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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... comment This is a demonstration of the working of elementary functions available in the Reduce system. It is not intended as an accuracy test. Other functions become available if certain library packages are loaded. ------- Integer functions that work in all domain modes, independent of switch NUMVAL, so long as their arguments evaluate to real numbers. Functions of one argument: FIX, SGN, ROUND, CEILING, FLOOR (The following functions are available only in symbolic mode, so they are not tested here: ISQRT, ICBRT, ILOG2, IROOTN); fix a; fix(a) % will be evaluated only if a evaluates to a real number. a := 27/4; 27 a := ---- 4 fix a; 6 fix 12.345; 12 sign (-15/2); -1 round 12.5; 13 ceiling 12.5; 13 floor 12.5; 12 % isqrt 12.5; % icbrt 12.5; % ilog2 130.7; % irootn(72,4); % irootn(72,3/2); % this will not evaluate. comment Functions which require arguments which evaluate to integers: Function of one argument: FACTORIAL Fumction of two arguments: PERM, CHOOSE; $ factorial 10; 3628800 perm(5,10); 30240 % permutations of 5 out of 10. choose(5,10); 252 % choose 5 out of 10; comment These functions are evaluated in dmodes ROUNDED and COMPLEX-ROUNDED (ON ROUNDED,COMPLEX) so long as their arguments and values evaluate to real numbers and NUMVAL (normally ON) is ON. Variable treated as function of no arguments: E, PI. Functions of one argument: EXP, LOG, LN, LOG10, NORM, ARG, SQRT, RAD2DEG, RAD2DMS, DEG2RAD, DEG2DMS, DMS2DEG, DMS2RAD, SIN, ASIN, COS, ACOS, TAN, ATAN, COT, ACOT, SEC, ASEC, CSC, ACSC, SINH, ASINH, COSH, ACOSH, TANH, ATANH, COTH, ACOTH, SECH, ASECH, CSCH, ACSCH. Functions of two arguments: EXPT, LOGB, HYPOT, ATAN2. Function evaluation is carried out to the precision specified in the latest PRECISION statement. (The following functions are available only in symbolic mode, so they are not tested here: SIND, ASIND, COSD, ACOSD, TAND, ATAND, COTD, ACOTD, SECD, ASECD, CSCD, ACSCD, ATAN2D, CBRT); on rounded; precision 6; 12 a := exp 3; a := 20.0855 log a; 3.0 ln a; 3.0 log10 1000; 3.0 norm (-12.345); 12.345 % for real x, this is equivalent to ABS x. arg (-12.345); 3.14159 % for real x, this -> if x<0 then pi else 0.0. sqrt 3; 1.73205 ws**2; 3.0 deg2rad 30; 0.523599 rad2deg ws; 30.0 a := deg2dms 12.345; a := {12,20,42.0} % a will be a list. dms2deg ws; 12.345 dms2rad a; 0.215461 rad2deg ws; 12.345 asin 0.5; 0.523599 sin ws; 0.5 acos 0.5; 1.0472 cos ws; 0.5 atan 0.5; 0.463648 tan ws; 0.5 acot 0.5; 1.10715 cot ws; 0.5 asec 3; 1.23096 sec ws; 3.0 acsc 3; 0.339837 csc ws; 3.0 asinh 0.5; 0.481212 sinh ws; 0.5 acosh 2; 1.31696 cosh ws; 2.0 atanh 0.5; 0.549306 tanh ws; 0.5 acoth 2; 0.549306 coth ws; 2.0 sech 1; 0.648054 asech ws; 1 csch 1; 0.850918 acsch ws; 1 expt(2,1.234); 2.35218 logb(ws,2); 1.234 hypot(3,4); 5.0 a := -3*pi/4; a := - 2.35619 % any -pi<a<=pi should work. atan2(sin a,cos a); - 2.35619 ws - a; 0 % should be 0. precision 20; 6 % functions will be computed to 20 places. sin 1.5; 0.99749498660405443094 asin ws; 1.5 precision 50; 20 % fuctions computed to 50 places. sin 1.5; 0.99749498660405443094172337114148732270665142592212 asin ws; 1.5 precision 6; 50 comment If argument or value are complex, functions are not computed when dmode is ROUNDED; $ sin(1+i); sin(i + 1) % complex argument. asin 2; asin(2) % value would be complex. on complex; *** Domain mode rounded changed to complex-rounded %now complex arguments and complex results will be handled. comment Complex functions of one argument: EXP, LOG, NORM, ARG, SQRT, SIN, ASIN, COS, ACOS, TAN, ATAN, COT, ACOT, SEC, ASEC, CSC, ACSC, SINH, ASINH, COSH. ACOSH, TANH, ATANH, COTH, ACOTH, SECH, ASECH, CSCH, ACSCH. (The following functions are available only in symbolic mode, so they are not tested here: SIND, ASIND, COSD, ACOSD, TAND, ATAND, COTD, ACOTD, SECD, ASECD, CSCD, ACSCD.) Complex function of two variables: EXPT, LOGB, ATAN2; e**(pi*i); - 1 + 1.22461e-16*i % should be -1 (except for computational error.) log(1+i); 0.346574 + 0.785398*i exp ws; 1 + i norm(5*exp(2i)); 5.0 arg(5*exp(2i)); 2.0 sqrt(1+i); 1.09868 + 0.45509*i ws**2; 1 + i asin 2; 1.5708 - 1.31696*i sin ws; 2.0 - 1.25983e-15*i acos 2; 1.31696*i cos ws; 2.0 atan(1+i); 1.01722 + 0.402359*i tan ws; 1 + i acot(1+i); 0.553574 - 0.402359*i cot ws; 1 + i asec 0.1; 2.99322*i sec ws; 0.1 acsc 0.1; 1.5708 - 2.99322*i csc ws; 0.1 + 7.23718e-17*i sinh(1+i); 0.634964 + 1.29846*i asinh ws; 1 + i cosh(1+i); 0.83373 + 0.988898*i acosh ws; 1 + i atanh 2; 0.549306 + 1.5708*i tanh ws; 2.0 + 1.83691e-16*i acoth 0.3; 0.30952 + 1.5708*i coth ws; 0.3 - 5.57196e-17*i asech(1-i); 0.530638 + 1.11852*i sech ws; 1 - i acsch(1-i); 0.530638 + 0.452278*i csch ws; 1 - i expt(1+i,1-i); 2.80788 + 1.31787*i logb(ws,1+i); 1 - i a := 1+i; a := 1 + i % any a such that - pi < repart a <= pi should work. atan2(sin a,cos a); 1 + i ws - a; 0 % should be 0; end; (TIME: elem 300 300)