Hyperbolic perplexity

[mecej4]

The intrinsic function ACOSH with complex argument was added to Fortran in Fortran 2008. Since this function is multi-valued for complex arguments, a convention is needed for what the return value should be. The F2008 (13.7.5) and F2018 (16.9.5) standards state:

5 Result Value. The result has a value equal to a processor-dependent approximation to the inverse hyperbolic cosine function of X. If the result is complex the imaginary part is expressed in radians and lies in the range 0 ≤ AIMAG (ACOSH (X)) ≤ π

In contrast, the Intel Fortran Development Reference Guide says:

If the result is complex, the real part is non-negative, and the imaginary part is expressed in radians and lies in the range -pi <= AIMAG (ACOSH (x)) <= pi.

The following program outputs a result with the imaginary part negative:

program xacosh
   implicit none
   complex :: z = (-0.01, -0.01)
   print 10, 'z', z, 'acosh(z)', acosh(z)
10 format(1x,A,T10,'( ',ES11.3,' , ',ES11.3,' )')
end

The result:

S:\LANG\ftn95>acosh
 z       (  -1.000E-02 ,  -1.000E-02 )
 acosh(z)(   1.000E-02 ,  -1.581E+00 )

Please clarify if the deviations ( w.r.t. (a) sign of the real part and (b) range of the imaginary part) from the standard are deliberately intended, and any reasons for the choices, if available. Thank you in advance.

[JohnNichols]

This is really fascinating,  the https://www.geeksforgeeks.org/acosh-function-for-complex-number-in-c/  site seems to agree with your Intel observations.  They provide a couple of quick samples.  

Running their sample but with the -0.000 I get -pi. 

-1.58 is just 335 degrees in the -PI to PI system.  

[Ron_Green]

This is an interesting one.  gfortran 12.0.1 is in agreement in values with ifort/ifx.   I will ask advice from someone more familiar with this function, and the F08 Standard.  I don't have the NAG compiler myself but I'll ask someone to try it with NAG.  

[Steve_Lionel · Accepted solution]

D:\Projects>nagfor -o t.exe t.f90
NAG Fortran Compiler Release 7.1(Hanzomon) Build 7114
[NAG Fortran Compiler normal termination]

D:\Projects>t.exe
 z       (  -1.000E-02 ,  -1.000E-02 )
 acosh(z)(   1.000E-02 ,  -1.581E+00 )

It's an error in the F2018 standard, corrected by interpretation request F18/0001. The corrected text says, "If the result is complex the real part is nonnegative, and the imaginary part is expressed in radians and lies in the range −π ≤ AIMAG (ACOSH (X)) ≤ π"

[mecej4]

The portion "the imaginary part is expressed in radians" strikes me as odd, since the imaginary part (with the i removed) does not represent an angle -- it is just a real number.

If one wishes to think about the Argand Diagram, the real part is the x-coordinate and the imaginary part is the y-coordinate.  It is only after the point (x, y), where z = x + i y, is plotted and a line drawn to it from the origin that we may speak of angles.