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% Test of Math Package. % Author: Stanley L. Kameny <stan%valley.uucp@rand.org>. % Copyright (c) 1987, 1988, 1989, 1991 Stanley L. Kameny. % All Rights Reserved. %********************************************************************* %** %** This math package will compute the floating point values of ** %** the usual elementary functions, namely: ** %** sin asin sind asind sinh asinh ** %** cos acos cosd acosd cosh acosh ** %** tan atan tand atand tanh atanh ** %** cot acot cotd acotd coth acoth ** %** sec asec secd asecd sech asech ** %** csc acsc cscd acscd csch acsch ** %** atan2 atan2d ** %** exp ln sqrt ** %** expt log cbrt ** %** logb hypot ** %** log10 floor ** %** ceiling ** %** round ** %** ** %** All functions are computed to the accuracy of the floating- ** %** point precision of the system set up at the time. ** %** ** %********************************************************************* %** File #1===Trig Function Tests=== %** Trig functions are tested in both degrees and radians modes. %********************************************************************* symbolic; math!!label; symbolic procedure terpr(i,j); if remainder(i,j)=0 then terpri()$ % #1: sind**2+cosd**2 test: ideal answers 1.0 for all args. for i:=0:45 do <<write " ",i,"->",sind float i**2+cosd float i**2; terpr(i,4)>>; % #2: quadrant test of sind, cosd: proper answers + +,+ -,- -,- +. begin scalar a; a:= sind 45.0; for i:= 0.0:3.0 do <<write " ",sind(i*90+45)/a," ", cosd (i*90+45)/a;terpri()>> end$ % #3: scaling test: all values should be 1 exactly. begin scalar a; a:= cosd 60.0; for i:= -10.0:10.0 do write fix(cosd(60+i*360)/a)," " end$ % #4: test of radians -> degrees evaluation: ideal values 1.0. array a(6)$ begin for i:=1:6 do a(i):=sind(15.0*i); for i:=1:6 do <<write sin(pii2*i/6.0)/a(i)," "; terpr(i,3)>> end$ % #5: test of tand*cotd: ideal values 1.0. begin for i:=5 step 5 until 85 do <<write tand float i*cotd float i," "; terpr(i,25)>>; terpri() end$ % #6: test of secd**2-tand**2: ideal values 1.0. begin for i:=5 step 5 until 85 do <<write secd float i**2-tand float i**2," "; terpr(i,25)>> end$ % #7: test of cscd**2-cotd**2: ideal values 1.0. begin for i:=5 step 5 until 85 do <<write cscd float i**2-cotd float i**2," "; terpr(i,25)>> end$ % #8: test of asind+acosd: ideal values 1.0. begin write "sind and cosd"; terpri(); for i:=-10:10 do <<write (asind(0.1*i)+acosd(0.1*i))/90," "; terpr(i,5)>>; write "sin and cos";terpri(); for i:=-10:10 do <<write (acos(0.1*i)+asin(0.1*i))/pii2," "; terpr(i,5)>> end$ % #9: test of atand+acotd: ideal values 1.0. begin scalar x; write "tand, atand and acotd"; terpri(); for i:=-80 step 10 until 80 do <<x:=tand float i; write (atand x+acotd x)/90," "; terpr(i,50)>>; terpri(); write "tan, atan and acot";terpri(); for i:=-80 step 10 until 80 do <<x:= tan (pii2*i/90.0); write (atan x+acot x)/pii2," "; terpr(i,50)>> end$ % #10: test of atand tand: ideal values i for i:=-9:89. begin for i:=-9:89 do <<write " ",i,"->",if i=0 then 1.0 else atand tand float i; terpr(i,4)>> end$ % #11: test of acot cotd: ideal values 10*i for i:=1:17. begin for i:=10 step 10 until 170 do <<write " ",i,"->",acotd cotd i; terpr(i,40)>>; terpri();terpri() end$ % #12: test of asind sind: ideal values 10*i for i:=-9:9. begin for i:=-90 step 10 until 90 do <<write " ",i,"->",asind sind float i; terpr(i,40)>> end$ % #13: test of acosd cosd: ideal values 10*i for i:=1:18. begin for i:=10 step 10 until 180 do <<write " ",i,"->",acosd cosd float i; terpr(i,40)>> end$ % #14: test of acscd cscd: ideal values 10*i for i:=-9:9, except % error for i=0. begin for i:=-90 step 10 until 90 do <<write " ",i,"->",if i=0 then "error" else acscd cscd float i; terpr(i,40)>> end$ % #15: test of asecd secd: ideal values 10*i for i :=0:18. except % error for i=9. begin for i:=0 step 10 until 180 do <<write" ",i,"->",if i=90 then "error" else asecd secd float i; terpr(i,40)>> end$ %********************************************************************* %** ===Exp,Log,Sqrt,Cbrt, and Expt Function tests=== %********************************************************************* % #16: test of properties of exp function: ideal results 1.0. array b(5)$ begin scalar x; x:=0; write "multiplicative property";terpri(); for i:=0:5 do b(i):=1+i/6.0; for i:=0:5 do for j:=i:5 do <<write " ",exp (b(i)+b(j))/exp(b(i))/exp(b(j)); terpr(x:=x+1,5)>> end$ % #17: various properties of exp: ideal results 1.0. begin write "inverse property"$ terpri()$ for i:=1:5 do write " ",exp(b(i))*exp(-b(i));terpri(); write "squares"; terpri(); for i:=-10:10 do <<write " ",sqrt(exp(0.2*i))/exp(0.1*i); terpr(i,5)>>; write "cubes"; terpri(); for i:=-10:10 do <<write " ",cbrt(exp(0.3*i))/exp(0.1*i); terpr(i,5)>> end$ % #18: test of log exp: ideal results 1.0. begin for i:=-5:5 do <<write if i=0 then "0/0" else (log exp float i)/i," "; terpr(i,5)>> end$ % #19: test of log10 expt(10,i): ideal results 1.0. begin scalar i; write "small values i:=-5:5"; terpri(); for j:=-5:5 do <<write if j neq 0 then log10 float expt(10,j)/j else "zero"," "; terpr(j,5)>>; write "large i=2**j where j:=0:6"; terpri(); for j:=0:5 do <<write (log10 float expt(10,2**j))/2**j," "; terpr(j,5)>>; terpri(); write "noninteger values of i=j/10.0 where j:=1:20";terpri(); for j:=1:20 do <<i:=j/10.0; write (log10 float expt(10,i))/i," "; terpr(j,5)>> end$ % #20: test of properties of expt(x,i)*(expt(x,-i). ideal result 1.0. begin integer j; for x:=2:6 do for i:=2:6 do <<write expt(x,i)*expt(x,-i)," "; terpr(j:=j+1,5)>> end$ % #21: test of expt(-x,i)/expt(x,i) for fractional i. begin integer j,k; write "odd numerator. ideal result -1.0"; terpri(); for i:=1:10 do <<k:=(2*i-1.0)/(8*i+1); write expt(-8,k)/expt(8,k)," "; terpr(j:=j+1,5)>>; write "even numerator. ideal result 1.0"; terpri(); for i:=1:10 do <<k:=(2.0*i)/(8*i+1); write expt(-8,k)/expt(8,k)," "; terpr(j:=j+1,5)>> end$ % #22: test of properties of ln or log or logb: % inverse argument: ideal result -1.0. begin integer x; for i:=2:5 do for j:= 2:10 do <<x:=x+1; write logb(float i,float j)/logb(float i,1.0/j)," "; terpr(x,5)>> end$ % #23: test of log(a*b) = log a+log b: ideal result 1.0. begin integer x; for i:=1:5 do for j:=i:5 do <<write log (i*j*0.01)/(log (i*0.1)+log(j*0.1))," "; terpr(x:=x+1,5)>> end$ % #24:test of sqrt x*sqrt x/x for x:=5i*(5i/3)**i where i:=1:20 % (test values strictly arbitrary): ideal results 1.0. begin scalar x,s; for i:=1:20 do <<x:= 5.0*i;s:=sqrt(x:=x*(expt(x/3,i))); write s*s/x," "; terpr(i,5)>> end$ % #25: test of cbrt x**3/x for x:=5i*(5i/3)**i where i:=-9:10 % (test values strictly arbitrary):ideal results 1.0. begin scalar x,s; for i:=-9:10 do <<x:= 5.0*i; if i neq 0 then s:= cbrt(x:=x*(expt(x/3,i))); write if i=0 then 1 else s*s*s/x," "; terpr(i,5)>> end$ %********************************************************************* %** ===Hyperbolic Function Tests=== %********************************************************************* % #26: test of sinh x+ cosh x= exp x: ideal results 1.0. begin scalar x; for i:=1:10 do <<x:=ln float i$ write (sinh x+cosh x)/exp x," "$ terpr(i,5)>> end$ % #27: test of cosh x-sinh x= exp(-x): ideal results 1.0. begin scalar x; for i:=1:10 do <<x:=ln float i$ write(cosh x-sinh x)*exp x," "$ terpr(i,5)>> end$ % #28: test of (cosh x)**2-(sinh x)**2: ideal results 1.0. begin scalar x$ for i:=1:10 do <<x:=ln float i$write(cosh x)**2-(sinh x)**2," "; terpr(i,5)>> end$ % #29: test of tanh*cosh/sinh: ideal results 1.0. begin scalar x; for i:=1:20 do <<x:=ln(i*0.1); write if i=10 then 1 else tanh x*cosh x/sinh x," "; terpr(i,5)>> end$ % #30: test of tanh*coth: ideal results 1.0. begin scalar x; for i:=1:20 do <<x:=ln(i*0.1); write if i=10 then 1 else tanh x*coth x," "; terpr(i,5)>> end$ % #31: test of sech*cosh: ideal results 1.0. begin scalar x; for i:=1:20 do <<x:=ln(i*0.1); write sech x*cosh x," "; terpr(i,5)>> end$ % #32: test of csch*sinh: ideal results 1.0. begin scalar x; for i:=1:20 do <<x:=ln(i*0.1); write if i=10 then 1 else csch x*sinh x," "; terpr(i,5)>> end$ % #33: test of asinh sinh: ideal results 1.0. begin scalar x; for i:=1:20 do <<x:=ln(i*0.1); write if i=10 then 1 else asinh sinh x/x," "; terpr(i,5)>> end$ % #34: test of acosh cosh: ideal results 1.0. However, acosh x % loses accuracy as x -> 1 since d/dx cosh x -> 0. begin scalar x; for i:=1:20 do <<x:=ln(1+i*0.05); write acosh cosh x/x," "; terpr(i,5)>> end$ % #35: test of cosh acosh:ideal results 1.0. begin scalar x; for i:=1:50 do <<x:=1+i/25.0; write (cosh acosh x)/x," "; terpr(i,5)>> end$ % #36: test of atanh tanh: ideal results 1.0. begin scalar x; for i:=1:20 do <<x:=ln(i*0.1); write if i=10 then 1 else (atanh tanh x)/x," "; terpr(i,5)>> end$ % #37: test of acoth coth: ideal results 1.0. begin scalar x; for i:=1:20 do <<x:=ln(i*0.1); write if i=10 then 1 else (acoth coth x)/x," "; terpr(i,5)>> end$ % #38: test of asech sech: ideal results 1.0. However, asech x % loses accuracy as x -> 1 since d/dx sech x -> 0. begin scalar x; for i:=1:20 do <<x:=ln(1+i*0.05); write (asech sech x)/x," "; terpr(i,5)>> end$ % #39: test of acsch csch: ideal results 1.0. begin scalar x; for i:=1:20 do <<x:=ln(i*0.1); write if i=10 then 1 else (acsch csch x)/x," "; terpr(i,5)>> end$ % End of Test. end;