.

                        EXPLANATION FILE OF PROGRAM INVHYPER
                        ====================================


  Inverse Hyperbolic Trigonometrie Functions
  ------------------------------------------

    This section conclu des the chapter on COROIC approximations. The following 
  discussion is actually not related to the CORDIC method itself, but rather
  merely completes the analysis of the hyperbolic trigonometrie functions.

   
    Recall from the previous section (see file Hyper.txt) the simple functional
  relations for the inverses  of the hyperbolic functions:

                         -1                  2
                      sinh  x = ln (x + sqrt(x  + 1)

                          -1                  2
                      cosh  X = ln (x + sqrt(x  - 1)                   (4.7.8)

                          -1             1 + x
                      tanh  x = (1/2) ln ----- 
	 	                         1 - x

  At first glance, it appears that the evaluation is straightforward. The modi-
  fied CORDIC natural-Iogarithm function of section 4.6 can be used in conjunc-
  tion with a good square-root function. (Even the square-root iteration algo-
  rithm discussed earlier can be included if no internaI square-root function is
  available.) As with the earlier discussion, however, there is a potential
  round-off error problem. For example, in the sinh-1 x case, large negative
  values of x lead to taking the difference between two large and nearly equal
  numbers. However, because sinh -x = -sinh x, the evaluation can be adjusted
  to always involve addition: 

                         -1                          2
                     sinh  x = sign x ln |x| + sqrt(x  + 1)            (4.8.1)
          -1            
  The cosh  x evaluation has potential round-off difficulty near x = +/- 1. The 
  function can be rewritten as

                         -1
                     cosh  x = ln (x + sqrt(x + 1).sqrt(x - 1) 
   
  The round-off error due to the x ± 1 factors is unavoidable.

            -1                                         -1
    The tanh  x evaluation has the same problem as cosh  x, but with an extra 
  twist. If x is so near unity that the x - 1 factor evaluates to zero, a
  divide-by-zero error will result. This must be guarded against. 

    In program INVHYPER, the accuracy of the calculated results is mainly limi-
  ted by the round-off error that occurs in the natural-Iogarithm subroutine.
   
    There are two situations to be wary of while using these subroutines. First,
  the argument transferred to the cosh-1 x subroutine is restricted to the range
  1 <= x < inf. If x < 1, an incorrect value will be returned. Second, the argu-
  ment in the tanh-1 x subroutine is tested to avoid a divide-by-zero error.
  Thus, aIl x values sufficiently close to unity will result in the same large
  value returned. Also, the valid argument range for  tanh-1 x is 0 <= x < 1.
  If x is outside this range, an incorrect value will be returnned.

From [BIBLI 01].
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End of file Invhyper.txt