Used new math lib of C to create new version of Mathlib
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ceriel
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@ -8,19 +8,27 @@ DEFINITION MODULE Mathlib;
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(* Some mathematical constants: *)
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CONST
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(* From: Handbook of Mathematical Functions
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Edited by M. Abramowitz and I.A. Stegun
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National Bureau of Standards
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Applied Mathematics Series 55
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(* From: Computer Approximations
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Hart, Cheney, e.a.
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The SIAM Series in Applied Mathematics
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John Wiley & Sons, INC. New York London Sydney, 1968
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*)
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pi = 3.141592653589793238462643;
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twicepi = 6.283185307179586476925286;
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halfpi = 1.570796326794896619231322;
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quartpi = 0.785398163397448309615661;
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e = 2.718281828459045235360287;
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ln2 = 0.693147180559945309417232;
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ln10 = 2.302585092994045684017992;
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pi = 3.14159265358979323846264338327950288;
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twicepi = 6.28318530717958647692528676655900576;
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halfpi = 1.57079632679489661923132169163975144;
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quartpi = 0.78539816339744830961566084581987572;
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e = 2.71828182845904523536028747135266250;
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ln2 = 0.69314718055994530941723212145817657;
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ln10 = 2.30258509299404568401799145468436421;
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longpi = 3.14159265358979323846264338327950288D;
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longtwicepi = 6.28318530717958647692528676655900576D;
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longhalfpi = 1.57079632679489661923132169163975144D;
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longquartpi = 0.78539816339744830961566084581987572D;
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longe = 2.71828182845904523536028747135266250D;
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longln2 = 0.69314718055994530941723212145817657D;
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longln10 = 2.30258509299404568401799145468436421D;
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(* basic functions *)
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@ -14,15 +14,11 @@ IMPLEMENTATION MODULE Mathlib;
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FROM EM IMPORT FIF, FEF;
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FROM Traps IMPORT Message;
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(* From: Handbook of Mathematical Functions
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Edited by M. Abramowitz and I.A. Stegun
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National Bureau of Standards
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Applied Mathematics Series 55
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*)
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CONST
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OneRadianInDegrees = 57.295779513082320876798155D;
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OneDegreeInRadians = 0.017453292519943295769237D;
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Sqrt2 = 1.41421356237309504880168872420969808D;
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OneOverSqrt2 = 0.70710678118654752440084436210484904D;
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(* basic functions *)
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@ -32,8 +28,7 @@ IMPLEMENTATION MODULE Mathlib;
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END pow;
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PROCEDURE longpow(x: LONGREAL; i: INTEGER): LONGREAL;
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VAR
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val: LONGREAL;
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VAR val: LONGREAL;
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ri: LONGREAL;
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BEGIN
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ri := FLOATD(i);
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@ -93,7 +88,7 @@ IMPLEMENTATION MODULE Mathlib;
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temp := temp / 2.0D;
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exp := exp + 2;
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END;
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FOR i := 0 TO 4 DO
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FOR i := 0 TO 5 DO
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temp := 0.5D*(temp + x/temp);
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END;
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RETURN temp;
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@ -105,42 +100,50 @@ IMPLEMENTATION MODULE Mathlib;
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END exp;
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PROCEDURE longexp(x: LONGREAL): LONGREAL;
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(*
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* n = floor(x / ln2), d = x / ln2 - n
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* exp(x) = exp((x / ln2) * ln2) = exp((n + d) * ln2) =
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* exp(n * ln2) * exp(d * ln2) = 2 ** n * exp(d * ln2)
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*)
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(* 2**x = (Q(x*x)+x*P(x*x))/(Q(x*x)-x*P(x*x)) for x in [0,0.5] *)
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(* Hart & Cheney #1069 *)
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CONST
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a1 = -0.9999999995D;
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a2 = 0.4999999206D;
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a3 = -0.1666653019D;
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a4 = 0.0416573475D;
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a5 = -0.0083013598D;
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a6 = 0.0013298820D;
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a7 = -0.0001413161D;
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p0 = 0.2080384346694663001443843411D+07;
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p1 = 0.3028697169744036299076048876D+05;
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p2 = 0.6061485330061080841615584556D+02;
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q0 = 0.6002720360238832528230907598D+07;
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q1 = 0.3277251518082914423057964422D+06;
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q2 = 0.1749287689093076403844945335D+04;
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q3 = 0.1000000000000000000000000000D+01;
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VAR
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neg: BOOLEAN;
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polval: LONGREAL;
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xPxx, Qxx: LONGREAL;
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n: LONGREAL;
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n1 : INTEGER;
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xsq : LONGREAL;
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large: BOOLEAN;
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BEGIN
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neg := x < 0.0D;
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IF neg THEN
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x := -x;
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END;
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x := FIF(x, 1.0D/LONG(ln2), n) * LONG(ln2);
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polval := 1.0D /(1.0D + x*(a1+x*(a2+x*(a3+x*(a4+x*(a5+x*(a6+x*a7)))))));
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x := FIF(x/longln2, 1.0D, n);
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large := x > 0.5D;
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IF large THEN x := x - 0.5D; END;
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xsq := x*x;
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xPxx := x*((p2*xsq+p1)*xsq+p0);
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Qxx := ((q3*xsq+q2)*xsq+q1)*xsq+q0;
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x := (Qxx + xPxx)/(Qxx - xPxx);
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IF large THEN
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x := x * Sqrt2;
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END;
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n1 := TRUNCD(n + 0.5D);
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WHILE n1 >= 16 DO
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polval := polval * 65536.0D;
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x := x * 65536.0D;
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n1 := n1 - 16;
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END;
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WHILE n1 > 0 DO
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polval := polval * 2.0D;
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x := x * 2.0D;
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DEC(n1);
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END;
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IF neg THEN RETURN 1.0D/polval; END;
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RETURN polval;
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IF neg THEN RETURN 1.0D/x; END;
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RETURN x;
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END longexp;
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PROCEDURE ln(x: REAL): REAL; (* natural log *)
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@ -149,18 +152,23 @@ IMPLEMENTATION MODULE Mathlib;
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END ln;
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PROCEDURE longln(x: LONGREAL): LONGREAL; (* natural log *)
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(* log(x) = z*P(z*z)/Q(z*z), z = (x-1)/(x+1), x in [1/sqrt(2), sqrt(2)]
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Hart & Cheney #2707
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*)
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CONST
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a1 = 0.9999964239D;
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a2 = -0.4998741238D;
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a3 = 0.3317990258D;
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a4 = -0.2407338084D;
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a5 = 0.1676540711D;
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a6 = -0.0953293897D;
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a7 = 0.0360884937D;
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a8 = -0.0064535442D;
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p0 = 0.7504094990777122217455611007D+02;
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p1 = -0.1345669115050430235318253537D+03;
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p2 = 0.7413719213248602512779336470D+02;
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p3 = -0.1277249755012330819984385000D+02;
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p4 = 0.3327108381087686938144000000D+00;
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q0 = 0.3752047495388561108727775374D+02;
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q1 = -0.7979028073715004879439951583D+02;
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q2 = 0.5616126132118257292058560360D+02;
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q3 = -0.1450868091858082685362325000D+02;
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q4 = 0.1000000000000000000000000000D+01;
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VAR
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exp: INTEGER;
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polval: LONGREAL;
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z, zsq: LONGREAL;
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BEGIN
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IF x <= 0.0D THEN
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@ -168,13 +176,15 @@ IMPLEMENTATION MODULE Mathlib;
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HALT
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END;
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x := FEF(x, exp);
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WHILE x < 1.0D DO
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WHILE x < OneOverSqrt2 DO
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x := x + x;
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DEC(exp);
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END;
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x := x - 1.0D;
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polval := x*(a1+x*(a2+x*(a3+x*(a4+x*(a5+x*(a6+x*(a7+a8*x)))))));
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RETURN polval + FLOATD(exp) * LONG(ln2);
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z := (x - 1.0D) / (x + 1.0D);
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zsq := z*z;
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RETURN z * ((((p4*zsq+p3)*zsq+p2)*zsq+p1)*zsq+p0) /
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((((q4*zsq+q3)*zsq+q2)*zsq+q1)*zsq+q0) +
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FLOATD(exp) * longln2;
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END longln;
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PROCEDURE log(x: REAL): REAL; (* log with base 10 *)
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@ -184,7 +194,7 @@ IMPLEMENTATION MODULE Mathlib;
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PROCEDURE longlog(x: LONGREAL): LONGREAL; (* log with base 10 *)
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BEGIN
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RETURN longln(x)/LONG(ln10);
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RETURN longln(x)/longln10;
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END longlog;
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(* trigonometric functions; arguments in radians *)
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@ -194,34 +204,80 @@ IMPLEMENTATION MODULE Mathlib;
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RETURN SHORT(longsin(LONG(x)));
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END sin;
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PROCEDURE longsin(x: LONGREAL): LONGREAL;
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PROCEDURE sinus(x: LONGREAL; quadrant: INTEGER) : LONGREAL;
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(* sin(0.5*pi*x) = x * P(x*x)/Q(x*x) for x in [0,1]
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Hart & Cheney # 3374
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*)
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CONST
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a2 = -0.1666666664D;
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a4 = 0.0083333315D;
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a6 = -0.0001984090D;
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a8 = 0.0000027526D;
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a10 = -0.0000000239D;
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p0 = 0.4857791909822798473837058825D+10;
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p1 = -0.1808816670894030772075877725D+10;
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p2 = 0.1724314784722489597789244188D+09;
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p3 = -0.6351331748520454245913645971D+07;
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p4 = 0.1002087631419532326179108883D+06;
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p5 = -0.5830988897678192576148973679D+03;
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q0 = 0.3092566379840468199410228418D+10;
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q1 = 0.1202384907680254190870913060D+09;
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q2 = 0.2321427631602460953669856368D+07;
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q3 = 0.2848331644063908832127222835D+05;
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q4 = 0.2287602116741682420054505174D+03;
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q5 = 0.1000000000000000000000000000D+01;
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A1 = 6.2822265625D;
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A2 = 0.00095874467958647692528676655900576D;
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VAR
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xsqr: LONGREAL;
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neg: BOOLEAN;
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xsq, x1, x2, n : LONGREAL;
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t : INTEGER;
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BEGIN
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neg := FALSE;
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IF x < 0.0D THEN
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neg := TRUE;
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INC(quadrant, 2);
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x := -x;
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END;
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x := FIF(x, 1.0D / LONG(twicepi), (* dummy *) xsqr) * LONG(twicepi);
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IF x >= LONG(pi) THEN
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neg := NOT neg;
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x := x - LONG(pi);
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IF longhalfpi - x = longhalfpi THEN
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CASE quadrant OF
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RETURN 0.0D;
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RETURN 1.0D;
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RETURN -1.0D;
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END;
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IF x > LONG(halfpi) THEN
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x := LONG(pi) - x;
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END;
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xsqr := x * x;
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x := x * (1.0D + xsqr*(a2+xsqr*(a4+xsqr*(a6+xsqr*(a8+xsqr*a10)))));
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IF neg THEN RETURN -x; END;
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RETURN x;
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IF x >= longtwicepi THEN
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IF x <= FLOATD(MAX(LONGINT)) THEN
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(* Use extended precision to calculate reduced argument.
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Split 2pi in 2 parts a1 and a2, of which the first only
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uses some bits of the mantissa, so that n * a1 is
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exactly representable, where n is the integer part of
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x/pi.
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Here we used 12 bits of the mantissa for a1.
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Also split x in integer part x1 and fraction part x2.
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We then compute x-n*2pi as ((x1 - n*a1) + x2) - n*a2.
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*)
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n := FLOATD(TRUNCD(x/longtwicepi));
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x1 := FLOATD(TRUNCD(x));
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x2 := x - x1;
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x := ((x1 - n * A1) + x2) - n * A2;
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ELSE
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x := FIF(x/longtwicepi, 1.0D, x1) * longtwicepi;
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END
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END;
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x := x / longhalfpi;
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t := TRUNC(x);
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x := x - FLOATD(t);
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quadrant := (quadrant + t MOD 4) MOD 4;
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IF ODD(quadrant) THEN
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x := 1.0D - x;
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END;
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IF quadrant > 1 THEN
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x := -x;
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END;
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xsq := x * x;
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RETURN x * (((((p5*xsq+p4)*xsq+p3)*xsq+p2)*xsq+p1)*xsq+p0) /
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(((((q5*xsq+q4)*xsq+q3)*xsq+q2)*xsq+q1)*xsq+q0);
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END sinus;
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PROCEDURE longsin(x: LONGREAL): LONGREAL;
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BEGIN
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RETURN sinus(x, 0);
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END longsin;
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PROCEDURE cos(x: REAL): REAL;
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@ -230,30 +286,9 @@ IMPLEMENTATION MODULE Mathlib;
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END cos;
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PROCEDURE longcos(x: LONGREAL): LONGREAL;
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CONST
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a2 = -0.4999999963D;
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a4 = 0.0416666418D;
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a6 = -0.0013888397D;
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a8 = 0.0000247609D;
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a10 = -0.0000002605D;
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VAR
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xsqr: LONGREAL;
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neg: BOOLEAN;
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BEGIN
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neg := FALSE;
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IF x < 0.0D THEN x := -x; END;
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x := FIF(x, 1.0D / LONG(twicepi), (* dummy *) xsqr) * LONG(twicepi);
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IF x >= LONG(pi) THEN
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x := LONG(twicepi) - x;
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END;
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IF x > LONG(halfpi) THEN
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neg := NOT neg;
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x := LONG(pi) - x;
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END;
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xsqr := x * x;
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x := 1.0D + xsqr*(a2+xsqr*(a4+xsqr*(a6+xsqr*(a8+xsqr*a10))));
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IF neg THEN RETURN -x; END;
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RETURN x;
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RETURN sinus(x, 1);
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END longcos;
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PROCEDURE tan(x: REAL): REAL;
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@ -277,24 +312,31 @@ IMPLEMENTATION MODULE Mathlib;
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RETURN SHORT(longarcsin(LONG(x)));
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END arcsin;
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PROCEDURE longarcsin(x: LONGREAL): LONGREAL;
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CONST
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a0 = 1.5707963050D;
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a1 = -0.2145988016D;
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a2 = 0.0889789874D;
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a3 = -0.0501743046D;
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a4 = 0.0308918810D;
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a5 = -0.0170881256D;
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a6 = 0.0066700901D;
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a7 = -0.0012624911D;
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PROCEDURE arcsincos(x: LONGREAL; cosfl: BOOLEAN): LONGREAL;
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VAR
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negative : BOOLEAN;
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BEGIN
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IF x < 0.0D THEN x := -x; END;
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negative := x <= 0.0D;
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IF negative THEN x := -x; END;
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IF x > 1.0D THEN
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Message("arcsin: argument > 1");
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Message("arcsin or arccos: argument > 1");
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HALT
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END;
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RETURN LONG(halfpi) -
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longsqrt(1.0D - x)*(a0+x*(a1+x*(a2+x*(a3+x*(a4+x*(a5+x*(a6+x*a7)))))));
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IF x = 1.0D THEN
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x := longhalfpi;
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ELSE
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x := longarctan(x/longsqrt(1.0D - x*x));
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END;
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IF negative THEN x := -x; END;
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IF cosfl THEN
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RETURN longhalfpi - x;
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END;
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RETURN x;
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END arcsincos;
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PROCEDURE longarcsin(x: LONGREAL): LONGREAL;
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BEGIN
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RETURN arcsincos(x, FALSE);
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END longarcsin;
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PROCEDURE arccos(x: REAL): REAL;
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@ -304,7 +346,7 @@ IMPLEMENTATION MODULE Mathlib;
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PROCEDURE longarccos(x: LONGREAL): LONGREAL;
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BEGIN
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RETURN LONG(halfpi) - longarcsin(x);
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RETURN arcsincos(x, TRUE);
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END longarccos;
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PROCEDURE arctan(x: REAL): REAL;
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@ -312,35 +354,109 @@ IMPLEMENTATION MODULE Mathlib;
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RETURN SHORT(longarctan(LONG(x)));
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END arctan;
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TYPE
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precomputed = RECORD
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X: LONGREAL; (* partition point *)
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arctan: LONGREAL; (* arctan of evaluation node *)
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OneOverXn: LONGREAL; (* 1/xn *)
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OneOverXnSquarePlusone: LONGREAL; (* ... *)
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END;
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VAR arctaninit: BOOLEAN;
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precomp : ARRAY[0..4] OF precomputed;
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PROCEDURE longarctan(x: LONGREAL): LONGREAL;
|
||||
(* The interval [0, infinity) is treated as follows:
|
||||
Define partition points Xi
|
||||
X0 = 0
|
||||
X1 = tan(pi/16)
|
||||
X2 = tan(3pi/16)
|
||||
X3 = tan(5pi/16)
|
||||
X4 = tan(7pi/16)
|
||||
X5 = infinity
|
||||
and evaluation nodes xi
|
||||
x2 = tan(2pi/16)
|
||||
x3 = tan(4pi/16)
|
||||
x4 = tan(6pi/16)
|
||||
x5 = infinity
|
||||
An argument x in [Xn-1, Xn] is now reduced to an argument
|
||||
t in [-X1, X1] by the following formulas:
|
||||
|
||||
t = 1/xn - (1/(xn*xn) + 1)/((1/xn) + x)
|
||||
|
||||
arctan(x) = arctan(xi) + arctan(t)
|
||||
|
||||
For the interval [0, p/16] an approximation is used:
|
||||
arctan(x) = x * P(x*x)/Q(x*x)
|
||||
*)
|
||||
(* Hart & Cheney # 5037 *)
|
||||
CONST
|
||||
a2 = -0.3333314528D;
|
||||
a4 = 0.1999355085D;
|
||||
a6 = -0.1420889944D;
|
||||
a8 = 0.1065626393D;
|
||||
a10 = -0.0752896400D;
|
||||
a12 = 0.0429096318D;
|
||||
a14 = -0.0161657367D;
|
||||
a16 = 0.0028662257D;
|
||||
p0 = 0.7698297257888171026986294745D+03;
|
||||
p1 = 0.1557282793158363491416585283D+04;
|
||||
p2 = 0.1033384651675161628243434662D+04;
|
||||
p3 = 0.2485841954911840502660889866D+03;
|
||||
p4 = 0.1566564964979791769948970100D+02;
|
||||
q0 = 0.7698297257888171026986294911D+03;
|
||||
q1 = 0.1813892701754635858982709369D+04;
|
||||
q2 = 0.1484049607102276827437401170D+04;
|
||||
q3 = 0.4904645326203706217748848797D+03;
|
||||
q4 = 0.5593479839280348664778328000D+02;
|
||||
q5 = 0.1000000000000000000000000000D+01;
|
||||
VAR
|
||||
xsqr: LONGREAL;
|
||||
rev: BOOLEAN;
|
||||
neg: BOOLEAN;
|
||||
i: INTEGER;
|
||||
BEGIN
|
||||
rev := FALSE;
|
||||
IF NOT arctaninit THEN
|
||||
arctaninit := TRUE;
|
||||
WITH precomp[0] DO
|
||||
X := 0.19891236737965800691159762264467622;
|
||||
arctan := 0.0D;
|
||||
OneOverXn := 0.0D;
|
||||
OneOverXnSquarePlusone := 0.0D;
|
||||
END;
|
||||
WITH precomp[1] DO
|
||||
X := 0.66817863791929891999775768652308076;
|
||||
arctan := longpi/8.0D;
|
||||
OneOverXn := 2.41421356237309504880168872420969808;
|
||||
OneOverXnSquarePlusone := 6.82842712474619009760337744841939616;
|
||||
END;
|
||||
WITH precomp[2] DO
|
||||
X := 1.49660576266548901760113513494247691;
|
||||
arctan := longquartpi;
|
||||
OneOverXn := 1.0;
|
||||
OneOverXnSquarePlusone := 2.0;
|
||||
END;
|
||||
WITH precomp[3] DO
|
||||
X := 5.02733949212584810451497507106407238;
|
||||
arctan := 3.0D*longpi/8.0D;
|
||||
OneOverXn := 0.41421356237309504880168872420969808;
|
||||
OneOverXnSquarePlusone := 1.17157287525380998659662255158060384;
|
||||
END;
|
||||
WITH precomp[4] DO
|
||||
X := 0.0;
|
||||
arctan := longhalfpi;
|
||||
OneOverXn := 0.0;
|
||||
OneOverXnSquarePlusone := 1.0;
|
||||
END;
|
||||
END;
|
||||
neg := FALSE;
|
||||
IF x < 0.0D THEN
|
||||
neg := TRUE;
|
||||
x := -x;
|
||||
END;
|
||||
IF x > 1.0D THEN
|
||||
rev := TRUE;
|
||||
x := 1.0D / x;
|
||||
i := 0;
|
||||
WHILE (i <= 3) AND (x <= precomp[i].X) DO
|
||||
INC(i);
|
||||
END;
|
||||
IF (i # 0) THEN
|
||||
WITH precomp[i] DO
|
||||
x := arctan + longarctan(OneOverXn-OneOverXnSquarePlusone/(OneOverXn+x));
|
||||
END
|
||||
ELSE
|
||||
xsqr := x * x;
|
||||
x := x * (1.0D + xsqr*(a2+xsqr*(a4+xsqr*(a6+xsqr*(a8+xsqr*(a10+xsqr*(a12+xsqr*(a14+xsqr*a16))))))));
|
||||
IF rev THEN
|
||||
x := LONG(quartpi) - x;
|
||||
x := x * ((((p4*xsqr+p3)*xsqr+p2)*xsqr+p1)*xsqr+p0) /
|
||||
(((((q5*xsqr+q4)*xsqr+q3)*xsqr+q2)*xsqr+q1)*xsqr+q0);
|
||||
END;
|
||||
IF neg THEN RETURN -x; END;
|
||||
RETURN x;
|
||||
|
|
@ -452,4 +568,6 @@ IMPLEMENTATION MODULE Mathlib;
|
|||
RETURN x * OneDegreeInRadians;
|
||||
END longDegreeToRadian;
|
||||
|
||||
BEGIN
|
||||
arctaninit := FALSE;
|
||||
END Mathlib.
|
||||
|
|
|
|||
Loading…
Reference in a new issue