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NAG C Library Manual

# NAG Library Function Documentnag_sinh (s10abc)

## 1  Purpose

nag_sinh (s10abc) returns the value of the hyperbolic sine, $\mathrm{sinh}x$.

## 2  Specification

 #include #include
 double nag_sinh (double x, NagError *fail)

## 3  Description

nag_sinh (s10abc) calculates an approximate value for the hyperbolic sine of its argument, $\mathrm{sinh}x$.
For $\left|x\right|\le 1$ the function is based on a Chebyshev expansion.
For $1<\left|x\right|\le {E}_{1}$, (where ${E}_{1}$ is a machine-dependent constant), $\mathrm{sinh}x=\frac{1}{2}\left({e}^{x}-{e}^{-x}\right)$.
For $\left|x\right|>{E}_{1}$, the function fails owing to the danger of setting overflow in calculating ${e}^{x}$. The result returned for such calls is $\mathrm{sinh}\left(\mathrm{sign}x{E}_{1}\right)$, i.e., it returns the result for the nearest valid argument.

## 4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5  Arguments

1:     xdoubleInput
On entry: the argument $x$ of the function.
2:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_REAL_ARG_GT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{x}}\right|\le 〈\mathit{\text{value}}〉$.
The function has been called with an argument too large in absolute magnitude. There is a danger of setting overflow. The result is the value of sinh at the closest argument for which a valid call could be made. (See Section 3 and the Users' Note for your implementation ).

## 7  Accuracy

If $\delta$ and $\epsilon$ are the relative errors in the argument and result, respectively, then in principle
 $ε ≃ x coth⁡x δ .$
That is, the relative error in the argument, $x$, is amplified by a factor, approximately $x\mathrm{coth}x$. The equality should hold if $\delta$ is greater than the machine precision ($\delta$ is a result of data errors etc.), but if $\delta$ is simply a result of round-off in the machine representation of $x$, then it is possible that an extra figure may be lost in internal calculation round-off.
It should be noted that for $\left|x\right|\ge 2$
 $ε ∼ x δ = Δ$
where $\Delta$ is the absolute error in the argument.

None.

## 9  Example

The following program reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

### 9.1  Program Text

Program Text (s10abce.c)

### 9.2  Program Data

Program Data (s10abce.d)

### 9.3  Program Results

Program Results (s10abce.r)