reduce-historical/r36/xlog/MATH.LOG

REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...


% 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;


nil


math!!label;


"Math package mod 1.7, 1 May 93"


symbolic procedure terpr(i,j); if remainder(i,j)=0 then terpri()$



on rounded;


nil
   % We need !!plumin, etc.

% #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)>>;

  0->1.0
  1->1.0  2->1.0  3->1.0  4->1.0
  5->1.0  6->1.0  7->1.0  8->1.0
  9->1.0  10->1.0  11->1.0  12->1.0
  13->1.0  14->1.0  15->1.0  16->1.0
  17->1.0  18->1.0  19->1.0  20->1.0
  21->1.0  22->1.0  23->1.0  24->1.0
  25->1.0  26->1.0  27->1.0  28->1.0
  29->1.0  30->1.0  31->1.0  32->1.0
  33->1.0  34->1.0  35->1.0  36->1.0
  37->1.0  38->1.0  39->1.0  40->1.0
  41->1.0  42->1.0  43->1.0  44->1.0
  45->1.0
nil


% #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$

 1.0 1.0
 1.0 -1.0
 -1.0 -1.0
 -1.0 1.0


% #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)," "
   for i:= -10.0:10.0 do write round(cosd(60+i*360)/a)," "
 end$

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

% #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$

1.0  1.0  1.0  
1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
0.99999999999999  1.0  

% #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$

1.0  0.99999999999999  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  

% #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$

sind and cosd
1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
sin and cos
1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

tand, atand and acotd
1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  
tan, atan and acot
1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  

% #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$

 -9->-9.0 -8->-8.0
 -7->-7.0 -6->-6.0 -5->-5.0 -4->-4.0
 -3->-3.0 -2->-2.0 -1->-1.0 0->1.0
 1->1.0 2->2.0 3->3.0 4->4.0
 5->5.0 6->6.0 7->7.0 8->8.0
 9->9.0 10->10.0 11->11.0 12->12.0
 13->13.0 14->14.0 15->15.0 16->16.0
 17->17.0 18->18.0 19->19.0 20->20.0
 21->21.0 22->22.0 23->23.0 24->24.0
 25->25.0 26->26.0 27->27.0 28->28.0
 29->29.0 30->30.0 31->31.0 32->32.0
 33->33.0 34->34.0 35->35.0 36->36.0
 37->37.0 38->38.0 39->39.0 40->40.0
 41->41.0 42->42.0 43->43.0 44->44.0
 45->45.0 46->46.0 47->47.0 48->48.0
 49->49.0 50->50.0 51->51.0 52->52.0
 53->53.0 54->54.0 55->55.0 56->56.0
 57->57.0 58->58.0 59->59.0 60->60.0
 61->61.0 62->62.0 63->63.0 64->64.0
 65->65.0 66->66.0 67->67.0 68->68.0
 69->69.0 70->70.0 71->71.0 72->72.0
 73->73.0 74->74.0 75->75.0 76->76.0
 77->77.0 78->78.0 79->79.0 80->80.0
 81->81.0 82->82.0 83->83.0 84->84.0
 85->85.0 86->86.0 87->87.0 88->88.0
 89->89.0

% #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$

 10->10.0 20->20.0 30->30.0 40->40.0
 50->50.0 60->60.0 70->70.0 80->80.0
 90->90.0 100->100.0 110->110.0 120->120.0
 130->130.0 140->140.0 150->150.0 160->160.0
 170->170.0



% #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$

 -90->-90.0 -80->-80.0
 -70->-70.0 -60->-60.0 -50->-50.0 -40->-40.0
 -30->-30.0 -20->-20.0 -10->-10.0 0->0.0e+000
 10->10.0 20->20.0 30->30.0 40->40.0
 50->50.0 60->60.0 70->70.0 80->80.0
 90->90.0

% #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$

 10->10.0 20->20.0 30->30.0 40->40.0
 50->50.0 60->60.0 70->70.0 80->80.0
 90->90.0 100->100.0 110->110.0 120->120.0
 130->130.0 140->140.0 150->150.0 160->160.0
 170->170.0 180->180.0

% #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$

 -90->-90.0 -80->-80.0
 -70->-70.0 -60->-60.0 -50->-50.0 -40->-40.0
 -30->-30.0 -20->-20.0 -10->-10.0 0->error
 10->10.0 20->20.0 30->30.0 40->40.0
 50->50.0 60->60.0 70->70.0 80->80.0
 90->90.0

% #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$

 0->0.0e+000
 10->10.0 20->20.0 30->30.0 40->40.0
 50->50.0 60->60.0 70->70.0 80->80.0
 90->error 100->100.0 110->110.0 120->120.0
 130->130.0 140->140.0 150->150.0 160->160.0
 170->170.0 180->180.0

%*********************************************************************
%**  ===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$

multiplicative property
  1.0  1.0  1.0  1.0  1.0
  1.0  1.0  1.0  1.0  1.0
  1.0  1.0  1.0  1.0  1.0
  1.0  1.0  1.0  1.0  1.0
  1.0

% #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$

inverse property
  1.0  1.0  1.0  1.0  1.0
squares
  1.0
  1.0  1.0  1.0  1.0  1.0
  1.0  1.0  1.0  1.0  1.0
  1.0  1.0  1.0  1.0  1.0
  1.0  1.0  1.0  1.0  1.0
cubes
  1.0
  1.0  1.0  1.0  1.0  1.0
  1.0  1.0  1.0  1.0  1.0
  1.0  1.0  1.0  1.0  1.0
  1.0  1.0  1.0  1.0  1.0


% #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$

1.0  
1.0  1.0  1.0  1.0  0/0  
1.0  1.0  1.0  1.0  1.0  


% #19: test of log10 expt(10.0,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(10.0,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(10.0,2**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$

small values i:=-5:5
1.0  
1.0  1.0  1.0  1.0  zero  
1.0  1.0  1.0  1.0  1.0  
large i=2**j where j:=0:6
1.0  
1.0  1.0  1.0  1.0  1.0  

noninteger values of i=j/10.0 where j:=1:20
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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(float x,i)*expt(float x,-i),"  "; terpr(j:=j+1,5)>>
 end$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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 rexpt(-8,k)/rexpt(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 rexpt(-8,k)/rexpt(8,k),"  ";
        terpr(j:=j+1,5)>>
 end$

odd numerator. ideal result -1.0
-1.0  -1.0  -1.0  -1.0  -1.0  
-1.0  -1.0  -1.0  -1.0  -1.0  
even numerator. ideal result 1.0
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

-1.0  -1.0  -1.0  -1.0  -1.0  
-1.0  -1.0  -1.0  -1.0  -1.0  
-1.0  -1.0  -1.0  -1.0  -1.0  
-1.0  -1.0  -1.0  -1.0  -1.0  
-1.0  -1.0  -1.0  -1.0  -1.0  
-1.0  -1.0  -1.0  -1.0  -1.0  
-1.0  -1.0  -1.0  -1.0  -1.0  
-1.0  

% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


%*********************************************************************
%**  ===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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$write(cosh x)**2-(sinh x)**2,"  "; terpr(i,5)>>
 end$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  0.99999999999999  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  0.99999999999999  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% #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$

1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1  
1.0  1.0  1.0  1.0  1.0  
1.0  1.0  1.0  1.0  1.0  


% End of Test.

end;
(TIME:  math 329 329)

nil