Add sinh(tanh(x)) and other similar functions.

[emsr]

Work on double exp integrals and recent fast math work in gcc indicates that sinh(tanh(x)), cosh(tanh(x)), sinh(atanh(x)), and cosh(atanh(x)). I'm not interested in optimizing but in accuracy.

[CaptainSifff]

Hi,
do you have a pointer on what is needed for the hyperbolic functions?
Are you interested in reliable series for e.g. sinh(tanh(x)) ?

[emsr]

On 6/21/19 9:01 AM, CaptainSifff wrote: Hi, do you have a pointer on what is needed for the hyperbolic functions? Are you interested in reliable series for e.g. sinh(tanh(x)) ? ??? You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub <#165>, or mute the thread <https://github.com/notifications/unsubscribe-auth/AAOYYX6J7PUPOLUZ77B3CNLP3TGKLANCNFSM4GHV4E4A>.

Yes, I would like a good series for sinh(tanh(x)). Maybe for sinh(a*tanh(x)) since these are used in double exponential quadrature rules (exp a = pi/2). Thanks!

[CaptainSifff]

And directly evaluating that function is not fast enough? or lacks precision for certain values since some overflow/underflow occurs?

[CaptainSifff]

Hi Ed,
I've got something to play for you...
We consider the functions
C(x) = Cosh(a Tanh(x))
and
S(x) = Sinh(a Tanh(x))
and consider the behaviour for large x.
Denoting real infinity as "inf"
We find
C(inf) = Cosh(a) and S(inf) = Sinh(a)
I think I can show that the large-x behaviour is given by
C(x) ~ C(inf) * F(a,x)
and
S(x) ~ S(inf) * F(a,x)
with the same function F(a,x). Now we have
F(a,x) = y(a,x) * K_1(y(a,x))
with y(a,x) = 2 * a * exp(-x)
and K_1 denoting the modified Bessel function of the second kind (http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html)
Since y(a,x) becomes exponentially small for large x we can use any Series around x=0 for K_1.
To give an approximation to first order that only relies on elementary functions we have:
F(a,x) ~ 1 + a^2 * exp(-2x) * (-1+2 \gamma + 2* ln(a) -x) + ...
Here \gamma is the Euler-Mascheroni constant(https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant)
I get good agreement in Mathematica between C(x) evaluated in Mathematica for a=1 and x>20.
If you have questions or my notation is unclear, just ask!

[CaptainSifff]

Forget what I have previously written...
I found a "minor" bug in my calculation that makes everything far simpler....
To begin again:
We consider the functions
C(x) = Cosh(a Tanh(x))
and
S(x) = Sinh(a Tanh(x))
and consider the behaviour for large x.
Denoting real infinity as "inf"
We find
C(inf) = Cosh(a) and S(inf) = Sinh(a)
I think I can show that the large-x behaviour is given by
C(x) ~ C(inf) * F(a,x)
and
S(x) ~ S(inf) * F(a,x)
with the same function F(a,x). Now we have
F(a,x) = Cosh(2 * a * exp(-2 * x))
Since exp(-2 * x) becomes exponentially small for large x we can use any Series around x=0 for Cosh(x).
To give an approximation to second order in exp(-2x) that only relies on elementary functions we have:
F(a,x) ~ 1 + 2 * a^2 * exp(-4x)
(more terms could be obtained from the Taylor series of cosh(x) but they are probably not required...)
I get good agreement in Mathematica of C(x) evaluated in Mathematica for a=1 and x>20.
If you have questions or my notation is unclear, just ask!

[emsr]

Very nice. I'm going to play with this myself.
I'm not sure what I'd call these ;-) but we'll think of something.

[CaptainSifff]

the most unimaginative I could come up with is:
C_a(x) := cosh(a * tanh(x))
S_a(x) := sinh( a * tanh(x))