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

> ## 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.

# atanh

*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) › Hyperbolic functions.* Returns the inverse hyperbolic tangent of `x`: the value whose hyperbolic tangent is `x`. `x` must be from -1 to 1.

## Syntax

```
atanh(x)
```

| Argument | Type | Description |
| --- | --- | --- |
| `x` | `DOUBLE` (any number) | A value from -1 to 1. Integers, `DECIMAL` and `FLOAT` values are converted to `DOUBLE`. |

**Returns:** `DOUBLE`.

## 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_atanh`. 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 seven correlation coefficients, each from -1 to 1. 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_atanh"'

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}.correlations"
run(f"CREATE SCHEMA IF NOT EXISTS {SCHEMA}")
run(f"CREATE TABLE IF NOT EXISTS {TABLE} (id BIGINT, r DOUBLE)")
run(f"INSERT INTO {TABLE} VALUES "
    "(1, 0.5), (2, -0.5), (3, NULL), (4, 0.0), (5, 1.0), (6, -1.0), (7, 0.9)")
```

```
[]
[]
[(7,)]
```

## What it does

`atanh(x)` is the inverse of [tanh](/develop/docs/dev-database-reference-sql-functions-numeric-tanh): for `x` between -1 and 1 it returns the value `y` with `tanh(y) = x`. It's an odd function, so `atanh(-0.5)` is `-atanh(0.5)`, `-0.5493061443340548`. Results are rounded to the nearest `DOUBLE`, so a round trip can be off in the last digit: `atanh(tanh(1))` is `0.9999999999999999`.

A common use is the Fisher z-transform of a correlation coefficient `r`: `atanh(r)` turns correlations into values you can average, and `tanh()` turns the average back into a correlation.

## NULLs and edge cases

| Input | Result |
| --- | --- |
| `NULL` | `NULL` |
| `0` | `0.0` |
| `0.5`, `-0.5` | `0.5493061443340548`, `-0.5493061443340548` |
| `-0.0::DOUBLE` | `-0.0` |
| `5e-324::DOUBLE` (the smallest `DOUBLE` above 0) | `5e-324` |
| `1` | `inf` |
| `-1` | `inf` (positive infinity, not `-inf`) |
| `&apos;nan&apos;::DOUBLE` | `nan` |
| `2`, `-2`, `1.0000000001`, `&apos;inf&apos;::DOUBLE` or `&apos;-inf&apos;::DOUBLE` | Error: `ATANH is undefined outside [-1,1]` |
| `&apos;12&apos;` (a string) | Converted to `DOUBLE`, then the same error |
| `&apos;abc&apos;` | 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;atanh(DATE)&apos;` |
| An `INTEGER`, `FLOAT` or `DECIMAL` value | Converted; the result is `DOUBLE` |

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

```
run("SELECT atanh(NULL), atanh(0), atanh(0.5), atanh(-0.5), atanh(-0.0::DOUBLE), atanh(5e-324::DOUBLE)")
run("SELECT atanh(1), atanh(-1), atanh('nan'::DOUBLE)")
run("SELECT typeof(atanh(1::INTEGER)), typeof(atanh(0.5::FLOAT)), typeof(atanh(0.5))")
run("SELECT atanh(2)")
run("SELECT atanh(1.0000000001)")
run("SELECT atanh('inf'::DOUBLE)")
run("SELECT atanh('12')")
run("SELECT atanh('abc')")
run("SELECT atanh(DATE '2026-10-16')")
```

```
[(None, 0.0, 0.5493061443340548, -0.5493061443340548, -0.0, 5e-324)]
[(inf, inf, nan)]
[('DOUBLE', 'DOUBLE', 'DOUBLE')]
OSError: ATANH is undefined outside [-1,1]
OSError: ATANH is undefined outside [-1,1]
OSError: ATANH is undefined outside [-1,1]
OSError: ATANH is undefined outside [-1,1]
OSError: Could not convert string 'abc' to DOUBLE
InternalError: Binder Error: No function matches the given name and argument types 'atanh(DATE)'. You might need to add explicit type casts.
```

## Examples

The transform of each correlation. Row 5 (`1.0`) and row 6 (`-1.0`) both give `inf`:

```
run(f"SELECT id, r, atanh(r) FROM {TABLE} ORDER BY id LIMIT 10")
```

```
[(1, 0.5, 0.5493061443340548), (2, -0.5, -0.5493061443340548), (3, None, None), (4, 0.0, 0.0), (5, 1.0, inf), (6, -1.0, inf), (7, 0.9, 1.4722194895832204)]
```

The average of the correlations strictly between -1 and 1, by the Fisher z-transform: transform, average, transform back:

```
run(f"SELECT tanh(avg(atanh(r))) FROM {TABLE} WHERE r > -1 AND r < 1")
```

```
[(0.3522891589092123,)]
```

## 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 `atanh()` of a column is always computed by the Query Engine:

- **In the `SELECT` list,** the scan reads the column and the Query Engine computes `atanh()` for each row it returns.
- **In `WHERE`,** the scan sends the column for every row, and a `FILTER` step in the Query Engine computes `atanh()` and keeps the rows that match.

The plan of a `WHERE` condition on `atanh(r)` shows the `FILTER` step above the scan:

```
plan(f"SELECT id FROM {TABLE} WHERE atanh(r) > 0.5")
```

```
┌───────────────────────────┐
│         PROJECTION        │
│    ────────────────────   │
│             #1            │
│                           │
└─────────────┬─────────────┘
┌─────────────┴─────────────┐
│           FILTER          │
│    ────────────────────   │
│      (atanh(r) > 0.5)     │
│                           │
└─────────────┬─────────────┘
┌─────────────┴─────────────┐
│         VAST_SCAN         │
│    ────────────────────   │
│    Function: VAST_SCAN    │
│                           │
│        Projections:       │
│             r             │
│             id            │
│                           │
└───────────────────────────┘
```

## Make it faster

- **Compare the raw column instead.** `atanh()` only grows, so `atanh(r) &gt; 0.5` keeps the same rows as `r &gt; tanh(0.5)`. A function of constants is computed once, before the scan, so that condition can be pushed down to VAST DataBase:

```
plan(f"SELECT id FROM {TABLE} WHERE r > tanh(0.5)")
run(f"SELECT id FROM {TABLE} WHERE r > tanh(0.5) ORDER BY id LIMIT 10")
```

```
┌───────────────────────────┐
│         VAST_SCAN         │
│    ────────────────────   │
│    Function: VAST_SCAN    │
│      Projections: id      │
│                           │
└───────────────────────────┘
[(1,), (5,), (7,)]
```

## Limitations and errors

- **One value outside -1 to 1 cancels the whole query.** The error is `ATANH is undefined outside [-1,1]`, and it happens on any row, here row 7, where `r * 2` is `1.8`. Return NULL for those rows with `CASE`:

```
run(f"SELECT id, atanh(r * 2) FROM {TABLE} ORDER BY id LIMIT 10")
run(f"SELECT id, CASE WHEN abs(r * 2) <= 1 THEN atanh(r * 2) END FROM {TABLE} ORDER BY id LIMIT 10")
```

```
OSError: ATANH is undefined outside [-1,1]
[(1, inf), (2, inf), (3, None), (4, 0.0), (5, None), (6, None), (7, None)]
```

- **`atanh(-1)` is `inf`, not `-inf`.** If the sign matters, test for -1 yourself:

```
run(f"SELECT id, CASE WHEN r = -1 THEN '-inf'::DOUBLE ELSE atanh(r) END FROM {TABLE} ORDER BY id LIMIT 10")
```

```
[(1, 0.5493061443340548), (2, -0.5493061443340548), (3, None), (4, 0.0), (5, inf), (6, -inf), (7, 1.4722194895832204)]
```

- **Infinity isn't accepted.** `atanh(&apos;inf&apos;::DOUBLE)` and `atanh(&apos;-inf&apos;::DOUBLE)` fail with the same error; NaN gives NaN.
- **One argument, numbers only.** A second argument, a date, a boolean, a `BLOB`, a list or an interval fails with `No function matches the given name and argument types`. A string is converted to `DOUBLE` first, and fails with `Could not convert string ... to DOUBLE` if it isn't a number.

## Similar functions

| Function | Computes | Relation to `atanh` |
| --- | --- | --- |
| [tanh](/develop/docs/dev-database-reference-sql-functions-numeric-tanh) | hyperbolic tangent | its inverse: `tanh(atanh(x))` gives back about `x` |
| [asinh](/develop/docs/dev-database-reference-sql-functions-numeric-asinh) | inverse hyperbolic sine | takes any number |
| [acosh](/develop/docs/dev-database-reference-sql-functions-numeric-acosh) | inverse hyperbolic cosine | takes values from 1 up |
| [ln](/develop/docs/dev-database-reference-sql-functions-numeric-ln) | natural logarithm | `atanh(x)` is, up to rounding, `0.5 * ln((1 + x) / (1 - x))` |

```
run("SELECT atanh(0.5), tanh(atanh(0.5)), 0.5 * ln((1 + 0.5) / (1 - 0.5)), asinh(0.5), acosh(1.5)")
```

```
[(0.5493061443340548, 0.49999999999999994, 0.5493061443340549, 0.48121182505960347, 0.9624236501192069)]
```

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#hyperbolic-functions): all the hyperbolic 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.