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2220 lines
57 KiB
C
2220 lines
57 KiB
C
/*
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!!DESCRIPTION!! a wellknown floating point test
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!!ORIGIN!! LCC 4.1 Testsuite
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!!LICENCE!! own, freely distributeable for non-profit. read CPYRIGHT.LCC
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*/
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#include "common.h"
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#ifdef NO_FLOATS
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main()
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{
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printf("NO_FLOATS\n\r");
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return 0;
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}
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#else
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#undef V9
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#define NOPAUSE
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/* A C version of Kahan's Floating Point Test "Paranoia"
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Thos Sumner, UCSF, Feb. 1985
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David Gay, BTL, Jan. 1986
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This is a rewrite from the Pascal version by
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B. A. Wichmann, 18 Jan. 1985
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(and does NOT exhibit good C programming style).
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(C) Apr 19 1983 in BASIC version by:
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Professor W. M. Kahan,
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567 Evans Hall
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Electrical Engineering & Computer Science Dept.
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University of California
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Berkeley, California 94720
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USA
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converted to Pascal by:
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B. A. Wichmann
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National Physical Laboratory
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Teddington Middx
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TW11 OLW
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UK
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converted to C by:
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David M. Gay and Thos Sumner
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AT&T Bell Labs Computer Center, Rm. U-76
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600 Mountain Avenue University of California
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Murray Hill, NJ 07974 San Francisco, CA 94143
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USA USA
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with simultaneous corrections to the Pascal source (reflected
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in the Pascal source available over netlib).
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[A couple of bug fixes from dgh = sun!dhough incorporated 31 July 1986.]
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Reports of results on various systems from all the versions
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of Paranoia are being collected by Richard Karpinski at the
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same address as Thos Sumner. This includes sample outputs,
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bug reports, and criticisms.
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You may copy this program freely if you acknowledge its source.
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Comments on the Pascal version to NPL, please.
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The C version catches signals from floating-point exceptions.
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If signal(SIGFPE,...) is unavailable in your environment, you may
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#define NOSIGNAL to comment out the invocations of signal.
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This source file is too big for some C compilers, but may be split
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into pieces. Comments containing "SPLIT" suggest convenient places
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for this splitting. At the end of these comments is an "ed script"
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(for the UNIX(tm) editor ed) that will do this splitting.
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By #defining Single when you compile this source, you may obtain
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a single-precision C version of Paranoia.
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The following is from the introductory commentary from Wichmann's work:
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The BASIC program of Kahan is written in Microsoft BASIC using many
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facilities which have no exact analogy in Pascal. The Pascal
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version below cannot therefore be exactly the same. Rather than be
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a minimal transcription of the BASIC program, the Pascal coding
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follows the conventional style of block-structured languages. Hence
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the Pascal version could be useful in producing versions in other
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structured languages.
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Rather than use identifiers of minimal length (which therefore have
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little mnemonic significance), the Pascal version uses meaningful
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identifiers as follows [Note: A few changes have been made for C]:
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BASIC C BASIC C BASIC C
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A J S StickyBit
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A1 AInverse J0 NoErrors T
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B Radix [Failure] T0 Underflow
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B1 BInverse J1 NoErrors T2 ThirtyTwo
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B2 RadixD2 [SeriousDefect] T5 OneAndHalf
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B9 BMinusU2 J2 NoErrors T7 TwentySeven
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C [Defect] T8 TwoForty
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C1 CInverse J3 NoErrors U OneUlp
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D [Flaw] U0 UnderflowThreshold
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D4 FourD K PageNo U1
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E0 L Milestone U2
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E1 M V
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E2 Exp2 N V0
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E3 N1 V8
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E5 MinSqEr O Zero V9
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E6 SqEr O1 One W
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E7 MaxSqEr O2 Two X
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E8 O3 Three X1
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E9 O4 Four X8
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F1 MinusOne O5 Five X9 Random1
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F2 Half O8 Eight Y
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F3 Third O9 Nine Y1
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F6 P Precision Y2
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F9 Q Y9 Random2
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G1 GMult Q8 Z
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G2 GDiv Q9 Z0 PseudoZero
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G3 GAddSub R Z1
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H R1 RMult Z2
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H1 HInverse R2 RDiv Z9
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I R3 RAddSub
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IO NoTrials R4 RSqrt
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I3 IEEE R9 Random9
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SqRWrng
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All the variables in BASIC are true variables and in consequence,
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the program is more difficult to follow since the "constants" must
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be determined (the glossary is very helpful). The Pascal version
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uses Real constants, but checks are added to ensure that the values
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are correctly converted by the compiler.
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The major textual change to the Pascal version apart from the
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identifiersis that named procedures are used, inserting parameters
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wherehelpful. New procedures are also introduced. The
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correspondence is as follows:
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BASIC Pascal
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lines
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90- 140 Pause
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170- 250 Instructions
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380- 460 Heading
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480- 670 Characteristics
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690- 870 History
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2940-2950 Random
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3710-3740 NewD
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4040-4080 DoesYequalX
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4090-4110 PrintIfNPositive
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4640-4850 TestPartialUnderflow
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=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=
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Below is an "ed script" that splits para.c into 10 files
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of the form part[1-8].c, subs.c, and msgs.c, plus a header
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file, paranoia.h, that these files require.
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r paranoia.c
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$
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?SPLIT
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+,$w msgs.c
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.,$d
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?SPLIT
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.d
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+d
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-,$w subs.c
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-,$d
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?part8
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+d
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?include
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.,$w part8.c
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.,$d
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-d
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?part7
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+d
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?include
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.,$w part7.c
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.,$d
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-d
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?part6
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+d
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?include
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.,$w part6.c
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.,$d
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-d
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?part5
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+d
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?include
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.,$w part5.c
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.,$d
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-d
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?part4
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+d
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?include
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.,$w part4.c
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.,$d
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-d
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?part3
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+d
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?include
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.,$w part3.c
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.,$d
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-d
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?part2
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+d
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?include
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.,$w part2.c
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.,$d
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?SPLIT
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.d
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1,/^#include/-1d
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1,$w part1.c
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/Computed constants/,$d
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1,$s/^int/extern &/
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1,$s/^FLOAT/extern &/
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1,$s/^char/extern &/
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1,$s! = .*!;!
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/^Guard/,/^Round/s/^/extern /
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/^jmp_buf/s/^/extern /
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/^Sig_type/s/^/extern /
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s/$/\
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extern void sigfpe();/
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w paranoia.h
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q
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*/
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#include <stdio.h>
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#ifndef NOSIGNAL
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#include <signal.h>
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#endif
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#include <setjmp.h>
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extern double fabs(), floor(), log(), pow(), sqrt();
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#ifdef Single
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#define FLOAT float
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#define FABS(x) (float)fabs((double)(x))
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#define FLOOR(x) (float)floor((double)(x))
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#define LOG(x) (float)log((double)(x))
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#define POW(x,y) (float)pow((double)(x),(double)(y))
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#define SQRT(x) (float)sqrt((double)(x))
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#else
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#define FLOAT double
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#define FABS(x) fabs(x)
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#define FLOOR(x) floor(x)
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#define LOG(x) log(x)
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#define POW(x,y) pow(x,y)
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#define SQRT(x) sqrt(x)
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#endif
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jmp_buf ovfl_buf;
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typedef void (*Sig_type)();
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Sig_type sigsave;
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#define KEYBOARD 0
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FLOAT Radix, BInvrse, RadixD2, BMinusU2;
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FLOAT Sign(), Random();
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/*Small floating point constants.*/
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FLOAT Zero = 0.0;
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FLOAT Half = 0.5;
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FLOAT One = 1.0;
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FLOAT Two = 2.0;
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FLOAT Three = 3.0;
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FLOAT Four = 4.0;
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FLOAT Five = 5.0;
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FLOAT Eight = 8.0;
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FLOAT Nine = 9.0;
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FLOAT TwentySeven = 27.0;
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FLOAT ThirtyTwo = 32.0;
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FLOAT TwoForty = 240.0;
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FLOAT MinusOne = -1.0;
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FLOAT OneAndHalf = 1.5;
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/*Integer constants*/
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int NoTrials = 20; /*Number of tests for commutativity. */
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#define False 0
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#define True 1
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/* Definitions for declared types
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Guard == (Yes, No);
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Rounding == (Chopped, Rounded, Other);
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Message == packed array [1..40] of char;
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Class == (Flaw, Defect, Serious, Failure);
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*/
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#define Yes 1
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#define No 0
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#define Chopped 2
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#define Rounded 1
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#define Other 0
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#define Flaw 3
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#define Defect 2
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#define Serious 1
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#define Failure 0
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typedef int Guard, Rounding, Class;
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typedef char Message;
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/* Declarations of Variables */
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int Indx;
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char ch[8];
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FLOAT AInvrse, A1;
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FLOAT C, CInvrse;
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FLOAT D, FourD;
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FLOAT E0, E1, Exp2, E3, MinSqEr;
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FLOAT SqEr, MaxSqEr, E9;
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FLOAT Third;
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FLOAT F6, F9;
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FLOAT H, HInvrse;
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int I;
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FLOAT StickyBit, J;
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FLOAT MyZero;
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FLOAT Precision;
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FLOAT Q, Q9;
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FLOAT R, Random9;
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FLOAT T, Underflow, S;
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FLOAT OneUlp, UfThold, U1, U2;
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FLOAT V, V0, V9;
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FLOAT W;
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FLOAT X, X1, X2, X8, Random1;
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FLOAT Y, Y1, Y2, Random2;
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FLOAT Z, PseudoZero, Z1, Z2, Z9;
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int ErrCnt[4];
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int fpecount;
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int Milestone;
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int PageNo;
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int M, N, N1;
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Guard GMult, GDiv, GAddSub;
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Rounding RMult, RDiv, RAddSub, RSqrt;
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int Break, Done, NotMonot, Monot, Anomaly, IEEE,
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SqRWrng, UfNGrad;
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/* Computed constants. */
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/*U1 gap below 1.0, i.e, 1.0-U1 is next number below 1.0 */
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/*U2 gap above 1.0, i.e, 1.0+U2 is next number above 1.0 */
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/* floating point exception receiver */
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void
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sigfpe(i)
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{
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fpecount++;
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printf("\n* * * FLOATING-POINT ERROR * * *\n");
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fflush(stdout);
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if (sigsave) {
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#ifndef NOSIGNAL
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signal(SIGFPE, sigsave);
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#endif
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sigsave = 0;
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longjmp(ovfl_buf, 1);
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}
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abort();
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}
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main()
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{
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#ifdef mc
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char *out;
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ieee_flags("set", "precision", "double", &out);
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#endif
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/* First two assignments use integer right-hand sides. */
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Zero = 0;
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One = 1;
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Two = One + One;
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Three = Two + One;
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Four = Three + One;
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Five = Four + One;
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Eight = Four + Four;
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Nine = Three * Three;
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TwentySeven = Nine * Three;
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ThirtyTwo = Four * Eight;
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TwoForty = Four * Five * Three * Four;
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MinusOne = -One;
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Half = One / Two;
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OneAndHalf = One + Half;
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ErrCnt[Failure] = 0;
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ErrCnt[Serious] = 0;
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ErrCnt[Defect] = 0;
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ErrCnt[Flaw] = 0;
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PageNo = 1;
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/*=============================================*/
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Milestone = 0;
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/*=============================================*/
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#ifndef NOSIGNAL
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signal(SIGFPE, sigfpe);
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#endif
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Instructions();
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Pause();
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Heading();
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Pause();
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Characteristics();
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Pause();
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History();
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Pause();
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/*=============================================*/
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Milestone = 7;
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/*=============================================*/
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printf("Program is now RUNNING tests on small integers:\n");
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TstCond (Failure, (Zero + Zero == Zero) && (One - One == Zero)
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&& (One > Zero) && (One + One == Two),
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"0+0 != 0, 1-1 != 0, 1 <= 0, or 1+1 != 2");
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Z = - Zero;
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if (Z != 0.0) {
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ErrCnt[Failure] = ErrCnt[Failure] + 1;
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printf("Comparison alleges that -0.0 is Non-zero!\n");
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U1 = 0.001;
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Radix = 1;
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TstPtUf();
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}
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TstCond (Failure, (Three == Two + One) && (Four == Three + One)
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&& (Four + Two * (- Two) == Zero)
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&& (Four - Three - One == Zero),
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"3 != 2+1, 4 != 3+1, 4+2*(-2) != 0, or 4-3-1 != 0");
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TstCond (Failure, (MinusOne == (0 - One))
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&& (MinusOne + One == Zero ) && (One + MinusOne == Zero)
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&& (MinusOne + FABS(One) == Zero)
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&& (MinusOne + MinusOne * MinusOne == Zero),
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"-1+1 != 0, (-1)+abs(1) != 0, or -1+(-1)*(-1) != 0");
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TstCond (Failure, Half + MinusOne + Half == Zero,
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"1/2 + (-1) + 1/2 != 0");
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/*=============================================*/
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/*SPLIT
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part2();
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part3();
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part4();
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part5();
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part6();
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part7();
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part8();
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}
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#include "paranoia.h"
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part2(){
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*/
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Milestone = 10;
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/*=============================================*/
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TstCond (Failure, (Nine == Three * Three)
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&& (TwentySeven == Nine * Three) && (Eight == Four + Four)
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&& (ThirtyTwo == Eight * Four)
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&& (ThirtyTwo - TwentySeven - Four - One == Zero),
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"9 != 3*3, 27 != 9*3, 32 != 8*4, or 32-27-4-1 != 0");
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TstCond (Failure, (Five == Four + One) &&
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(TwoForty == Four * Five * Three * Four)
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&& (TwoForty / Three - Four * Four * Five == Zero)
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&& ( TwoForty / Four - Five * Three * Four == Zero)
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&& ( TwoForty / Five - Four * Three * Four == Zero),
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"5 != 4+1, 240/3 != 80, 240/4 != 60, or 240/5 != 48");
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if (ErrCnt[Failure] == 0) {
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printf("-1, 0, 1/2, 1, 2, 3, 4, 5, 9, 27, 32 & 240 are O.K.\n");
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printf("\n");
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}
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printf("Searching for Radix and Precision.\n");
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W = One;
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do {
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W = W + W;
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Y = W + One;
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Z = Y - W;
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Y = Z - One;
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} while (MinusOne + FABS(Y) < Zero);
|
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/*.. now W is just big enough that |((W+1)-W)-1| >= 1 ...*/
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Precision = Zero;
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Y = One;
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do {
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Radix = W + Y;
|
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Y = Y + Y;
|
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Radix = Radix - W;
|
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} while ( Radix == Zero);
|
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if (Radix < Two) Radix = One;
|
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printf("Radix = %f .\n", Radix);
|
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if (Radix != 1) {
|
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W = One;
|
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do {
|
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Precision = Precision + One;
|
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W = W * Radix;
|
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Y = W + One;
|
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} while ((Y - W) == One);
|
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}
|
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/*... now W == Radix^Precision is barely too big to satisfy (W+1)-W == 1
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...*/
|
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U1 = One / W;
|
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U2 = Radix * U1;
|
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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)));
|
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|
|
/*... now U2 == 1 ulp of 1 + ... */
|
|
X = Two / Three;
|
|
F6 = X - Half;
|
|
Third = F6 + F6;
|
|
X = Third - Half;
|
|
X = FABS(X + F6);
|
|
if (X < U1) X = U1;
|
|
|
|
/*... now X == (unknown no.) ulps of 1 -... */
|
|
do {
|
|
U1 = X;
|
|
Y = Half * U1 + ThirtyTwo * U1 * U1;
|
|
Y = Half - Y;
|
|
X = Half + Y;
|
|
Y = Half - X;
|
|
X = Half + Y;
|
|
} while ( ! ((U1 <= X) || (X <= Zero)));
|
|
/*... now U1 == 1 ulp of 1 - ... */
|
|
if (U1 == E1) printf("confirms closest relative separation U1 .\n");
|
|
else printf("gets better closest relative separation U1 = %.7e .\n", U1);
|
|
W = One / U1;
|
|
F9 = (Half - U1) + Half;
|
|
Radix = FLOOR(0.01 + U2 / U1);
|
|
if (Radix == E0) printf("Radix confirmed.\n");
|
|
else printf("MYSTERY: recalculated Radix = %.7e .\n", Radix);
|
|
TstCond (Defect, Radix <= Eight + Eight,
|
|
"Radix is too big: roundoff problems");
|
|
TstCond (Flaw, (Radix == Two) || (Radix == 10)
|
|
|| (Radix == One), "Radix is not as good as 2 or 10");
|
|
/*=============================================*/
|
|
Milestone = 20;
|
|
/*=============================================*/
|
|
TstCond (Failure, F9 - Half < Half,
|
|
"(1-U1)-1/2 < 1/2 is FALSE, prog. fails?");
|
|
X = F9;
|
|
I = 1;
|
|
Y = X - Half;
|
|
Z = Y - Half;
|
|
TstCond (Failure, (X != One)
|
|
|| (Z == Zero), "Comparison is fuzzy,X=1 but X-1/2-1/2 != 0");
|
|
X = One + U2;
|
|
I = 0;
|
|
/*=============================================*/
|
|
Milestone = 25;
|
|
/*=============================================*/
|
|
/*... BMinusU2 = nextafter(Radix, 0) */
|
|
BMinusU2 = Radix - One;
|
|
BMinusU2 = (BMinusU2 - U2) + One;
|
|
/* Purify Integers */
|
|
if (Radix != One) {
|
|
X = - TwoForty * LOG(U1) / LOG(Radix);
|
|
Y = FLOOR(Half + X);
|
|
if (FABS(X - Y) * Four < One) X = Y;
|
|
Precision = X / TwoForty;
|
|
Y = FLOOR(Half + Precision);
|
|
if (FABS(Precision - Y) * TwoForty < Half) Precision = Y;
|
|
}
|
|
if ((Precision != FLOOR(Precision)) || (Radix == One)) {
|
|
printf("Precision cannot be characterized by an Integer number\n");
|
|
printf("of significant digits but, by itself, this is a minor flaw.\n");
|
|
}
|
|
if (Radix == One)
|
|
printf("logarithmic encoding has precision characterized solely by U1.\n");
|
|
else printf("The number of significant digits of the Radix is %f .\n",
|
|
Precision);
|
|
TstCond (Serious, U2 * Nine * Nine * TwoForty < One,
|
|
"Precision worse than 5 decimal figures ");
|
|
/*=============================================*/
|
|
Milestone = 30;
|
|
/*=============================================*/
|
|
/* Test for extra-precise subepressions */
|
|
X = FABS(((Four / Three - One) - One / Four) * Three - One / Four);
|
|
do {
|
|
Z2 = X;
|
|
X = (One + (Half * Z2 + ThirtyTwo * Z2 * Z2)) - One;
|
|
} while ( ! ((Z2 <= X) || (X <= Zero)));
|
|
X = Y = Z = FABS((Three / Four - Two / Three) * Three - One / Four);
|
|
do {
|
|
Z1 = Z;
|
|
Z = (One / Two - ((One / Two - (Half * Z1 + ThirtyTwo * Z1 * Z1))
|
|
+ One / Two)) + One / Two;
|
|
} while ( ! ((Z1 <= Z) || (Z <= Zero)));
|
|
do {
|
|
do {
|
|
Y1 = Y;
|
|
Y = (Half - ((Half - (Half * Y1 + ThirtyTwo * Y1 * Y1)) + Half
|
|
)) + Half;
|
|
} while ( ! ((Y1 <= Y) || (Y <= Zero)));
|
|
X1 = X;
|
|
X = ((Half * X1 + ThirtyTwo * X1 * X1) - F9) + F9;
|
|
} while ( ! ((X1 <= X) || (X <= Zero)));
|
|
if ((X1 != Y1) || (X1 != Z1)) {
|
|
BadCond(Serious, "Disagreements among the values X1, Y1, Z1,\n");
|
|
printf("respectively %.7e, %.7e, %.7e,\n", X1, Y1, Z1);
|
|
printf("are symptoms of inconsistencies introduced\n");
|
|
printf("by extra-precise evaluation of arithmetic subexpressions.\n");
|
|
notify("Possibly some part of this");
|
|
if ((X1 == U1) || (Y1 == U1) || (Z1 == U1)) printf(
|
|
"That feature is not tested further by this program.\n") ;
|
|
}
|
|
else {
|
|
if ((Z1 != U1) || (Z2 != U2)) {
|
|
if ((Z1 >= U1) || (Z2 >= U2)) {
|
|
BadCond(Failure, "");
|
|
notify("Precision");
|
|
printf("\tU1 = %.7e, Z1 - U1 = %.7e\n",U1,Z1-U1);
|
|
printf("\tU2 = %.7e, Z2 - U2 = %.7e\n",U2,Z2-U2);
|
|
}
|
|
else {
|
|
if ((Z1 <= Zero) || (Z2 <= Zero)) {
|
|
printf("Because of unusual Radix = %f", Radix);
|
|
printf(", or exact rational arithmetic a result\n");
|
|
printf("Z1 = %.7e, or Z2 = %.7e ", Z1, Z2);
|
|
notify("of an\nextra-precision");
|
|
}
|
|
if (Z1 != Z2 || Z1 > Zero) {
|
|
X = Z1 / U1;
|
|
Y = Z2 / U2;
|
|
if (Y > X) X = Y;
|
|
Q = - LOG(X);
|
|
printf("Some subexpressions appear to be calculated extra\n");
|
|
printf("precisely with about %g extra B-digits, i.e.\n",
|
|
(Q / LOG(Radix)));
|
|
printf("roughly %g extra significant decimals.\n",
|
|
Q / LOG(10.));
|
|
}
|
|
printf("That feature is not tested further by this program.\n");
|
|
}
|
|
}
|
|
}
|
|
Pause();
|
|
/*=============================================*/
|
|
/*SPLIT
|
|
}
|
|
#include "paranoia.h"
|
|
part3(){
|
|
*/
|
|
Milestone = 35;
|
|
/*=============================================*/
|
|
if (Radix >= Two) {
|
|
X = W / (Radix * Radix);
|
|
Y = X + One;
|
|
Z = Y - X;
|
|
T = Z + U2;
|
|
X = T - Z;
|
|
TstCond (Failure, X == U2,
|
|
"Subtraction is not normalized X=Y,X+Z != Y+Z!");
|
|
if (X == U2) printf(
|
|
"Subtraction appears to be normalized, as it should be.");
|
|
}
|
|
printf("\nChecking for guard digit in *, /, and -.\n");
|
|
Y = F9 * One;
|
|
Z = One * F9;
|
|
X = F9 - Half;
|
|
Y = (Y - Half) - X;
|
|
Z = (Z - Half) - X;
|
|
X = One + U2;
|
|
T = X * Radix;
|
|
R = Radix * X;
|
|
X = T - Radix;
|
|
X = X - Radix * U2;
|
|
T = R - Radix;
|
|
T = T - Radix * U2;
|
|
X = X * (Radix - One);
|
|
T = T * (Radix - One);
|
|
if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero)) GMult = Yes;
|
|
else {
|
|
GMult = No;
|
|
TstCond (Serious, False,
|
|
"* lacks a Guard Digit, so 1*X != X");
|
|
}
|
|
Z = Radix * U2;
|
|
X = One + Z;
|
|
Y = FABS((X + Z) - X * X) - U2;
|
|
X = One - U2;
|
|
Z = FABS((X - U2) - X * X) - U1;
|
|
TstCond (Failure, (Y <= Zero)
|
|
&& (Z <= Zero), "* gets too many final digits wrong.\n");
|
|
Y = One - U2;
|
|
X = One + U2;
|
|
Z = One / Y;
|
|
Y = Z - X;
|
|
X = One / Three;
|
|
Z = Three / Nine;
|
|
X = X - Z;
|
|
T = Nine / TwentySeven;
|
|
Z = Z - T;
|
|
TstCond(Defect, X == Zero && Y == Zero && Z == Zero,
|
|
"Division lacks a Guard Digit, so error can exceed 1 ulp\nor 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"
|
|
part4(){
|
|
*/
|
|
Milestone = 50;
|
|
/*=============================================*/
|
|
TstCond (Failure, ((F9 + U1) - Half == Half)
|
|
&& ((BMinusU2 + U2 ) - One == Radix - One),
|
|
"Incomplete carry-propagation in Addition");
|
|
X = One - U1 * U1;
|
|
Y = One + U2 * (One - U2);
|
|
Z = F9 - Half;
|
|
X = (X - Half) - Z;
|
|
Y = Y - One;
|
|
if ((X == Zero) && (Y == Zero)) {
|
|
RAddSub = Chopped;
|
|
printf("Add/Subtract appears to be chopped.\n");
|
|
}
|
|
if (GAddSub == Yes) {
|
|
X = (Half + U2) * U2;
|
|
Y = (Half - U2) * U2;
|
|
X = One + X;
|
|
Y = One + Y;
|
|
X = (One + U2) - X;
|
|
Y = One - Y;
|
|
if ((X == Zero) && (Y == Zero)) {
|
|
X = (Half + U2) * U1;
|
|
Y = (Half - U2) * U1;
|
|
X = One - X;
|
|
Y = One - Y;
|
|
X = F9 - X;
|
|
Y = One - Y;
|
|
if ((X == Zero) && (Y == Zero)) {
|
|
RAddSub = Rounded;
|
|
printf("Addition/Subtraction appears to round correctly.\n");
|
|
if (GAddSub == No) notify("Add/Subtract");
|
|
}
|
|
else printf("Addition/Subtraction neither rounds nor chops.\n");
|
|
}
|
|
else printf("Addition/Subtraction neither rounds nor chops.\n");
|
|
}
|
|
else printf("Addition/Subtraction neither rounds nor chops.\n");
|
|
S = One;
|
|
X = One + Half * (One + Half);
|
|
Y = (One + U2) * Half;
|
|
Z = X - Y;
|
|
T = Y - X;
|
|
StickyBit = Z + T;
|
|
if (StickyBit != Zero) {
|
|
S = Zero;
|
|
BadCond(Flaw, "(X - Y) + (Y - X) is non zero!\n");
|
|
}
|
|
StickyBit = Zero;
|
|
if ((GMult == Yes) && (GDiv == Yes) && (GAddSub == Yes)
|
|
&& (RMult == Rounded) && (RDiv == Rounded)
|
|
&& (RAddSub == Rounded) && (FLOOR(RadixD2) == RadixD2)) {
|
|
printf("Checking for sticky bit.\n");
|
|
X = (Half + U1) * U2;
|
|
Y = Half * U2;
|
|
Z = One + Y;
|
|
T = One + X;
|
|
if ((Z - One <= Zero) && (T - One >= U2)) {
|
|
Z = T + Y;
|
|
Y = Z - X;
|
|
if ((Z - T >= U2) && (Y - T == Zero)) {
|
|
X = (Half + U1) * U1;
|
|
Y = Half * U1;
|
|
Z = One - Y;
|
|
T = One - X;
|
|
if ((Z - One == Zero) && (T - F9 == Zero)) {
|
|
Z = (Half - U1) * U1;
|
|
T = F9 - Z;
|
|
Q = F9 - Y;
|
|
if ((T - F9 == Zero) && (F9 - U1 - Q == Zero)) {
|
|
Z = (One + U2) * OneAndHalf;
|
|
T = (OneAndHalf + U2) - Z + U2;
|
|
X = One + Half / Radix;
|
|
Y = One + Radix * U2;
|
|
Z = X * Y;
|
|
if (T == Zero && X + Radix * U2 - Z == Zero) {
|
|
if (Radix != Two) {
|
|
X = Two + U2;
|
|
Y = X / Two;
|
|
if ((Y - One == Zero)) StickyBit = S;
|
|
}
|
|
else StickyBit = S;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
}
|
|
if (StickyBit == One) printf("Sticky bit apparently used correctly.\n");
|
|
else printf("Sticky bit used incorrectly or not at all.\n");
|
|
TstCond (Flaw, !(GMult == No || GDiv == No || GAddSub == No ||
|
|
RMult == Other || RDiv == Other || RAddSub == Other),
|
|
"lack(s) of guard digits or failure(s) to correctly round or chop\n(noted above) count as one flaw in the final tally below");
|
|
/*=============================================*/
|
|
Milestone = 60;
|
|
/*=============================================*/
|
|
printf("\n");
|
|
printf("Does Multiplication commute? ");
|
|
printf("Testing on %d random pairs.\n", NoTrials);
|
|
Random9 = SQRT(3.0);
|
|
Random1 = Third;
|
|
I = 1;
|
|
do {
|
|
X = Random();
|
|
Y = Random();
|
|
Z9 = Y * X;
|
|
Z = X * Y;
|
|
Z9 = Z - Z9;
|
|
I = I + 1;
|
|
} while ( ! ((I > NoTrials) || (Z9 != Zero)));
|
|
if (I == NoTrials) {
|
|
Random1 = One + Half / Three;
|
|
Random2 = (U2 + U1) + One;
|
|
Z = Random1 * Random2;
|
|
Y = Random2 * Random1;
|
|
Z9 = (One + Half / Three) * ((U2 + U1) + One) - (One + Half /
|
|
Three) * ((U2 + U1) + One);
|
|
}
|
|
if (! ((I == NoTrials) || (Z9 == Zero)))
|
|
BadCond(Defect, "X * Y == Y * X trial fails.\n");
|
|
else printf(" No failures found in %d integer pairs.\n", NoTrials);
|
|
/*=============================================*/
|
|
Milestone = 70;
|
|
/*=============================================*/
|
|
printf("\nRunning test of square root(x).\n");
|
|
TstCond (Failure, (Zero == SQRT(Zero))
|
|
&& (- Zero == SQRT(- Zero))
|
|
&& (One == SQRT(One)), "Square root of 0.0, -0.0 or 1.0 wrong");
|
|
MinSqEr = Zero;
|
|
MaxSqEr = Zero;
|
|
J = Zero;
|
|
X = Radix;
|
|
OneUlp = U2;
|
|
SqXMinX (Serious);
|
|
X = BInvrse;
|
|
OneUlp = BInvrse * U1;
|
|
SqXMinX (Serious);
|
|
X = U1;
|
|
OneUlp = U1 * U1;
|
|
SqXMinX (Serious);
|
|
if (J != Zero) Pause();
|
|
printf("Testing if sqrt(X * X) == X for %d Integers X.\n", NoTrials);
|
|
J = Zero;
|
|
X = Two;
|
|
Y = Radix;
|
|
if ((Radix != One)) do {
|
|
X = Y;
|
|
Y = Radix * Y;
|
|
} while ( ! ((Y - X >= NoTrials)));
|
|
OneUlp = X * U2;
|
|
I = 1;
|
|
while (I <= 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"
|
|
part5(){
|
|
*/
|
|
Milestone = 80;
|
|
/*=============================================*/
|
|
MinSqEr = MinSqEr + Half;
|
|
MaxSqEr = MaxSqEr - Half;
|
|
Y = (SQRT(One + U2) - One) / U2;
|
|
SqEr = (Y - One) + U2 / Eight;
|
|
if (SqEr > MaxSqEr) MaxSqEr = SqEr;
|
|
SqEr = Y + U2 / Eight;
|
|
if (SqEr < MinSqEr) MinSqEr = SqEr;
|
|
Y = ((SQRT(F9) - U2) - (One - U2)) / U1;
|
|
SqEr = Y + U1 / Eight;
|
|
if (SqEr > MaxSqEr) MaxSqEr = SqEr;
|
|
SqEr = (Y + One) + U1 / Eight;
|
|
if (SqEr < MinSqEr) MinSqEr = SqEr;
|
|
OneUlp = U2;
|
|
X = OneUlp;
|
|
for( Indx = 1; Indx <= 3; ++Indx) {
|
|
Y = SQRT((X + U1 + X) + F9);
|
|
Y = ((Y - U2) - ((One - U2) + X)) / OneUlp;
|
|
Z = ((U1 - X) + F9) * Half * X * X / OneUlp;
|
|
SqEr = (Y + Half) + Z;
|
|
if (SqEr < MinSqEr) MinSqEr = SqEr;
|
|
SqEr = (Y - Half) + Z;
|
|
if (SqEr > MaxSqEr) MaxSqEr = SqEr;
|
|
if (((Indx == 1) || (Indx == 3)))
|
|
X = OneUlp * Sign (X) * FLOOR(Eight / (Nine * SQRT(OneUlp)));
|
|
else {
|
|
OneUlp = U1;
|
|
X = - OneUlp;
|
|
}
|
|
}
|
|
/*=============================================*/
|
|
Milestone = 85;
|
|
/*=============================================*/
|
|
SqRWrng = False;
|
|
Anomaly = False;
|
|
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 = 3.0;
|
|
Break = False;
|
|
do {
|
|
X = One;
|
|
SR3980();
|
|
if (I <= 10) {
|
|
I = 1023;
|
|
SR3980();
|
|
}
|
|
if (Z == MinusOne) Break = True;
|
|
else {
|
|
Z = MinusOne;
|
|
PrintIfNPositive();
|
|
N = 0;
|
|
/* .. if(-1)^N is invalid, replace MinusOne by One. */
|
|
I = - 4;
|
|
}
|
|
} while ( ! Break);
|
|
PrintIfNPositive();
|
|
N1 = N;
|
|
N = 0;
|
|
Z = A1;
|
|
M = FLOOR(Two * LOG(W) / LOG(A1));
|
|
Break = False;
|
|
do {
|
|
X = Z;
|
|
I = 1;
|
|
SR3980();
|
|
if (Z == AInvrse) Break = True;
|
|
else Z = AInvrse;
|
|
} while ( ! (Break));
|
|
/*=============================================*/
|
|
Milestone = 100;
|
|
/*=============================================*/
|
|
/* Powers of Radix have been tested, */
|
|
/* next try a few primes */
|
|
M = NoTrials;
|
|
Z = Three;
|
|
do {
|
|
X = Z;
|
|
I = 1;
|
|
SR3980();
|
|
do {
|
|
Z = Z + Two;
|
|
} while ( Three * FLOOR(Z / Three) == Z );
|
|
} while ( Z < Eight * Three );
|
|
if (N > 0) {
|
|
printf("Errors like this may invalidate financial calculations\n");
|
|
printf("\tinvolving interest rates.\n");
|
|
}
|
|
PrintIfNPositive();
|
|
N += N1;
|
|
if (N == 0) printf("... no discrepancis found.\n");
|
|
if (N > 0) Pause();
|
|
else printf("\n");
|
|
/*=============================================*/
|
|
/*SPLIT
|
|
}
|
|
#include "paranoia.h"
|
|
part6(){
|
|
*/
|
|
Milestone = 110;
|
|
/*=============================================*/
|
|
printf("Seeking Underflow thresholds UfThold and E0.\n");
|
|
D = U1;
|
|
if (Precision != FLOOR(Precision)) {
|
|
D = BInvrse;
|
|
X = Precision;
|
|
do {
|
|
D = D * BInvrse;
|
|
X = X - One;
|
|
} while ( X > Zero);
|
|
}
|
|
Y = One;
|
|
Z = D;
|
|
/* ... D is power of 1/Radix < 1. */
|
|
do {
|
|
C = Y;
|
|
Y = Z;
|
|
Z = Y * Y;
|
|
} 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"
|
|
part7(){
|
|
*/
|
|
Milestone = 130;
|
|
/*=============================================*/
|
|
Y = - FLOOR(Half - TwoForty * LOG(UfThold) / LOG(HInvrse)) / TwoForty;
|
|
Y2 = Y + Y;
|
|
printf("Since underflow occurs below the threshold\n");
|
|
printf("UfThold = (%.17e) ^ (%.17e)\nonly underflow ", HInvrse, Y);
|
|
printf("should afflict the expression\n\t(%.17e) ^ (%.17e);\n", HInvrse, Y);
|
|
V9 = POW(HInvrse, Y2);
|
|
printf("actually calculating yields: %.17e .\n", V9);
|
|
if (! ((V9 >= Zero) && (V9 <= (Radix + Radix + E9) * UfThold))) {
|
|
BadCond(Serious, "this is not between 0 and underflow\n");
|
|
printf(" threshold = %.17e .\n", UfThold);
|
|
}
|
|
else if (! (V9 > UfThold * (One + E9)))
|
|
printf("This computed value is O.K.\n");
|
|
else {
|
|
BadCond(Defect, "this is not between 0 and underflow\n");
|
|
printf(" threshold = %.17e .\n", UfThold);
|
|
}
|
|
/*=============================================*/
|
|
Milestone = 140;
|
|
/*=============================================*/
|
|
printf("\n");
|
|
/* ...calculate Exp2 == exp(2) == 7.389056099... */
|
|
X = Zero;
|
|
I = 2;
|
|
Y = Two * Three;
|
|
Q = Zero;
|
|
N = 0;
|
|
do {
|
|
Z = X;
|
|
I = I + 1;
|
|
Y = Y / (I + I);
|
|
R = Y + Q;
|
|
X = Z + R;
|
|
Q = (Z - X) + R;
|
|
} while(X > Z);
|
|
Z = (OneAndHalf + One / Eight) + X / (OneAndHalf * ThirtyTwo);
|
|
X = Z * Z;
|
|
Exp2 = X * X;
|
|
X = F9;
|
|
Y = X - U1;
|
|
printf("Testing X^((X + 1) / (X - 1)) vs. exp(2) = %.17e as X -> 1.\n",
|
|
Exp2);
|
|
for(I = 1;;) {
|
|
Z = X - BInvrse;
|
|
Z = (X + One) / (Z - (One - BInvrse));
|
|
Q = POW(X, Z) - Exp2;
|
|
if (FABS(Q) > TwoForty * U2) {
|
|
N = 1;
|
|
V9 = (X - BInvrse) - (One - BInvrse);
|
|
BadCond(Defect, "Calculated");
|
|
printf(" %.17e for\n", POW(X,Z));
|
|
printf("\t(1 + (%.17e) ^ (%.17e);\n", V9, Z);
|
|
printf("\tdiffers from correct value by %.17e .\n", Q);
|
|
printf("\tThis much error may spoil financial\n");
|
|
printf("\tcalculations involving tiny interest rates.\n");
|
|
break;
|
|
}
|
|
else {
|
|
Z = (Y - X) * Two + Y;
|
|
X = Y;
|
|
Y = Z;
|
|
Z = One + (X - F9)*(X - F9);
|
|
if (Z > One && I < NoTrials) I++;
|
|
else {
|
|
if (X > One) {
|
|
if (N == 0)
|
|
printf("Accuracy seems adequate.\n");
|
|
break;
|
|
}
|
|
else {
|
|
X = One + U2;
|
|
Y = U2 + U2;
|
|
Y += X;
|
|
I = 1;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
/*=============================================*/
|
|
Milestone = 150;
|
|
/*=============================================*/
|
|
printf("Testing powers Z^Q at four nearly extreme values.\n");
|
|
N = 0;
|
|
Z = A1;
|
|
Q = FLOOR(Half - LOG(C) / LOG(A1));
|
|
Break = False;
|
|
do {
|
|
X = CInvrse;
|
|
Y = POW(Z, Q);
|
|
IsYeqX();
|
|
Q = - Q;
|
|
X = C;
|
|
Y = POW(Z, Q);
|
|
IsYeqX();
|
|
if (Z < One) Break = True;
|
|
else Z = AInvrse;
|
|
} while ( ! (Break));
|
|
PrintIfNPositive();
|
|
if (N == 0) printf(" ... no discrepancies found.\n");
|
|
printf("\n");
|
|
|
|
/*=============================================*/
|
|
Milestone = 160;
|
|
/*=============================================*/
|
|
Pause();
|
|
printf("Searching for Overflow threshold:\n");
|
|
printf("This may generate an error.\n");
|
|
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"
|
|
part8(){
|
|
*/
|
|
Milestone = 190;
|
|
/*=============================================*/
|
|
Pause();
|
|
X = UfThold * V;
|
|
Y = Radix * Radix;
|
|
if (X*Y < One || X > Y) {
|
|
if (X * Y < U1 || X > Y/U1) BadCond(Defect, "Badly");
|
|
else BadCond(Flaw, "");
|
|
|
|
printf(" unbalanced range; UfThold * V = %.17e\n\t%s\n",
|
|
X, "is too far from 1.\n");
|
|
}
|
|
/*=============================================*/
|
|
Milestone = 200;
|
|
/*=============================================*/
|
|
for (Indx = 1; Indx <= 5; ++Indx) {
|
|
X = F9;
|
|
switch (Indx) {
|
|
case 2: X = One + U2; break;
|
|
case 3: X = V; break;
|
|
case 4: X = UfThold; break;
|
|
case 5: X = Radix;
|
|
}
|
|
Y = X;
|
|
sigsave = sigfpe;
|
|
if (setjmp(ovfl_buf))
|
|
printf(" X / X traps when X = %g\n", X);
|
|
else {
|
|
V9 = (Y / X - Half) - Half;
|
|
if (V9 == Zero) continue;
|
|
if (V9 == - U1 && Indx < 5) BadCond(Flaw, "");
|
|
else BadCond(Serious, "");
|
|
printf(" X / X differs from 1 when X = %.17e\n", X);
|
|
printf(" instead, X / X - 1/2 - 1/2 = %.17e .\n", V9);
|
|
}
|
|
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"
|
|
*/
|
|
|
|
/* Sign */
|
|
|
|
FLOAT Sign (X)
|
|
FLOAT X;
|
|
{ return X >= 0. ? 1.0 : -1.0; }
|
|
|
|
/* Pause */
|
|
|
|
Pause()
|
|
{
|
|
#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;
|
|
}
|
|
|
|
/* TstCond */
|
|
|
|
TstCond (K, Valid, T)
|
|
int K, Valid;
|
|
char *T;
|
|
{ if (! Valid) { BadCond(K,T); printf(".\n"); } }
|
|
|
|
BadCond(K, T)
|
|
int K;
|
|
char *T;
|
|
{
|
|
static char *msg[] = { "FAILURE", "SERIOUS DEFECT", "DEFECT", "FLAW" };
|
|
|
|
ErrCnt [K] = ErrCnt [K] + 1;
|
|
printf("%s: %s", msg[K], T);
|
|
}
|
|
|
|
/* Random */
|
|
/* Random computes
|
|
X = (Random1 + Random9)^5
|
|
Random1 = X - FLOOR(X) + 0.000005 * X;
|
|
and returns the new value of Random1
|
|
*/
|
|
|
|
FLOAT Random()
|
|
{
|
|
FLOAT X, Y;
|
|
|
|
X = Random1 + Random9;
|
|
Y = X * X;
|
|
Y = Y * Y;
|
|
X = X * Y;
|
|
Y = X - FLOOR(X);
|
|
Random1 = Y + X * 0.000005;
|
|
return(Random1);
|
|
}
|
|
|
|
/* SqXMinX */
|
|
|
|
SqXMinX (ErrKind)
|
|
int ErrKind;
|
|
{
|
|
FLOAT XA, XB;
|
|
|
|
XB = X * BInvrse;
|
|
XA = X - XB;
|
|
SqEr = ((SQRT(X * X) - XB) - XA) / OneUlp;
|
|
if (SqEr != Zero) {
|
|
if (SqEr < MinSqEr) MinSqEr = SqEr;
|
|
if (SqEr > MaxSqEr) MaxSqEr = SqEr;
|
|
J = J + 1.0;
|
|
BadCond(ErrKind, "\n");
|
|
printf("sqrt( %.17e) - %.17e = %.17e\n", X * X, X, OneUlp * SqEr);
|
|
printf("\tinstead of correct value 0 .\n");
|
|
}
|
|
}
|
|
|
|
/* NewD */
|
|
|
|
NewD()
|
|
{
|
|
X = Z1 * Q;
|
|
X = FLOOR(Half - X / Radix) * Radix + X;
|
|
Q = (Q - X * Z) / Radix + X * X * (D / Radix);
|
|
Z = Z - Two * X * D;
|
|
if (Z <= Zero) {
|
|
Z = - Z;
|
|
Z1 = - Z1;
|
|
}
|
|
D = Radix * D;
|
|
}
|
|
|
|
/* SR3750 */
|
|
|
|
SR3750()
|
|
{
|
|
if (! ((X - Radix < Z2 - Radix) || (X - Z2 > W - Z2))) {
|
|
I = I + 1;
|
|
X2 = SQRT(X * D);
|
|
Y2 = (X2 - Z2) - (Y - Z2);
|
|
X2 = X8 / (Y - Half);
|
|
X2 = X2 - Half * X2 * X2;
|
|
SqEr = (Y2 + Half) + (Half - X2);
|
|
if (SqEr < MinSqEr) MinSqEr = SqEr;
|
|
SqEr = Y2 - X2;
|
|
if (SqEr > MaxSqEr) MaxSqEr = SqEr;
|
|
}
|
|
}
|
|
|
|
/* IsYeqX */
|
|
|
|
IsYeqX()
|
|
{
|
|
if (Y != X) {
|
|
if (N <= 0) {
|
|
if (Z == Zero && Q <= Zero)
|
|
printf("WARNING: computing\n");
|
|
else BadCond(Defect, "computing\n");
|
|
printf("\t(%.17e) ^ (%.17e)\n", Z, Q);
|
|
printf("\tyielded %.17e;\n", Y);
|
|
printf("\twhich compared unequal to correct %.17e ;\n",
|
|
X);
|
|
printf("\t\tthey differ by %.17e .\n", Y - X);
|
|
}
|
|
N = N + 1; /* ... count discrepancies. */
|
|
}
|
|
}
|
|
|
|
/* SR3980 */
|
|
|
|
SR3980()
|
|
{
|
|
do {
|
|
Q = (FLOAT) I;
|
|
Y = POW(Z, Q);
|
|
IsYeqX();
|
|
if (++I > M) break;
|
|
X = Z * X;
|
|
} while ( X < W );
|
|
}
|
|
|
|
/* PrintIfNPositive */
|
|
|
|
PrintIfNPositive()
|
|
{
|
|
if (N > 0) printf("Similar discrepancies have occurred %d times.\n", N);
|
|
}
|
|
|
|
/* TstPtUf */
|
|
|
|
TstPtUf()
|
|
{
|
|
N = 0;
|
|
if (Z != Zero) {
|
|
printf("Since comparison denies Z = 0, evaluating ");
|
|
printf("(Z + Z) / Z should be safe.\n");
|
|
sigsave = sigfpe;
|
|
if (setjmp(ovfl_buf)) goto very_serious;
|
|
Q9 = (Z + Z) / Z;
|
|
printf("What the machine gets for (Z + Z) / Z is %.17e .\n",
|
|
Q9);
|
|
if (FABS(Q9 - Two) < Radix * U2) {
|
|
printf("This is O.K., provided Over/Underflow");
|
|
printf(" has NOT just been signaled.\n");
|
|
}
|
|
else {
|
|
if ((Q9 < One) || (Q9 > Two)) {
|
|
very_serious:
|
|
N = 1;
|
|
ErrCnt [Serious] = ErrCnt [Serious] + 1;
|
|
printf("This is a VERY SERIOUS DEFECT!\n");
|
|
}
|
|
else {
|
|
N = 1;
|
|
ErrCnt [Defect] = ErrCnt [Defect] + 1;
|
|
printf("This is a DEFECT!\n");
|
|
}
|
|
}
|
|
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();
|
|
}
|
|
}
|
|
}
|
|
|
|
notify(s)
|
|
char *s;
|
|
{
|
|
printf("%s test appears to be inconsistent...\n", s);
|
|
printf(" PLEASE NOTIFY KARPINKSI!\n");
|
|
}
|
|
|
|
/*SPLIT msgs.c */
|
|
|
|
/* Instructions */
|
|
|
|
msglist(s)
|
|
char **s;
|
|
{ while(*s) printf("%s\n", *s++); }
|
|
|
|
Instructions()
|
|
{
|
|
static char *instr[] = {
|
|
"Lest this program stop prematurely, i.e. before displaying\n",
|
|
" 'END OF TEST',\n",
|
|
"try to persuade the computer NOT to terminate execution when an",
|
|
"error like Over/Underflow or Division by Zero occurs, but rather",
|
|
"to persevere with a surrogate value after, perhaps, displaying some",
|
|
"warning. If persuasion avails naught, don't despair but run this",
|
|
"program anyway to see how many milestones it passes, and then",
|
|
"amend it to make further progress.\n",
|
|
"Answer questions with Y, y, N or n (unless otherwise indicated).\n",
|
|
0};
|
|
|
|
msglist(instr);
|
|
}
|
|
|
|
/* Heading */
|
|
|
|
Heading()
|
|
{
|
|
static char *head[] = {
|
|
"Users are invited to help debug and augment this program so it will",
|
|
"cope with unanticipated and newly uncovered arithmetic pathologies.\n",
|
|
"Please send suggestions and interesting results to",
|
|
"\tRichard Karpinski",
|
|
"\tComputer Center U-76",
|
|
"\tUniversity of California",
|
|
"\tSan Francisco, CA 94143-0704, USA\n",
|
|
"In doing so, please include the following information:",
|
|
#ifdef Single
|
|
"\tPrecision:\tsingle;",
|
|
#else
|
|
"\tPrecision:\tdouble;",
|
|
#endif
|
|
"\tVersion:\t10 February 1989;",
|
|
"\tComputer:\n",
|
|
"\tCompiler:\n",
|
|
"\tOptimization level:\n",
|
|
"\tOther relevant compiler options:",
|
|
0};
|
|
|
|
msglist(head);
|
|
}
|
|
|
|
/* Characteristics */
|
|
|
|
Characteristics()
|
|
{
|
|
static char *chars[] = {
|
|
"Running this program should reveal these characteristics:",
|
|
" Radix = 1, 2, 4, 8, 10, 16, 100, 256 ...",
|
|
" Precision = number of significant digits carried.",
|
|
" U2 = Radix/Radix^Precision = One Ulp",
|
|
"\t(OneUlpnit in the Last Place) of 1.000xxx .",
|
|
" U1 = 1/Radix^Precision = One Ulp of numbers a little less than 1.0 .",
|
|
" Adequacy of guard digits for Mult., Div. and Subt.",
|
|
" Whether arithmetic is chopped, correctly rounded, or something else",
|
|
"\tfor Mult., Div., Add/Subt. and Sqrt.",
|
|
" Whether a Sticky Bit used correctly for rounding.",
|
|
" UnderflowThreshold = an underflow threshold.",
|
|
" E0 and PseudoZero tell whether underflow is abrupt, gradual, or fuzzy.",
|
|
" V = an overflow threshold, roughly.",
|
|
" V0 tells, roughly, whether Infinity is represented.",
|
|
" Comparisions are checked for consistency with subtraction",
|
|
"\tand for contamination with pseudo-zeros.",
|
|
" Sqrt is tested. Y^X is not tested.",
|
|
" Extra-precise subexpressions are revealed but NOT YET tested.",
|
|
" Decimal-Binary conversion is NOT YET tested for accuracy.",
|
|
0};
|
|
|
|
msglist(chars);
|
|
}
|
|
|
|
History()
|
|
|
|
{ /* History */
|
|
/* Converted from Brian Wichmann's Pascal version to C by Thos Sumner,
|
|
with further massaging by David M. Gay. */
|
|
|
|
static char *hist[] = {
|
|
"The program attempts to discriminate among",
|
|
" FLAWs, like lack of a sticky bit,",
|
|
" Serious DEFECTs, like lack of a guard digit, and",
|
|
" FAILUREs, like 2+2 == 5 .",
|
|
"Failures may confound subsequent diagnoses.\n",
|
|
"The diagnostic capabilities of this program go beyond an earlier",
|
|
"program called 'MACHAR', which can be found at the end of the",
|
|
"book 'Software Manual for the Elementary Functions' (1980) by",
|
|
"W. J. Cody and W. Waite. Although both programs try to discover",
|
|
"the Radix, Precision and range (over/underflow thresholds)",
|
|
"of the arithmetic, this program tries to cope with a wider variety",
|
|
"of pathologies, and to say how well the arithmetic is implemented.",
|
|
"\nThe program is based upon a conventional radix representation for",
|
|
"floating-point numbers, but also allows logarithmic encoding",
|
|
"as used by certain early WANG machines.\n",
|
|
"BASIC version of this program (C) 1983 by Prof. W. M. Kahan;",
|
|
"see source comments for more history.",
|
|
0};
|
|
|
|
msglist(hist);
|
|
}
|
|
|
|
double
|
|
pow(x, y) /* return x ^ y (exponentiation) */
|
|
double x, y;
|
|
{
|
|
extern double exp(), frexp(), ldexp(), log(), modf();
|
|
double xy, ye;
|
|
long i;
|
|
int ex, ey = 0, flip = 0;
|
|
|
|
if (!y) return 1.0;
|
|
|
|
if ((y < -1100. || y > 1100.) && x != -1.) return exp(y * log(x));
|
|
|
|
if (y < 0.) { y = -y; flip = 1; }
|
|
y = modf(y, &ye);
|
|
if (y) xy = exp(y * log(x));
|
|
else xy = 1.0;
|
|
/* next several lines assume >= 32 bit integers */
|
|
x = frexp(x, &ex);
|
|
if (i = ye) for(;;) {
|
|
if (i & 1) { xy *= x; ey += ex; }
|
|
if (!(i >>= 1)) break;
|
|
x *= x;
|
|
ex *= 2;
|
|
if (x < .5) { x *= 2.; ex -= 1; }
|
|
}
|
|
if (flip) { xy = 1. / xy; ey = -ey; }
|
|
return ldexp(xy, ey);
|
|
}
|
|
|
|
#endif /* NO_FLOATS */
|