[linux-cirrus] Yet another MaverickCrunch hardware bug?
- From: Martin Guy <martinwguy@xxxxxxxx>
- To: linux-cirrus@xxxxxxxxxxxxx
- Date: Sun, 15 Feb 2009 12:38:40 +0000
Hi!
You probably know I have some patches to gcc that fix Maverick code
generation and even add a flag -mieee to disable all the
non-strictly-conformant instructions.
I'm now going mad trying to nail the last rare Maverick bug and am
seeing such crazy results that I must consider there being something
dodgy about my particular hardware or kernel.
Briefly, would someone with a Cirrus EP93XX chip running an EABI OS
run the attached program and tell me whether for them too it gives
different results at each run, then let me know together with their
silicon revision number (5th and 6th chars of the second line of text
on the chip housing).
Here's what I am seeing:
Every test program I try works perfectly except for paranoia, which
gives different results each time I run it, from failure to segfault
to illegal instruction to infinite loops.
It's worse than that. If I have a freshly-compiled binary and go:
$ cp paranoia2 foo
$ ./paranoia2
$ cmp -l foo paranoia2
I find that the binary itself appears to have been modified in its
text segment, and any further runs "modify" it more, and comparing
"objdump -d -z" output for the two files shows which words of program
code have been replaced by zeroes or junk.
It's not really modified. If I run a thrashing program that allocates
a buffer the size of the RAM and accesses every page repeatedly so as
to flush the kernel's disk cache, the paranoia2 binary reverts back to
its original contents when it is reloaded from SD card or over NFS. If
I am working in a RAM disk or tmpfs instead, it stays modified.
So it's looking like the Maverick unit is managing to twitch so badly
as to write to the in-core copy of its text segment (which the MMU
maps read-only!) thereby corrupting the kernel's in-core cache. It's
not poisoning the i-cache or d-cache; I've tried disabling those and
the same thing happens.
Before I go the whole hog of characterizing this and finding a
compiler workaround, would somebody try this on another EP chip with
EABI OS to check that this really is another factory defect, not just
a fault of this individual unit or of my kernel?
Or if you have my patched gcc-4.3.2 compiler or the futaris 4.[12]
ones, try compiling the attached source
-mcpu=ep9312 -mfpu=maverick -mfloat-abi=softfp
(-mieee -mcirrus-fix-invalid-insns with mine, -mfix-crunch-d1 with
futaris/cirrus compilers) at any level of optimization and try the
resulting binary? With the other compilers the program won't report
success, but it should fail the same way from one run to the next!
Sum of the attached binary is 27575
Cheers!
M
#define NOPAUSE
/* 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).
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 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
.d
+d
-,$w msgs.c
-,$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,$s/^char/extern &/
1,$s! = .*!;!
/^Guard/,/^Round/s/^/extern /
/^jmp_buf/s/^/extern /
/^Sig_type/s/^/extern /
s/$/\
extern void sigfpe(INT);/
w paranoia.h
q
*/
#include <stdio.h>
#ifndef NOSIGNAL
#include <signal.h>
#endif
#include <setjmp.h>
#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;
#ifdef KR_headers
#define VOID /* void */
#define INT /* int */
#define FP /* FLOAT */
#define CHARP /* char * */
#define CHARPP /* char ** */
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();
#else
#define VOID void
#define INT int
#define FP FLOAT
#define CHARP char *
#define CHARPP char **
#ifdef __STDC__
#include <stdlib.h>
#include <math.h>
#else
#ifdef __cplusplus
extern "C" {
#endif
extern double fabs(double), floor(double), log(double);
extern double pow(double,double), sqrt(double);
extern void exit(INT);
#ifdef __cplusplus
}
#endif
#endif
typedef void (*Sig_type)(int);
FLOAT Sign(FLOAT), Random(void);
extern void BadCond(int, char*);
extern void SqXMinX(int);
extern void TstCond(int, int, char*);
extern void notify(char*);
extern int read(int, char*, int);
#endif
#undef V9
extern void Characteristics(VOID);
extern void Heading(VOID);
extern void History(VOID);
extern void Instructions(VOID);
extern void IsYeqX(VOID);
extern void NewD(VOID);
extern void Pause(VOID);
extern void PrintIfNPositive(VOID);
extern void SR3750(VOID);
extern void SR3980(VOID);
extern void TstPtUf(VOID);
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 */
/* floating point exception receiver */
void
sigfpe(INT x)
{
fpecount++;
printf("\n* * * FLOATING-POINT ERROR %d * * *\n", x);
fflush(stdout);
if (sigsave) {
#ifndef NOSIGNAL
signal(SIGFPE, sigsave);
#endif
sigsave = 0;
longjmp(ovfl_buf, 1);
}
exit(1);
}
main(VOID)
{
setbuf(stdout, NULL);
/* 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");
TstCond (Failure, (Zero + Zero == Zero), "0+0 != 0");
TstCond (Failure, (One - One == Zero), "1-1 != 0");
TstCond (Failure, (One > Zero), "1 <= 0");
TstCond (Failure, (One + One == Two), "1+1 != 2");
Z = - Zero;
if (Z != 0.0) {
ErrCnt[Failure] = ErrCnt[Failure] + 1;
printf("Comparison alleges that -0.0 is Non-zero!\n");
printf("Zero = %g, Z = %g\n", Zero, Z);
U2 = 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");
/*=============================================*/
/*SPLIT
{
extern void part2(VOID), part3(VOID), part4(VOID),
part5(VOID), part6(VOID), part7(VOID);
int part8(VOID);
part2();
part3();
part4();
part5();
part6();
part7();
return part8();
}
}
#include "paranoia.h"
void part2(VOID){
*/
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,
"5 != 4+1");
TstCond (Failure, TwoForty == Four * Five * Three * Four,
"240 != 4*5*3*4");
TstCond (Failure, TwoForty / Three - Four * Four * Five == Zero,
"240/3-4*4*4 != 0");
TstCond (Failure, TwoForty / Four - Five * Three * Four == Zero,
"240/4-5*3*4 != 0");
TstCond (Failure, TwoForty / Five - Four * Three * Four == Zero,
"240/5-4*3*4 != 0");
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();
/*=============================================*/
/*SPLIT
}
#include "paranoia.h"
void part3(VOID){
*/
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");
/*=============================================*/
/*SPLIT
}
#include "paranoia.h"
void part4(VOID){
*/
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);
}
/*=============================================*/
/*SPLIT
}
#include "paranoia.h"
void part5(VOID){
*/
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");
/*=============================================*/
/*SPLIT
}
#include "paranoia.h"
void part6(VOID){
*/
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);
}
/*=============================================*/
/*SPLIT
}
#include "paranoia.h"
void part7(VOID){
*/
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);
}
}
/*=============================================*/
/*SPLIT
}
#include "paranoia.h"
int part8(VOID){
*/
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 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);
sigsave = 0;
#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);
sigsave = 0;
#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("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;
}
/*SPLIT subs.c
#include "paranoia.h"
*/
FLOAT
Sign (FP X)
#ifdef KR_headers
FLOAT X;
#endif
{ return X >= 0. ? 1.0 : -1.0; }
void
Pause(VOID)
{
#ifndef NOPAUSE
char ch[8];
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;
}
void
TstCond (INT K, INT Valid, CHARP T)
#ifdef KR_headers
int K, Valid;
char *T;
#endif
{ if (! Valid) { BadCond(K,T); printf(".\n"); } }
void
BadCond(INT K, CHARP T)
#ifdef KR_headers
int K;
char *T;
#endif
{
static char *msg[] = { "FAILURE", "SERIOUS DEFECT", "DEFECT", "FLAW" };
ErrCnt [K] = ErrCnt [K] + 1;
printf("%s: %s", msg[K], T);
}
FLOAT
Random(VOID)
/* Random computes
X = (Random1 + Random9)^5
Random1 = X - FLOOR(X) + 0.000005 * X;
and returns the new value of Random1
*/
{
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 (INT ErrKind)
#ifdef KR_headers
int ErrKind;
#endif
{
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(VOID)
{
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(VOID)
{
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(VOID)
{
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(VOID)
{
do {
Q = (FLOAT) I;
Y = POW(Z, Q);
IsYeqX();
if (++I > M) break;
X = Z * X;
} while ( X < W );
}
void
PrintIfNPositive(VOID)
{
if (N > 0) printf("Similar discrepancies have occurred %d times.\n", N);
}
void
TstPtUf(VOID)
{
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(CHARP s)
#ifdef KR_headers
char *s;
#endif
{
printf("%s test appears to be inconsistent...\n", s);
printf(" PLEASE NOTIFY KARPINKSI!\n");
}
/*SPLIT msgs.c
#include "paranoia.h"
*/
void
msglist(CHARPP s)
#ifdef KR_headers
char **s;
#endif
{ while(*s) printf("%s\n", *s++); }
void
Instructions(VOID)
{
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(VOID)
{
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:\t10 February 1989;",
"\tComputer:\n",
"\tCompiler:\n",
"\tOptimization level:\n",
"\tOther relevant compiler options:",
0};
msglist(head);
}
void
Characteristics(VOID)
{
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(VOID)
{ /* 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);
}
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