NAG Library Routine Document

s10abf (sinh)


    1  Purpose
    7  Accuracy


s10abf returns the value of the hyperbolic sine, sinhx, via the function name.


Fortran Interface
Function s10abf ( x, ifail)
Real (Kind=nag_wp):: s10abf
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: x
C Header Interface
#include nagmk26.h
double  s10abf_ (const double *x, Integer *ifail)


s10abf calculates an approximate value for the hyperbolic sine of its argument, sinhx.
For x1 it uses the Chebyshev expansion
where t=2x2-1.
For 1<xE1,  sinhx=12ex-e-x
where E1 is a machine-dependent constant, details of which are given in the Users' Note for your implementation.
For x>E1, the routine fails owing to the danger of setting overflow in calculating ex. The result returned for such calls is sinhsignxE1, i.e., it returns the result for the nearest valid argument.


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


1:     x – Real (Kind=nag_wp)Input
On entry: the argument x of the function.
2:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
The routine has been called with an argument too large in absolute magnitude. There is a danger of setting overflow. The result is the value of sinhx at the closest argument for which a valid call could be made. (See Section 3 and the Users' Note for your implementation.)
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


If δ and ε are the relative errors in the argument and result, respectively, then in principle
ε xcothx×δ.  
That is the relative error in the argument, x, is amplified by a factor, approximately xcothx. The equality should hold if δ is greater than the machine precision (δ is a result of data errors etc.) but, if δ 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.
The behaviour of the error amplification factor can be seen in the following graph:
Figure 1
Figure 1
It should be noted that for x2 
where Δ is the absolute error in the argument.

Parallelism and Performance

s10abf is not threaded in any implementation.

Further Comments



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.

Program Text

Program Text (s10abfe.f90)

Program Data

Program Data (s10abfe.d)

Program Results

Program Results (s10abfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017