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(* ------------------- VERSION 2 06.03.86 ------------------- *)
PACKET mathlib DEFINES sqrt, **, exp, ln, log2, log10, e, pi,
sin, cos, tan, sind, cosd, tand,
arctan, arctand, random, initializerandom :
LET pii = 3.141592653589793238462,
pi2 = 1.570796326794896619231,
pi3 = 1.047197551196597746154,
pi6 = 0.523598775598298873077,
pi4 = 1.273239544735162686151,
ln2 = 0.693147180559945309417,
lg2 = 0.301029995663981195213,
ln10 = 2.302585092994045684018,
lge = 0.434294481903251827651,
ei = 2.718281828459045235360,
pi180 = 57.295779513082320876798,
sqrt3 = 1.732050807568877293527,
sqr3 = 0.577350269189625764509,
sqr3p2= 3.732050807568877293527,
sqr3m2= 0.267949192431122706473,
sqr2 = 0.707106781186547524400;
REAL VAR rdg::0.4711;
REAL PROC pi: pii END PROC pi;
REAL PROC e : ei END PROC e;
REAL PROC ln ( REAL CONST x ):
log2(x) * ln2
END PROC ln;
REAL PROC log10( REAL CONST x ):
log2(x) * lg2
END PROC log10;
REAL PROC log2 ( REAL CONST z ):
REAL VAR t, summe::0.0, x::z;
IF x=1.0 THEN 0.0
ELIF x>0.0 THEN normal
ELSE errorstop("log2: " + text (x,20)); 0.0 FI.
normal:
IF x >= 0.5 THEN normalise downwards
ELSE normalise upwards FI;
IF x < sqr2 THEN summe := summe - 0.75; t := trans8
ELSE summe := summe - 0.25; t := trans2 FI;
summe + reihenentwicklung.
normalise downwards:
WHILE x >= 8.0 REP x := 0.0625 * x; summe:=summe+4.0 PER;
WHILE x >= 1.0 REP x := 0.5 * x; summe:=summe+1.0 PER.
normalise upwards:
WHILE x<=0.0625 REP x := 16.0 * x; summe:=summe-4.0 PER;
WHILE x<= 0.5 REP x := 2.0 * x; summe:=summe-1.0 PER.
trans8: (x - 0.5946035575013605)/(x + 0.5946035575013605).
trans2: (x - 0.8408964152537145)/(x + 0.8408964152537145).
reihenentwicklung: x := t * t; t * 0.06405572387119384648 *
((((((3.465*x+4.095)*x+5.005)*x+6.435)*x+9.009)*x+15.015)*x+45.045)
END PROC log2;
REAL PROC sqrt ( REAL CONST z ):
REAL VAR y0, y1, x::z;
INT VAR p :: decimal exponent(x) DIV 2;
IF p <= -64 THEN 0.0
ELIF x < 0.0 THEN errorstop("sqrt: " + text (x,20)); 0.0
ELSE nontrivial FI.
nontrivial:
set exp (decimal exponent (x) -p-p, x);
IF x<10.0 THEN x := 5.3176703 - 40.760905/( 8.408065 + x )
ELSE x := 16.81595 - 1288.973 /( 84.08065 + x ) FI;
y0 := x;
set exp (decimal exponent (x) + p, y0);
y1 := 0.5 * ( y0 + z/y0 );
y0 := 0.5 * ( y1 + z/y1 );
y1 := 0.5 * ( y0 + z/y0 );
0.5 * ( y1 + z/y1 )
END PROC sqrt;
REAL PROC exp ( REAL CONST z ):
REAL VAR x::z, a::1.0; BOOL VAR negativ :: x<0.0;
IF negativ THEN x := -x FI;
IF x>292.42830676
THEN IF NOT negativ THEN errorstop ("REAL-Ueberlauf") FI ; 0.0
ELIF x<=0.0001
THEN ( 0.5*z + 1.0 ) * z + 1.0
ELSE approx
FI.
approx:
IF x > ln10
THEN x := lge*x;
a := 1.0;
set exp (int(x), a);
x := frac(x)*ln10
FI;
IF x >= 2.0 THEN a := 7.389056098930650227230*a; x := x-2.0 FI;
IF x >= 1.0 THEN a := 2.718281828459045235360*a; x := x-1.0 FI;
IF x >= 0.5 THEN a := 1.648721270700128146848*a; x := x-0.5 FI;
IF x >= 0.25 THEN a := 1.284025416687741484073*a; x := x-0.25 FI;
IF x >= 0.125 THEN a := 1.133148453066826316829*a; x := x-0.125 FI;
IF x >= 0.0625THEN a := 1.064494458917859429563*a; x := x-0.0625FI;
a:=a/50.4*(((((((0.01*x+0.07)*x+0.42)*x+2.1)*x+8.4)*x+25.2)*x+50.4)*x+50.4);
IF negativ THEN 1.0/a ELSE a FI .
ENDPROC exp ;
REAL PROC tan (REAL CONST x):
IF x < 0.0 THEN - tg( -x * pi4)
ELSE tg( x * pi4) FI
END PROC tan;
REAL PROC tand (REAL CONST x):
IF x < 0.0 THEN - tg( -x / 45.0)
ELSE tg( x / 45.0) FI
END PROC tand;
REAL PROC tg (REAL CONST x ):
REAL VAR q::floor(x), s::x-q; INT VAR n;
q := q - floor(0.25*q) * 4.0 ;
IF q < 2.0
THEN IF q < 1.0
THEN n:=0;
ELSE n:=1; s := 1.0 - s FI
ELSE IF q < 3.0
THEN n:=2;
ELSE n:=3; s := 1.0 - s FI
FI;
q := s * s;
q := (((((((((-5.116186989653120e-11*q-5.608325022830701e-10)*q-
9.526170109403018e-9)*q-1.517906721393745e-7)*q-2.430939946375515e-6)*q-
3.901461426385464e-5)*q-6.324811612385572e-4)*q-1.076606829172646e-2)*q-
0.2617993877991508)*q+pi4);
SELECT n OF
CASE 0 : s/q
CASE 1 : q/s
CASE 2 : -q/s
OTHERWISE : -s/q ENDSELECT .
END PROC tg;
REAL PROC sin ( REAL CONST x ):
REAL VAR y, r, q;
IF x < 0.0 THEN y := -x; q := 4.0 ELSE y := x; q := 0.0 FI;
y := y * pi4;
r := floor(y);
sincos( q+r , y-r )
END PROC sin;
REAL PROC sind ( REAL CONST x ):
REAL VAR y, r, q;
IF x < 0.0 THEN y := -x; q := 4.0 ELSE y := x; q := 0.0 FI;
y := y / 45.0;
r := floor(y);
sincos( q+r , y-r )
END PROC sind;
REAL PROC cos ( REAL CONST x ):
REAL VAR y, q;
IF x < 0.0 THEN y := -x ELSE y := x FI;
y := y * pi4;
q := floor(y);
sincos( q+2.0, y-q )
END PROC cos;
REAL PROC cosd ( REAL CONST x ):
REAL VAR y, q;
IF x < 0.0 THEN y := -x ELSE y := x FI;
y := y / 45.0;
q := floor(y);
sincos( q+2.0, y-q )
END PROC cosd;
REAL PROC sincos ( REAL CONST q, y ):
REAL VAR r :: q - floor( 0.125*q + 0.1 ) * 8.0;
IF r >= 4.0 THEN IF r >= 6.0 THEN IF r >= 7.0 THEN - sin approx(1.0-y)
ELSE - cos approx(y) FI
ELSE IF r >= 5.0 THEN - cos approx(1.0-y)
ELSE - sin approx(y) FI FI
ELSE IF r >= 2.0 THEN IF r >= 3.0 THEN sin approx(1.0-y)
ELSE cos approx(y) FI
ELSE IF r >= 1.0 THEN cos approx(1.0-y)
ELSE sin approx(y) FI FI FI
END PROC sincos;
REAL PROC sin approx ( REAL CONST x ):
REAL VAR z::x*x;
x*((((((0.6877101540593035e-11*z-0.1757149296873372e-8)*z+0.3133616216672568
e-6)*z-0.3657620415845891e-4)*z+0.2490394570188737e-2)*z-0.807455121882e-1)*
z+0.7853981633974483)
END PROC sin approx;
REAL PROC cos approx ( REAL CONST x ):
REAL VAR z::x*x;
((((((-0.3857761864560276e-12*z+0.115004970178141e-9)*z-0.246113638267419e-7
)*z+0.3590860445885748e-5)*z-0.3259918869266875e-3)*z+0.1585434424381541e-1)
*z-0.3084251375340425)*z+1.0
END PROC cos approx;
REAL PROC arctan ( REAL CONST y ):
REAL VAR f, z, x; BOOL VAR neg :: y < 0.0;
IF neg THEN x := -y ELSE x := y FI;
IF x>1.0 THEN f := a ELSE f := -b; neg := NOT neg FI;
z := x * x;
x := x/(((((((0.0107090276046822*z-0.01647757182108040)*z
+0.02177846332482151)*z-0.03019339673273880)*z+0.04656083561183398)*z
-0.0888888888888888)*z+0.3333333333333333)*z+1.0);
IF neg THEN x - f ELSE f - x FI.
a:IF x>sqr3p2 THEN x := 1.0/x; pi2 ELSE x := 4.0/(sqrt3+x+x+x)-sqr3; pi3 FI.
b:IF x<sqr3m2 THEN 0.0 ELSE x := sqrt3 - 4.0/(sqrt3+x); pi6 FI
END PROC arctan;
REAL PROC arctand ( REAL CONST x ):
arctan(x) * pi180
END PROC arctand;
REAL OP ** ( REAL CONST b, e ):
IF b=0.0
THEN IF e=0.0 THEN 1.0 ELSE 0.0 FI
ELIF b < 0.0
THEN errorstop("("+text(b,20)+") ** "+text(e)); (-b) ** e
ELSE exp( e * log2( b ) * ln2 )
FI
END OP **;
REAL OP ** ( REAL CONST a, INT CONST b ) :
REAL VAR p := 1.0 ,
r := a ;
INT VAR n := ABS b ,
m ;
IF (a = 0.0 OR a = -0.0)
THEN IF b = 0
THEN 1.0
ELSE 0.0
FI
ELSE WHILE n>0 REP
m := n DIV 2 ;
IF m + m = n
THEN n := m ;
r := r*r
ELSE n DECR 1 ;
p := p*r
FI
END REP ;
IF b>0
THEN p
ELSE 1.0 / p
FI
FI .
END OP ** ;
REAL PROC random:
rdg:=rdg+pii;rdg:=rdg*rdg;rdg:=rdg*rdg;rdg:=rdg*rdg;rdg:=frac(rdg);rdg
END PROC random;
PROC initializerandom ( REAL CONST z ):
rdg := frac(z)
END PROC initializerandom;
END PACKET mathlib;
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