00100	          SAITRG  -  SAIL ACCESSIBLE TRIGNOMETRY FUNCTIONS

00200	
00300	                          Bruce g. Baumgart
00400	
00500	
00600	The require source file statement for declaring the SAITRG routines is:
00700	
00800		REQUIRE "SAITRG[SYS,BGB]" SOURCE_FILE;
00900	
01000	The SAIL declarations for the SAITRG routines are:
01100	
01200		REQUIRE "SAITRG[SYS,BGB]" LOAD_MODULE;
01300		EXTERNAL REAL PROCEDURE SIN  (REAL X);
01400		EXTERNAL REAL PROCEDURE COS  (REAL X);
01500		EXTERNAL REAL PROCEDURE SIND (INTEGER I);
01600		EXTERNAL REAL PROCEDURE COSD (INTEGER I);
01700		EXTERNAL REAL PROCEDURE ASIN (REAL X);
01800		EXTERNAL REAL PROCEDURE ACOS (REAL X);
01900		EXTERNAL REAL PROCEDURE SQRT (REAL X);
02000		EXTERNAL REAL PROCEDURE ATAN (REAL X);
02100		EXTERNAL REAL PROCEDURE ATAN2(REAL Y,X);
02200	
02300		These routines were  plagarized  from  the  FORTRAN  library,
02400	translated  from  MACRO into FAIL, and slightly altered for entry and
02500	return  under  SAIL  conventions  which  don't  require  saving   and
02600	restoring accumulators.
02700	
02800		SIN and COS expect their argument in radians as real numbers.
02900	SIND and COSD take their argument in degrees as integers.
03000	
03100		The arc sine and arc cosine are mathematically  only  defined
03200	for  arguments  on  the interval from -1 to +1 inclusive, however the
03300	ASIN and ACOS compute the following
03400		ASIN(X)	=	ATAN(X/SQRT(1-X↑2));
03500		ACOS(X)	=	π/2 - ASIN(X);
03600	...and  consequently  will always return values that could be angles.
03700	With proper arguments ASIN returns values between -π/2  and  π/2  and
03800	ACOS returns values between 0 and π.
03900	
04000		SQRT is the square root function.
04100	
04200		The  argument  for  ATAN  can  be  any  real number the value
04300	returned will be on the interval from -π/2 to +π/2 inclusive.
04400	
04500		ATAN2 takes two real numbers and returns the angle in radians
04600	between the vector to (X,Y) from the orgin and  the  positive  x-axis
04700	unit vector. ATAN2 values lie between -π and +π.
     

00100	TIMING
00200		
00300	The timings given by DEC for 1 micro second core are:
00400	
00500		SIN	91		ASIN	149
00600		COS	148		ACOS	166
00700		SQRT	62
00800		ATAN	55		ATAN2	94
00900	
01000		The DEC timings are in the ball park with  the  exception  of
01100	COS  which  only  differs  from SIN by a move, fadr, jrst which don't
01200	amount to 57 micro seconds of execution time.
01300	
01400		I estimate timing as follows:
01500	
01600	50 micro seconds:	ATAN and SQRT.
01700	100 micro seconds:	ATAN2, SIN and COS.
01800	150 micro seconds:	ASIN and ACOS.
01900	
02000		The SAITRG routines are fully documented in  the  DEC  manual
02100	titled "Science Library and Fortran Utility Routines".