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+/* A C version of Kahan's Floating Point Test "Paranoia"
+
+ Thos Sumner, UCSF, Feb. 1985
+ David Gay, BTL, Jan. 1986
+
+ This is a rewrite from the Pascal version by
+
+ B. A. Wichmann, 18 Jan. 1985
+
+ (and does NOT exhibit good C programming style).
+
+(C) Apr 19 1983 in BASIC version by:
+ Professor W. M. Kahan,
+ 567 Evans Hall
+ Electrical Engineering & Computer Science Dept.
+ University of California
+ Berkeley, California 94720
+ USA
+
+converted to Pascal by:
+ B. A. Wichmann
+ National Physical Laboratory
+ Teddington Middx
+ TW11 OLW
+ UK
+
+converted to C by:
+
+ David M. Gay and Thos Sumner
+ AT&T Bell Labs Computer Center, Rm. U-76
+ 600 Mountainn Avenue University of California
+ Murray Hill, NJ 07974 San Francisco, CA 94143
+ USA USA
+
+with simultaneous corrections to the Pascal source (reflected
+in the Pascal source available over netlib).
+
+Reports of results on various systems from all the versions
+of Paranoia are being collected by Richard Karpinski at the
+same address as Thos Sumner. This includes sample outputs,
+bug reports, and criticisms.
+
+You may copy this program freely if you acknowledge its source.
+Comments on the Pascal version to NPL, please.
+
+
+The C version catches signals from floating-point exceptions.
+If signal(SIGFPE,...) is unavailable in your environment, you may
+#define NOSIGNAL to comment out the invocations of signal.
+
+This source file is too big for some C compilers, but may be split
+into pieces. Comments containing "SPLIT" suggest convenient places
+for this splitting. At the end of these comments is an "ed script"
+(for the UNIX(tm) editor ed) that will do this splitting.
+
+By #defining Single when you compile this source, you may obtain
+a single-precision C version of Paranoia.
+
+
+The following is from the introductory commentary from Wichmann's work:
+
+The BASIC program of Kahan is written in Microsoft BASIC using many
+facilities which have no exact analogy in Pascal. The Pascal
+version below cannot therefore be exactly the same. Rather than be
+a minimal transcription of the BASIC program, the Pascal coding
+follows the conventional style of block-structured languages. Hence
+the Pascal version could be useful in producing versions in other
+structured languages.
+
+Rather than use identifiers of minimal length (which therefore have
+little mnemonic significance), the Pascal version uses meaningful
+identifiers as follows [Note: A few changes have been made for C]:
+
+
+BASIC C BASIC C BASIC C
+
+ A J S StickyBit
+ A1 AInverse J0 NoErrors T
+ B Radix [Failure] T0 Underflow
+ B1 BInverse J1 NoErrors T2 ThirtyTwo
+ B2 RadixD2 [SeriousDefect] T5 OneAndHalf
+ B9 BMinusU2 J2 NoErrors T7 TwentySeven
+ C [Defect] T8 TwoForty
+ C1 CInverse J3 NoErrors U OneUlp
+ D [Flaw] U0 UnderflowThreshold
+ D4 FourD K PageNo U1
+ E0 L Milestone U2
+ E1 M V
+ E2 Exp2 N V0
+ E3 N1 V8
+ E5 MinSqEr O Zero V9
+ E6 SqEr O1 One W
+ E7 MaxSqEr O2 Two X
+ E8 O3 Three X1
+ E9 O4 Four X8
+ F1 MinusOne O5 Five X9 Random1
+ F2 Half O8 Eight Y
+ F3 Third O9 Nine Y1
+ F6 P Precision Y2
+ F9 Q Y9 Random2
+ G1 GMult Q8 Z
+ G2 GDiv Q9 Z0 PseudoZero
+ G3 GAddSub R Z1
+ H R1 RMult Z2
+ H1 HInverse R2 RDiv Z9
+ I R3 RAddSub
+ IO NoTrials R4 RSqrt
+ I3 IEEE R9 Random9
+
+ SqRWrng
+
+All the variables in BASIC are true variables and in consequence,
+the program is more difficult to follow since the "constants" must
+be determined (the glossary is very helpful). The Pascal version
+uses Real constants, but checks are added to ensure that the values
+are correctly converted by the compiler.
+
+The major textual change to the Pascal version apart from the
+identifiersis that named procedures are used, inserting parameters
+wherehelpful. New procedures are also introduced. The
+correspondence is as follows:
+
+
+BASIC Pascal
+lines
+
+ 90- 140 Pause
+ 170- 250 Instructions
+ 380- 460 Heading
+ 480- 670 Characteristics
+ 690- 870 History
+2940-2950 Random
+3710-3740 NewD
+4040-4080 DoesYequalX
+4090-4110 PrintIfNPositive
+4640-4850 TestPartialUnderflow
+
+=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=
+
+Below is an "ed script" that splits para.c into 10 files
+of the form part[1-8].c, subs.c, and msgs.c, plus a header
+file, paranoia.h, that these files require.
+r paranoia.c
+$
+?SPLIT
++,$w msgs.c
+.,$d
+?SPLIT
+.d
++d
+-,$w subs.c
+-,$d
+?part8
++d
+?include
+.,$w part8.c
+.,$d
+-d
+?part7
++d
+?include
+.,$w part7.c
+.,$d
+-d
+?part6
++d
+?include
+.,$w part6.c
+.,$d
+-d
+?part5
++d
+?include
+.,$w part5.c
+.,$d
+-d
+?part4
++d
+?include
+.,$w part4.c
+.,$d
+-d
+?part3
++d
+?include
+.,$w part3.c
+.,$d
+-d
+?part2
++d
+?include
+.,$w part2.c
+.,$d
+?SPLIT
+.d
+1,/^#include/-1d
+1,$w part1.c
+/Computed constants/,$d
+1,$s/^int/extern &/
+1,$s/^FLOAT/extern &/
+1,$s! = .*!;!
+/^Guard/,/^Round/s/^/extern /
+/^jmp_buf/s/^/extern /
+/^Sig_type/s/^/extern /
+a
+extern int sigfpe();
+.
+w paranoia.h
+q
+
+*/
+
+#include <stdio.h>
+#ifndef NOSIGNAL
+#include <signal.h>
+#endif
+#include <setjmp.h>
+
+extern double fabs(), floor(), log(), pow(), sqrt();
+
+#ifdef Single
+#define FLOAT float
+#define FABS(x) (float)fabs((double)(x))
+#define FLOOR(x) (float)floor((double)(x))
+#define LOG(x) (float)log((double)(x))
+#define POW(x,y) (float)pow((double)(x),(double)(y))
+#define SQRT(x) (float)sqrt((double)(x))
+#else
+#define FLOAT double
+#define FABS(x) fabs(x)
+#define FLOOR(x) floor(x)
+#define LOG(x) log(x)
+#define POW(x,y) pow(x,y)
+#define SQRT(x) sqrt(x)
+#endif
+
+jmp_buf ovfl_buf;
+typedef int (*Sig_type)();
+Sig_type sigsave;
+
+#define KEYBOARD 0
+
+FLOAT Radix, BInvrse, RadixD2, BMinusU2;
+FLOAT Sign(), Random();
+
+/*Small floating point constants.*/
+FLOAT Zero = 0.0;
+FLOAT Half = 0.5;
+FLOAT One = 1.0;
+FLOAT Two = 2.0;
+FLOAT Three = 3.0;
+FLOAT Four = 4.0;
+FLOAT Five = 5.0;
+FLOAT Eight = 8.0;
+FLOAT Nine = 9.0;
+FLOAT TwentySeven = 27.0;
+FLOAT ThirtyTwo = 32.0;
+FLOAT TwoForty = 240.0;
+FLOAT MinusOne = -1.0;
+FLOAT OneAndHalf = 1.5;
+/*Integer constants*/
+int NoTrials = 20; /*Number of tests for commutativity. */
+#define False 0
+#define True 1
+
+/* Definitions for declared types
+ Guard == (Yes, No);
+ Rounding == (Chopped, Rounded, Other);
+ Message == packed array [1..40] of char;
+ Class == (Flaw, Defect, Serious, Failure);
+ */
+#define Yes 1
+#define No 0
+#define Chopped 2
+#define Rounded 1
+#define Other 0
+#define Flaw 3
+#define Defect 2
+#define Serious 1
+#define Failure 0
+typedef int Guard, Rounding, Class;
+typedef char Message;
+
+/* Declarations of Variables */
+int Indx;
+char ch[8];
+FLOAT AInvrse, A1;
+FLOAT C, CInvrse;
+FLOAT D, FourD;
+FLOAT E0, E1, Exp2, E3, MinSqEr;
+FLOAT SqEr, MaxSqEr, E9;
+FLOAT Third;
+FLOAT F6, F9;
+FLOAT H, HInvrse;
+int I;
+FLOAT StickyBit, J;
+FLOAT MyZero;
+FLOAT Precision;
+FLOAT Q, Q9;
+FLOAT R, Random9;
+FLOAT T, Underflow, S;
+FLOAT OneUlp, UfThold, U1, U2;
+FLOAT V, V0, V9;
+FLOAT W;
+FLOAT X, X1, X2, X8, Random1;
+FLOAT Y, Y1, Y2, Random2;
+FLOAT Z, PseudoZero, Z1, Z2, Z9;
+volatile FLOAT VV;
+int ErrCnt[4];
+int fpecount;
+int Milestone;
+int PageNo;
+int M, N, N1;
+Guard GMult, GDiv, GAddSub;
+Rounding RMult, RDiv, RAddSub, RSqrt;
+int Break, Done, NotMonot, Monot, Anomaly, IEEE,
+ SqRWrng, UfNGrad;
+/* Computed constants. */
+/*U1 gap below 1.0, i.e, 1.0-U1 is next number below 1.0 */
+/*U2 gap above 1.0, i.e, 1.0+U2 is next number above 1.0 */
+
+/* floating point exception receiver */
+sigfpe()
+{
+ fpecount++;
+ printf("\n* * * FLOATING-POINT ERROR * * *\n");
+ fflush(stdout);
+ if (sigsave) {
+#ifndef NOSIGNAL
+ signal(SIGFPE, sigsave);
+#endif
+ sigsave = 0;
+ longjmp(ovfl_buf, 1);
+ }
+ abort();
+}
+
+main()
+{
+ /* Set coprocessor to double precision, no arith traps. */
+ /* __setfpucw(0x127f);*/
+ dprec();
+ /* First two assignments use integer right-hand sides. */
+ Zero = 0;
+ One = 1;
+ Two = One + One;
+ Three = Two + One;
+ Four = Three + One;
+ Five = Four + One;
+ Eight = Four + Four;
+ Nine = Three * Three;
+ TwentySeven = Nine * Three;
+ ThirtyTwo = Four * Eight;
+ TwoForty = Four * Five * Three * Four;
+ MinusOne = -One;
+ Half = One / Two;
+ OneAndHalf = One + Half;
+ ErrCnt[Failure] = 0;
+ ErrCnt[Serious] = 0;
+ ErrCnt[Defect] = 0;
+ ErrCnt[Flaw] = 0;
+ PageNo = 1;
+ /*=============================================*/
+ Milestone = 0;
+ /*=============================================*/
+#ifndef NOSIGNAL
+ signal(SIGFPE, sigfpe);
+#endif
+ Instructions();
+ Pause();
+ Heading();
+ Pause();
+ Characteristics();
+ Pause();
+ History();
+ Pause();
+ /*=============================================*/
+ Milestone = 7;
+ /*=============================================*/
+ printf("Program is now RUNNING tests on small integers:\n");
+
+ TstCond (Failure, (Zero + Zero == Zero) && (One - One == Zero)
+ && (One > Zero) && (One + One == Two),
+ "0+0 != 0, 1-1 != 0, 1 <= 0, or 1+1 != 2");
+ Z = - Zero;
+ if (Z == 0.0) {
+ U1 = 0.001;
+ Radix = 1;
+ TstPtUf();
+ }
+ else {
+ ErrCnt[Failure] = ErrCnt[Failure] + 1;
+ printf("Comparison alleges that -0.0 is Non-zero!\n");
+ }
+ TstCond (Failure, (Three == Two + One) && (Four == Three + One)
+ && (Four + Two * (- Two) == Zero)
+ && (Four - Three - One == Zero),
+ "3 != 2+1, 4 != 3+1, 4+2*(-2) != 0, or 4-3-1 != 0");
+ TstCond (Failure, (MinusOne == (0 - One))
+ && (MinusOne + One == Zero ) && (One + MinusOne == Zero)
+ && (MinusOne + FABS(One) == Zero)
+ && (MinusOne + MinusOne * MinusOne == Zero),
+ "-1+1 != 0, (-1)+abs(1) != 0, or -1+(-1)*(-1) != 0");
+ TstCond (Failure, Half + MinusOne + Half == Zero,
+ "1/2 + (-1) + 1/2 != 0");
+ /*=============================================*/
+ /*SPLIT
+ part2();
+ part3();
+ part4();
+ part5();
+ part6();
+ part7();
+ part8();
+ }
+#include "paranoia.h"
+part2(){
+*/
+ Milestone = 10;
+ /*=============================================*/
+ TstCond (Failure, (Nine == Three * Three)
+ && (TwentySeven == Nine * Three) && (Eight == Four + Four)
+ && (ThirtyTwo == Eight * Four)
+ && (ThirtyTwo - TwentySeven - Four - One == Zero),
+ "9 != 3*3, 27 != 9*3, 32 != 8*4, or 32-27-4-1 != 0");
+ TstCond (Failure, (Five == Four + One) &&
+ (TwoForty == Four * Five * Three * Four)
+ && (TwoForty / Three - Four * Four * Five == Zero)
+ && ( TwoForty / Four - Five * Three * Four == Zero)
+ && ( TwoForty / Five - Four * Three * Four == Zero),
+ "5 != 4+1, 240/3 != 80, 240/4 != 60, or 240/5 != 48");
+ if (ErrCnt[Failure] == 0) {
+ printf("-1, 0, 1/2, 1, 2, 3, 4, 5, 9, 27, 32 & 240 are O.K.\n");
+ printf("\n");
+ }
+ printf("Searching for Radix and Precision.\n");
+ W = One;
+ do {
+ W = W + W;
+ Y = W + One;
+ Z = Y - W;
+ Y = Z - One;
+ } while (MinusOne + FABS(Y) < Zero);
+ /*.. now W is just big enough that |((W+1)-W)-1| >= 1 ...*/
+ Precision = Zero;
+ Y = One;
+ do {
+ Radix = W + Y;
+ Y = Y + Y;
+ Radix = Radix - W;
+ } while ( Radix == Zero);
+ if (Radix < Two) Radix = One;
+ printf("Radix = %f .\n", Radix);
+ if (Radix != 1) {
+ W = One;
+ do {
+ Precision = Precision + One;
+ W = W * Radix;
+ Y = W + One;
+ } while ((Y - W) == One);
+ }
+ /*... now W == Radix^Precision is barely too big to satisfy (W+1)-W == 1
+ ...*/
+ U1 = One / W;
+ U2 = Radix * U1;
+ printf("Closest relative separation found is U1 = %.7e .\n\n", U1);
+ printf("Recalculating radix and precision.");
+
+ /*save old values*/
+ E0 = Radix;
+ E1 = U1;
+ E9 = U2;
+ E3 = Precision;
+
+ X = Four / Three;
+ Third = X - One;
+ F6 = Half - Third;
+ X = F6 + F6;
+ X = FABS(X - Third);
+ if (X < U2) X = U2;
+
+ /*... now X = (unknown no.) ulps of 1+...*/
+ do {
+ U2 = X;
+ Y = Half * U2 + ThirtyTwo * U2 * U2;
+ Y = One + Y;
+ X = Y - One;
+ } while ( ! ((U2 <= X) || (X <= Zero)));
+
+ /*... now U2 == 1 ulp of 1 + ... */
+ X = Two / Three;
+ F6 = X - Half;
+ Third = F6 + F6;
+ X = Third - Half;
+ X = FABS(X + F6);
+ if (X < U1) X = U1;
+
+ /*... now X == (unknown no.) ulps of 1 -... */
+ do {
+ U1 = X;
+ Y = Half * U1 + ThirtyTwo * U1 * U1;
+ Y = Half - Y;
+ X = Half + Y;
+ Y = Half - X;
+ X = Half + Y;
+ } while ( ! ((U1 <= X) || (X <= Zero)));
+ /*... now U1 == 1 ulp of 1 - ... */
+ if (U1 == E1) printf("confirms closest relative separation U1 .\n");
+ else printf("gets better closest relative separation U1 = %.7e .\n", U1);
+ W = One / U1;
+ F9 = (Half - U1) + Half;
+ Radix = FLOOR(0.01 + U2 / U1);
+ if (Radix == E0) printf("Radix confirmed.\n");
+ else printf("MYSTERY: recalculated Radix = %.7e .\n", Radix);
+ TstCond (Defect, Radix <= Eight + Eight,
+ "Radix is too big: roundoff problems");
+ TstCond (Flaw, (Radix == Two) || (Radix == 10)
+ || (Radix == One), "Radix is not as good as 2 or 10");
+ /*=============================================*/
+ Milestone = 20;
+ /*=============================================*/
+ TstCond (Failure, F9 - Half < Half,
+ "(1-U1)-1/2 < 1/2 is FALSE, prog. fails?");
+ X = F9;
+ I = 1;
+ Y = X - Half;
+ Z = Y - Half;
+ TstCond (Failure, (X != One)
+ || (Z == Zero), "Comparison is fuzzy,X=1 but X-1/2-1/2 != 0");
+ X = One + U2;
+ I = 0;
+ /*=============================================*/
+ Milestone = 25;
+ /*=============================================*/
+ /*... BMinusU2 = nextafter(Radix, 0) */
+ BMinusU2 = Radix - One;
+ BMinusU2 = (BMinusU2 - U2) + One;
+ /* Purify Integers */
+ if (Radix != One) {
+ X = - TwoForty * LOG(U1) / LOG(Radix);
+ Y = FLOOR(Half + X);
+ if (FABS(X - Y) * Four < One) X = Y;
+ Precision = X / TwoForty;
+ Y = FLOOR(Half + Precision);
+ if (FABS(Precision - Y) * TwoForty < Half) Precision = Y;
+ }
+ if ((Precision != FLOOR(Precision)) || (Radix == One)) {
+ printf("Precision cannot be characterized by an Integer number\n");
+ printf("of significant digits but, by itself, this is a minor flaw.\n");
+ }
+ if (Radix == One)
+ printf("logarithmic encoding has precision characterized solely by U1.\n");
+ else printf("The number of significant digits of the Radix is %f .\n",
+ Precision);
+ TstCond (Serious, U2 * Nine * Nine * TwoForty < One,
+ "Precision worse than 5 decimal figures ");
+ /*=============================================*/
+ Milestone = 30;
+ /*=============================================*/
+ /* Test for extra-precise subepressions */
+ X = FABS(((Four / Three - One) - One / Four) * Three - One / Four);
+ do {
+ Z2 = X;
+ X = (One + (Half * Z2 + ThirtyTwo * Z2 * Z2)) - One;
+ } while ( ! ((Z2 <= X) || (X <= Zero)));
+ X = Y = Z = FABS((Three / Four - Two / Three) * Three - One / Four);
+ do {
+ Z1 = Z;
+ Z = (One / Two - ((One / Two - (Half * Z1 + ThirtyTwo * Z1 * Z1))
+ + One / Two)) + One / Two;
+ } while ( ! ((Z1 <= Z) || (Z <= Zero)));
+ do {
+ do {
+ Y1 = Y;
+ Y = (Half - ((Half - (Half * Y1 + ThirtyTwo * Y1 * Y1)) + Half
+ )) + Half;
+ } while ( ! ((Y1 <= Y) || (Y <= Zero)));
+ X1 = X;
+ X = ((Half * X1 + ThirtyTwo * X1 * X1) - F9) + F9;
+ } while ( ! ((X1 <= X) || (X <= Zero)));
+ if ((X1 != Y1) || (X1 != Z1)) {
+ BadCond(Serious, "Disagreements among the values X1, Y1, Z1,\n");
+ printf("respectively %.7e, %.7e, %.7e,\n", X1, Y1, Z1);
+ printf("are symptoms of inconsistencies introduced\n");
+ printf("by extra-precise evaluation of arithmetic subexpressions.\n");
+ notify("Possibly some part of this");
+ if ((X1 == U1) || (Y1 == U1) || (Z1 == U1)) printf(
+ "That feature is not tested further by this program.\n") ;
+ }
+ else {
+ if ((Z1 != U1) || (Z2 != U2)) {
+ if ((Z1 >= U1) || (Z2 >= U2)) {
+ BadCond(Failure, "");
+ notify("Precision");
+ printf("\tU1 = %.7e, Z1 - U1 = %.7e\n",U1,Z1-U1);
+ printf("\tU2 = %.7e, Z2 - U2 = %.7e\n",U2,Z2-U2);
+ }
+ else {
+ if ((Z1 <= Zero) || (Z2 <= Zero)) {
+ printf("Because of unusual Radix = %f", Radix);
+ printf(", or exact rational arithmetic a result\n");
+ printf("Z1 = %.7e, or Z2 = %.7e ", Z1, Z2);
+ notify("of an\nextra-precision");
+ }
+ if (Z1 != Z2 || Z1 > Zero) {
+ X = Z1 / U1;
+ Y = Z2 / U2;
+ if (Y > X) X = Y;
+ Q = - LOG(X);
+ printf("Some subexpressions appear to be calculated extra\n");
+ printf("precisely with about %g extra B-digits, i.e.\n",
+ (Q / LOG(Radix)));
+ printf("roughly %g extra significant decimals.\n",
+ Q / LOG(10.));
+ }
+ printf("That feature is not tested further by this program.\n");
+ }
+ }
+ }
+ Pause();
+ /*=============================================*/
+ /*SPLIT
+ }
+#include "paranoia.h"
+part3(){
+*/
+ Milestone = 35;
+ /*=============================================*/
+ if (Radix >= Two) {
+ X = W / (Radix * Radix);
+ Y = X + One;
+ Z = Y - X;
+ T = Z + U2;
+ X = T - Z;
+ TstCond (Failure, X == U2,
+ "Subtraction is not normalized X=Y,X+Z != Y+Z!");
+ if (X == U2) printf(
+ "Subtraction appears to be normalized, as it should be.");
+ }
+ printf("\nChecking for guard digit in *, /, and -.\n");
+ Y = F9 * One;
+ Z = One * F9;
+ X = F9 - Half;
+ Y = (Y - Half) - X;
+ Z = (Z - Half) - X;
+ X = One + U2;
+ T = X * Radix;
+ R = Radix * X;
+ X = T - Radix;
+ X = X - Radix * U2;
+ T = R - Radix;
+ T = T - Radix * U2;
+ X = X * (Radix - One);
+ T = T * (Radix - One);
+ if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero)) GMult = Yes;
+ else {
+ GMult = No;
+ TstCond (Serious, False,
+ "* lacks a Guard Digit, so 1*X != X");
+ }
+ Z = Radix * U2;
+ X = One + Z;
+ Y = FABS((X + Z) - X * X) - U2;
+ X = One - U2;
+ Z = FABS((X - U2) - X * X) - U1;
+ TstCond (Failure, (Y <= Zero)
+ && (Z <= Zero), "* gets too many final digits wrong.\n");
+ Y = One - U2;
+ X = One + U2;
+ Z = One / Y;
+ Y = Z - X;
+ X = One / Three;
+ Z = Three / Nine;
+ X = X - Z;
+ T = Nine / TwentySeven;
+ Z = Z - T;
+ TstCond(Defect, X == Zero && Y == Zero && Z == Zero,
+ "Division lacks a Guard Digit, so error can exceed 1 ulp\n\
+or 1/3 and 3/9 and 9/27 may disagree");
+ Y = F9 / One;
+ X = F9 - Half;
+ Y = (Y - Half) - X;
+ X = One + U2;
+ T = X / One;
+ X = T - X;
+ if ((X == Zero) && (Y == Zero) && (Z == Zero)) GDiv = Yes;
+ else {
+ GDiv = No;
+ TstCond (Serious, False,
+ "Division lacks a Guard Digit, so X/1 != X");
+ }
+ X = One / (One + U2);
+ Y = X - Half - Half;
+ TstCond (Serious, Y < Zero,
+ "Computed value of 1/1.000..1 >= 1");
+ X = One - U2;
+ Y = One + Radix * U2;
+ Z = X * Radix;
+ T = Y * Radix;
+ R = Z / Radix;
+ StickyBit = T / Radix;
+ X = R - X;
+ Y = StickyBit - Y;
+ TstCond (Failure, X == Zero && Y == Zero,
+ "* and/or / gets too many last digits wrong");
+ Y = One - U1;
+ X = One - F9;
+ Y = One - Y;
+ T = Radix - U2;
+ Z = Radix - BMinusU2;
+ T = Radix - T;
+ if ((X == U1) && (Y == U1) && (Z == U2) && (T == U2)) GAddSub = Yes;
+ else {
+ GAddSub = No;
+ TstCond (Serious, False,
+ "- lacks Guard Digit, so cancellation is obscured");
+ }
+ if (F9 != One && F9 - One >= Zero) {
+ BadCond(Serious, "comparison alleges (1-U1) < 1 although\n");
+ printf(" subtration yields (1-U1) - 1 = 0 , thereby vitiating\n");
+ printf(" such precautions against division by zero as\n");
+ printf(" ... if (X == 1.0) {.....} else {.../(X-1.0)...}\n");
+ }
+ if (GMult == Yes && GDiv == Yes && GAddSub == Yes) printf(
+ " *, /, and - appear to have guard digits, as they should.\n");
+ /*=============================================*/
+ Milestone = 40;
+ /*=============================================*/
+ Pause();
+ printf("Checking rounding on multiply, divide and add/subtract.\n");
+ RMult = Other;
+ RDiv = Other;
+ RAddSub = Other;
+ RadixD2 = Radix / Two;
+ A1 = Two;
+ Done = False;
+ do {
+ AInvrse = Radix;
+ do {
+ X = AInvrse;
+ AInvrse = AInvrse / A1;
+ } while ( ! (FLOOR(AInvrse) != AInvrse));
+ Done = (X == One) || (A1 > Three);
+ if (! Done) A1 = Nine + One;
+ } while ( ! (Done));
+ if (X == One) A1 = Radix;
+ AInvrse = One / A1;
+ X = A1;
+ Y = AInvrse;
+ Done = False;
+ do {
+ Z = X * Y - Half;
+ TstCond (Failure, Z == Half,
+ "X * (1/X) differs from 1");
+ Done = X == Radix;
+ X = Radix;
+ Y = One / X;
+ } while ( ! (Done));
+ Y2 = One + U2;
+ Y1 = One - U2;
+ X = OneAndHalf - U2;
+ Y = OneAndHalf + U2;
+ Z = (X - U2) * Y2;
+ T = Y * Y1;
+ Z = Z - X;
+ T = T - X;
+ X = X * Y2;
+ Y = (Y + U2) * Y1;
+ X = X - OneAndHalf;
+ Y = Y - OneAndHalf;
+ if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T <= Zero)) {
+ X = (OneAndHalf + U2) * Y2;
+ Y = OneAndHalf - U2 - U2;
+ Z = OneAndHalf + U2 + U2;
+ T = (OneAndHalf - U2) * Y1;
+ X = X - (Z + U2);
+ StickyBit = Y * Y1;
+ S = Z * Y2;
+ T = T - Y;
+ Y = (U2 - Y) + StickyBit;
+ Z = S - (Z + U2 + U2);
+ StickyBit = (Y2 + U2) * Y1;
+ Y1 = Y2 * Y1;
+ StickyBit = StickyBit - Y2;
+ Y1 = Y1 - Half;
+ if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero)
+ && ( StickyBit == Zero) && (Y1 == Half)) {
+ RMult = Rounded;
+ printf("Multiplication appears to round correctly.\n");
+ }
+ else if ((X + U2 == Zero) && (Y < Zero) && (Z + U2 == Zero)
+ && (T < Zero) && (StickyBit + U2 == Zero)
+ && (Y1 < Half)) {
+ RMult = Chopped;
+ printf("Multiplication appears to chop.\n");
+ }
+ else printf("* is neither chopped nor correctly rounded.\n");
+ if ((RMult == Rounded) && (GMult == No)) notify("Multiplication");
+ }
+ else printf("* is neither chopped nor correctly rounded.\n");
+ /*=============================================*/
+ Milestone = 45;
+ /*=============================================*/
+ Y2 = One + U2;
+ Y1 = One - U2;
+ Z = OneAndHalf + U2 + U2;
+ X = Z / Y2;
+ T = OneAndHalf - U2 - U2;
+ Y = (T - U2) / Y1;
+ Z = (Z + U2) / Y2;
+ X = X - OneAndHalf;
+ Y = Y - T;
+ T = T / Y1;
+ Z = Z - (OneAndHalf + U2);
+ T = (U2 - OneAndHalf) + T;
+ if (! ((X > Zero) || (Y > Zero) || (Z > Zero) || (T > Zero))) {
+ X = OneAndHalf / Y2;
+ Y = OneAndHalf - U2;
+ Z = OneAndHalf + U2;
+ X = X - Y;
+ T = OneAndHalf / Y1;
+ Y = Y / Y1;
+ T = T - (Z + U2);
+ Y = Y - Z;
+ Z = Z / Y2;
+ Y1 = (Y2 + U2) / Y2;
+ Z = Z - OneAndHalf;
+ Y2 = Y1 - Y2;
+ Y1 = (F9 - U1) / F9;
+ if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero)
+ && (Y2 == Zero) && (Y2 == Zero)
+ && (Y1 - Half == F9 - Half )) {
+ RDiv = Rounded;
+ printf("Division appears to round correctly.\n");
+ if (GDiv == No) notify("Division");
+ }
+ else if ((X < Zero) && (Y < Zero) && (Z < Zero) && (T < Zero)
+ && (Y2 < Zero) && (Y1 - Half < F9 - Half)) {
+ RDiv = Chopped;
+ printf("Division appears to chop.\n");
+ }
+ }
+ if (RDiv == Other) printf("/ is neither chopped nor correctly rounded.\n");
+ BInvrse = One / Radix;
+ TstCond (Failure, (BInvrse * Radix - Half == Half),
+ "Radix * ( 1 / Radix ) differs from 1");
+ /*=============================================*/
+ /*SPLIT
+ }
+#include "paranoia.h"
+part4(){
+*/
+ Milestone = 50;
+ /*=============================================*/
+ TstCond (Failure, ((F9 + U1) - Half == Half)
+ && ((BMinusU2 + U2 ) - One == Radix - One),
+ "Incomplete carry-propagation in Addition");
+ X = One - U1 * U1;
+ Y = One + U2 * (One - U2);
+ Z = F9 - Half;
+ X = (X - Half) - Z;
+ Y = Y - One;
+ if ((X == Zero) && (Y == Zero)) {
+ RAddSub = Chopped;
+ printf("Add/Subtract appears to be chopped.\n");
+ }
+ if (GAddSub == Yes) {
+ X = (Half + U2) * U2;
+ Y = (Half - U2) * U2;
+ X = One + X;
+ Y = One + Y;
+ X = (One + U2) - X;
+ Y = One - Y;
+ if ((X == Zero) && (Y == Zero)) {
+ X = (Half + U2) * U1;
+ Y = (Half - U2) * U1;
+ X = One - X;
+ Y = One - Y;
+ X = F9 - X;
+ Y = One - Y;
+ if ((X == Zero) && (Y == Zero)) {
+ RAddSub = Rounded;
+ printf("Addition/Subtraction appears to round correctly.\n");
+ if (GAddSub == No) notify("Add/Subtract");
+ }
+ else printf("Addition/Subtraction neither rounds nor chops.\n");
+ }
+ else printf("Addition/Subtraction neither rounds nor chops.\n");
+ }
+ else printf("Addition/Subtraction neither rounds nor chops.\n");
+ S = One;
+ X = One + Half * (One + Half);
+ Y = (One + U2) * Half;
+ Z = X - Y;
+ T = Y - X;
+ StickyBit = Z + T;
+ if (StickyBit != Zero) {
+ S = Zero;
+ BadCond(Flaw, "(X - Y) + (Y - X) is non zero!\n");
+ }
+ StickyBit = Zero;
+ if ((GMult == Yes) && (GDiv == Yes) && (GAddSub == Yes)
+ && (RMult == Rounded) && (RDiv == Rounded)
+ && (RAddSub == Rounded) && (FLOOR(RadixD2) == RadixD2)) {
+ printf("Checking for sticky bit.\n");
+ X = (Half + U1) * U2;
+ Y = Half * U2;
+ Z = One + Y;
+ T = One + X;
+ if ((Z - One <= Zero) && (T - One >= U2)) {
+ Z = T + Y;
+ Y = Z - X;
+ if ((Z - T >= U2) && (Y - T == Zero)) {
+ X = (Half + U1) * U1;
+ Y = Half * U1;
+ Z = One - Y;
+ T = One - X;
+ if ((Z - One == Zero) && (T - F9 == Zero)) {
+ Z = (Half - U1) * U1;
+ T = F9 - Z;
+ Q = F9 - Y;
+ if ((T - F9 == Zero) && (F9 - U1 - Q == Zero)) {
+ Z = (One + U2) * OneAndHalf;
+ T = (OneAndHalf + U2) - Z + U2;
+ X = One + Half / Radix;
+ Y = One + Radix * U2;
+ Z = X * Y;
+ if (T == Zero && X + Radix * U2 - Z == Zero) {
+ if (Radix != Two) {
+ X = Two + U2;
+ Y = X / Two;
+ if ((Y - One == Zero)) StickyBit = S;
+ }
+ else StickyBit = S;
+ }
+ }
+ }
+ }
+ }
+ }
+ if (StickyBit == One) printf("Sticky bit apparently used correctly.\n");
+ else printf("Sticky bit used incorrectly or not at all.\n");
+ TstCond (Flaw, !(GMult == No || GDiv == No || GAddSub == No ||
+ RMult == Other || RDiv == Other || RAddSub == Other),
+ "lack(s) of guard digits or failure(s) to correctly round or chop\n\
+(noted above) count as one flaw in the final tally below");
+ /*=============================================*/
+ Milestone = 60;
+ /*=============================================*/
+ printf("\n");
+ printf("Does Multiplication commute? ");
+ printf("Testing on %d random pairs.\n", NoTrials);
+ Random9 = SQRT(3.0);
+ Random1 = Third;
+ I = 1;
+ do {
+ X = Random();
+ Y = Random();
+ Z9 = Y * X;
+ Z = X * Y;
+ Z9 = Z - Z9;
+ I = I + 1;
+ } while ( ! ((I > NoTrials) || (Z9 != Zero)));
+ if (I == NoTrials) {
+ Random1 = One + Half / Three;
+ Random2 = (U2 + U1) + One;
+ Z = Random1 * Random2;
+ Y = Random2 * Random1;
+ Z9 = (One + Half / Three) * ((U2 + U1) + One) - (One + Half /
+ Three) * ((U2 + U1) + One);
+ }
+ if (! ((I == NoTrials) || (Z9 == Zero)))
+ BadCond(Defect, "X * Y == Y * X trial fails.\n");
+ else printf(" No failures found in %d integer pairs.\n", NoTrials);
+ /*=============================================*/
+ Milestone = 70;
+ /*=============================================*/
+ printf("\nRunning test of square root(x).\n");
+ TstCond (Failure, (Zero == SQRT(Zero))
+ && (- Zero == SQRT(- Zero))
+ && (One == SQRT(One)), "Square root of 0.0, -0.0 or 1.0 wrong");
+ MinSqEr = Zero;
+ MaxSqEr = Zero;
+ J = Zero;
+ X = Radix;
+ OneUlp = U2;
+ SqXMinX (Serious);
+ X = BInvrse;
+ OneUlp = BInvrse * U1;
+ SqXMinX (Serious);
+ X = U1;
+ OneUlp = U1 * U1;
+ SqXMinX (Serious);
+ if (J != Zero) Pause();
+ printf("Testing if sqrt(X * X) == X for %d Integers X.\n", NoTrials);
+ J = Zero;
+ X = Two;
+ Y = Radix;
+ if ((Radix != One)) do {
+ X = Y;
+ Y = Radix * Y;
+ } while ( ! ((Y - X >= NoTrials)));
+ OneUlp = X * U2;
+ I = 1;
+ while (I < 10) {
+ X = X + One;
+ SqXMinX (Defect);
+ if (J > Zero) break;
+ I = I + 1;
+ }
+ printf("Test for sqrt monotonicity.\n");
+ I = - 1;
+ X = BMinusU2;
+ Y = Radix;
+ Z = Radix + Radix * U2;
+ NotMonot = False;
+ Monot = False;
+ while ( ! (NotMonot || Monot)) {
+ I = I + 1;
+ X = SQRT(X);
+ Q = SQRT(Y);
+ Z = SQRT(Z);
+ if ((X > Q) || (Q > Z)) NotMonot = True;
+ else {
+ Q = FLOOR(Q + Half);
+ if ((I > 0) || (Radix == Q * Q)) Monot = True;
+ else if (I > 0) {
+ if (I > 1) Monot = True;
+ else {
+ Y = Y * BInvrse;
+ X = Y - U1;
+ Z = Y + U1;
+ }
+ }
+ else {
+ Y = Q;
+ X = Y - U2;
+ Z = Y + U2;
+ }
+ }
+ }
+ if (Monot) printf("sqrt has passed a test for Monotonicity.\n");
+ else {
+ BadCond(Defect, "");
+ printf("sqrt(X) is non-monotonic for X near %.7e .\n", Y);
+ }
+ /*=============================================*/
+ /*SPLIT
+ }
+#include "paranoia.h"
+part5(){
+*/
+ Milestone = 80;
+ /*=============================================*/
+ MinSqEr = MinSqEr + Half;
+ MaxSqEr = MaxSqEr - Half;
+ Y = (SQRT(One + U2) - One) / U2;
+ SqEr = (Y - One) + U2 / Eight;
+ if (SqEr > MaxSqEr) MaxSqEr = SqEr;
+ SqEr = Y + U2 / Eight;
+ if (SqEr < MinSqEr) MinSqEr = SqEr;
+ Y = ((SQRT(F9) - U2) - (One - U2)) / U1;
+ SqEr = Y + U1 / Eight;
+ if (SqEr > MaxSqEr) MaxSqEr = SqEr;
+ SqEr = (Y + One) + U1 / Eight;
+ if (SqEr < MinSqEr) MinSqEr = SqEr;
+ OneUlp = U2;
+ X = OneUlp;
+ for( Indx = 1; Indx <= 3; ++Indx) {
+ Y = SQRT((X + U1 + X) + F9);
+ Y = ((Y - U2) - ((One - U2) + X)) / OneUlp;
+ Z = ((U1 - X) + F9) * Half * X * X / OneUlp;
+ SqEr = (Y + Half) + Z;
+ if (SqEr < MinSqEr) MinSqEr = SqEr;
+ SqEr = (Y - Half) + Z;
+ if (SqEr > MaxSqEr) MaxSqEr = SqEr;
+ if (((Indx == 1) || (Indx == 3)))
+ X = OneUlp * Sign (X) * FLOOR(Eight / (Nine * SQRT(OneUlp)));
+ else {
+ OneUlp = U1;
+ X = - OneUlp;
+ }
+ }
+ /*=============================================*/
+ Milestone = 85;
+ /*=============================================*/
+ SqRWrng = False;
+ Anomaly = False;
+ if (Radix != One) {
+ printf("Testing whether sqrt is rounded or chopped.\n");
+ D = FLOOR(Half + POW(Radix, One + Precision - FLOOR(Precision)));
+ /* ... == Radix^(1 + fract) if (Precision == Integer + fract. */
+ X = D / Radix;
+ Y = D / A1;
+ if ((X != FLOOR(X)) || (Y != FLOOR(Y))) {
+ Anomaly = True;
+ }
+ else {
+ X = Zero;
+ Z2 = X;
+ Y = One;
+ Y2 = Y;
+ Z1 = Radix - One;
+ FourD = Four * D;
+ do {
+ if (Y2 > Z2) {
+ Q = Radix;
+ Y1 = Y;
+ do {
+ X1 = FABS(Q + FLOOR(Half - Q / Y1) * Y1);
+ Q = Y1;
+ Y1 = X1;
+ } while ( ! (X1 <= Zero));
+ if (Q <= One) {
+ Z2 = Y2;
+ Z = Y;
+ }
+ }
+ Y = Y + Two;
+ X = X + Eight;
+ Y2 = Y2 + X;
+ if (Y2 >= FourD) Y2 = Y2 - FourD;
+ } while ( ! (Y >= D));
+ X8 = FourD - Z2;
+ Q = (X8 + Z * Z) / FourD;
+ X8 = X8 / Eight;
+ if (Q != FLOOR(Q)) Anomaly = True;
+ else {
+ Break = False;
+ do {
+ X = Z1 * Z;
+ X = X - FLOOR(X / Radix) * Radix;
+ if (X == One)
+ Break = True;
+ else
+ Z1 = Z1 - One;
+ } while ( ! (Break || (Z1 <= Zero)));
+ if ((Z1 <= Zero) && (! Break)) Anomaly = True;
+ else {
+ if (Z1 > RadixD2) Z1 = Z1 - Radix;
+ do {
+ NewD();
+ } while ( ! (U2 * D >= F9));
+ if (D * Radix - D != W - D) Anomaly = True;
+ else {
+ Z2 = D;
+ I = 0;
+ Y = D + (One + Z) * Half;
+ X = D + Z + Q;
+ SR3750();
+ Y = D + (One - Z) * Half + D;
+ X = D - Z + D;
+ X = X + Q + X;
+ SR3750();
+ NewD();
+ if (D - Z2 != W - Z2) Anomaly = True;
+ else {
+ Y = (D - Z2) + (Z2 + (One - Z) * Half);
+ X = (D - Z2) + (Z2 - Z + Q);
+ SR3750();
+ Y = (One + Z) * Half;
+ X = Q;
+ SR3750();
+ if (I == 0) Anomaly = True;
+ }
+ }
+ }
+ }
+ }
+ if ((I == 0) || Anomaly) {
+ BadCond(Failure, "Anomalous arithmetic with Integer < ");
+ printf("Radix^Precision = %.7e\n", W);
+ printf(" fails test whether sqrt rounds or chops.\n");
+ SqRWrng = True;
+ }
+ }
+ if (! Anomaly) {
+ if (! ((MinSqEr < Zero) || (MaxSqEr > Zero))) {
+ RSqrt = Rounded;
+ printf("Square root appears to be correctly rounded.\n");
+ }
+ else {
+ if ((MaxSqEr + U2 > U2 - Half) || (MinSqEr > Half)
+ || (MinSqEr + Radix < Half)) SqRWrng = True;
+ else {
+ RSqrt = Chopped;
+ printf("Square root appears to be chopped.\n");
+ }
+ }
+ }
+ if (SqRWrng) {
+ printf("Square root is neither chopped nor correctly rounded.\n");
+ printf("Observed errors run from %.7e ", MinSqEr - Half);
+ printf("to %.7e ulps.\n", Half + MaxSqEr);
+ TstCond (Serious, MaxSqEr - MinSqEr < Radix * Radix,
+ "sqrt gets too many last digits wrong");
+ }
+ /*=============================================*/
+ Milestone = 90;
+ /*=============================================*/
+ Pause();
+ printf("Testing powers Z^i for small Integers Z and i.\n");
+ N = 0;
+ /* ... test powers of zero. */
+ I = 0;
+ Z = -Zero;
+ M = 3.0;
+ Break = False;
+ do {
+ X = One;
+ SR3980();
+ if (I <= 10) {
+ I = 1023;
+ SR3980();
+ }
+ if (Z == MinusOne) Break = True;
+ else {
+ Z = MinusOne;
+ PrintIfNPositive();
+ N = 0;
+ /* .. if(-1)^N is invalid, replace MinusOne by One. */
+ I = - 4;
+ }
+ } while ( ! Break);
+ PrintIfNPositive();
+ N1 = N;
+ N = 0;
+ Z = A1;
+ M = FLOOR(Two * LOG(W) / LOG(A1));
+ Break = False;
+ do {
+ X = Z;
+ I = 1;
+ SR3980();
+ if (Z == AInvrse) Break = True;
+ else Z = AInvrse;
+ } while ( ! (Break));
+ /*=============================================*/
+ Milestone = 100;
+ /*=============================================*/
+ /* Powers of Radix have been tested, */
+ /* next try a few primes */
+ M = NoTrials;
+ Z = Three;
+ do {
+ X = Z;
+ I = 1;
+ SR3980();
+ do {
+ Z = Z + Two;
+ } while ( Three * FLOOR(Z / Three) == Z );
+ } while ( Z < Eight * Three );
+ if (N > 0) {
+ printf("Errors like this may invalidate financial calculations\n");
+ printf("\tinvolving interest rates.\n");
+ }
+ PrintIfNPositive();
+ N += N1;
+ if (N == 0) printf("... no discrepancis found.\n");
+ if (N > 0) Pause();
+ else printf("\n");
+ /*=============================================*/
+ /*SPLIT
+ }
+#include "paranoia.h"
+part6(){
+*/
+ Milestone = 110;
+ /*=============================================*/
+ printf("Seeking Underflow thresholds UfThold and E0.\n");
+ D = U1;
+ if (Precision != FLOOR(Precision)) {
+ D = BInvrse;
+ X = Precision;
+ do {
+ D = D * BInvrse;
+ X = X - One;
+ } while ( X > Zero);
+ }
+ Y = One;
+ Z = D;
+ /* ... D is power of 1/Radix < 1. */
+ do {
+ C = Y;
+ Y = Z;
+ Z = Y * Y;
+ VV = Z;
+ } while ((Y > Z) && (VV + VV > VV));
+ Y = C;
+ Z = Y * D;
+ do {
+ C = Y;
+ Y = Z;
+ Z = Y * D;
+ VV = Z;
+ } while ((Y > Z) && (VV + VV > VV));
+ if (Radix < Two) HInvrse = Two;
+ else HInvrse = Radix;
+ H = One / HInvrse;
+ /* ... 1/HInvrse == H == Min(1/Radix, 1/2) */
+ CInvrse = One / C;
+ E0 = C;
+ Z = E0 * H;
+ /* ...1/Radix^(BIG Integer) << 1 << CInvrse == 1/C */
+ do {
+ Y = E0;
+ E0 = Z;
+ Z = E0 * H;
+ VV = Z;
+ } while ((E0 > VV) && (VV + VV > VV));
+ UfThold = E0;
+ E1 = Zero;
+ Q = Zero;
+ E9 = U2;
+ S = One + E9;
+ D = C * S;
+ if (D <= C) {
+ E9 = Radix * U2;
+ S = One + E9;
+ D = C * S;
+ if (D <= C) {
+ BadCond(Failure, "multiplication gets too many last digits wrong.\n");
+ Underflow = E0;
+ Y1 = Zero;
+ PseudoZero = Z;
+ Pause();
+ }
+ }
+ else {
+ Underflow = D;
+ PseudoZero = Underflow * H;
+ UfThold = Zero;
+ do {
+ Y1 = Underflow;
+ Underflow = PseudoZero;
+ if (E1 + E1 <= E1) {
+ Y2 = Underflow * HInvrse;
+ E1 = FABS(Y1 - Y2);
+ Q = Y1;
+ if ((UfThold == Zero) && (Y1 != Y2)) UfThold = Y1;
+ }
+ PseudoZero = PseudoZero * H;
+ VV = PseudoZero;
+ } while ((Underflow > VV)
+ && (VV + VV > VV));
+ }
+ /* Comment line 4530 .. 4560 */
+ if (PseudoZero != Zero) {
+ printf("\n");
+ Z = PseudoZero;
+ /* ... Test PseudoZero for "phoney- zero" violates */
+ /* ... PseudoZero < Underflow or PseudoZero < PseudoZero + PseudoZero
+ ... */
+ if (PseudoZero <= Zero) {
+ BadCond(Failure, "Positive expressions can underflow to an\n");
+ printf("allegedly negative value\n");
+ printf("PseudoZero that prints out as: %g .\n", PseudoZero);
+ X = - PseudoZero;
+ if (X <= Zero) {
+ printf("But -PseudoZero, which should be\n");
+ printf("positive, isn't; it prints out as %g .\n", X);
+ }
+ }
+ else {
+ BadCond(Flaw, "Underflow can stick at an allegedly positive\n");
+ printf("value PseudoZero that prints out as %g .\n", PseudoZero);
+ }
+ TstPtUf();
+ }
+ /*=============================================*/
+ Milestone = 120;
+ /*=============================================*/
+ if (CInvrse * Y > CInvrse * Y1) {
+ S = H * S;
+ E0 = Underflow;
+ }
+ if (! ((E1 == Zero) || (E1 == E0))) {
+ BadCond(Defect, "");
+ if (E1 < E0) {
+ printf("Products underflow at a higher");
+ printf(" threshold than differences.\n");
+ if (PseudoZero == Zero)
+ E0 = E1;
+ }
+ else {
+ printf("Difference underflows at a higher");
+ printf(" threshold than products.\n");
+ }
+ }
+ printf("Smallest strictly positive number found is E0 = %g .\n", E0);
+ Z = E0;
+ TstPtUf();
+ Underflow = E0;
+ if (N == 1) Underflow = Y;
+ I = 4;
+ if (E1 == Zero) I = 3;
+ if (UfThold == Zero) I = I - 2;
+ UfNGrad = True;
+ switch (I) {
+ case 1:
+ UfThold = Underflow;
+ if ((CInvrse * Q) != ((CInvrse * Y) * S)) {
+ UfThold = Y;
+ BadCond(Failure, "Either accuracy deteriorates as numbers\n");
+ printf("approach a threshold = %.17e\n", UfThold);;
+ printf(" coming down from %.17e\n", C);
+ printf(" or else multiplication gets too many last digits wrong.\n");
+ }
+ Pause();
+ break;
+
+ case 2:
+ BadCond(Failure, "Underflow confuses Comparison which alleges that\n");
+ printf("Q == Y while denying that |Q - Y| == 0; these values\n");
+ printf("print out as Q = %.17e, Y = %.17e .\n", Q, Y);
+ printf ("|Q - Y| = %.17e .\n" , FABS(Q - Y2));
+ UfThold = Q;
+ break;
+
+ case 3:
+ X = X;
+ break;
+
+ case 4:
+ if ((Q == UfThold) && (E1 == E0)
+ && (FABS( UfThold - E1 / E9) <= E1)) {
+ UfNGrad = False;
+ printf("Underflow is gradual; it incurs Absolute Error =\n");
+ printf("(roundoff in UfThold) < E0.\n");
+ Y = E0 * CInvrse;
+ Y = Y * (OneAndHalf + U2);
+ X = CInvrse * (One + U2);
+ Y = Y / X;
+ IEEE = (Y == E0);
+ }
+ }
+ if (UfNGrad) {
+ printf("\n");
+ R = SQRT(Underflow / UfThold);
+ if (R <= H) {
+ Z = R * UfThold;
+ X = Z * (One + R * H * (One + H));
+ }
+ else {
+ Z = UfThold;
+ X = Z * (One + H * H * (One + H));
+ }
+ if (! ((X == Z) || (X - Z != Zero))) {
+ BadCond(Flaw, "");
+ printf("X = %.17e\n\tis not equal to Z = %.17e .\n", X, Z);
+ Z9 = X - Z;
+ printf("yet X - Z yields %.17e .\n", Z9);
+ printf(" Should this NOT signal Underflow, ");
+ printf("this is a SERIOUS DEFECT\nthat causes ");
+ printf("confusion when innocent statements like\n");;
+ printf(" if (X == Z) ... else");
+ printf(" ... (f(X) - f(Z)) / (X - Z) ...\n");
+ printf("encounter Division by Zero although actually\n");
+ printf("X / Z = 1 + %g .\n", (X / Z - Half) - Half);
+ }
+ }
+ printf("The Underflow threshold is %.17e, %s\n", UfThold,
+ " below which");
+ printf("calculation may suffer larger Relative error than ");
+ printf("merely roundoff.\n");
+ Y2 = U1 * U1;
+ Y = Y2 * Y2;
+ Y2 = Y * U1;
+ if (Y2 <= UfThold) {
+ if (Y > E0) {
+ BadCond(Defect, "");
+ I = 5;
+ }
+ else {
+ BadCond(Serious, "");
+ I = 4;
+ }
+ printf("Range is too narrow; U1^%d Underflows.\n", I);
+ }
+ /*=============================================*/
+ /*SPLIT
+ }
+#include "paranoia.h"
+part7(){
+*/
+ Milestone = 130;
+ /*=============================================*/
+ Y = - FLOOR(Half - TwoForty * LOG(UfThold) / LOG(HInvrse)) / TwoForty;
+ Y2 = Y + Y;
+ printf("Since underflow occurs below the threshold\n");
+ printf("UfThold = (%.17e) ^ (%.17e)\nonly underflow ", HInvrse, Y);
+ printf("should afflict the expression\n\t(%.17e) ^ (%.17e);\n", HInvrse, Y);
+ V9 = POW(HInvrse, Y2);
+ printf("actually calculating yields: %.17e .\n", V9);
+ if (! ((V9 >= Zero) && (V9 <= (Radix + Radix + E9) * UfThold))) {
+ BadCond(Serious, "this is not between 0 and underflow\n");
+ printf(" threshold = %.17e .\n", UfThold);
+ }
+ else if (! (V9 > UfThold * (One + E9)))
+ printf("This computed value is O.K.\n");
+ else {
+ BadCond(Defect, "this is not between 0 and underflow\n");
+ printf(" threshold = %.17e .\n", UfThold);
+ }
+ /*=============================================*/
+ Milestone = 140;
+ /*=============================================*/
+ printf("\n");
+ /* ...calculate Exp2 == exp(2) == 7.389056099... */
+ X = Zero;
+ I = 2;
+ Y = Two * Three;
+ Q = Zero;
+ N = 0;
+ do {
+ Z = X;
+ I = I + 1;
+ Y = Y / (I + I);
+ R = Y + Q;
+ X = Z + R;
+ Q = (Z - X) + R;
+ } while(X > Z);
+ Z = (OneAndHalf + One / Eight) + X / (OneAndHalf * ThirtyTwo);
+ X = Z * Z;
+ Exp2 = X * X;
+ X = F9;
+ Y = X - U1;
+ printf("Testing X^((X + 1) / (X - 1)) vs. exp(2) = %.17e as X -> 1.\n",
+ Exp2);
+ for(I = 1;;) {
+ Z = X - BInvrse;
+ Z = (X + One) / (Z - (One - BInvrse));
+ Q = POW(X, Z) - Exp2;
+ if (FABS(Q) > TwoForty * U2) {
+ N = 1;
+ V9 = (X - BInvrse) - (One - BInvrse);
+ BadCond(Defect, "Calculated");
+ printf(" %.17e for\n", POW(X,Z));
+ printf("\t(1 + (%.17e) ^ (%.17e);\n", V9, Z);
+ printf("\tdiffers from correct value by %.17e .\n", Q);
+ printf("\tThis much error may spoil financial\n");
+ printf("\tcalculations involving tiny interest rates.\n");
+ break;
+ }
+ else {
+ Z = (Y - X) * Two + Y;
+ X = Y;
+ Y = Z;
+ Z = One + (X - F9)*(X - F9);
+ if (Z > One && I < NoTrials) I++;
+ else {
+ if (X > One) {
+ if (N == 0)
+ printf("Accuracy seems adequate.\n");
+ break;
+ }
+ else {
+ X = One + U2;
+ Y = U2 + U2;
+ Y += X;
+ I = 1;
+ }
+ }
+ }
+ }
+ /*=============================================*/
+ Milestone = 150;
+ /*=============================================*/
+ printf("Testing powers Z^Q at four nearly extreme values.\n");
+ N = 0;
+ Z = A1;
+ Q = FLOOR(Half - LOG(C) / LOG(A1));
+ Break = False;
+ do {
+ X = CInvrse;
+ Y = POW(Z, Q);
+ IsYeqX();
+ Q = - Q;
+ X = C;
+ Y = POW(Z, Q);
+ IsYeqX();
+ if (Z < One) Break = True;
+ else Z = AInvrse;
+ } while ( ! (Break));
+ PrintIfNPositive();
+ if (N == 0) printf(" ... no discrepancies found.\n");
+ printf("\n");
+
+ /*=============================================*/
+ Milestone = 160;
+ /*=============================================*/
+ Pause();
+ printf("Searching for Overflow threshold:\n");
+ printf("This may generate an error.\n");
+ sigsave = sigfpe;
+ I = 0;
+ Y = - CInvrse;
+ V9 = HInvrse * Y;
+ if (setjmp(ovfl_buf)) goto overflow;
+ do {
+ V = Y;
+ Y = V9;
+ V9 = HInvrse * Y;
+ } while(V9 < Y);
+ I = 1;
+overflow:
+ Z = V9;
+ printf("Can `Z = -Y' overflow?\n");
+ printf("Trying it on Y = %.17e .\n", Y);
+ V9 = - Y;
+ V0 = V9;
+ if (V - Y == V + V0) printf("Seems O.K.\n");
+ else {
+ printf("finds a ");
+ BadCond(Flaw, "-(-Y) differs from Y.\n");
+ }
+ if (Z != Y) {
+ BadCond(Serious, "");
+ printf("overflow past %.17e\n\tshrinks to %.17e .\n", Y, Z);
+ }
+ Y = V * (HInvrse * U2 - HInvrse);
+ Z = Y + ((One - HInvrse) * U2) * V;
+ if (Z < V0) Y = Z;
+ if (Y < V0) V = Y;
+ if (V0 - V < V0) V = V0;
+ printf("Overflow threshold is V = %.17e .\n", V);
+ if (I) printf("Overflow saturates at V0 = %.17e .\n", V0);
+ else printf("There is no saturation value because \
+the system traps on overflow.\n");
+ V9 = V * One;
+ printf("No Overflow should be signaled for V * 1 = %.17e\n", V9);
+ V9 = V / One;
+ printf(" nor for V / 1 = %.17e .\n", V9);
+ printf("Any overflow signal separating this * from the one\n");
+ printf("above is a DEFECT.\n");
+ /*=============================================*/
+ Milestone = 170;
+ /*=============================================*/
+ if (!(-V < V && -V0 < V0 && -UfThold < V && UfThold < V)) {
+ BadCond(Failure, "Comparisons involving ");
+ printf("+-%g, +-%g\nand +-%g are confused by Overflow.",
+ V, V0, UfThold);
+ }
+ /*=============================================*/
+ Milestone = 175;
+ /*=============================================*/
+ printf("\n");
+ for(Indx = 1; Indx <= 3; ++Indx) {
+ switch (Indx) {
+ case 1: Z = UfThold; break;
+ case 2: Z = E0; break;
+ case 3: Z = PseudoZero; break;
+ }
+ if (Z != Zero) {
+ V9 = SQRT(Z);
+ Y = V9 * V9;
+ if (Y / (One - Radix * E9) < Z
+ || Y > (One + Radix + E9) * Z) {
+ if (V9 > U1) BadCond(Serious, "");
+ else BadCond(Defect, "");
+ printf("Comparison alleges that what prints as Z = %.17e\n", Z);
+ printf(" is too far from sqrt(Z) ^ 2 = %.17e .\n", Y);
+ }
+ }
+ }
+ /*=============================================*/
+ Milestone = 180;
+ /*=============================================*/
+ for(Indx = 1; Indx <= 2; ++Indx) {
+ if (Indx == 1) Z = V;
+ else Z = V0;
+ V9 = SQRT(Z);
+ X = (One - Radix * E9) * V9;
+ V9 = V9 * X;
+ if (((V9 < (One - Two * Radix * E9) * Z) || (V9 > Z))) {
+ Y = V9;
+ if (X < W) BadCond(Serious, "");
+ else BadCond(Defect, "");
+ printf("Comparison alleges that Z = %17e\n", Z);
+ printf(" is too far from sqrt(Z) ^ 2 (%.17e) .\n", Y);
+ }
+ }
+ /*=============================================*/
+ /*SPLIT
+ }
+#include "paranoia.h"
+part8(){
+*/
+ Milestone = 190;
+ /*=============================================*/
+ Pause();
+ X = UfThold * V;
+ Y = Radix * Radix;
+ if (X*Y < One || X > Y) {
+ if (X * Y < U1 || X > Y/U1) BadCond(Defect, "Badly");
+ else BadCond(Flaw, "");
+
+ printf(" unbalanced range; UfThold * V = %.17e\n\t%s\n",
+ X, "is too far from 1.\n");
+ }
+ /*=============================================*/
+ Milestone = 200;
+ /*=============================================*/
+ for (Indx = 1; Indx <= 5; ++Indx) {
+ X = F9;
+ switch (Indx) {
+ case 2: X = One + U2; break;
+ case 3: X = V; break;
+ case 4: X = UfThold; break;
+ case 5: X = Radix;
+ }
+ Y = X;
+ sigsave = sigfpe;
+ if (setjmp(ovfl_buf))
+ printf(" X / X traps when X = %g\n", X);
+ else {
+ V9 = (Y / X - Half) - Half;
+ if (V9 == Zero) continue;
+ if (V9 == - U1 && Indx < 5) BadCond(Flaw, "");
+ else BadCond(Serious, "");
+ printf(" X / X differs from 1 when X = %.17e\n", X);
+ printf(" instead, X / X - 1/2 - 1/2 = %.17e .\n", V9);
+ }
+ }
+ /*=============================================*/
+ Milestone = 210;
+ /*=============================================*/
+ MyZero = Zero;
+ printf("\n");
+ printf("What message and/or values does Division by Zero produce?\n") ;
+#ifndef NOPAUSE
+ printf("This can interupt your program. You can ");
+ printf("skip this part if you wish.\n");
+ printf("Do you wish to compute 1 / 0? ");
+ fflush(stdout);
+ read (KEYBOARD, ch, 8);
+ if ((ch[0] == 'Y') || (ch[0] == 'y')) {
+#endif
+ sigsave = sigfpe;
+ printf(" Trying to compute 1 / 0 produces ...");
+ if (!setjmp(ovfl_buf)) printf(" %.7e .\n", One / MyZero);
+#ifndef NOPAUSE
+ }
+ else printf("O.K.\n");
+ printf("\nDo you wish to compute 0 / 0? ");
+ fflush(stdout);
+ read (KEYBOARD, ch, 80);
+ if ((ch[0] == 'Y') || (ch[0] == 'y')) {
+#endif
+ sigsave = sigfpe;
+ printf("\n Trying to compute 0 / 0 produces ...");
+ if (!setjmp(ovfl_buf)) printf(" %.7e .\n", Zero / MyZero);
+#ifndef NOPAUSE
+ }
+ else printf("O.K.\n");
+#endif
+ /*=============================================*/
+ Milestone = 220;
+ /*=============================================*/
+ Pause();
+ printf("\n");
+ {
+ static char *msg[] = {
+ "FAILUREs encountered =",
+ "SERIOUS DEFECTs discovered =",
+ "DEFECTs discovered =",
+ "FLAWs discovered =" };
+ int i;
+ for(i = 0; i < 4; i++) if (ErrCnt[i])
+ printf("The number of %-29s %d.\n",
+ msg[i], ErrCnt[i]);
+ }
+ printf("\n");
+ if ((ErrCnt[Failure] + ErrCnt[Serious] + ErrCnt[Defect]
+ + ErrCnt[Flaw]) > 0) {
+ if ((ErrCnt[Failure] + ErrCnt[Serious] + ErrCnt[
+ Defect] == 0) && (ErrCnt[Flaw] > 0)) {
+ printf("The arithmetic diagnosed seems ");
+ printf("satisfactory though flawed.\n");
+ }
+ if ((ErrCnt[Failure] + ErrCnt[Serious] == 0)
+ && ( ErrCnt[Defect] > 0)) {
+ printf("The arithmetic diagnosed may be acceptable\n");
+ printf("despite inconvenient Defects.\n");
+ }
+ if ((ErrCnt[Failure] + ErrCnt[Serious]) > 0) {
+ printf("The arithmetic diagnosed has ");
+ printf("unacceptable serious defects.\n");
+ }
+ if (ErrCnt[Failure] > 0) {
+ printf("Fatal FAILURE may have spoiled this");
+ printf(" program's subsequent diagnoses.\n");
+ }
+ }
+ else {
+ printf("No failures, defects nor flaws have been discovered.\n");
+ if (! ((RMult == Rounded) && (RDiv == Rounded)
+ && (RAddSub == Rounded) && (RSqrt == Rounded)))
+ printf("The arithmetic diagnosed seems satisfactory.\n");
+ else {
+ if (StickyBit >= One &&
+ (Radix - Two) * (Radix - Nine - One) == Zero) {
+ printf("Rounding appears to conform to ");
+ printf("the proposed IEEE standard P");
+ if ((Radix == Two) &&
+ ((Precision - Four * Three * Two) *
+ ( Precision - TwentySeven -
+ TwentySeven + One) == Zero))
+ printf("754");
+ else printf("854");
+ if (IEEE) printf(".\n");
+ else {
+ printf(",\nexcept for possibly Double Rounding");
+ printf(" during Gradual Underflow.\n");
+ }
+ }
+ printf("The arithmetic diagnosed appears to be excellent!\n");
+ }
+ }
+ if (fpecount)
+ printf("\nA total of %d floating point exceptions were registered.\n",
+ fpecount);
+ printf("END OF TEST.\n");
+ }
+
+/*SPLIT subs.c
+#include "paranoia.h"
+*/
+
+/* Sign */
+
+FLOAT Sign (X)
+FLOAT X;
+{ return X >= 0. ? 1.0 : -1.0; }
+
+/* Pause */
+
+Pause()
+{
+ char ch[8];
+
+#ifndef NOPAUSE
+ printf("\nTo continue, press RETURN");
+ fflush(stdout);
+ read(KEYBOARD, ch, 8);
+#endif
+ printf("\nDiagnosis resumes after milestone Number %d", Milestone);
+ printf(" Page: %d\n\n", PageNo);
+ ++Milestone;
+ ++PageNo;
+ }
+
+ /* TstCond */
+
+TstCond (K, Valid, T)
+int K, Valid;
+char *T;
+{ if (! Valid) { BadCond(K,T); printf(".\n"); } }
+
+BadCond(K, T)
+int K;
+char *T;
+{
+ static char *msg[] = { "FAILURE", "SERIOUS DEFECT", "DEFECT", "FLAW" };
+
+ ErrCnt [K] = ErrCnt [K] + 1;
+ printf("%s: %s", msg[K], T);
+ }
+
+/* Random */
+/* Random computes
+ X = (Random1 + Random9)^5
+ Random1 = X - FLOOR(X) + 0.000005 * X;
+ and returns the new value of Random1
+*/
+
+FLOAT Random()
+{
+ FLOAT X, Y;
+
+ X = Random1 + Random9;
+ Y = X * X;
+ Y = Y * Y;
+ X = X * Y;
+ Y = X - FLOOR(X);
+ Random1 = Y + X * 0.000005;
+ return(Random1);
+ }
+
+/* SqXMinX */
+
+SqXMinX (ErrKind)
+int ErrKind;
+{
+ FLOAT XA, XB;
+
+ XB = X * BInvrse;
+ XA = X - XB;
+ SqEr = ((SQRT(X * X) - XB) - XA) / OneUlp;
+ if (SqEr != Zero) {
+ if (SqEr < MinSqEr) MinSqEr = SqEr;
+ if (SqEr > MaxSqEr) MaxSqEr = SqEr;
+ J = J + 1.0;
+ BadCond(ErrKind, "\n");
+ printf("sqrt( %.17e) - %.17e = %.17e\n", X * X, X, OneUlp * SqEr);
+ printf("\tinstead of correct value 0 .\n");
+ }
+ }
+
+/* NewD */
+
+NewD()
+{
+ X = Z1 * Q;
+ X = FLOOR(Half - X / Radix) * Radix + X;
+ Q = (Q - X * Z) / Radix + X * X * (D / Radix);
+ Z = Z - Two * X * D;
+ if (Z <= Zero) {
+ Z = - Z;
+ Z1 = - Z1;
+ }
+ D = Radix * D;
+ }
+
+/* SR3750 */
+
+SR3750()
+{
+ if (! ((X - Radix < Z2 - Radix) || (X - Z2 > W - Z2))) {
+ I = I + 1;
+ X2 = SQRT(X * D);
+ Y2 = (X2 - Z2) - (Y - Z2);
+ X2 = X8 / (Y - Half);
+ X2 = X2 - Half * X2 * X2;
+ SqEr = (Y2 + Half) + (Half - X2);
+ if (SqEr < MinSqEr) MinSqEr = SqEr;
+ SqEr = Y2 - X2;
+ if (SqEr > MaxSqEr) MaxSqEr = SqEr;
+ }
+ }
+
+/* IsYeqX */
+
+IsYeqX()
+{
+ if (Y != X) {
+ if (N <= 0) {
+ if (Z == Zero && Q <= Zero)
+ printf("WARNING: computing\n");
+ else BadCond(Defect, "computing\n");
+ printf("\t(%.17e) ^ (%.17e)\n", Z, Q);
+ printf("\tyielded %.17e;\n", Y);
+ printf("\twhich compared unequal to correct %.17e ;\n",
+ X);
+ printf("\t\tthey differ by %.17e .\n", Y - X);
+ }
+ N = N + 1; /* ... count discrepancies. */
+ }
+ }
+
+/* SR3980 */
+
+SR3980()
+{
+ do {
+ Q = (FLOAT) I;
+ Y = POW(Z, Q);
+ IsYeqX();
+ if (++I > M) break;
+ X = Z * X;
+ } while ( X < W );
+ }
+
+/* PrintIfNPositive */
+
+PrintIfNPositive()
+{
+ if (N > 0) printf("Similar discrepancies have occurred %d times.\n", N);
+ }
+
+/* TstPtUf */
+
+TstPtUf()
+{
+ N = 0;
+ if (Z != Zero) {
+ printf("Since comparison denies Z = 0, evaluating ");
+ printf("(Z + Z) / Z should be safe.\n");
+ sigsave = sigfpe;
+ if (setjmp(ovfl_buf)) goto very_serious;
+ Q9 = (Z + Z) / Z;
+ printf("What the machine gets for (Z + Z) / Z is %.17e .\n",
+ Q9);
+ if (FABS(Q9 - Two) < Radix * U2) {
+ printf("This is O.K., provided Over/Underflow");
+ printf(" has NOT just been signaled.\n");
+ }
+ else {
+ if ((Q9 < One) || (Q9 > Two)) {
+very_serious:
+ N = 1;
+ ErrCnt [Serious] = ErrCnt [Serious] + 1;
+ printf("This is a VERY SERIOUS DEFECT!\n");
+ }
+ else {
+ N = 1;
+ ErrCnt [Defect] = ErrCnt [Defect] + 1;
+ printf("This is a DEFECT!\n");
+ }
+ }
+ V9 = Z * One;
+ Random1 = V9;
+ V9 = One * Z;
+ Random2 = V9;
+ V9 = Z / One;
+ if ((Z == Random1) && (Z == Random2) && (Z == V9)) {
+ if (N > 0) Pause();
+ }
+ else {
+ N = 1;
+ BadCond(Defect, "What prints as Z = ");
+ printf("%.17e\n\tcompares different from ", Z);
+ if (Z != Random1) printf("Z * 1 = %.17e ", Random1);
+ if (! ((Z == Random2)
+ || (Random2 == Random1)))
+ printf("1 * Z == %g\n", Random2);
+ if (! (Z == V9)) printf("Z / 1 = %.17e\n", V9);
+ if (Random2 != Random1) {
+ ErrCnt [Defect] = ErrCnt [Defect] + 1;
+ BadCond(Defect, "Multiplication does not commute!\n");
+ printf("\tComparison alleges that 1 * Z = %.17e\n",
+ Random2);
+ printf("\tdiffers from Z * 1 = %.17e\n", Random1);
+ }
+ Pause();
+ }
+ }
+ }
+
+notify(s)
+char *s;
+{
+ printf("%s test appears to be inconsistent...\n", s);
+ printf(" PLEASE NOTIFY KARPINKSI!\n");
+ }
+
+/*SPLIT msgs.c */
+
+/* Instructions */
+
+msglist(s)
+char **s;
+{ while(*s) printf("%s\n", *s++); }
+
+Instructions()
+{
+ static char *instr[] = {
+ "Lest this program stop prematurely, i.e. before displaying\n",
+ " `END OF TEST',\n",
+ "try to persuade the computer NOT to terminate execution when an",
+ "error like Over/Underflow or Division by Zero occurs, but rather",
+ "to persevere with a surrogate value after, perhaps, displaying some",
+ "warning. If persuasion avails naught, don't despair but run this",
+ "program anyway to see how many milestones it passes, and then",
+ "amend it to make further progress.\n",
+ "Answer questions with Y, y, N or n (unless otherwise indicated).\n",
+ 0};
+
+ msglist(instr);
+ }
+
+/* Heading */
+
+Heading()
+{
+ static char *head[] = {
+ "Users are invited to help debug and augment this program so it will",
+ "cope with unanticipated and newly uncovered arithmetic pathologies.\n",
+ "Please send suggestions and interesting results to",
+ "\tRichard Karpinski",
+ "\tComputer Center U-76",
+ "\tUniversity of California",
+ "\tSan Francisco, CA 94143-0704, USA\n",
+ "In doing so, please include the following information:",
+#ifdef Single
+ "\tPrecision:\tsingle;",
+#else
+ "\tPrecision:\tdouble;",
+#endif
+ "\tVersion:\t27 January 1986;",
+ "\tComputer:\n",
+ "\tCompiler:\n",
+ "\tOptimization level:\n",
+ "\tOther relevant compiler options:",
+ 0};
+
+ msglist(head);
+ }
+
+/* Characteristics */
+
+Characteristics()
+{
+ static char *chars[] = {
+ "Running this program should reveal these characteristics:",
+ " Radix = 1, 2, 4, 8, 10, 16, 100, 256 ...",
+ " Precision = number of significant digits carried.",
+ " U2 = Radix/Radix^Precision = One Ulp",
+ "\t(OneUlpnit in the Last Place) of 1.000xxx .",
+ " U1 = 1/Radix^Precision = One Ulp of numbers a little less than 1.0 .",
+ " Adequacy of guard digits for Mult., Div. and Subt.",
+ " Whether arithmetic is chopped, correctly rounded, or something else",
+ "\tfor Mult., Div., Add/Subt. and Sqrt.",
+ " Whether a Sticky Bit used correctly for rounding.",
+ " UnderflowThreshold = an underflow threshold.",
+ " E0 and PseudoZero tell whether underflow is abrupt, gradual, or fuzzy.",
+ " V = an overflow threshold, roughly.",
+ " V0 tells, roughly, whether Infinity is represented.",
+ " Comparisions are checked for consistency with subtraction",
+ "\tand for contamination with pseudo-zeros.",
+ " Sqrt is tested. Y^X is not tested.",
+ " Extra-precise subexpressions are revealed but NOT YET tested.",
+ " Decimal-Binary conversion is NOT YET tested for accuracy.",
+ 0};
+
+ msglist(chars);
+ }
+
+History()
+
+{ /* History */
+ /* Converted from Brian Wichmann's Pascal version to C by Thos Sumner,
+ with further massaging by David M. Gay. */
+
+ static char *hist[] = {
+ "The program attempts to discriminate among",
+ " FLAWs, like lack of a sticky bit,",
+ " Serious DEFECTs, like lack of a guard digit, and",
+ " FAILUREs, like 2+2 == 5 .",
+ "Failures may confound subsequent diagnoses.\n",
+ "The diagnostic capabilities of this program go beyond an earlier",
+ "program called `MACHAR', which can be found at the end of the",
+ "book `Software Manual for the Elementary Functions' (1980) by",
+ "W. J. Cody and W. Waite. Although both programs try to discover",
+ "the Radix, Precision and range (over/underflow thresholds)",
+ "of the arithmetic, this program tries to cope with a wider variety",
+ "of pathologies, and to say how well the arithmetic is implemented.",
+ "\nThe program is based upon a conventional radix representation for",
+ "floating-point numbers, but also allows logarithmic encoding",
+ "as used by certain early WANG machines.\n",
+ "BASIC version of this program (C) 1983 by Prof. W. M. Kahan;",
+ "see source comments for more history.",
+ 0};
+
+ msglist(hist);
+ }
generated by cgit v1.2.3 (git 2.25.1) at 2024-11-05 18:01:25 +0000