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-/*
- * RTEMS RCS Id: $Id$
- *
- * 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).
- *
- * Sun May 16 18:21:51 MDT 1993 Jeffrey Wheat (cassidy@cygnus.com)
- * Removed KR_headers defines, removed ANSI prototyping
- * Cleaned up and reformated code. Added special CYGNUS
- * "verbose" mode type messages (-DCYGNUS).
- * Note: This code is VERY NASTY.
- *
- * Adjusted to use Standard C headers 19 Jan. 1992 (dmg);
- * compile with -DKR_headers or insert
- * #define KR_headers
- * at the beginning if you have an old-style C compiler.
- *
- * (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 Mountain 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).
- * [A couple of bug fixes from dgh = sun!dhough incorporated 31 July 1986.]
- *
- * 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_PRECISION 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
- *
-*/
-
-#include <stdio.h>
-#include <string.h>
-
-/*
- * To compile this on host using only libm from newlib (and using host libc)
- * do:
- * gcc -g -DNEED_REENT -DCYGNUS paranoia.c .../newlib-1.6/newlib/libm.a
- */
-
-#ifdef NEED_REENT
-#include "reent.h"
-struct _reent libm_reent = _REENT_INIT(libm_reent);
-struct _reent *_impure_ptr = &libm_reent;
-#endif
-
-#ifndef NOSIGNAL
-#include <signal.h>
-#include <setjmp.h>
-#else /* NOSIGNAL */
-#define longjmp(e,v)
-#define setjmp(e) 0
-#define jmp_buf int
-#endif /* NOSIGNAL */
-
-#ifdef SINGLE_PRECISION
-#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 /* !SINGLE_PRECISION */
-#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 /* SINGLE_PRECISION */
-
-jmp_buf ovfl_buf;
-extern double fabs (), floor (), log (), pow (), sqrt ();
-extern void exit ();
-typedef void (*Sig_type) ();
-FLOAT Sign (), Random ();
-extern void BadCond ();
-extern void SqXMinX ();
-extern void TstCond ();
-extern void notify ();
-extern int read ();
-extern void Characteristics ();
-extern void Heading ();
-extern void History ();
-extern void Instructions ();
-extern void IsYeqX ();
-extern void NewD ();
-extern void Pause ();
-extern void PrintIfNPositive ();
-extern void SR3750 ();
-extern void SR3980 ();
-extern void TstPtUf ();
-
-Sig_type sigsave;
-
-#define KEYBOARD 0
-
-FLOAT Radix, BInvrse, RadixD2, BMinusU2;
-
-/*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;
-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
- */
-
-int batchmode; /* global batchmode test */
-
-/* program name and version variables and macro */
-char *temp;
-char *program_name;
-char *program_vers;
-
-#ifndef VERSION
-#define VERSION "1.1 [cygnus]"
-#endif /* VERSION */
-
-#define basename(cp) ((temp=(char *)strrchr((cp), '/')) ? temp+1 : (cp))
-
-#ifndef BATCHMODE
-# ifdef CYGNUS
-# define BATCHMODE
-# endif
-#endif
-
-/* floating point exception receiver */
-void
-_sigfpe (x)
-int x;
-{
- fpecount++;
- printf ("\n* * * FLOATING-POINT ERROR %d * * *\n", x);
- fflush (stdout);
- if (sigsave) {
-#ifndef NOSIGNAL
- signal (SIGFPE, sigsave);
-#endif /* NOSIGNAL */
- sigsave = 0;
- longjmp (ovfl_buf, 1);
- }
- exit (1);
-}
-
-#ifdef NOMAIN
-#define main paranoia
-#endif
-
-int
-main (argc, argv)
-int argc;
-char **argv;
-{
- /* 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;
-
-#ifdef BATCHMODE
- batchmode = 1; /* run test in batchmode? */
-#else /* !BATCHMODE */
- batchmode = 0; /* run test interactively */
-#endif /* BATCHMODE */
-
- program_name = basename (argv[0]);
- program_vers = VERSION;
-
- printf ("%s version %s\n", program_name, program_vers);
-
- /*=============================================*/
- Milestone = 0;
- /*=============================================*/
-#ifndef NOSIGNAL
- signal (SIGFPE, _sigfpe);
-#endif
-
- if (!batchmode) {
- Instructions ();
- Pause ();
- Heading ();
- 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) {
- ErrCnt[Failure] = ErrCnt[Failure] + 1;
- printf ("Comparison alleges that -0.0 is Non-zero!\n");
- U1 = 0.001;
- Radix = 1;
- TstPtUf ();
- }
- 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");
- /*=============================================*/
- 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\n ");
-
- /*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 ();
- /*=============================================*/
- 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 (" subtraction 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");
- /*=============================================*/
- 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 <= NoTrials) {
- 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);
- }
- /*=============================================*/
- 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;
- RSqrt = Other; /* ~dgh */
- 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;
- Break = False;
- do {
- X = One;
- SR3980 ();
- if (I <= 10) {
- I = 1023;
- SR3980 ();
- }
- if (Z == MinusOne)
- Break = True;
- else {
- Z = MinusOne;
- /* .. if(-1)^N is invalid, replace MinusOne by One. */
- I = -4;
- }
- }
- while (!Break);
- PrintIfNPositive ();
- N1 = N;
- N = 0;
- Z = A1;
- M = (int) 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 discrepancies found.\n");
- if (N > 0)
- Pause ();
- else
- printf ("\n");
- /*=============================================*/
- 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;
- }
- while ((Y > Z) && (Z + Z > Z));
- Y = C;
- Z = Y * D;
- do {
- C = Y;
- Y = Z;
- Z = Y * D;
- }
- while ((Y > Z) && (Z + Z > Z));
- 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;
- }
- while ((E0 > Z) && (Z + Z > Z));
- 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;
- }
- while ((Underflow > PseudoZero)
- && (PseudoZero + PseudoZero > PseudoZero));
- }
- /* 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, Y2);
- 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");
- sigsave = _sigfpe;
- if (setjmp (ovfl_buf)) {
- printf ("Underflow / UfThold failed!\n");
- R = H + H;
- } else
- R = SQRT (Underflow / UfThold);
- sigsave = 0;
- 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");
- sigsave = _sigfpe;
- if (setjmp (ovfl_buf))
- printf ("X / Z fails!\n");
- else
- printf ("X / Z = 1 + %g .\n", (X / Z - Half) - Half);
- sigsave = 0;
- }
- }
- 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);
- }
- /*=============================================*/
- 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, Y2);
- printf ("actually calculating yields:");
- if (setjmp (ovfl_buf)) {
- sigsave = 0;
- BadCond (Serious, "trap on underflow.\n");
- } else {
- sigsave = _sigfpe;
- V9 = POW (HInvrse, Y2);
- sigsave = 0;
- printf (" %.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");
- Y = -CInvrse;
- V9 = HInvrse * Y;
- sigsave = _sigfpe;
- if (setjmp (ovfl_buf)) {
- I = 0;
- V9 = Y;
- goto overflow;
- }
- do {
- V = Y;
- Y = V9;
- V9 = HInvrse * Y;
- }
- while (V9 < Y);
- I = 1;
- overflow:
- sigsave = 0;
- 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);
- }
- if (I) {
- 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;
- } else {
- V = Y * (HInvrse * U2 - HInvrse);
- V = V + ((One - HInvrse) * U2) * Y;
- }
- 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) { /* dgh: + E9 --> * E9 */
- 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);
- }
- }
- /*=============================================*/
- 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);
- }
- sigsave = 0;
- }
- /*=============================================*/
- Milestone = 210;
- /*=============================================*/
- MyZero = Zero;
- printf ("\n");
- printf ("What message and/or values does Division by Zero produce?\n");
-#ifndef BATCHMODE
- 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 /* !BATCHMODE */
- sigsave = _sigfpe;
- printf (" Trying to compute 1 / 0 produces ...");
- if (!setjmp (ovfl_buf))
- printf (" %.7e .\n", One / MyZero);
- sigsave = 0;
-#ifndef BATCHMODE
- } 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 /* !BATCHMODE */
- sigsave = _sigfpe;
- printf ("\n Trying to compute 0 / 0 produces ...");
- if (!setjmp (ovfl_buf))
- printf (" %.7e .\n", Zero / MyZero);
- sigsave = 0;
-#ifndef BATCHMODE
- } else
- printf ("O.K.\n");
-#endif /* !BATCHMODE */
- /*=============================================*/
- 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 ("Potentially 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");
- return 0;
-}
-
-FLOAT
-Sign (X)
- FLOAT X;
-{
- return X >= 0. ? 1.0 : -1.0;
-}
-
-void
-Pause ()
-{
-#ifndef BATCHMODE
- char ch[8];
-
- printf ("\nTo continue, press RETURN");
- fflush (stdout);
- read (KEYBOARD, ch, 8);
-#endif /* !BATCHMODE */
-#ifndef CYGNUS
- printf ("\nDiagnosis resumes after milestone Number %d", Milestone);
- printf (" Page: %d\n\n", PageNo);
- ++Milestone;
- ++PageNo;
-#endif /* !CYGNUS */
-}
-
-void
-TstCond (K, Valid, T)
- int K, Valid;
- char *T;
-{
-#ifdef CYGNUS
- printf ("TEST: %s\n", T);
-#endif /* CYGNUS */
- if (!Valid) {
- BadCond (K, T);
- printf (".\n");
- }
-#ifdef CYGNUS
- printf ("PASS: %s\n", T);
-#endif /* CYGNUS */
-}
-
-void
-BadCond (K, T)
- int K;
- char *T;
-{
- static char *msg[] =
- {"FAILURE", "SERIOUS DEFECT", "DEFECT", "FLAW"};
-
- ErrCnt[K] = ErrCnt[K] + 1;
-#ifndef CYGNUS
- printf ("%s: %s", msg[K], T);
-#else
- printf ("ERROR: Severity: %s: %s", msg[K], T);
-#endif /* CYGNUS */
-}
-
-/*
- * 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);
-}
-
-void
-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");
- }
-}
-
-void
-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;
-}
-
-void
-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;
- }
-}
-
-void
-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. */
- }
-}
-
-void
-SR3980 ()
-{
- do {
- Q = (FLOAT) I;
- Y = POW (Z, Q);
- IsYeqX ();
- if (++I > M)
- break;
- X = Z * X;
- }
- while (X < W);
-}
-
-void
-PrintIfNPositive ()
-{
- if (N > 0)
- printf ("Similar discrepancies have occurred %d times.\n", N);
-}
-
-void
-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");
- }
- }
- sigsave = 0;
- 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 ();
- }
- }
-}
-
-void
-notify (s)
- char *s;
-{
- printf ("%s test appears to be inconsistent...\n", s);
- printf (" PLEASE NOTIFY KARPINKSI!\n");
-}
-
-void
-msglist (s)
- char **s;
-{
- while (*s)
- printf ("%s\n", *s++);
-}
-
-void
-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);
-}
-
-void
-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_PRECISION
- "\tPrecision:\tsingle;",
-#else /* !SINGLE_PRECISION */
- "\tPrecision:\tdouble;",
-#endif /* SINGLE_PRECISION */
- "\tVersion:\t10 February 1989;",
- "\tComputer:\n",
- "\tCompiler:\n",
- "\tOptimization level:\n",
- "\tOther relevant compiler options:",
- 0};
-
- msglist (head);
-}
-
-void
-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);
-}
-
-void
-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);
-}