Re: fixed point vs floating point

[rav@goanna.cs.rmit.edu.au (robin)]

	>Matthew Heaney <mheaney@ni.net> wrote:

	>Anyway, I'll accept that fixed point operations are slower, but the
	>question remains: Why do most programmers insist on using floating point
	>when a fixed point better models the abstraction?  Is the reason solely
	>based on efficiency?

	>By the way, are there off-the-shelf subprograms (in Ada preferably, but I'd
	>take any language) for computing the elementary functions (sin, cos, etc)
	>of a fixed point angle?

The following evaluates sine using fixed point, for a limited range of
angles.
_____________________________________________________

/* Copyright (c) 1996, 1997 by R. A. Vowels. */
/* This subroutine computes the sine of an angle using fixed-point binary arithmetic. */
SINE:
	PROCEDURE(XX) OPTIONS (REORDER) RETURNS (FIXED BINARY(31,28));
		DECLARE XX FIXED BINARY(31,24);
		DECLARE X		FIXED BINARY (31,28);
		DECLARE Sum		FIXED BINARY (31,28);
		DECLARE Term		FIXED BINARY (31,28);
		DECLARE (J, K)		FIXED BINARY;
		DECLARE Neg		BIT(1);

   DECLARE (HiX, HiY) FIXED BINARY (31,14);
   DECLARE (LoX, LoY) FIXED BINARY (31,28);
   DECLARE (ProdHiXLoY, ProdHiYLoX, Prod) FIXED BINARY (31,28);

/* This macro procedure forms the product of the fixed-point arguments X1 and Y1. */
%MULT: PROCEDURE(LHS, X1, Y1);
		DECLARE (LHS, X1, Y1) CHARACTER;
			ANSWER ('HiX = ' || X1 || ';') SKIP; /* Extract the upper 14 bits of X. */
			ANSWER ('HiY = ' || Y1 || ';') SKIP; /* Extract the upper 14 bits of X. */
			ANSWER ('LoX = ' || X1 || ' - HiX;' ) SKIP; /* Extract the lower 14 bits of X. */
			ANSWER ('LoY = ' || Y1 || ' - HiY;' ) SKIP; /* Extract the lower 14 bits of Y. */

			ANSWER ('Prod = HiX * HiY;') SKIP;  /* the product of the high bits. */
			ANSWER ('ProdHiXLoY = HiX * LoY;') SKIP; /* Product of the upper bits of X, lower bits of Y. */
			ANSWER ('ProdHiYLoX = HiY * LoX;') SKIP; /* Product of the upper bits of Y, lower bits of X. */
			ANSWER (LHS || ' = Prod + ProdHiXLoY + ProdHiYLoX') SKIP; /* Sum them all. */

%END MULT;
%ACTIVATE MULT NORESCAN;


			/* The following code segment evaluates the sine of an angle x using the   */
			/* first 6 terms of the Taylor series:                                     */
			/* sine x = x - x**3/3! + x**5/5! - x**7/7! + x**9/9! - x**11/11!          */
			/* Range: -pi <= x <= pi                                                   */
			/* Accuracy: 8 decimal places (typically better than 7 significant digits).*/
			X = XX;

			Sum = X;
			Neg = X < 0;
			IF Neg THEN X = -X;
			IF X > 1.5707963 THEN /*sin(x) = sin(180-x). */
				X = 3.1415927 - X;

			/* For small angles, the radian angle is closer to the sine than this approximation. */
			/* Therefore, we only execute the following DO group for larger angles. */
			IF X > 0.002 THEN
				DO;


					Sum = X;
					Term = X / 6;
					MULT(Term, Term, X);
					MULT(Term, Term, X);   /* To give x**3/3! */
					Sum = Sum - Term;

					Term = Term / 20;
					MULT(Term, Term, X);
					MULT(Term, Term, X);   /* To give x**5/5! */
					Sum = Sum + Term;

					Term = Term / 42;
					MULT(Term, Term, X);
					MULT(Term, Term, X);   /* To give x**7/7! */
					Sum = Sum - Term;

					Term = Term / 72;
					MULT(Term, Term, X);
					MULT(Term, Term, X);   /* To give x**9/9! */
					Sum = Sum + Term;

					Term = Term / 110;
					HiX = X;
					Term = Term * HiX;
					Term = Term * HiX;   /* To give x**11/11! */
					Sum = Sum - Term;

					IF Neg THEN Sum = -Sum;
				END;
   RETURN (Sum);
END SINE;