ASINH

Compute the inverse hyperbolic sine of a real number.

Category: mathReturns: DOUBLEDialect: Standard

Syntax

ASINH(expr)

Description

## Overview Returns the inverse hyperbolic sine of the input. Unlike ASIN, ASINH has no domain restriction: every real number is a valid input. It is the algebraic inverse of SINH across the entire real line and is equivalent to LN(x + SQRT(x*x + 1)). ASINH is popular in statistics and machine learning as a 'signed log' transform. Unlike LN, which is undefined for non-positive values, ASINH accepts the full real line while still producing a roughly logarithmic growth pattern for large magnitudes. This makes it useful for visualizing data with heavy tails in both directions (for example, profit-and-loss columns). ## Behavior - Domain: all finite DOUBLE values. - Range: all real numbers. - Returns NULL if the argument is NULL. - Is anti-symmetric: ASINH(-x) = -ASINH(x). - Grows like LN(2*|x|) for large |x|. - ASINH(SINH(x)) equals x for all real x. ## Numeric precision - Uses the IEEE 754 double-precision asinh routine, accurate to within one ULP. - Does not suffer the catastrophic cancellation that the naive closed form LN(x + SQRT(x*x + 1)) exhibits for large negative x; call ASINH directly rather than recomputing. - Near zero, ASINH(x) is numerically close to x. ## Compatibility - Conforms to the SQL standard definition of ASINH as a scalar DOUBLE function. - Matches typical mathematical library implementations.

Parameters

Name	Type	Description
`expr`		Specifies the real-valued input. Accepts any finite DOUBLE value; no domain restriction.

Examples

-- Basic literal
SELECT ASINH(0);
-- Result: 0.0

-- Inverse hyperbolic sine of 1
SELECT ASINH(1);
-- Result: 0.8813736198295702

-- Negative input
SELECT ASINH(-2.5);
-- Result: -1.6472311463710958

-- Round trip: ASINH(SINH(x)) = x
SELECT ASINH(SINH(4.0));
-- Result: 4.0

-- Closed form: ASINH(x) = LN(x + SQRT(x*x + 1))
SELECT ASINH(2), LN(2 + SQRT(5));
-- Result: 1.4436354751788103, 1.4436354751788103

-- Column use: signed log-like transform for features with both signs
SELECT customer_id, ASINH(net_balance) AS signed_log_balance
FROM finance.ledger.accounts;

Pitfalls

The naive formula LN(x + SQRT(x*x + 1)) loses precision for large negative x because x + SQRT(x*x + 1) is close to zero. Use ASINH directly, not the hand-coded closed form.
ASINH has no bounded range, so it cannot be used as a similarity score. Combine with TANH if you need [-1, 1] output.
Although ASINH is defined for all reals, converting a very large BIGINT through DOUBLE loses precision before the call. Cast to DECIMAL or divide first if precision matters.
ASINH(NaN) returns NaN and propagates silently through aggregates. Guard with a finiteness filter if NaN can appear.