From: Kbwms <Kbwms AT aol DOT com>
Message-ID: <643f6e90.3558c708@aol.com>
Date: Tue, 12 May 1998 18:02:47 EDT
To: j DOT bischoff AT airbus DOT dasa DOT de
Cc: djgpp AT delorie DOT com
Mime-Version: 1.0
Subject: Re: Code to Fix sinh() in libm.a
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Subj:	 Re: Code to Fix sinh() in libm.a
To:	j DOT bischoff AT airbus DOT dasa DOT de (Jens Bischoff):

Dear Jens Bischoff,

On 05-11-98 at 03:42:34 EST you wrote:
> CC:	djgpp AT delorie DOT com
>
> Hi!
>
> Thank you for your work!
> However, I believe that the coefficients are slightly inexact.
> The correct values should be:
>
> long double C[] = { 1.L,
>                     1/6.L,
>                     1/120.L,
>                     1/5040.L,
>                     1/362880.L,
>                     1/39916800.L,
>                     1/6227020800.L};
>
> if they are derived from the series for sinhx(x):
>
> sinh(x) = 1 + x^3/3! + x^5/5! + x^7/7! + ...
>
> Besides, this approximation (that uses 7 coefficients) is only
> accurate with 64bit (double) arithmetic.
> With long doubles a 8th coefficient should be added:
>
> C[7] = 1/1307674368000.L
>
/* ------------------------------------------ */
/* sinh - Returns hyperbolic sine of argument */
/* ------------------------------------------ */
    /* --------------------------------------- */
    /* Approxination no. 1906 from Hart, et al */
    /* --------------------------------------- */
    long double C[] = {
	    .10000000000000000000428964e1,
	    .16666666666666666026158977e0,
	    .83333333333336099945076e-2,
	    .19841269840742957068490e-3,
	    .27557319739041248550000e-5,
	    .25051838782995770000000e-7,
	    .16130886652601000000000e-9
    };


The coefficients were taken from Hart, et al, *Computer Approximations*,
Wiley and Sons, New York, 1968, pages 104 & 213, approximation # 1906.
For arguments in (0, .5), it is accurate to 20.84 decimal digits which
is about 69 bits, certainly enough for a double precision function.  The
coefficients are stated in long double and the arithmetic is carried out
in long double to ensure that the precision is maintained.

I hasten to remind you that the subject at hand is an approximation
to double precision function sinh().  For an approximation to sinhl(),
formula # 1907, with 8 coefficients, yields an accuracy of about 81 bits.

Thanks for the note.

Regards,

K.B. Williams