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## Issue COMPLEX-ATANH-BOGUS-FORMULA Writeup

    Date: Wed, 20 Sep 89 11:24:48 EDT
    From: gls@Think.COM (Guy Steele)
    Message-Id: <8909201524.AA21857@verdi.think.com>
    To: x3j13@sail.stanford.edu
    Cc: gls@Think.COM, cl-cleanup@sail.stanford.edu
    Subject: Issue COMPLEX-ATANH-BOGUS-FORMULA

    I hate to bring *anything* up at this late date, but while working over the
    numbers chapter second edition I have been going over this branch cut stuff
    one more time, with even greater care, and have discovered that the formula
    for ATANH on page 209 and again on page 213 is completely bogus.  What that
    computes is not anything like a hyperbolic arc tangent.  It would seem that
    I must have mistranscribed the APL formula in Penfield's article.

    CLtL has:   arctanh z = log ((1+z) sqrt(1 - (1 / z^2)))

    Should be:  arctanh z = log ((1+z) sqrt(1 / (1 - z^2)))

    Note that they differ in the transposition of two operators.  (Boy, am I
    embarrassed.)

    Clearly this must be corrected.  In the meantime I have found a more
    definitive treatment of complex branch cuts by W. Kahan, and I propose to
    follow his recommendations.  This involves correcting the formula for
    ATANH, and adopting new formulas for ACOS and ACOSH that are equivalent to
    the ones we have now but more perspicuous.

    I would appreciate knowing very soon on an informal basis whether anyone
    objects to this change, so that I can include some discussion of it in the
    second edition.  (Of course I'm not asking for a vote until we have an
    official meeting.)

    --Guy
    ----------------------------------------------------------------
    Status:	       New proposal
    Forum:         Cleanup
    Issue:         COMPLEX-ATANH-BOGUS-FORMULA
    References:    CLtL pp. 209, 212, 213
    	       Penfield, P. "Principal Values and Branch Cuts in
    		Complex APL", Proc. APL 81 Conference Proceedings,
    		Association for Computing Machinery, 1981
    	       Kahan, W. "Branch Cuts for Complex Elementary
    		Functions, or Much Ado About Nothing's Sign Bit"
    		in Iserles and Powell (eds.) "The State of the Art
    		in Numerical Analysis", pp. 165-211, Clarendon
    		Press, 1987
    Related issues: COMPLEX-ATAN-BRANCH-CUT, IEEE-ATAN-BRANCH-CUT
    Category:      CHANGE

    Edit history:  Version 1, 20-SEP-89, Steele

    Problem description:

    The formula that defines ATANH in CLtL is incorrect, apparently
    because of a mistranscription of a formula from Penfield's article.

    CLtL has:   arctanh z = log ((1+z) sqrt(1 - (1 / z^2)))

    Should be:  arctanh z = log ((1+z) sqrt(1 / (1 - z^2)))

    However, given the change to ATAN in issue COMPLEX-ATAN-BRANCH-CUT,
    it seems simpler to follow Kahan's recommendation and define

    	arctanh z = (log(1+z) - log(1-z))/2

    thereby preserving the identity  i arctan z = arctanh iz .

    Kahan also notes that Penfield's formula for arccosh (CLtL p. 213)

    	arccosh z = log(z + (z + 1) sqrt((z-1)/(z+1)))

    has a gratuitous removable singularity at z=-1 and recommends

    	arccosh z = 2 log(sqrt((z+1)/2) + sqrt((z-1)/2))

    which has the same values and is also well-defined at z=-1.

    Finally, Kahan recommends a different defining formula for acos
    that is more similar to that of acosh (but less similar to that
    of asin).

    Proposal (COMPLEX-ATANH-BRANCH-CUT:TWEAK-MORE):

    (1) Replace the erroneous formula

    	arctanh z = log ((1+z) sqrt(1 - (1 / z^2)))
    with
    	arctanh z = (log(1+z) - log(1-z))/2

    (2) Note that  i arctan z = arctanh iz .

    (3) Replace the gratuitously singular formula

    	arccosh z = log(z + (z + 1) sqrt((z-1)/(z+1)))
    with
    	arccosh z = 2 log(sqrt((z+1)/2) + sqrt((z-1)/2))

    (4) Adopt the formula (already in CLtL)

    	arccos z = (pi / 2) - arcsin z

    as the official definition of arccos, and also note that the
    formulas

    	arccos z = -i log(z + i sqrt(1 - z^2))

    (already in CLtL) and

    	arccos z = 2 log(sqrt((1+z)/2) + i sqrt((1-z)/2)) / i

    (recommended by Kahan) are equivalent.

    Rationale:

    Compatibility with what seems to be becoming standard practice.

    Current practice:

    Implementations I have checked have a correct implementation
    of ATANH rather than slavishly following the bogus CLtL formula.

    Cost to Implementors:

    ATANH must be rewritten.  It is not a very difficult fix.

    Possibly ACOSH must be rewritten.  It is not a very difficult fix.

    Cost to Users:

    The compatibility note on p. 210 of CLtL gave users fair warning that
    a change of this kind might be adopted.

    Cost of non-adoption:

    Possible incorrect implementations of ATANH.

    Incompatibility with HP calculators.

    Benefits:

    Numerical analysts may find the new definition easier to use.

    Esthetics:

    A toss-up, except to those who care.

    Discussion:

    Kahan's article not only discussed formulas but also gives specific
    implementation techniques for use with IEEE 754 arithmetic.

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