@@ -1,701 +1,701 @@
-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
+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