [Set up the examples](/develop/docs/dev-database-reference-sql-functions-numeric-acos#set-up-the-examples):

```
run("SELECT acos(NULL), acos('nan'::DOUBLE), acos(-0.0::DOUBLE), acos(5e-324::DOUBLE)")
run("SELECT acos('12')")
run("SELECT acos(1.0000000001)")
run("SELECT acos(-2)")
run("SELECT acos('inf'::DOUBLE)")
run("SELECT acos('-inf'::DOUBLE)")
run("SELECT acos('abc')")
run("SELECT acos(DATE '2026-10-16')")
```

```
[(None, nan, 1.5707963267948966, 1.5707963267948966)]
OSError: ACOS is undefined outside [-1,1]
OSError: ACOS is undefined outside [-1,1]
OSError: ACOS is undefined outside [-1,1]
OSError: input value inf is out of range for numeric function
OSError: input value -inf is out of range for numeric function
OSError: Could not convert string 'abc' to DOUBLE
InternalError: Binder Error: No function matches the given name and argument types 'acos(DATE)'. You might need to add explicit type casts.
```

## Examples

The angle between each pair of vectors, in radians and in degrees. A similarity of 1 means the vectors point the same way (0 degrees), 0 means they're at right angles, and -1 means they point in opposite directions:

```
run(f"SELECT id, similarity, acos(similarity), degrees(acos(similarity)) FROM {TABLE} ORDER BY id LIMIT 10")
```

```
[(1, 0.5, 1.0471975511965979, 60.00000000000001), (2, -0.5, 2.0943951023931957, 120.00000000000001), (3, None, None, None), (4, 0.0, 1.5707963267948966, 90.0), (5, 1.0, 0.0, 0.0), (6, -1.0, 3.141592653589793, 180.0)]
```

Keep only the pairs less than 1 radian apart:

```
run(f"SELECT id, similarity FROM {TABLE} WHERE acos(similarity) < 1.0 ORDER BY id LIMIT 10")
```

```
[(5, 1.0)]
```

## How it runs in VAST

The Query Engine runs on the cluster's CNodes, and a query on a table starts with a scan in VAST DataBase (`VAST_SCAN`). VAST DataBase can evaluate a comparison of a column with a constant, but not a function of a column, so `acos()` is always computed by the Query Engine:

- **In the `SELECT` list,** the scan reads the column and the Query Engine computes `acos()` for each row it returns.
- **In `WHERE`,** the scan sends the column for every row, and a `FILTER` step in the Query Engine computes `acos()` and keeps the rows that match. No rows are skipped in storage.

A value outside -1 to 1 in any row the Query Engine computes `acos()` for cancels the whole query.

The plan of the `WHERE` example shows the `FILTER` step above the scan:

```
plan(f"SELECT id FROM {TABLE} WHERE acos(similarity) < 1.0")
```

```
┌───────────────────────────┐
│         PROJECTION        │
│    ────────────────────   │
│             #1            │
│                           │
└─────────────┬─────────────┘
┌─────────────┴─────────────┐
│           FILTER          │
│    ────────────────────   │
│  (acos(similarity) < 1.0) │
│                           │
└─────────────┬─────────────┘
┌─────────────┴─────────────┐
│         VAST_SCAN         │
│    ────────────────────   │
│    Function: VAST_SCAN    │
│                           │
│        Projections:       │
│         similarity        │
│             id            │
│                           │
└───────────────────────────┘
```

## Make it faster

- **Compare the raw column with the cosine of the angle.** `acos()` gets smaller as its argument gets larger, so `acos(similarity) &lt; 1.0` is the same condition as `similarity &gt; cos(1.0)`. A function on constants is computed once, before the scan, so the second form is a comparison of the column with a constant, which is pushed down to VAST DataBase. Its plan has no `FILTER` step:

```
plan(f"SELECT id FROM {TABLE} WHERE similarity > cos(1.0)")
run(f"SELECT id, similarity FROM {TABLE} WHERE similarity > cos(1.0) ORDER BY id LIMIT 10")
```

```
┌───────────────────────────┐
│         VAST_SCAN         │
│    ────────────────────   │
│    Function: VAST_SCAN    │
│      Projections: id      │
│                           │
└───────────────────────────┘
[(5, 1.0)]
```

- **Store the angle if you filter on it often.** Compute it when you write the rows and keep it in its own column: a condition on that column is pushed down, a condition on `acos()` isn't.

## Limitations and errors

- **The argument must be from -1 to 1.** Anything else fails with `ACOS is undefined outside [-1,1]`, and infinity fails with `input value inf is out of range for numeric function`. One such row cancels the query. A cosine similarity computed in floating point can come out a tiny amount above 1 or below -1; clamp it into range first with [greatest](/develop/docs/dev-database-reference-sql-functions-conditional-and-cast-greatest) and [least](/develop/docs/dev-database-reference-sql-functions-conditional-and-cast-least), or keep those rows out with a pushed-down range on the raw column:

```
run("SELECT acos(1.0000000001)")
run("SELECT acos(least(greatest(1.0000000001, -1.0), 1.0))")
run(f"SELECT id, acos(similarity) FROM {TABLE} WHERE similarity BETWEEN -1 AND 1 ORDER BY id LIMIT 10")
```

```
OSError: ACOS is undefined outside [-1,1]
[(0.0,)]
[(1, 1.0471975511965979), (2, 2.0943951023931957), (4, 1.5707963267948966), (5, 0.0), (6, 3.141592653589793)]
```

- **Numbers only.** A `BLOB`, a date, a boolean, a list or an interval fails with `No function matches the given name and argument types`; the error lists the one form `acos` has, `acos(DOUBLE) -&gt; DOUBLE`. A string literal is converted to `DOUBLE` if it holds a number, and otherwise fails with `Could not convert string &apos;abc&apos; to DOUBLE`.
- **One argument.** `acos(1, 2)` fails with `No function matches the given name and argument types &apos;acos(INTEGER_LITERAL, INTEGER_LITERAL)&apos;`.
- **Radians only.** There's no degrees variant; use `degrees(acos(x))`.

## Similar functions

| Function | Returns | Result for `0.5` |
| --- | --- | --- |
| `acos(x)` | the angle whose cosine is `x`, from 0 to pi | `1.0471975511965979` |
| [asin](/develop/docs/dev-database-reference-sql-functions-numeric-asin) | the angle whose sine is `x`, from -pi/2 to pi/2 | `0.5235987755982989` |
| [atan](/develop/docs/dev-database-reference-sql-functions-numeric-atan) | the angle whose tangent is `x`, from -pi/2 to pi/2 | `0.4636476090008061` |
| [cos](/develop/docs/dev-database-reference-sql-functions-numeric-cos) | the cosine of an angle: the inverse of `acos` | `cos(acos(0.5))` gives back about `0.5` |
| [degrees](/develop/docs/dev-database-reference-sql-functions-numeric-degrees) | radians converted to degrees | `degrees(acos(0.5))` is about `60` |
| [pi](/develop/docs/dev-database-reference-sql-functions-numeric-pi) | pi; `pi() = acos(-1)` is true |  |

```
run("SELECT acos(0.5), asin(0.5), atan(0.5), cos(acos(0.5)), degrees(acos(0.5))")
run("SELECT pi() = acos(-1)")
```

```
[(1.0471975511965979, 0.5235987755982989, 0.4636476090008061, 0.4999999999999999, 60.00000000000001)]
[(True,)]
```

Clean up:

```
run(f"DROP TABLE IF EXISTS {TABLE}")
run(f"DROP SCHEMA IF EXISTS {SCHEMA}")
conn.close()
```

```
[]
[]
```

## See also

- [Numeric and math functions](/develop/docs/dev-database-reference-sql-functions-numeric): all the numeric functions.
- [Where filters run](/develop/docs/dev-database-reference-sql-select#where-filters-run) and [pushdowns](/develop/docs/dev-database-reference-pushdowns).
- [EXPLAIN](/develop/docs/dev-database-reference-sql-explain): how to read a plan.