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/* * $Id: paranoia.c,v 1.2 2001-09-27 12:02:29 chris Exp $ * * A C version of Kahan's Floating Point Test "Paranoia" * * Thos Sumner, UCSF, Feb. 1985 * David Gay, BTL, Jan. 1986 * * This is a rewrite from the Pascal version by * * B. A. Wichmann, 18 Jan. 1985 * * (and does NOT exhibit good C programming style). * * Sun May 16 18:21:51 MDT 1993 Jeffrey Wheat (cassidy@cygnus.com) * Removed KR_headers defines, removed ANSI prototyping * Cleaned up and reformated code. Added special CYGNUS * "verbose" mode type messages (-DCYGNUS). * Note: This code is VERY NASTY. * * Adjusted to use Standard C headers 19 Jan. 1992 (dmg); * compile with -DKR_headers or insert * #define KR_headers * at the beginning if you have an old-style C compiler. * * (C) Apr 19 1983 in BASIC version by: * Professor W. M. Kahan, * 567 Evans Hall * Electrical Engineering & Computer Science Dept. * University of California * Berkeley, California 94720 * USA * * converted to Pascal by: * B. A. Wichmann * National Physical Laboratory * Teddington Middx * TW11 OLW * UK * * converted to C by: * * David M. Gay and Thos Sumner * AT&T Bell Labs Computer Center, Rm. U-76 * 600 Mountain Avenue University of California * Murray Hill, NJ 07974 San Francisco, CA 94143 * USA USA * * with simultaneous corrections to the Pascal source (reflected * in the Pascal source available over netlib). * [A couple of bug fixes from dgh = sun!dhough incorporated 31 July 1986.] * * Reports of results on various systems from all the versions * of Paranoia are being collected by Richard Karpinski at the * same address as Thos Sumner. This includes sample outputs, * bug reports, and criticisms. * * You may copy this program freely if you acknowledge its source. * Comments on the Pascal version to NPL, please. * * * The C version catches signals from floating-point exceptions. * If signal(SIGFPE,...) is unavailable in your environment, you may * #define NOSIGNAL to comment out the invocations of signal. * * This source file is too big for some C compilers, but may be split * into pieces. Comments containing "SPLIT" suggest convenient places * for this splitting. At the end of these comments is an "ed script" * (for the UNIX(tm) editor ed) that will do this splitting. * * By #defining SINGLE_PRECISION when you compile this source, you may * obtain a single-precision C version of Paranoia. * * The following is from the introductory commentary from Wichmann's work: * * The BASIC program of Kahan is written in Microsoft BASIC using many * facilities which have no exact analogy in Pascal. The Pascal * version below cannot therefore be exactly the same. Rather than be * a minimal transcription of the BASIC program, the Pascal coding * follows the conventional style of block-structured languages. Hence * the Pascal version could be useful in producing versions in other * structured languages. * * Rather than use identifiers of minimal length (which therefore have * little mnemonic significance), the Pascal version uses meaningful * identifiers as follows [Note: A few changes have been made for C]: * * * BASIC C BASIC C BASIC C * * A J S StickyBit * A1 AInverse J0 NoErrors T * B Radix [Failure] T0 Underflow * B1 BInverse J1 NoErrors T2 ThirtyTwo * B2 RadixD2 [SeriousDefect] T5 OneAndHalf * B9 BMinusU2 J2 NoErrors T7 TwentySeven * C [Defect] T8 TwoForty * C1 CInverse J3 NoErrors U OneUlp * D [Flaw] U0 UnderflowThreshold * D4 FourD K PageNo U1 * E0 L Milestone U2 * E1 M V * E2 Exp2 N V0 * E3 N1 V8 * E5 MinSqEr O Zero V9 * E6 SqEr O1 One W * E7 MaxSqEr O2 Two X * E8 O3 Three X1 * E9 O4 Four X8 * F1 MinusOne O5 Five X9 Random1 * F2 Half O8 Eight Y * F3 Third O9 Nine Y1 * F6 P Precision Y2 * F9 Q Y9 Random2 * G1 GMult Q8 Z * G2 GDiv Q9 Z0 PseudoZero * G3 GAddSub R Z1 * H R1 RMult Z2 * H1 HInverse R2 RDiv Z9 * I R3 RAddSub * IO NoTrials R4 RSqrt * I3 IEEE R9 Random9 * * SqRWrng * * All the variables in BASIC are true variables and in consequence, * the program is more difficult to follow since the "constants" must * be determined (the glossary is very helpful). The Pascal version * uses Real constants, but checks are added to ensure that the values * are correctly converted by the compiler. * * The major textual change to the Pascal version apart from the * identifiersis that named procedures are used, inserting parameters * wherehelpful. New procedures are also introduced. The * correspondence is as follows: * * * BASIC Pascal * lines * * 90- 140 Pause * 170- 250 Instructions * 380- 460 Heading * 480- 670 Characteristics * 690- 870 History * 2940-2950 Random * 3710-3740 NewD * 4040-4080 DoesYequalX * 4090-4110 PrintIfNPositive * 4640-4850 TestPartialUnderflow * */ #include <stdio.h> #include <string.h> #if defined(solaris2) #include <math.h> #endif /* * To compile this on host using only libm from newlib (and using host libc) * do: * gcc -g -DNEED_REENT -DCYGNUS paranoia.c .../newlib-1.6/newlib/libm.a */ #ifdef NEED_REENT #include <reent.h> struct _reent libm_reent = _REENT_INIT(libm_reent); struct _reent *_impure_ptr = &libm_reent; #endif #ifndef NOSIGNAL #include <signal.h> #include <setjmp.h> #else /* NOSIGNAL */ #define longjmp(e,v) #define setjmp(e) 0 #define jmp_buf int #endif /* NOSIGNAL */ #ifdef SINGLE_PRECISION #define FLOAT float #define FABS(x) (float)fabs((double)(x)) #define FLOOR(x) (float)floor((double)(x)) #define LOG(x) (float)log((double)(x)) #define POW(x,y) (float)pow((double)(x),(double)(y)) #define SQRT(x) (float)sqrt((double)(x)) #else /* !SINGLE_PRECISION */ #define FLOAT double #define FABS(x) fabs(x) #define FLOOR(x) floor(x) #define LOG(x) log(x) #define POW(x,y) pow(x,y) #define SQRT(x) sqrt(x) #endif /* SINGLE_PRECISION */ jmp_buf ovfl_buf; extern double fabs (), floor (), log (), pow (), sqrt (); extern void exit (); typedef void (*Sig_type) (); FLOAT Sign (), Random (); extern void BadCond (); extern void SqXMinX (); extern void TstCond (); extern void notify (); extern int read (); extern void Characteristics (); extern void Heading (); extern void History (); extern void Instructions (); extern void IsYeqX (); extern void NewD (); extern void Pause (); extern void PrintIfNPositive (); extern void SR3750 (); extern void SR3980 (); extern void TstPtUf (); Sig_type sigsave; #define KEYBOARD 0 FLOAT Radix, BInvrse, RadixD2, BMinusU2; /*Small floating point constants.*/ FLOAT Zero = 0.0; FLOAT Half = 0.5; FLOAT One = 1.0; FLOAT Two = 2.0; FLOAT Three = 3.0; FLOAT Four = 4.0; FLOAT Five = 5.0; FLOAT Eight = 8.0; FLOAT Nine = 9.0; FLOAT TwentySeven = 27.0; FLOAT ThirtyTwo = 32.0; FLOAT TwoForty = 240.0; FLOAT MinusOne = -1.0; FLOAT OneAndHalf = 1.5; /*Integer constants*/ int NoTrials = 20; /*Number of tests for commutativity. */ #define False 0 #define True 1 /* * Definitions for declared types * Guard == (Yes, No); * Rounding == (Chopped, Rounded, Other); * Message == packed array [1..40] of char; * Class == (Flaw, Defect, Serious, Failure); */ #define Yes 1 #define No 0 #define Chopped 2 #define Rounded 1 #define Other 0 #define Flaw 3 #define Defect 2 #define Serious 1 #define Failure 0 typedef int Guard, Rounding, Class; typedef char Message; /* Declarations of Variables */ int Indx; char ch[8]; FLOAT AInvrse, A1; FLOAT C, CInvrse; FLOAT D, FourD; FLOAT E0, E1, Exp2, E3, MinSqEr; FLOAT SqEr, MaxSqEr, E9; FLOAT Third; FLOAT F6, F9; FLOAT HVar, 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 WVar; FLOAT X, X1, X2, X8, Random1; FLOAT Y, Y1, Y2, Random2; FLOAT Z, PseudoZero, Z1, Z2, Z9; int ErrCnt[4]; int fpecount; int Milestone; int PageNo; int M, N, N1; Guard GMult, GDiv, GAddSub; Rounding RMult, RDiv, RAddSub, RSqrt; int Break, Done, NotMonot, Monot, Anomaly, IEEE, SqRWrng, UfNGrad; /* Computed constants. * U1 gap below 1.0, i.e, 1.0 - U1 is next number below 1.0 * U2 gap above 1.0, i.e, 1.0 + U2 is next number above 1.0 */ int batchmode; /* global batchmode test */ /* program name and version variables and macro */ char *temp; char *program_name; char *program_vers; #ifndef VERSION #define VERSION "1.1 [cygnus]" #endif /* VERSION */ #define basename(cp) ((temp=(char *)strrchr((cp), '/')) ? temp+1 : (cp)) #ifndef BATCHMODE # ifdef CYGNUS # define BATCHMODE # endif #endif /* floating point exception receiver */ void _sigfpe (x) int x; { fpecount++; printf ("\n* * * FLOATING-POINT ERROR %d * * *\n", x); fflush (stdout); if (sigsave) { #ifndef NOSIGNAL signal (SIGFPE, sigsave); #endif /* NOSIGNAL */ sigsave = 0; longjmp (ovfl_buf, 1); } exit (1); } #ifdef NOMAIN #define main paranoia #endif int main (argc, argv) int argc; char **argv; { /* First two assignments use integer right-hand sides. */ Zero = 0; One = 1; Two = One + One; Three = Two + One; Four = Three + One; Five = Four + One; Eight = Four + Four; Nine = Three * Three; TwentySeven = Nine * Three; ThirtyTwo = Four * Eight; TwoForty = Four * Five * Three * Four; MinusOne = -One; Half = One / Two; OneAndHalf = One + Half; ErrCnt[Failure] = 0; ErrCnt[Serious] = 0; ErrCnt[Defect] = 0; ErrCnt[Flaw] = 0; PageNo = 1; #ifdef BATCHMODE batchmode = 1; /* run test in batchmode? */ #else /* !BATCHMODE */ batchmode = 0; /* run test interactively */ #endif /* BATCHMODE */ program_name = basename (argv[0]); program_vers = VERSION; printf ("%s version %s\n", program_name, program_vers); /*=============================================*/ Milestone = 0; /*=============================================*/ #ifndef NOSIGNAL signal (SIGFPE, _sigfpe); #endif if (!batchmode) { Instructions (); Pause (); Heading (); Instructions (); Pause (); Heading (); Pause (); Characteristics (); Pause (); History (); Pause (); } /*=============================================*/ Milestone = 7; /*=============================================*/ printf ("Program is now RUNNING tests on small integers:\n"); TstCond (Failure, (Zero + Zero == Zero) && (One - One == Zero) && (One > Zero) && (One + One == Two), "0+0 != 0, 1-1 != 0, 1 <= 0, or 1+1 != 2"); Z = -Zero; if (Z != 0.0) { ErrCnt[Failure] = ErrCnt[Failure] + 1; printf ("Comparison alleges that -0.0 is Non-zero!\n"); U1 = 0.001; Radix = 1; TstPtUf (); } TstCond (Failure, (Three == Two + One) && (Four == Three + One) && (Four + Two * (-Two) == Zero) && (Four - Three - One == Zero), "3 != 2+1, 4 != 3+1, 4+2*(-2) != 0, or 4-3-1 != 0"); TstCond (Failure, (MinusOne == (0 - One)) && (MinusOne + One == Zero) && (One + MinusOne == Zero) && (MinusOne + FABS (One) == Zero) && (MinusOne + MinusOne * MinusOne == Zero), "-1+1 != 0, (-1)+abs(1) != 0, or -1+(-1)*(-1) != 0"); TstCond (Failure, Half + MinusOne + Half == Zero, "1/2 + (-1) + 1/2 != 0"); /*=============================================*/ Milestone = 10; /*=============================================*/ TstCond (Failure, (Nine == Three * Three) && (TwentySeven == Nine * Three) && (Eight == Four + Four) && (ThirtyTwo == Eight * Four) && (ThirtyTwo - TwentySeven - Four - One == Zero), "9 != 3*3, 27 != 9*3, 32 != 8*4, or 32-27-4-1 != 0"); TstCond (Failure, (Five == Four + One) && (TwoForty == Four * Five * Three * Four) && (TwoForty / Three - Four * Four * Five == Zero) && (TwoForty / Four - Five * Three * Four == Zero) && (TwoForty / Five - Four * Three * Four == Zero), "5 != 4+1, 240/3 != 80, 240/4 != 60, or 240/5 != 48"); if (ErrCnt[Failure] == 0) { printf ("-1, 0, 1/2, 1, 2, 3, 4, 5, 9, 27, 32 & 240 are O.K.\n"); printf ("\n"); } printf ("Searching for Radix and Precision.\n"); WVar = One; do { WVar = WVar + WVar; Y = WVar + One; Z = Y - WVar; Y = Z - One; } while (MinusOne + FABS (Y) < Zero); /*.. now WVar is just big enough that |((WVar+1)-WVar)-1| >= 1 ...*/ Precision = Zero; Y = One; do { Radix = WVar + Y; Y = Y + Y; Radix = Radix - WVar; } while (Radix == Zero); if (Radix < Two) Radix = One; printf ("Radix = %f .\n", Radix); if (Radix != 1) { WVar = One; do { Precision = Precision + One; WVar = WVar * Radix; Y = WVar + One; } while ((Y - WVar) == One); } /*... now WVar == Radix^Precision is barely too big to satisfy (WVar+1)-WVar == 1 ...*/ U1 = One / WVar; 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); WVar = One / U1; F9 = (Half - U1) + Half; Radix = FLOOR (0.01 + U2 / U1); if (Radix == E0) printf ("Radix confirmed.\n"); else printf ("MYSTERY: recalculated Radix = %.7e .\n", Radix); TstCond (Defect, Radix <= Eight + Eight, "Radix is too big: roundoff problems"); TstCond (Flaw, (Radix == Two) || (Radix == 10) || (Radix == One), "Radix is not as good as 2 or 10"); /*=============================================*/ Milestone = 20; /*=============================================*/ TstCond (Failure, F9 - Half < Half, "(1-U1)-1/2 < 1/2 is FALSE, prog. fails?"); X = F9; I = 1; Y = X - Half; Z = Y - Half; TstCond (Failure, (X != One) || (Z == Zero), "Comparison is fuzzy,X=1 but X-1/2-1/2 != 0"); X = One + U2; I = 0; /*=============================================*/ Milestone = 25; /*=============================================*/ /*... BMinusU2 = nextafter(Radix, 0) */ BMinusU2 = Radix - One; BMinusU2 = (BMinusU2 - U2) + One; /* Purify Integers */ if (Radix != One) { X = -TwoForty * LOG (U1) / LOG (Radix); Y = FLOOR (Half + X); if (FABS (X - Y) * Four < One) X = Y; Precision = X / TwoForty; Y = FLOOR (Half + Precision); if (FABS (Precision - Y) * TwoForty < Half) Precision = Y; } if ((Precision != FLOOR (Precision)) || (Radix == One)) { printf ("Precision cannot be characterized by an Integer number\n"); printf ("of significant digits but, by itself, this is a minor flaw.\n"); } if (Radix == One) printf ("logarithmic encoding has precision characterized solely by U1.\n"); else printf ("The number of significant digits of the Radix is %f .\n", Precision); TstCond (Serious, U2 * Nine * Nine * TwoForty < One, "Precision worse than 5 decimal figures "); /*=============================================*/ Milestone = 30; /*=============================================*/ /* Test for extra-precise subepressions */ X = FABS (((Four / Three - One) - One / Four) * Three - One / Four); do { Z2 = X; X = (One + (Half * Z2 + ThirtyTwo * Z2 * Z2)) - One; } while (!((Z2 <= X) || (X <= Zero))); X = Y = Z = FABS ((Three / Four - Two / Three) * Three - One / Four); do { Z1 = Z; Z = (One / Two - ((One / Two - (Half * Z1 + ThirtyTwo * Z1 * Z1)) + One / Two)) + One / Two; } while (!((Z1 <= Z) || (Z <= Zero))); do { do { Y1 = Y; Y = (Half - ((Half - (Half * Y1 + ThirtyTwo * Y1 * Y1)) + Half )) + Half; } while (!((Y1 <= Y) || (Y <= Zero))); X1 = X; X = ((Half * X1 + ThirtyTwo * X1 * X1) - F9) + F9; } while (!((X1 <= X) || (X <= Zero))); if ((X1 != Y1) || (X1 != Z1)) { BadCond (Serious, "Disagreements among the values X1, Y1, Z1,\n"); printf ("respectively %.7e, %.7e, %.7e,\n", X1, Y1, Z1); printf ("are symptoms of inconsistencies introduced\n"); printf ("by extra-precise evaluation of arithmetic subexpressions.\n"); notify ("Possibly some part of this"); if ((X1 == U1) || (Y1 == U1) || (Z1 == U1)) printf ( "That feature is not tested further by this program.\n"); } else { if ((Z1 != U1) || (Z2 != U2)) { if ((Z1 >= U1) || (Z2 >= U2)) { BadCond (Failure, ""); notify ("Precision"); printf ("\tU1 = %.7e, Z1 - U1 = %.7e\n", U1, Z1 - U1); printf ("\tU2 = %.7e, Z2 - U2 = %.7e\n", U2, Z2 - U2); } else { if ((Z1 <= Zero) || (Z2 <= Zero)) { printf ("Because of unusual Radix = %f", Radix); printf (", or exact rational arithmetic a result\n"); printf ("Z1 = %.7e, or Z2 = %.7e ", Z1, Z2); notify ("of an\nextra-precision"); } if (Z1 != Z2 || Z1 > Zero) { X = Z1 / U1; Y = Z2 / U2; if (Y > X) X = Y; Q = -LOG (X); printf ("Some subexpressions appear to be calculated extra\n"); printf ("precisely with about %g extra B-digits, i.e.\n", (Q / LOG (Radix))); printf ("roughly %g extra significant decimals.\n", Q / LOG (10.)); } printf ("That feature is not tested further by this program.\n"); } } } Pause (); /*=============================================*/ Milestone = 35; /*=============================================*/ if (Radix >= Two) { X = WVar / (Radix * Radix); Y = X + One; Z = Y - X; T = Z + U2; X = T - Z; TstCond (Failure, X == U2, "Subtraction is not normalized X=Y,X+Z != Y+Z!"); if (X == U2) printf ( "Subtraction appears to be normalized, as it should be."); } printf ("\nChecking for guard digit in *, /, and -.\n"); Y = F9 * One; Z = One * F9; X = F9 - Half; Y = (Y - Half) - X; Z = (Z - Half) - X; X = One + U2; T = X * Radix; R = Radix * X; X = T - Radix; X = X - Radix * U2; T = R - Radix; T = T - Radix * U2; X = X * (Radix - One); T = T * (Radix - One); if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero)) GMult = Yes; else { GMult = No; TstCond (Serious, False, "* lacks a Guard Digit, so 1*X != X"); } Z = Radix * U2; X = One + Z; Y = FABS ((X + Z) - X * X) - U2; X = One - U2; Z = FABS ((X - U2) - X * X) - U1; TstCond (Failure, (Y <= Zero) && (Z <= Zero), "* gets too many final digits wrong.\n"); Y = One - U2; X = One + U2; Z = One / Y; Y = Z - X; X = One / Three; Z = Three / Nine; X = X - Z; T = Nine / TwentySeven; Z = Z - T; TstCond (Defect, X == Zero && Y == Zero && Z == Zero, "Division lacks a Guard Digit, so error can exceed 1 ulp\n\ or 1/3 and 3/9 and 9/27 may disagree"); Y = F9 / One; X = F9 - Half; Y = (Y - Half) - X; X = One + U2; T = X / One; X = T - X; if ((X == Zero) && (Y == Zero) && (Z == Zero)) GDiv = Yes; else { GDiv = No; TstCond (Serious, False, "Division lacks a Guard Digit, so X/1 != X"); } X = One / (One + U2); Y = X - Half - Half; TstCond (Serious, Y < Zero, "Computed value of 1/1.000..1 >= 1"); X = One - U2; Y = One + Radix * U2; Z = X * Radix; T = Y * Radix; R = Z / Radix; StickyBit = T / Radix; X = R - X; Y = StickyBit - Y; TstCond (Failure, X == Zero && Y == Zero, "* and/or / gets too many last digits wrong"); Y = One - U1; X = One - F9; Y = One - Y; T = Radix - U2; Z = Radix - BMinusU2; T = Radix - T; if ((X == U1) && (Y == U1) && (Z == U2) && (T == U2)) GAddSub = Yes; else { GAddSub = No; TstCond (Serious, False, "- lacks Guard Digit, so cancellation is obscured"); } if (F9 != One && F9 - One >= Zero) { BadCond (Serious, "comparison alleges (1-U1) < 1 although\n"); printf (" subtraction yields (1-U1) - 1 = 0 , thereby vitiating\n"); printf (" such precautions against division by zero as\n"); printf (" ... if (X == 1.0) {.....} else {.../(X-1.0)...}\n"); } if (GMult == Yes && GDiv == Yes && GAddSub == Yes) printf ( " *, /, and - appear to have guard digits, as they should.\n"); /*=============================================*/ Milestone = 40; /*=============================================*/ Pause (); printf ("Checking rounding on multiply, divide and add/subtract.\n"); RMult = Other; RDiv = Other; RAddSub = Other; RadixD2 = Radix / Two; A1 = Two; Done = False; do { AInvrse = Radix; do { X = AInvrse; AInvrse = AInvrse / A1; } while (!(FLOOR (AInvrse) != AInvrse)); Done = (X == One) || (A1 > Three); if (!Done) A1 = Nine + One; } while (!(Done)); if (X == One) A1 = Radix; AInvrse = One / A1; X = A1; Y = AInvrse; Done = False; do { Z = X * Y - Half; TstCond (Failure, Z == Half, "X * (1/X) differs from 1"); Done = X == Radix; X = Radix; Y = One / X; } while (!(Done)); Y2 = One + U2; Y1 = One - U2; X = OneAndHalf - U2; Y = OneAndHalf + U2; Z = (X - U2) * Y2; T = Y * Y1; Z = Z - X; T = T - X; X = X * Y2; Y = (Y + U2) * Y1; X = X - OneAndHalf; Y = Y - OneAndHalf; if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T <= Zero)) { X = (OneAndHalf + U2) * Y2; Y = OneAndHalf - U2 - U2; Z = OneAndHalf + U2 + U2; T = (OneAndHalf - U2) * Y1; X = X - (Z + U2); StickyBit = Y * Y1; S = Z * Y2; T = T - Y; Y = (U2 - Y) + StickyBit; Z = S - (Z + U2 + U2); StickyBit = (Y2 + U2) * Y1; Y1 = Y2 * Y1; StickyBit = StickyBit - Y2; Y1 = Y1 - Half; if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero) && (StickyBit == Zero) && (Y1 == Half)) { RMult = Rounded; printf ("Multiplication appears to round correctly.\n"); } else if ((X + U2 == Zero) && (Y < Zero) && (Z + U2 == Zero) && (T < Zero) && (StickyBit + U2 == Zero) && (Y1 < Half)) { RMult = Chopped; printf ("Multiplication appears to chop.\n"); } else printf ("* is neither chopped nor correctly rounded.\n"); if ((RMult == Rounded) && (GMult == No)) notify ("Multiplication"); } else printf ("* is neither chopped nor correctly rounded.\n"); /*=============================================*/ Milestone = 45; /*=============================================*/ Y2 = One + U2; Y1 = One - U2; Z = OneAndHalf + U2 + U2; X = Z / Y2; T = OneAndHalf - U2 - U2; Y = (T - U2) / Y1; Z = (Z + U2) / Y2; X = X - OneAndHalf; Y = Y - T; T = T / Y1; Z = Z - (OneAndHalf + U2); T = (U2 - OneAndHalf) + T; if (!((X > Zero) || (Y > Zero) || (Z > Zero) || (T > Zero))) { X = OneAndHalf / Y2; Y = OneAndHalf - U2; Z = OneAndHalf + U2; X = X - Y; T = OneAndHalf / Y1; Y = Y / Y1; T = T - (Z + U2); Y = Y - Z; Z = Z / Y2; Y1 = (Y2 + U2) / Y2; Z = Z - OneAndHalf; Y2 = Y1 - Y2; Y1 = (F9 - U1) / F9; if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero) && (Y2 == Zero) && (Y2 == Zero) && (Y1 - Half == F9 - Half)) { RDiv = Rounded; printf ("Division appears to round correctly.\n"); if (GDiv == No) notify ("Division"); } else if ((X < Zero) && (Y < Zero) && (Z < Zero) && (T < Zero) && (Y2 < Zero) && (Y1 - Half < F9 - Half)) { RDiv = Chopped; printf ("Division appears to chop.\n"); } } if (RDiv == Other) printf ("/ is neither chopped nor correctly rounded.\n"); BInvrse = One / Radix; TstCond (Failure, (BInvrse * Radix - Half == Half), "Radix * ( 1 / Radix ) differs from 1"); /*=============================================*/ Milestone = 50; /*=============================================*/ TstCond (Failure, ((F9 + U1) - Half == Half) && ((BMinusU2 + U2) - One == Radix - One), "Incomplete carry-propagation in Addition"); X = One - U1 * U1; Y = One + U2 * (One - U2); Z = F9 - Half; X = (X - Half) - Z; Y = Y - One; if ((X == Zero) && (Y == Zero)) { RAddSub = Chopped; printf ("Add/Subtract appears to be chopped.\n"); } if (GAddSub == Yes) { X = (Half + U2) * U2; Y = (Half - U2) * U2; X = One + X; Y = One + Y; X = (One + U2) - X; Y = One - Y; if ((X == Zero) && (Y == Zero)) { X = (Half + U2) * U1; Y = (Half - U2) * U1; X = One - X; Y = One - Y; X = F9 - X; Y = One - Y; if ((X == Zero) && (Y == Zero)) { RAddSub = Rounded; printf ("Addition/Subtraction appears to round correctly.\n"); if (GAddSub == No) notify ("Add/Subtract"); } else printf ("Addition/Subtraction neither rounds nor chops.\n"); } else printf ("Addition/Subtraction neither rounds nor chops.\n"); } else printf ("Addition/Subtraction neither rounds nor chops.\n"); S = One; X = One + Half * (One + Half); Y = (One + U2) * Half; Z = X - Y; T = Y - X; StickyBit = Z + T; if (StickyBit != Zero) { S = Zero; BadCond (Flaw, "(X - Y) + (Y - X) is non zero!\n"); } StickyBit = Zero; if ((GMult == Yes) && (GDiv == Yes) && (GAddSub == Yes) && (RMult == Rounded) && (RDiv == Rounded) && (RAddSub == Rounded) && (FLOOR (RadixD2) == RadixD2)) { printf ("Checking for sticky bit.\n"); X = (Half + U1) * U2; Y = Half * U2; Z = One + Y; T = One + X; if ((Z - One <= Zero) && (T - One >= U2)) { Z = T + Y; Y = Z - X; if ((Z - T >= U2) && (Y - T == Zero)) { X = (Half + U1) * U1; Y = Half * U1; Z = One - Y; T = One - X; if ((Z - One == Zero) && (T - F9 == Zero)) { Z = (Half - U1) * U1; T = F9 - Z; Q = F9 - Y; if ((T - F9 == Zero) && (F9 - U1 - Q == Zero)) { Z = (One + U2) * OneAndHalf; T = (OneAndHalf + U2) - Z + U2; X = One + Half / Radix; Y = One + Radix * U2; Z = X * Y; if (T == Zero && X + Radix * U2 - Z == Zero) { if (Radix != Two) { X = Two + U2; Y = X / Two; if ((Y - One == Zero)) StickyBit = S; } else StickyBit = S; } } } } } } if (StickyBit == One) printf ("Sticky bit apparently used correctly.\n"); else printf ("Sticky bit used incorrectly or not at all.\n"); TstCond (Flaw, !(GMult == No || GDiv == No || GAddSub == No || RMult == Other || RDiv == Other || RAddSub == Other), "lack(s) of guard digits or failure(s) to correctly round or chop\n\ (noted above) count as one flaw in the final tally below"); /*=============================================*/ Milestone = 60; /*=============================================*/ printf ("\n"); printf ("Does Multiplication commute? "); printf ("Testing on %d random pairs.\n", NoTrials); Random9 = SQRT (3.0); Random1 = Third; I = 1; do { X = Random (); Y = Random (); Z9 = Y * X; Z = X * Y; Z9 = Z - Z9; I = I + 1; } while (!((I > NoTrials) || (Z9 != Zero))); if (I == NoTrials) { Random1 = One + Half / Three; Random2 = (U2 + U1) + One; Z = Random1 * Random2; Y = Random2 * Random1; Z9 = (One + Half / Three) * ((U2 + U1) + One) - (One + Half / Three) * ((U2 + U1) + One); } if (!((I == NoTrials) || (Z9 == Zero))) BadCond (Defect, "X * Y == Y * X trial fails.\n"); else printf (" No failures found in %d integer pairs.\n", NoTrials); /*=============================================*/ Milestone = 70; /*=============================================*/ printf ("\nRunning test of square root(x).\n"); TstCond (Failure, (Zero == SQRT (Zero)) && (-Zero == SQRT (-Zero)) && (One == SQRT (One)), "Square root of 0.0, -0.0 or 1.0 wrong"); MinSqEr = Zero; MaxSqEr = Zero; J = Zero; X = Radix; OneUlp = U2; SqXMinX (Serious); X = BInvrse; OneUlp = BInvrse * U1; SqXMinX (Serious); X = U1; OneUlp = U1 * U1; SqXMinX (Serious); if (J != Zero) Pause (); printf ("Testing if sqrt(X * X) == X for %d Integers X.\n", NoTrials); J = Zero; X = Two; Y = Radix; if ((Radix != One)) do { X = Y; Y = Radix * Y; } while (!((Y - X >= NoTrials))); OneUlp = X * U2; I = 1; while (I <= NoTrials) { X = X + One; SqXMinX (Defect); if (J > Zero) break; I = I + 1; } printf ("Test for sqrt monotonicity.\n"); I = -1; X = BMinusU2; Y = Radix; Z = Radix + Radix * U2; NotMonot = False; Monot = False; while (!(NotMonot || Monot)) { I = I + 1; X = SQRT (X); Q = SQRT (Y); Z = SQRT (Z); if ((X > Q) || (Q > Z)) NotMonot = True; else { Q = FLOOR (Q + Half); if ((I > 0) || (Radix == Q * Q)) Monot = True; else if (I > 0) { if (I > 1) Monot = True; else { Y = Y * BInvrse; X = Y - U1; Z = Y + U1; } } else { Y = Q; X = Y - U2; Z = Y + U2; } } } if (Monot) printf ("sqrt has passed a test for Monotonicity.\n"); else { BadCond (Defect, ""); printf ("sqrt(X) is non-monotonic for X near %.7e .\n", Y); } /*=============================================*/ Milestone = 80; /*=============================================*/ MinSqEr = MinSqEr + Half; MaxSqEr = MaxSqEr - Half; Y = (SQRT (One + U2) - One) / U2; SqEr = (Y - One) + U2 / Eight; if (SqEr > MaxSqEr) MaxSqEr = SqEr; SqEr = Y + U2 / Eight; if (SqEr < MinSqEr) MinSqEr = SqEr; Y = ((SQRT (F9) - U2) - (One - U2)) / U1; SqEr = Y + U1 / Eight; if (SqEr > MaxSqEr) MaxSqEr = SqEr; SqEr = (Y + One) + U1 / Eight; if (SqEr < MinSqEr) MinSqEr = SqEr; OneUlp = U2; X = OneUlp; for (Indx = 1; Indx <= 3; ++Indx) { Y = SQRT ((X + U1 + X) + F9); Y = ((Y - U2) - ((One - U2) + X)) / OneUlp; Z = ((U1 - X) + F9) * Half * X * X / OneUlp; SqEr = (Y + Half) + Z; if (SqEr < MinSqEr) MinSqEr = SqEr; SqEr = (Y - Half) + Z; if (SqEr > MaxSqEr) MaxSqEr = SqEr; if (((Indx == 1) || (Indx == 3))) X = OneUlp * Sign (X) * FLOOR (Eight / (Nine * SQRT (OneUlp))); else { OneUlp = U1; X = -OneUlp; } } /*=============================================*/ Milestone = 85; /*=============================================*/ SqRWrng = False; Anomaly = False; RSqrt = Other; /* ~dgh */ if (Radix != One) { printf ("Testing whether sqrt is rounded or chopped.\n"); D = FLOOR (Half + POW (Radix, One + Precision - FLOOR (Precision))); /* ... == Radix^(1 + fract) if (Precision == Integer + fract. */ X = D / Radix; Y = D / A1; if ((X != FLOOR (X)) || (Y != FLOOR (Y))) { Anomaly = True; } else { X = Zero; Z2 = X; Y = One; Y2 = Y; Z1 = Radix - One; FourD = Four * D; do { if (Y2 > Z2) { Q = Radix; Y1 = Y; do { X1 = FABS (Q + FLOOR (Half - Q / Y1) * Y1); Q = Y1; Y1 = X1; } while (!(X1 <= Zero)); if (Q <= One) { Z2 = Y2; Z = Y; } } Y = Y + Two; X = X + Eight; Y2 = Y2 + X; if (Y2 >= FourD) Y2 = Y2 - FourD; } while (!(Y >= D)); X8 = FourD - Z2; Q = (X8 + Z * Z) / FourD; X8 = X8 / Eight; if (Q != FLOOR (Q)) Anomaly = True; else { Break = False; do { X = Z1 * Z; X = X - FLOOR (X / Radix) * Radix; if (X == One) Break = True; else Z1 = Z1 - One; } while (!(Break || (Z1 <= Zero))); if ((Z1 <= Zero) && (!Break)) Anomaly = True; else { if (Z1 > RadixD2) Z1 = Z1 - Radix; do { NewD (); } while (!(U2 * D >= F9)); if (D * Radix - D != WVar - 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 != WVar - 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", WVar); 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 (WVar) / LOG (A1)); Break = False; do { X = Z; I = 1; SR3980 (); if (Z == AInvrse) Break = True; else Z = AInvrse; } while (!(Break)); /*=============================================*/ Milestone = 100; /*=============================================*/ /* Powers of Radix have been tested, */ /* next try a few primes */ M = NoTrials; Z = Three; do { X = Z; I = 1; SR3980 (); do { Z = Z + Two; } while (Three * FLOOR (Z / Three) == Z); } while (Z < Eight * Three); if (N > 0) { printf ("Errors like this may invalidate financial calculations\n"); printf ("\tinvolving interest rates.\n"); } PrintIfNPositive (); N += N1; if (N == 0) printf ("... no discrepancies found.\n"); if (N > 0) Pause (); else printf ("\n"); /*=============================================*/ Milestone = 110; /*=============================================*/ printf ("Seeking Underflow thresholds UfThold and E0.\n"); D = U1; if (Precision != FLOOR (Precision)) { D = BInvrse; X = Precision; do { D = D * BInvrse; X = X - One; } while (X > Zero); } Y = One; Z = D; /* ... D is power of 1/Radix < 1. */ do { C = Y; Y = Z; Z = Y * Y; } while ((Y > Z) && (Z + Z > Z)); Y = C; Z = Y * D; do { C = Y; Y = Z; Z = Y * D; } while ((Y > Z) && (Z + Z > Z)); if (Radix < Two) HInvrse = Two; else HInvrse = Radix; HVar = One / HInvrse; /* ... 1/HInvrse == HVar == Min(1/Radix, 1/2) */ CInvrse = One / C; E0 = C; Z = E0 * HVar; /* ...1/Radix^(BIG Integer) << 1 << CInvrse == 1/C */ do { Y = E0; E0 = Z; Z = E0 * HVar; } 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 * HVar; 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 * HVar; } 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 = HVar * 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 = HVar + HVar; } else R = SQRT (Underflow / UfThold); sigsave = 0; if (R <= HVar) { Z = R * UfThold; X = Z * (One + R * HVar * (One + HVar)); } else { Z = UfThold; X = Z * (One + HVar * HVar * (One + HVar)); } if (!((X == Z) || (X - Z != Zero))) { BadCond (Flaw, ""); printf ("X = %.17e\n\tis not equal to Z = %.17e .\n", X, Z); Z9 = X - Z; printf ("yet X - Z yields %.17e .\n", Z9); printf (" Should this NOT signal Underflow, "); printf ("this is a SERIOUS DEFECT\nthat causes "); printf ("confusion when innocent statements like\n");; printf (" if (X == Z) ... else"); printf (" ... (f(X) - f(Z)) / (X - Z) ...\n"); printf ("encounter Division by Zero although actually\n"); sigsave = _sigfpe; if (setjmp (ovfl_buf)) printf ("X / Z fails!\n"); else printf ("X / Z = 1 + %g .\n", (X / Z - Half) - Half); sigsave = 0; } } printf ("The Underflow threshold is %.17e, %s\n", UfThold, " below which"); printf ("calculation may suffer larger Relative error than "); printf ("merely roundoff.\n"); Y2 = U1 * U1; Y = Y2 * Y2; Y2 = Y * U1; if (Y2 <= UfThold) { if (Y > E0) { BadCond (Defect, ""); I = 5; } else { BadCond (Serious, ""); I = 4; } printf ("Range is too narrow; U1^%d Underflows.\n", I); } /*=============================================*/ Milestone = 130; /*=============================================*/ Y = -FLOOR (Half - TwoForty * LOG (UfThold) / LOG (HInvrse)) / TwoForty; Y2 = Y + Y; printf ("Since underflow occurs below the threshold\n"); printf ("UfThold = (%.17e) ^ (%.17e)\nonly underflow ", HInvrse, Y); printf ("should afflict the expression\n\t(%.17e) ^ (%.17e);\n", HInvrse, Y2); printf ("actually calculating yields:"); if (setjmp (ovfl_buf)) { sigsave = 0; BadCond (Serious, "trap on underflow.\n"); } else { sigsave = _sigfpe; V9 = POW (HInvrse, Y2); sigsave = 0; printf (" %.17e .\n", V9); if (!((V9 >= Zero) && (V9 <= (Radix + Radix + E9) * UfThold))) { BadCond (Serious, "this is not between 0 and underflow\n"); printf (" threshold = %.17e .\n", UfThold); } else if (!(V9 > UfThold * (One + E9))) printf ("This computed value is O.K.\n"); else { BadCond (Defect, "this is not between 0 and underflow\n"); printf (" threshold = %.17e .\n", UfThold); } } /*=============================================*/ Milestone = 140; /*=============================================*/ printf ("\n"); /* ...calculate Exp2 == exp(2) == 7.389056099... */ X = Zero; I = 2; Y = Two * Three; Q = Zero; N = 0; do { Z = X; I = I + 1; Y = Y / (I + I); R = Y + Q; X = Z + R; Q = (Z - X) + R; } while (X > Z); Z = (OneAndHalf + One / Eight) + X / (OneAndHalf * ThirtyTwo); X = Z * Z; Exp2 = X * X; X = F9; Y = X - U1; printf ("Testing X^((X + 1) / (X - 1)) vs. exp(2) = %.17e as X -> 1.\n", Exp2); for (I = 1;;) { Z = X - BInvrse; Z = (X + One) / (Z - (One - BInvrse)); Q = POW (X, Z) - Exp2; if (FABS (Q) > TwoForty * U2) { N = 1; V9 = (X - BInvrse) - (One - BInvrse); BadCond (Defect, "Calculated"); printf (" %.17e for\n", POW (X, Z)); printf ("\t(1 + (%.17e) ^ (%.17e);\n", V9, Z); printf ("\tdiffers from correct value by %.17e .\n", Q); printf ("\tThis much error may spoil financial\n"); printf ("\tcalculations involving tiny interest rates.\n"); break; } else { Z = (Y - X) * Two + Y; X = Y; Y = Z; Z = One + (X - F9) * (X - F9); if (Z > One && I < NoTrials) I++; else { if (X > One) { if (N == 0) printf ("Accuracy seems adequate.\n"); break; } else { X = One + U2; Y = U2 + U2; Y += X; I = 1; } } } } /*=============================================*/ Milestone = 150; /*=============================================*/ printf ("Testing powers Z^Q at four nearly extreme values.\n"); N = 0; Z = A1; Q = FLOOR (Half - LOG (C) / LOG (A1)); Break = False; do { X = CInvrse; Y = POW (Z, Q); IsYeqX (); Q = -Q; X = C; Y = POW (Z, Q); IsYeqX (); if (Z < One) Break = True; else Z = AInvrse; } while (!(Break)); PrintIfNPositive (); if (N == 0) printf (" ... no discrepancies found.\n"); printf ("\n"); /*=============================================*/ Milestone = 160; /*=============================================*/ Pause (); printf ("Searching for Overflow threshold:\n"); printf ("This may generate an error.\n"); Y = -CInvrse; V9 = HInvrse * Y; sigsave = _sigfpe; if (setjmp (ovfl_buf)) { I = 0; V9 = Y; goto overflow; } do { V = Y; Y = V9; V9 = HInvrse * Y; } while (V9 < Y); I = 1; overflow: sigsave = 0; Z = V9; printf ("Can `Z = -Y' overflow?\n"); printf ("Trying it on Y = %.17e .\n", Y); V9 = -Y; V0 = V9; if (V - Y == V + V0) printf ("Seems O.K.\n"); else { printf ("finds a "); BadCond (Flaw, "-(-Y) differs from Y.\n"); } if (Z != Y) { BadCond (Serious, ""); printf ("overflow past %.17e\n\tshrinks to %.17e .\n", Y, Z); } if (I) { Y = V * (HInvrse * U2 - HInvrse); Z = Y + ((One - HInvrse) * U2) * V; if (Z < V0) Y = Z; if (Y < V0) V = Y; if (V0 - V < V0) V = V0; } else { V = Y * (HInvrse * U2 - HInvrse); V = V + ((One - HInvrse) * U2) * Y; } printf ("Overflow threshold is V = %.17e .\n", V); if (I) printf ("Overflow saturates at V0 = %.17e .\n", V0); else printf ("There is no saturation value because \ the system traps on overflow.\n"); V9 = V * One; printf ("No Overflow should be signaled for V * 1 = %.17e\n", V9); V9 = V / One; printf (" nor for V / 1 = %.17e .\n", V9); printf ("Any overflow signal separating this * from the one\n"); printf ("above is a DEFECT.\n"); /*=============================================*/ Milestone = 170; /*=============================================*/ if (!(-V < V && -V0 < V0 && -UfThold < V && UfThold < V)) { BadCond (Failure, "Comparisons involving "); printf ("+-%g, +-%g\nand +-%g are confused by Overflow.", V, V0, UfThold); } /*=============================================*/ Milestone = 175; /*=============================================*/ printf ("\n"); for (Indx = 1; Indx <= 3; ++Indx) { switch (Indx) { case 1: Z = UfThold; break; case 2: Z = E0; break; case 3: Z = PseudoZero; break; } if (Z != Zero) { V9 = SQRT (Z); Y = V9 * V9; if (Y / (One - Radix * E9) < Z || Y > (One + Radix * E9) * Z) { /* dgh: + E9 --> * E9 */ if (V9 > U1) BadCond (Serious, ""); else BadCond (Defect, ""); printf ("Comparison alleges that what prints as Z = %.17e\n", Z); printf (" is too far from sqrt(Z) ^ 2 = %.17e .\n", Y); } } } /*=============================================*/ Milestone = 180; /*=============================================*/ for (Indx = 1; Indx <= 2; ++Indx) { if (Indx == 1) Z = V; else Z = V0; V9 = SQRT (Z); X = (One - Radix * E9) * V9; V9 = V9 * X; if (((V9 < (One - Two * Radix * E9) * Z) || (V9 > Z))) { Y = V9; if (X < WVar) BadCond (Serious, ""); else BadCond (Defect, ""); printf ("Comparison alleges that Z = %17e\n", Z); printf (" is too far from sqrt(Z) ^ 2 (%.17e) .\n", Y); } } /*=============================================*/ Milestone = 190; /*=============================================*/ Pause (); X = UfThold * V; Y = Radix * Radix; if (X * Y < One || X > Y) { if (X * Y < U1 || X > Y / U1) BadCond (Defect, "Badly"); else BadCond (Flaw, ""); printf (" unbalanced range; UfThold * V = %.17e\n\t%s\n", X, "is too far from 1.\n"); } /*=============================================*/ Milestone = 200; /*=============================================*/ for (Indx = 1; Indx <= 5; ++Indx) { X = F9; switch (Indx) { case 2: X = One + U2; break; case 3: X = V; break; case 4: X = UfThold; break; case 5: X = Radix; } Y = X; sigsave = _sigfpe; if (setjmp (ovfl_buf)) printf (" X / X traps when X = %g\n", X); else { V9 = (Y / X - Half) - Half; if (V9 == Zero) continue; if (V9 == -U1 && Indx < 5) BadCond (Flaw, ""); else BadCond (Serious, ""); printf (" X / X differs from 1 when X = %.17e\n", X); printf (" instead, X / X - 1/2 - 1/2 = %.17e .\n", V9); } sigsave = 0; } /*=============================================*/ Milestone = 210; /*=============================================*/ MyZero = Zero; printf ("\n"); printf ("What message and/or values does Division by Zero produce?\n"); #ifndef BATCHMODE printf ("This can interupt your program. You can "); printf ("skip this part if you wish.\n"); printf ("Do you wish to compute 1 / 0? "); fflush (stdout); read (KEYBOARD, ch, 8); if ((ch[0] == 'Y') || (ch[0] == 'y')) { #endif /* !BATCHMODE */ sigsave = _sigfpe; printf (" Trying to compute 1 / 0 produces ..."); if (!setjmp (ovfl_buf)) printf (" %.7e .\n", One / MyZero); sigsave = 0; #ifndef BATCHMODE } else printf ("O.K.\n"); printf ("\nDo you wish to compute 0 / 0? "); fflush (stdout); read (KEYBOARD, ch, 80); if ((ch[0] == 'Y') || (ch[0] == 'y')) { #endif /* !BATCHMODE */ sigsave = _sigfpe; printf ("\n Trying to compute 0 / 0 produces ..."); if (!setjmp (ovfl_buf)) printf (" %.7e .\n", Zero / MyZero); sigsave = 0; #ifndef BATCHMODE } else printf ("O.K.\n"); #endif /* !BATCHMODE */ /*=============================================*/ Milestone = 220; /*=============================================*/ Pause (); printf ("\n"); { static char *msg[] = { "FAILUREs encountered =", "SERIOUS DEFECTs discovered =", "DEFECTs discovered =", "FLAWs discovered ="}; int i; for (i = 0; i < 4; i++) if (ErrCnt[i]) printf ("The number of %-29s %d.\n", msg[i], ErrCnt[i]); } printf ("\n"); if ((ErrCnt[Failure] + ErrCnt[Serious] + ErrCnt[Defect] + ErrCnt[Flaw]) > 0) { if ((ErrCnt[Failure] + ErrCnt[Serious] + ErrCnt[ Defect] == 0) && (ErrCnt[Flaw] > 0)) { printf ("The arithmetic diagnosed seems "); printf ("Satisfactory though flawed.\n"); } if ((ErrCnt[Failure] + ErrCnt[Serious] == 0) && (ErrCnt[Defect] > 0)) { printf ("The arithmetic diagnosed may be Acceptable\n"); printf ("despite inconvenient Defects.\n"); } if ((ErrCnt[Failure] + ErrCnt[Serious]) > 0) { printf ("The arithmetic diagnosed has "); printf ("unacceptable Serious Defects.\n"); } if (ErrCnt[Failure] > 0) { printf ("Potentially fatal FAILURE may have spoiled this"); printf (" program's subsequent diagnoses.\n"); } } else { printf ("No failures, defects nor flaws have been discovered.\n"); if (!((RMult == Rounded) && (RDiv == Rounded) && (RAddSub == Rounded) && (RSqrt == Rounded))) printf ("The arithmetic diagnosed seems Satisfactory.\n"); else { if (StickyBit >= One && (Radix - Two) * (Radix - Nine - One) == Zero) { printf ("Rounding appears to conform to "); printf ("the proposed IEEE standard P"); if ((Radix == Two) && ((Precision - Four * Three * Two) * (Precision - TwentySeven - TwentySeven + One) == Zero)) printf ("754"); else printf ("854"); if (IEEE) printf (".\n"); else { printf (",\nexcept for possibly Double Rounding"); printf (" during Gradual Underflow.\n"); } } printf ("The arithmetic diagnosed appears to be Excellent!\n"); } } if (fpecount) printf ("\nA total of %d floating point exceptions were registered.\n", fpecount); printf ("END OF TEST.\n"); return 0; } FLOAT Sign (X) FLOAT X; { return X >= 0. ? 1.0 : -1.0; } void Pause () { #ifndef BATCHMODE char ch[8]; printf ("\nTo continue, press RETURN"); fflush (stdout); read (KEYBOARD, ch, 8); #endif /* !BATCHMODE */ #ifndef CYGNUS printf ("\nDiagnosis resumes after milestone Number %d", Milestone); printf (" Page: %d\n\n", PageNo); ++Milestone; ++PageNo; #endif /* !CYGNUS */ } void TstCond (K, Valid, T) int K, Valid; char *T; { #ifdef CYGNUS printf ("TEST: %s\n", T); #endif /* CYGNUS */ if (!Valid) { BadCond (K, T); printf (".\n"); } #ifdef CYGNUS printf ("PASS: %s\n", T); #endif /* CYGNUS */ } void BadCond (K, T) int K; char *T; { static char *msg[] = {"FAILURE", "SERIOUS DEFECT", "DEFECT", "FLAW"}; ErrCnt[K] = ErrCnt[K] + 1; #ifndef CYGNUS printf ("%s: %s", msg[K], T); #else printf ("ERROR: Severity: %s: %s", msg[K], T); #endif /* CYGNUS */ } /* * Random computes * X = (Random1 + Random9)^5 * Random1 = X - FLOOR(X) + 0.000005 * X; * and returns the new value of Random1 */ FLOAT Random () { FLOAT X, Y; X = Random1 + Random9; Y = X * X; Y = Y * Y; X = X * Y; Y = X - FLOOR (X); Random1 = Y + X * 0.000005; return (Random1); } void SqXMinX (ErrKind) int ErrKind; { FLOAT XA, XB; XB = X * BInvrse; XA = X - XB; SqEr = ((SQRT (X * X) - XB) - XA) / OneUlp; if (SqEr != Zero) { if (SqEr < MinSqEr) MinSqEr = SqEr; if (SqEr > MaxSqEr) MaxSqEr = SqEr; J = J + 1.0; BadCond (ErrKind, "\n"); printf ("sqrt( %.17e) - %.17e = %.17e\n", X * X, X, OneUlp * SqEr); printf ("\tinstead of correct value 0 .\n"); } } void NewD () { X = Z1 * Q; X = FLOOR (Half - X / Radix) * Radix + X; Q = (Q - X * Z) / Radix + X * X * (D / Radix); Z = Z - Two * X * D; if (Z <= Zero) { Z = -Z; Z1 = -Z1; } D = Radix * D; } void SR3750 () { if (!((X - Radix < Z2 - Radix) || (X - Z2 > WVar - Z2))) { I = I + 1; X2 = SQRT (X * D); Y2 = (X2 - Z2) - (Y - Z2); X2 = X8 / (Y - Half); X2 = X2 - Half * X2 * X2; SqEr = (Y2 + Half) + (Half - X2); if (SqEr < MinSqEr) MinSqEr = SqEr; SqEr = Y2 - X2; if (SqEr > MaxSqEr) MaxSqEr = SqEr; } } void IsYeqX () { if (Y != X) { if (N <= 0) { if (Z == Zero && Q <= Zero) printf ("WARNING: computing\n"); else BadCond (Defect, "computing\n"); printf ("\t(%.17e) ^ (%.17e)\n", Z, Q); printf ("\tyielded %.17e;\n", Y); printf ("\twhich compared unequal to correct %.17e ;\n", X); printf ("\t\tthey differ by %.17e .\n", Y - X); } N = N + 1; /* ... count discrepancies. */ } } void SR3980 () { do { Q = (FLOAT) I; Y = POW (Z, Q); IsYeqX (); if (++I > M) break; X = Z * X; } while (X < WVar); } void PrintIfNPositive () { if (N > 0) printf ("Similar discrepancies have occurred %d times.\n", N); } void TstPtUf () { N = 0; if (Z != Zero) { printf ("Since comparison denies Z = 0, evaluating "); printf ("(Z + Z) / Z should be safe.\n"); sigsave = _sigfpe; if (setjmp (ovfl_buf)) goto very_serious; Q9 = (Z + Z) / Z; printf ("What the machine gets for (Z + Z) / Z is %.17e .\n", Q9); if (FABS (Q9 - Two) < Radix * U2) { printf ("This is O.K., provided Over/Underflow"); printf (" has NOT just been signaled.\n"); } else { if ((Q9 < One) || (Q9 > Two)) { very_serious: N = 1; ErrCnt[Serious] = ErrCnt[Serious] + 1; printf ("This is a VERY SERIOUS DEFECT!\n"); } else { N = 1; ErrCnt[Defect] = ErrCnt[Defect] + 1; printf ("This is a DEFECT!\n"); } } sigsave = 0; V9 = Z * One; Random1 = V9; V9 = One * Z; Random2 = V9; V9 = Z / One; if ((Z == Random1) && (Z == Random2) && (Z == V9)) { if (N > 0) Pause (); } else { N = 1; BadCond (Defect, "What prints as Z = "); printf ("%.17e\n\tcompares different from ", Z); if (Z != Random1) printf ("Z * 1 = %.17e ", Random1); if (!((Z == Random2) || (Random2 == Random1))) printf ("1 * Z == %g\n", Random2); if (!(Z == V9)) printf ("Z / 1 = %.17e\n", V9); if (Random2 != Random1) { ErrCnt[Defect] = ErrCnt[Defect] + 1; BadCond (Defect, "Multiplication does not commute!\n"); printf ("\tComparison alleges that 1 * Z = %.17e\n", Random2); printf ("\tdiffers from Z * 1 = %.17e\n", Random1); } Pause (); } } } void notify (s) char *s; { printf ("%s test appears to be inconsistent...\n", s); printf (" PLEASE NOTIFY KARPINKSI!\n"); } void msglist (s) char **s; { while (*s) printf ("%s\n", *s++); } void Instructions () { static char *instr[] = { "Lest this program stop prematurely, i.e. before displaying\n", " `END OF TEST',\n", "try to persuade the computer NOT to terminate execution when an", "error like Over/Underflow or Division by Zero occurs, but rather", "to persevere with a surrogate value after, perhaps, displaying some", "warning. If persuasion avails naught, don't despair but run this", "program anyway to see how many milestones it passes, and then", "amend it to make further progress.\n", "Answer questions with Y, y, N or n (unless otherwise indicated).\n", 0}; msglist (instr); } void Heading () { static char *head[] = { "Users are invited to help debug and augment this program so it will", "cope with unanticipated and newly uncovered arithmetic pathologies.\n", "Please send suggestions and interesting results to", "\tRichard Karpinski", "\tComputer Center U-76", "\tUniversity of California", "\tSan Francisco, CA 94143-0704, USA\n", "In doing so, please include the following information:", #ifdef SINGLE_PRECISION "\tPrecision:\tsingle;", #else /* !SINGLE_PRECISION */ "\tPrecision:\tdouble;", #endif /* SINGLE_PRECISION */ "\tVersion:\t10 February 1989;", "\tComputer:\n", "\tCompiler:\n", "\tOptimization level:\n", "\tOther relevant compiler options:", 0}; msglist (head); } void Characteristics () { static char *chars[] = { "Running this program should reveal these characteristics:", " Radix = 1, 2, 4, 8, 10, 16, 100, 256 ...", " Precision = number of significant digits carried.", " U2 = Radix/Radix^Precision = One Ulp", "\t(OneUlpnit in the Last Place) of 1.000xxx .", " U1 = 1/Radix^Precision = One Ulp of numbers a little less than 1.0 .", " Adequacy of guard digits for Mult., Div. and Subt.", " Whether arithmetic is chopped, correctly rounded, or something else", "\tfor Mult., Div., Add/Subt. and Sqrt.", " Whether a Sticky Bit used correctly for rounding.", " UnderflowThreshold = an underflow threshold.", " E0 and PseudoZero tell whether underflow is abrupt, gradual, or fuzzy.", " V = an overflow threshold, roughly.", " V0 tells, roughly, whether Infinity is represented.", " Comparisions are checked for consistency with subtraction", "\tand for contamination with pseudo-zeros.", " Sqrt is tested. Y^X is not tested.", " Extra-precise subexpressions are revealed but NOT YET tested.", " Decimal-Binary conversion is NOT YET tested for accuracy.", 0}; msglist (chars); } void History () { /* History */ /* Converted from Brian Wichmann's Pascal version to C by Thos Sumner, with further massaging by David M. Gay. */ static char *hist[] = { "The program attempts to discriminate among", " FLAWs, like lack of a sticky bit,", " Serious DEFECTs, like lack of a guard digit, and", " FAILUREs, like 2+2 == 5 .", "Failures may confound subsequent diagnoses.\n", "The diagnostic capabilities of this program go beyond an earlier", "program called `MACHAR', which can be found at the end of the", "book `Software Manual for the Elementary Functions' (1980) by", "W. J. Cody and W. Waite. Although both programs try to discover", "the Radix, Precision and range (over/underflow thresholds)", "of the arithmetic, this program tries to cope with a wider variety", "of pathologies, and to say how well the arithmetic is implemented.", "\nThe program is based upon a conventional radix representation for", "floating-point numbers, but also allows logarithmic encoding", "as used by certain early WANG machines.\n", "BASIC version of this program (C) 1983 by Prof. W. M. Kahan;", "see source comments for more history.", 0}; msglist (hist); }