[_CDMiwok_01_]<bonsai2.0>cedar>release>FloatingPoint>RealFns.mesa!1

RealFns.mesa

Stewart, August 27, 1982 11:21 am

E. McCreight, November 16, 1984 11:47:34 am PST (RealFnsExtras)

Russ Atkinson (RRA) November 14, 1986 7:06:17 pm PST

Doug Wyatt, January 15, 1987 1:14:03 pm PST

Willie-s, August 6, 1991 1:05 pm PDT

RealFns: CEDAR DEFINITIONS

= BEGIN

Exponent and logarithm functions

Exp: PROC [REAL] RETURNS [REAL];

For an input argument n, returns e^n (e=2.718...).

Computed by continued fractions.

Ln: PROC [REAL] RETURNS [REAL];

Computes the natural logarithm (base e) of the input argument.

Log: PROC [base, arg: REAL] RETURNS [REAL];

Computes logarithm to the base base of arg.

Computed by Ln(arg)/Ln(base).

Power: PROC [base, exponent: REAL] RETURNS [REAL];

Calculates base to the exponent power by e(exponent*Ln(base)).

Root: PROC [index, arg: REAL] RETURNS [REAL];

Calculates the index root of arg by e(Ln(arg)/index).

SqRt: PROC [REAL] RETURNS [REAL];

Calculates the square root of the input value by Newton's iteration.

Trigonometric functions

Sin: PROC [radians: REAL] RETURNS [sin: REAL];

SinDeg: PROC [degrees: REAL] RETURNS [sin: REAL];

Cos: PROC [radians: REAL] RETURNS [cos: REAL];

CosDeg: PROC [degrees: REAL] RETURNS [cos: REAL];

Tan: PROC [radians: REAL] RETURNS [tan: REAL];

TanDeg: PROC [degrees: REAL] RETURNS [tan: REAL];

Computes the trigonometric functions by polynomial;

good to 7.33 decimal places.

ArcTan: PROC [y, x: REAL] RETURNS [radians: REAL];

ArcTanDeg: PROC [y, x: REAL] RETURNS [degrees: REAL];

Good to 8.7 decimal places.

CoTan: PROC [radians: REAL] RETURNS [cotan: REAL];

CoTanDeg: PROC [degrees: REAL] RETURNS [cotan: REAL];

ArcSin: PROC [x: REAL] RETURNS [ REAL];

ArcSinDeg: PUBLIC PROC [x: REAL] RETURNS [degrees: REAL];

ArcCos: PROC[x: REAL] RETURNS [REAL];

ArcCosDeg: PUBLIC PROC [x: REAL] RETURNS [degrees: REAL];

Hyperbolic circle functions

SinH: PROC [x: REAL] RETURNS [REAL];

Hyperbolic sine

CosH: PROC [x: REAL] RETURNS [REAL];

Hyperbolic cosine

TanH: PROC [x: REAL] RETURNS [REAL];

Hyperbolic tangent

CotH: PROC [x: REAL] RETURNS [REAL];

Hyperbolic cotangent

InvSinH: PROC [x: REAL] RETURNS [REAL];

InvCosH: PROC [x: REAL] RETURNS [REAL];

...defined for x >= 1.0, returns non-negative result

InvTanH: PROC [x: REAL] RETURNS [REAL];

...defined for x IN (-1.0..1.0)

InvCotH: PROC [x: REAL] RETURNS [REAL];

...defined for x NOT IN [-1.0..1.0]

Gamma functions

LnGamma: PROC [x: REAL] RETURNS [REAL];

.. defined for x>0

Gamma: PROC [x: REAL] RETURNS [REAL];

.. defined for x>0; computed by Exp[LnGamma[x]]

Gamma[x+1] = x! for integer x.

Bessel functions

J0: PROC [x: REAL] RETURNS [REAL];

Bessel function of first kind of order 0

J1: PROC [x: REAL] RETURNS [REAL];

Bessel function of first kind of order 1

Jn: PROC [n: INT, x: REAL] RETURNS [REAL];

Bessel function of first kind of order n

Rational approximation

Rational: TYPE ~ RECORD [numerator: INT, denominator: INT];

RationalFromReal: PROC [x: REAL, limit: INT ¬ INT.LAST] RETURNS [Rational];

... computes the best rational approximation to a real using continued fractions.

|numerator| <= limit, 0 < denominator <= limit.

Approximate equality

AlmostZero: PROC [x: REAL, distance: [-126..127]] RETURNS [BOOL];

AlmostZero[x, distance] ==

ABS[x] < 2^distance

AlmostEqual: PROC [y, x: REAL, distance: [-126..0]] RETURNS [BOOL];

AlmostEqual[y, x, distance] ==

NotNaN[x] AND NotNaN[y] AND

ABS[x-y] <= MAX[ABS[x], ABS[y]] * 2^distance

Notes:

When either number is a Nan (not-a-number), then equality is never guaranteed, so FALSE is a conservative answer.

Comparing anything against 0.0 can give surprising results using the above definition, since ABS[x-0] = MAX[ABS[x], ABS[0]], which implies that AlmostEqual[0, x, distance] is only true for distance = 0.

END.