ACOSH

Compute the inverse hyperbolic cosine of a real number at least 1.

Category: mathReturns: DOUBLEDialect: Standard

Syntax

ACOSH(expr)

Description

## Overview Returns the inverse hyperbolic cosine of the input. ACOSH is defined only for inputs greater than or equal to 1, and returns a non-negative angle (technically a rapidity, not a geometric angle). It is the algebraic inverse of COSH on the non-negative branch and is equivalent to LN(x + SQRT(x*x - 1)). ACOSH appears most often in special relativity (recovering rapidity from the Lorentz gamma factor), catenary cable analysis, and numerical inversions of hyperbolic functions. The domain restriction (x >= 1) mirrors COSH's range. ## Behavior - Domain: [1, infinity). Values strictly less than 1 return NULL. - Range: [0, infinity). - Returns NULL if the argument is NULL. - Is strictly increasing: ACOSH preserves order on its domain. - ACOSH(COSH(x)) equals |x|, not x. The inverse returns only the non-negative branch. ## Numeric precision - Uses the IEEE 754 double-precision acosh routine, accurate to within one ULP. - The naive closed form LN(x + SQRT(x*x - 1)) loses precision for x just above 1 because of cancellation in the radical. Call ACOSH directly rather than expanding. - Near x = 1, the derivative is unbounded, so ACOSH is sensitive to input noise in that regime. ## Compatibility - Conforms to the SQL standard definition of ACOSH as a scalar DOUBLE function with domain x >= 1. - Matches typical mathematical library implementations.

Parameters

Name	Type	Description
`expr`		Specifies the real-valued input. Must be greater than or equal to 1. Values less than 1 yield NULL.

Examples

-- Basic literal: ACOSH of 1 is 0
SELECT ACOSH(1);
-- Result: 0.0

-- Inverse hyperbolic cosine of 2
SELECT ACOSH(2);
-- Result: 1.3169578969248166

-- Round trip: ACOSH(COSH(x)) = |x|
SELECT ACOSH(COSH(3.0));
-- Result: 3.0

-- Out-of-domain input returns NULL
SELECT ACOSH(0.5);
-- Result: NULL

-- Closed form: ACOSH(x) = LN(x + SQRT(x*x - 1))
SELECT ACOSH(3), LN(3 + SQRT(8));
-- Result: 1.7627471740390859, 1.7627471740390859

-- Column use: recover rapidity from relativistic gamma factor
SELECT track_id, ACOSH(gamma) AS rapidity
FROM physics.simulation.tracks
WHERE gamma >= 1;

Pitfalls

Inputs less than 1 silently return NULL. Validate upstream or clamp with GREATEST(x, 1.0) when rounding can push a legitimately-unit value slightly below 1.
ACOSH(COSH(x)) equals |x|, not x. The function cannot recover the sign of the original rapidity.
The naive closed form LN(x + SQRT(x*x - 1)) is numerically unstable near x = 1. Use ACOSH directly to retain precision.
Near x = 1 the derivative diverges, so tiny noise in the input produces large noise in the output. Smooth or filter upstream when working with near-unit values.