extern "C" {
#include <stdio.h>
#include <stddef.h>
#include <limits.h>
#include <string.h>
#include <stdlib.h>
#include <math.h>
#include <unistd.h>
#include <float.h>
#define GTY(x)
#include "ansidecl.h"
#include "auto-host.h"
#include "hwint.h"
#undef EXTRA_MODES_FILE
struct rtx_def;
typedef struct rtx_def *rtx;
struct rtvec_def;
typedef struct rtvec_def *rtvec;
union tree_node;
typedef union tree_node *tree;
#define DEFTREECODE(SYM, STRING, TYPE, NARGS) SYM,
enum tree_code {
#include "tree.def"
LAST_AND_UNUSED_TREE_CODE
};
#undef DEFTREECODE
#define ENUM_BITFIELD(X) enum X
#define class klass
#include "real.h"
#undef class
}
#undef setjmp
#define setjmp(x) (0)
static bool verbose = false;
static int verbose_index = 0;
class real_c_float
{
public:
static const enum machine_mode MODE = SFmode;
private:
static const int external_max = 128 / 32;
static const int internal_max
= (sizeof (REAL_VALUE_TYPE) + sizeof (long) + 1) / sizeof (long);
long image[external_max < internal_max ? internal_max : external_max];
void from_long(long);
void from_str(const char *);
void binop(int code, const real_c_float&);
void unop(int code);
bool cmp(int code, const real_c_float&) const;
public:
real_c_float()
{ }
real_c_float(long l)
{ from_long(l); }
real_c_float(const char *s)
{ from_str(s); }
real_c_float(const real_c_float &b)
{ memcpy(image, b.image, sizeof(image)); }
const real_c_float& operator= (long l)
{ from_long(l); return *this; }
const real_c_float& operator= (const char *s)
{ from_str(s); return *this; }
const real_c_float& operator= (const real_c_float &b)
{ memcpy(image, b.image, sizeof(image)); return *this; }
const real_c_float& operator+= (const real_c_float &b)
{ binop(PLUS_EXPR, b); return *this; }
const real_c_float& operator-= (const real_c_float &b)
{ binop(MINUS_EXPR, b); return *this; }
const real_c_float& operator*= (const real_c_float &b)
{ binop(MULT_EXPR, b); return *this; }
const real_c_float& operator/= (const real_c_float &b)
{ binop(RDIV_EXPR, b); return *this; }
real_c_float operator- () const
{ real_c_float r(*this); r.unop(NEGATE_EXPR); return r; }
real_c_float abs () const
{ real_c_float r(*this); r.unop(ABS_EXPR); return r; }
bool operator < (const real_c_float &b) const { return cmp(LT_EXPR, b); }
bool operator <= (const real_c_float &b) const { return cmp(LE_EXPR, b); }
bool operator == (const real_c_float &b) const { return cmp(EQ_EXPR, b); }
bool operator != (const real_c_float &b) const { return cmp(NE_EXPR, b); }
bool operator >= (const real_c_float &b) const { return cmp(GE_EXPR, b); }
bool operator > (const real_c_float &b) const { return cmp(GT_EXPR, b); }
const char * str () const;
const char * hex () const;
long integer () const;
int exp () const;
void ldexp (int);
};
void
real_c_float::from_long (long l)
{
REAL_VALUE_TYPE f;
real_from_integer (&f, MODE, l, l < 0 ? -1 : 0, 0);
real_to_target (image, &f, MODE);
}
void
real_c_float::from_str (const char *s)
{
REAL_VALUE_TYPE f;
const char *p = s;
if (*p == '-' || *p == '+')
p++;
if (strcasecmp(p, "inf") == 0)
{
real_inf (&f);
if (*s == '-')
real_arithmetic (&f, NEGATE_EXPR, &f, NULL);
}
else if (strcasecmp(p, "nan") == 0)
real_nan (&f, "", 1, MODE);
else
real_from_string (&f, s);
real_to_target (image, &f, MODE);
}
void
real_c_float::binop (int code, const real_c_float &b)
{
REAL_VALUE_TYPE ai, bi, ri;
real_from_target (&ai, image, MODE);
real_from_target (&bi, b.image, MODE);
real_arithmetic (&ri, code, &ai, &bi);
real_to_target (image, &ri, MODE);
if (verbose)
{
char ab[64], bb[64], rb[64];
const real_format *fmt = real_format_for_mode[MODE - QFmode];
const int digits = (fmt->p * fmt->log2_b + 3) / 4;
char symbol_for_code;
real_from_target (&ri, image, MODE);
real_to_hexadecimal (ab, &ai, sizeof(ab), digits, 0);
real_to_hexadecimal (bb, &bi, sizeof(bb), digits, 0);
real_to_hexadecimal (rb, &ri, sizeof(rb), digits, 0);
switch (code)
{
case PLUS_EXPR:
symbol_for_code = '+';
break;
case MINUS_EXPR:
symbol_for_code = '-';
break;
case MULT_EXPR:
symbol_for_code = '*';
break;
case RDIV_EXPR:
symbol_for_code = '/';
break;
default:
abort ();
}
fprintf (stderr, "%6d: %s %c %s = %s\n", verbose_index++,
ab, symbol_for_code, bb, rb);
}
}
void
real_c_float::unop (int code)
{
REAL_VALUE_TYPE ai, ri;
real_from_target (&ai, image, MODE);
real_arithmetic (&ri, code, &ai, NULL);
real_to_target (image, &ri, MODE);
if (verbose)
{
char ab[64], rb[64];
const real_format *fmt = real_format_for_mode[MODE - QFmode];
const int digits = (fmt->p * fmt->log2_b + 3) / 4;
const char *symbol_for_code;
real_from_target (&ri, image, MODE);
real_to_hexadecimal (ab, &ai, sizeof(ab), digits, 0);
real_to_hexadecimal (rb, &ri, sizeof(rb), digits, 0);
switch (code)
{
case NEGATE_EXPR:
symbol_for_code = "-";
break;
case ABS_EXPR:
symbol_for_code = "abs ";
break;
default:
abort ();
}
fprintf (stderr, "%6d: %s%s = %s\n", verbose_index++,
symbol_for_code, ab, rb);
}
}
bool
real_c_float::cmp (int code, const real_c_float &b) const
{
REAL_VALUE_TYPE ai, bi;
bool ret;
real_from_target (&ai, image, MODE);
real_from_target (&bi, b.image, MODE);
ret = real_compare (code, &ai, &bi);
if (verbose)
{
char ab[64], bb[64];
const real_format *fmt = real_format_for_mode[MODE - QFmode];
const int digits = (fmt->p * fmt->log2_b + 3) / 4;
const char *symbol_for_code;
real_to_hexadecimal (ab, &ai, sizeof(ab), digits, 0);
real_to_hexadecimal (bb, &bi, sizeof(bb), digits, 0);
switch (code)
{
case LT_EXPR:
symbol_for_code = "<";
break;
case LE_EXPR:
symbol_for_code = "<=";
break;
case EQ_EXPR:
symbol_for_code = "==";
break;
case NE_EXPR:
symbol_for_code = "!=";
break;
case GE_EXPR:
symbol_for_code = ">=";
break;
case GT_EXPR:
symbol_for_code = ">";
break;
default:
abort ();
}
fprintf (stderr, "%6d: %s %s %s = %s\n", verbose_index++,
ab, symbol_for_code, bb, (ret ? "true" : "false"));
}
return ret;
}
const char *
real_c_float::str() const
{
REAL_VALUE_TYPE f;
const real_format *fmt = real_format_for_mode[MODE - QFmode];
const int digits = int(fmt->p * fmt->log2_b * .30102999566398119521 + 1);
real_from_target (&f, image, MODE);
char *buf = new char[digits + 10];
real_to_decimal (buf, &f, digits+10, digits, 0);
return buf;
}
const char *
real_c_float::hex() const
{
REAL_VALUE_TYPE f;
const real_format *fmt = real_format_for_mode[MODE - QFmode];
const int digits = (fmt->p * fmt->log2_b + 3) / 4;
real_from_target (&f, image, MODE);
char *buf = new char[digits + 10];
real_to_hexadecimal (buf, &f, digits+10, digits, 0);
return buf;
}
long
real_c_float::integer() const
{
REAL_VALUE_TYPE f;
real_from_target (&f, image, MODE);
return real_to_integer (&f);
}
int
real_c_float::exp() const
{
REAL_VALUE_TYPE f;
real_from_target (&f, image, MODE);
return real_exponent (&f);
}
void
real_c_float::ldexp (int exp)
{
REAL_VALUE_TYPE ai;
real_from_target (&ai, image, MODE);
real_ldexp (&ai, &ai, exp);
real_to_target (image, &ai, MODE);
}
template<typename T>
class native_float
{
private:
volatile T image;
static T from_str (const char *);
static T do_abs (T);
static T verbose_binop (T, char, T, T);
static T verbose_unop (const char *, T, T);
static bool verbose_cmp (T, const char *, T, bool);
public:
native_float()
{ }
native_float(long l)
{ image = l; }
native_float(const char *s)
{ image = from_str(s); }
native_float(const native_float &b)
{ image = b.image; }
const native_float& operator= (long l)
{ image = l; return *this; }
const native_float& operator= (const char *s)
{ image = from_str(s); return *this; }
const native_float& operator= (const native_float &b)
{ image = b.image; return *this; }
const native_float& operator+= (const native_float &b)
{
image = verbose_binop(image, '+', b.image, image + b.image);
return *this;
}
const native_float& operator-= (const native_float &b)
{
image = verbose_binop(image, '-', b.image, image - b.image);
return *this;
}
const native_float& operator*= (const native_float &b)
{
image = verbose_binop(image, '*', b.image, image * b.image);
return *this;
}
const native_float& operator/= (const native_float &b)
{
image = verbose_binop(image, '/', b.image, image / b.image);
return *this;
}
native_float operator- () const
{
native_float r;
r.image = verbose_unop("-", image, -image);
return r;
}
native_float abs () const
{
native_float r;
r.image = verbose_unop("abs ", image, do_abs(image));
return r;
}
bool operator < (const native_float &b) const
{ return verbose_cmp(image, "<", b.image, image < b.image); }
bool operator <= (const native_float &b) const
{ return verbose_cmp(image, "<=", b.image, image <= b.image); }
bool operator == (const native_float &b) const
{ return verbose_cmp(image, "==", b.image, image == b.image); }
bool operator != (const native_float &b) const
{ return verbose_cmp(image, "!=", b.image, image != b.image); }
bool operator >= (const native_float &b) const
{ return verbose_cmp(image, ">=", b.image, image >= b.image); }
bool operator > (const native_float &b) const
{ return verbose_cmp(image, ">", b.image, image > b.image); }
const char * str () const;
const char * hex () const;
long integer () const
{ return long(image); }
int exp () const;
void ldexp (int);
};
template<typename T>
inline T
native_float<T>::from_str (const char *s)
{
return strtold (s, NULL);
}
template<>
inline float
native_float<float>::from_str (const char *s)
{
return strtof (s, NULL);
}
template<>
inline double
native_float<double>::from_str (const char *s)
{
return strtod (s, NULL);
}
template<typename T>
inline T
native_float<T>::do_abs (T image)
{
return fabsl (image);
}
template<>
inline float
native_float<float>::do_abs (float image)
{
return fabsf (image);
}
template<>
inline double
native_float<double>::do_abs (double image)
{
return fabs (image);
}
template<typename T>
T
native_float<T>::verbose_binop (T a, char symbol, T b, T r)
{
if (verbose)
{
const int digits = int(sizeof(T) * CHAR_BIT / 4) - 1;
#ifdef NO_LONG_DOUBLE
fprintf (stderr, "%6d: %.*a %c %.*a = %.*a\n", verbose_index++,
digits, (double)a, symbol,
digits, (double)b, digits, (double)r);
#else
fprintf (stderr, "%6d: %.*La %c %.*La = %.*La\n", verbose_index++,
digits, (long double)a, symbol,
digits, (long double)b, digits, (long double)r);
#endif
}
return r;
}
template<typename T>
T
native_float<T>::verbose_unop (const char *symbol, T a, T r)
{
if (verbose)
{
const int digits = int(sizeof(T) * CHAR_BIT / 4) - 1;
#ifdef NO_LONG_DOUBLE
fprintf (stderr, "%6d: %s%.*a = %.*a\n", verbose_index++,
symbol, digits, (double)a, digits, (double)r);
#else
fprintf (stderr, "%6d: %s%.*La = %.*La\n", verbose_index++,
symbol, digits, (long double)a, digits, (long double)r);
#endif
}
return r;
}
template<typename T>
bool
native_float<T>::verbose_cmp (T a, const char *symbol, T b, bool r)
{
if (verbose)
{
const int digits = int(sizeof(T) * CHAR_BIT / 4) - 1;
#ifndef NO_LONG_DOUBLE
fprintf (stderr, "%6d: %.*a %s %.*a = %s\n", verbose_index++,
digits, (double)a, symbol,
digits, (double)b, (r ? "true" : "false"));
#else
fprintf (stderr, "%6d: %.*La %s %.*La = %s\n", verbose_index++,
digits, (long double)a, symbol,
digits, (long double)b, (r ? "true" : "false"));
#endif
}
return r;
}
template<typename T>
const char *
native_float<T>::str() const
{
char *buf = new char[50];
const int digits = int(sizeof(T) * CHAR_BIT * .30102999566398119521 + 1);
#ifndef NO_LONG_DOUBLE
sprintf (buf, "%.*e", digits - 1, (double) image);
#else
sprintf (buf, "%.*Le", digits - 1, (long double) image);
#endif
return buf;
}
template<typename T>
const char *
native_float<T>::hex() const
{
char *buf = new char[50];
const int digits = int(sizeof(T) * CHAR_BIT / 4);
#ifndef NO_LONG_DOUBLE
sprintf (buf, "%.*a", digits - 1, (double) image);
#else
sprintf (buf, "%.*La", digits - 1, (long double) image);
#endif
return buf;
}
template<typename T>
int
native_float<T>::exp() const
{
int e;
frexp (image, &e);
return e;
}
template<typename T>
void
native_float<T>::ldexp (int exp)
{
image = ldexpl (image, exp);
}
template<>
void
native_float<float>::ldexp (int exp)
{
image = ldexpf (image, exp);
}
template<>
void
native_float<double>::ldexp (int exp)
{
image = ::ldexp (image, exp);
}
template<typename FLOAT>
inline FLOAT
FABS (const FLOAT &f)
{
return f.abs();
}
template<typename FLOAT, typename RHS>
inline FLOAT
operator+ (const FLOAT &a, const RHS &b)
{
return FLOAT(a) += FLOAT(b);
}
template<typename FLOAT, typename RHS>
inline FLOAT
operator- (const FLOAT &a, const RHS &b)
{
return FLOAT(a) -= FLOAT(b);
}
template<typename FLOAT, typename RHS>
inline FLOAT
operator* (const FLOAT &a, const RHS &b)
{
return FLOAT(a) *= FLOAT(b);
}
template<typename FLOAT, typename RHS>
inline FLOAT
operator/ (const FLOAT &a, const RHS &b)
{
return FLOAT(a) /= FLOAT(b);
}
template<typename FLOAT>
FLOAT
FLOOR (const FLOAT &f)
{
long i = f.integer();
FLOAT r;
r = i;
if (i < 0 && f != r)
r = i - 1;
return r;
}
template<typename FLOAT>
FLOAT
SQRT (const FLOAT &f)
{
#if 0
FLOAT zero = long(0);
FLOAT two = 2;
FLOAT one = 1;
FLOAT diff, diff2;
FLOAT z, t;
if (f == zero)
return zero;
if (f < zero)
return zero / zero;
if (f == one)
return f;
z = f;
z.ldexp (-f.exp() / 2);
diff2 = FABS (z * z - f);
if (diff2 > zero)
while (1)
{
t = (f / (two * z)) + (z / two);
diff = FABS (t * t - f);
if (diff >= diff2)
break;
z = t;
diff2 = diff;
}
return z;
#elif defined(NO_LONG_DOUBLE)
double d;
char buf[64];
d = strtod (f.hex(), NULL);
d = sqrt (d);
sprintf(buf, "%.35a", d);
return FLOAT(buf);
#else
long double ld;
char buf[64];
ld = strtold (f.hex(), NULL);
ld = sqrtl (ld);
sprintf(buf, "%.35La", ld);
return FLOAT(buf);
#endif
}
template<typename FLOAT>
FLOAT
LOG (FLOAT x)
{
#if 0
FLOAT zero = long(0);
FLOAT one = 1;
if (x <= zero)
return zero / zero;
if (x == one)
return zero;
int exp = x.exp() - 1;
x.ldexp(-exp);
FLOAT xm1 = x - one;
FLOAT y = xm1;
long n = 2;
FLOAT sum = xm1;
while (1)
{
y *= xm1;
FLOAT term = y / FLOAT (n);
FLOAT next = sum + term;
if (next == sum)
break;
sum = next;
if (++n == 1000)
break;
}
if (exp)
sum += FLOAT (exp) * FLOAT(".69314718055994530941");
return sum;
#elif defined (NO_LONG_DOUBLE)
double d;
char buf[64];
d = strtod (x.hex(), NULL);
d = log (d);
sprintf(buf, "%.35a", d);
return FLOAT(buf);
#else
long double ld;
char buf[64];
ld = strtold (x.hex(), NULL);
ld = logl (ld);
sprintf(buf, "%.35La", ld);
return FLOAT(buf);
#endif
}
template<typename FLOAT>
FLOAT
EXP (const FLOAT &x)
{
#ifdef NO_LONG_DOUBLE
double d;
char buf[64];
d = strtod (x.hex(), NULL);
d = exp (d);
sprintf(buf, "%.35a", d);
return FLOAT(buf);
#else
long double ld;
char buf[64];
ld = strtold (x.hex(), NULL);
ld = expl (ld);
sprintf(buf, "%.35La", ld);
return FLOAT(buf);
#endif
}
template<typename FLOAT>
FLOAT
POW (const FLOAT &base, const FLOAT &exp)
{
#ifdef NO_LONG_DOUBLE
double d1, d2;
char buf[64];
d1 = strtod (base.hex(), NULL);
d2 = strtod (exp.hex(), NULL);
d1 = pow (d1, d2);
sprintf(buf, "%.35a", d1);
return FLOAT(buf);
#else
long double ld1, ld2;
char buf[64];
ld1 = strtold (base.hex(), NULL);
ld2 = strtold (exp.hex(), NULL);
ld1 = powl (ld1, ld2);
sprintf(buf, "%.35La", ld1);
return FLOAT(buf);
#endif
}
int NoTrials = 20;
bool do_pause = false;
enum Guard { No, Yes };
enum Rounding { Other, Rounded, Chopped };
enum Class { Failure, Serious, Defect, Flaw };
template<typename FLOAT>
struct Paranoia
{
FLOAT Radix, BInvrse, RadixD2, BMinusU2;
FLOAT Zero;
FLOAT Half;
FLOAT One;
FLOAT Two;
FLOAT Three;
FLOAT Four;
FLOAT Five;
FLOAT Eight;
FLOAT Nine;
FLOAT TwentySeven;
FLOAT ThirtyTwo;
FLOAT TwoForty;
FLOAT MinusOne;
FLOAT OneAndHalf;
int Indx;
char ch[8];
FLOAT AInvrse, A1;
FLOAT C, CInvrse;
FLOAT D, FourD;
FLOAT E0, E1, Exp2, E3, MinSqEr;
FLOAT SqEr, MaxSqEr, E9;
FLOAT Third;
FLOAT F6, F9;
FLOAT H, HInvrse;
int I;
FLOAT StickyBit, J;
FLOAT MyZero;
FLOAT Precision;
FLOAT Q, Q9;
FLOAT R, Random9;
FLOAT T, Underflow, S;
FLOAT OneUlp, UfThold, U1, U2;
FLOAT V, V0, V9;
FLOAT W;
FLOAT X, X1, X2, X8, Random1;
FLOAT Y, Y1, Y2, Random2;
FLOAT Z, PseudoZero, Z1, Z2, Z9;
int ErrCnt[4];
int 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;
int main ();
FLOAT Sign (FLOAT);
FLOAT Random ();
void Pause ();
void BadCond (int, const char *);
void SqXMinX (int);
void TstCond (int, int, const char *);
void notify (const char *);
void IsYeqX ();
void NewD ();
void PrintIfNPositive ();
void SR3750 ();
void TstPtUf ();
Paranoia() { memset(this, 0, sizeof (*this)); }
};
template<typename FLOAT>
int
Paranoia<FLOAT>::main()
{
Zero = long(0);
One = long(1);
Two = long(2);
Three = long(3);
Four = long(4);
Five = long(5);
Eight = long(8);
Nine = long(9);
TwentySeven = long(27);
ThirtyTwo = long(32);
TwoForty = long(240);
MinusOne = long(-1);
Half = "0x1p-1";
OneAndHalf = "0x3p-1";
ErrCnt[Failure] = 0;
ErrCnt[Serious] = 0;
ErrCnt[Defect] = 0;
ErrCnt[Flaw] = 0;
PageNo = 1;
Milestone = 7;
printf ("Program is now RUNNING tests on small integers:\n");
TstCond (Failure, (Zero + Zero == Zero), "0+0 != 0");
TstCond (Failure, (One - One == Zero), "1-1 != 0");
TstCond (Failure, (One > Zero), "1 <= 0");
TstCond (Failure, (One + One == Two), "1+1 != 2");
Z = -Zero;
if (Z != Zero)
{
ErrCnt[Failure] = ErrCnt[Failure] + 1;
printf ("Comparison alleges that -0.0 is Non-zero!\n");
U2 = "0.001";
Radix = 1;
TstPtUf ();
}
TstCond (Failure, (Three == Two + One), "3 != 2+1");
TstCond (Failure, (Four == Three + One), "4 != 3+1");
TstCond (Failure, (Four + Two * (-Two) == Zero), "4 + 2*(-2) != 0");
TstCond (Failure, (Four - Three - One == Zero), "4-3-1 != 0");
TstCond (Failure, (MinusOne == (Zero - One)), "-1 != 0-1");
TstCond (Failure, (MinusOne + One == Zero), "-1+1 != 0");
TstCond (Failure, (One + MinusOne == Zero), "1+(-1) != 0");
TstCond (Failure, (MinusOne + FABS (One) == Zero), "-1+abs(1) != 0");
TstCond (Failure, (MinusOne + MinusOne * MinusOne == Zero),
"-1+(-1)*(-1) != 0");
TstCond (Failure, Half + MinusOne + Half == Zero, "1/2 + (-1) + 1/2 != 0");
Milestone = 10;
TstCond (Failure, (Nine == Three * Three), "9 != 3*3");
TstCond (Failure, (TwentySeven == Nine * Three), "27 != 9*3");
TstCond (Failure, (Eight == Four + Four), "8 != 4+4");
TstCond (Failure, (ThirtyTwo == Eight * Four), "32 != 8*4");
TstCond (Failure, (ThirtyTwo - TwentySeven - Four - One == Zero),
"32-27-4-1 != 0");
TstCond (Failure, Five == Four + One, "5 != 4+1");
TstCond (Failure, TwoForty == Four * Five * Three * Four, "240 != 4*5*3*4");
TstCond (Failure, TwoForty / Three - Four * Four * Five == Zero,
"240/3 - 4*4*5 != 0");
TstCond (Failure, TwoForty / Four - Five * Three * Four == Zero,
"240/4 - 5*3*4 != 0");
TstCond (Failure, TwoForty / Five - Four * Three * Four == Zero,
"240/5 - 4*3*4 != 0");
if (ErrCnt[Failure] == 0)
{
printf ("-1, 0, 1/2, 1, 2, 3, 4, 5, 9, 27, 32 & 240 are O.K.\n");
printf ("\n");
}
printf ("Searching for Radix and Precision.\n");
W = One;
do
{
W = W + W;
Y = W + One;
Z = Y - W;
Y = Z - One;
}
while (MinusOne + FABS (Y) < Zero);
Precision = Zero;
Y = One;
do
{
Radix = W + Y;
Y = Y + Y;
Radix = Radix - W;
}
while (Radix == Zero);
if (Radix < Two)
Radix = One;
printf ("Radix = %s .\n", Radix.str());
if (Radix != One)
{
W = One;
do
{
Precision = Precision + One;
W = W * Radix;
Y = W + One;
}
while ((Y - W) == One);
}
U1 = One / W;
U2 = Radix * U1;
printf ("Closest relative separation found is U1 = %s .\n\n", U1.str());
printf ("Recalculating radix and precision\n ");
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;
do
{
U2 = X;
Y = Half * U2 + ThirtyTwo * U2 * U2;
Y = One + Y;
X = Y - One;
}
while (!((U2 <= X) || (X <= Zero)));
X = Two / Three;
F6 = X - Half;
Third = F6 + F6;
X = Third - Half;
X = FABS (X + F6);
if (X < U1)
X = U1;
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)));
if (U1 == E1)
printf ("confirms closest relative separation U1 .\n");
else
printf ("gets better closest relative separation U1 = %s .\n", U1.str());
W = One / U1;
F9 = (Half - U1) + Half;
Radix = FLOOR (FLOAT ("0.01") + U2 / U1);
if (Radix == E0)
printf ("Radix confirmed.\n");
else
printf ("MYSTERY: recalculated Radix = %s .\n", Radix.str());
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 = Radix - One;
BMinusU2 = (BMinusU2 - U2) + One;
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 %s .\n",
Precision.str());
TstCond (Serious, U2 * Nine * Nine * TwoForty < One,
"Precision worse than 5 decimal figures ");
Milestone = 30;
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 %s, %s, %s,\n", X1.str(), Y1.str(), Z1.str());
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 = %s, Z1 - U1 = %s\n", U1.str(), (Z1 - U1).str());
printf ("\tU2 = %s, Z2 - U2 = %s\n", U2.str(), (Z2 - U2).str());
}
else
{
if ((Z1 <= Zero) || (Z2 <= Zero))
{
printf ("Because of unusual Radix = %s", Radix.str());
printf (", or exact rational arithmetic a result\n");
printf ("Z1 = %s, or Z2 = %s ", Z1.str(), Z2.str());
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 precisely\n");
printf ("with about %s extra B-digits, i.e.\n",
(Q / LOG (Radix)).str());
printf ("roughly %s extra significant decimals.\n",
(Q / LOG (FLOAT (10))).str());
}
printf
("That feature is not tested further by this program.\n");
}
}
}
Pause ();
Milestone = 35;
if (Radix >= Two)
{
X = W / (Radix * Radix);
Y = X + One;
Z = Y - X;
T = Z + U2;
X = T - Z;
TstCond (Failure, X == U2,
"Subtraction is not normalized X=Y,X+Z != Y+Z!");
if (X == U2)
printf ("Subtraction appears to be normalized, as it should be.");
}
printf ("\nChecking for guard digit in *, /, and -.\n");
Y = F9 * One;
Z = One * F9;
X = F9 - Half;
Y = (Y - Half) - X;
Z = (Z - Half) - X;
X = One + U2;
T = X * Radix;
R = Radix * X;
X = T - Radix;
X = X - Radix * U2;
T = R - Radix;
T = T - Radix * U2;
X = X * (Radix - One);
T = T * (Radix - One);
if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero))
GMult = Yes;
else
{
GMult = No;
TstCond (Serious, false, "* lacks a Guard Digit, so 1*X != X");
}
Z = Radix * U2;
X = One + Z;
Y = FABS ((X + Z) - X * X) - U2;
X = One - U2;
Z = FABS ((X - U2) - X * X) - U1;
TstCond (Failure, (Y <= Zero)
&& (Z <= Zero), "* gets too many final digits wrong.\n");
Y = One - U2;
X = One + U2;
Z = One / Y;
Y = Z - X;
X = One / Three;
Z = Three / Nine;
X = X - Z;
T = Nine / TwentySeven;
Z = Z - T;
TstCond (Defect, X == Zero && Y == Zero && Z == Zero,
"Division lacks a Guard Digit, so error can exceed 1 ulp\n"
"or 1/3 and 3/9 and 9/27 may disagree");
Y = F9 / One;
X = F9 - Half;
Y = (Y - Half) - X;
X = One + U2;
T = X / One;
X = T - X;
if ((X == Zero) && (Y == Zero) && (Z == Zero))
GDiv = Yes;
else
{
GDiv = No;
TstCond (Serious, false, "Division lacks a Guard Digit, so X/1 != X");
}
X = One / (One + U2);
Y = X - Half - Half;
TstCond (Serious, Y < Zero, "Computed value of 1/1.000..1 >= 1");
X = One - U2;
Y = One + Radix * U2;
Z = X * Radix;
T = Y * Radix;
R = Z / Radix;
StickyBit = T / Radix;
X = R - X;
Y = StickyBit - Y;
TstCond (Failure, X == Zero && Y == Zero,
"* and/or / gets too many last digits wrong");
Y = One - U1;
X = One - F9;
Y = One - Y;
T = Radix - U2;
Z = Radix - BMinusU2;
T = Radix - T;
if ((X == U1) && (Y == U1) && (Z == U2) && (T == U2))
GAddSub = Yes;
else
{
GAddSub = No;
TstCond (Serious, false,
"- lacks Guard Digit, so cancellation is obscured");
}
if (F9 != One && F9 - One >= Zero)
{
BadCond (Serious, "comparison alleges (1-U1) < 1 although\n");
printf (" subtraction yields (1-U1) - 1 = 0 , thereby vitiating\n");
printf (" such precautions against division by zero as\n");
printf (" ... if (X == 1.0) {.....} else {.../(X-1.0)...}\n");
}
if (GMult == Yes && GDiv == Yes && GAddSub == Yes)
printf
(" *, /, and - appear to have guard digits, as they should.\n");
Milestone = 40;
Pause ();
printf ("Checking rounding on multiply, divide and add/subtract.\n");
RMult = Other;
RDiv = Other;
RAddSub = Other;
RadixD2 = Radix / Two;
A1 = Two;
Done = false;
do
{
AInvrse = Radix;
do
{
X = AInvrse;
AInvrse = AInvrse / A1;
}
while (!(FLOOR (AInvrse) != AInvrse));
Done = (X == One) || (A1 > Three);
if (!Done)
A1 = Nine + One;
}
while (!(Done));
if (X == One)
A1 = Radix;
AInvrse = One / A1;
X = A1;
Y = AInvrse;
Done = false;
do
{
Z = X * Y - Half;
TstCond (Failure, Z == Half, "X * (1/X) differs from 1");
Done = X == Radix;
X = Radix;
Y = One / X;
}
while (!(Done));
Y2 = One + U2;
Y1 = One - U2;
X = OneAndHalf - U2;
Y = OneAndHalf + U2;
Z = (X - U2) * Y2;
T = Y * Y1;
Z = Z - X;
T = T - X;
X = X * Y2;
Y = (Y + U2) * Y1;
X = X - OneAndHalf;
Y = Y - OneAndHalf;
if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T <= Zero))
{
X = (OneAndHalf + U2) * Y2;
Y = OneAndHalf - U2 - U2;
Z = OneAndHalf + U2 + U2;
T = (OneAndHalf - U2) * Y1;
X = X - (Z + U2);
StickyBit = Y * Y1;
S = Z * Y2;
T = T - Y;
Y = (U2 - Y) + StickyBit;
Z = S - (Z + U2 + U2);
StickyBit = (Y2 + U2) * Y1;
Y1 = Y2 * Y1;
StickyBit = StickyBit - Y2;
Y1 = Y1 - Half;
if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero)
&& (StickyBit == Zero) && (Y1 == Half))
{
RMult = Rounded;
printf ("Multiplication appears to round correctly.\n");
}
else if ((X + U2 == Zero) && (Y < Zero) && (Z + U2 == Zero)
&& (T < Zero) && (StickyBit + U2 == Zero) && (Y1 < Half))
{
RMult = Chopped;
printf ("Multiplication appears to chop.\n");
}
else
printf ("* is neither chopped nor correctly rounded.\n");
if ((RMult == Rounded) && (GMult == No))
notify ("Multiplication");
}
else
printf ("* is neither chopped nor correctly rounded.\n");
Milestone = 45;
Y2 = One + U2;
Y1 = One - U2;
Z = OneAndHalf + U2 + U2;
X = Z / Y2;
T = OneAndHalf - U2 - U2;
Y = (T - U2) / Y1;
Z = (Z + U2) / Y2;
X = X - OneAndHalf;
Y = Y - T;
T = T / Y1;
Z = Z - (OneAndHalf + U2);
T = (U2 - OneAndHalf) + T;
if (!((X > Zero) || (Y > Zero) || (Z > Zero) || (T > Zero)))
{
X = OneAndHalf / Y2;
Y = OneAndHalf - U2;
Z = OneAndHalf + U2;
X = X - Y;
T = OneAndHalf / Y1;
Y = Y / Y1;
T = T - (Z + U2);
Y = Y - Z;
Z = Z / Y2;
Y1 = (Y2 + U2) / Y2;
Z = Z - OneAndHalf;
Y2 = Y1 - Y2;
Y1 = (F9 - U1) / F9;
if ((X == Zero) && (Y == Zero) && (Z == Zero) && (T == Zero)
&& (Y2 == Zero) && (Y2 == Zero) && (Y1 - Half == F9 - Half))
{
RDiv = Rounded;
printf ("Division appears to round correctly.\n");
if (GDiv == No)
notify ("Division");
}
else if ((X < Zero) && (Y < Zero) && (Z < Zero) && (T < Zero)
&& (Y2 < Zero) && (Y1 - Half < F9 - Half))
{
RDiv = Chopped;
printf ("Division appears to chop.\n");
}
}
if (RDiv == Other)
printf ("/ is neither chopped nor correctly rounded.\n");
BInvrse = One / Radix;
TstCond (Failure, (BInvrse * Radix - Half == Half),
"Radix * ( 1 / Radix ) differs from 1");
Milestone = 50;
TstCond (Failure, ((F9 + U1) - Half == Half)
&& ((BMinusU2 + U2) - One == Radix - One),
"Incomplete carry-propagation in Addition");
X = One - U1 * U1;
Y = One + U2 * (One - U2);
Z = F9 - Half;
X = (X - Half) - Z;
Y = Y - One;
if ((X == Zero) && (Y == Zero))
{
RAddSub = Chopped;
printf ("Add/Subtract appears to be chopped.\n");
}
if (GAddSub == Yes)
{
X = (Half + U2) * U2;
Y = (Half - U2) * U2;
X = One + X;
Y = One + Y;
X = (One + U2) - X;
Y = One - Y;
if ((X == Zero) && (Y == Zero))
{
X = (Half + U2) * U1;
Y = (Half - U2) * U1;
X = One - X;
Y = One - Y;
X = F9 - X;
Y = One - Y;
if ((X == Zero) && (Y == Zero))
{
RAddSub = Rounded;
printf ("Addition/Subtraction appears to round correctly.\n");
if (GAddSub == No)
notify ("Add/Subtract");
}
else
printf ("Addition/Subtraction neither rounds nor chops.\n");
}
else
printf ("Addition/Subtraction neither rounds nor chops.\n");
}
else
printf ("Addition/Subtraction neither rounds nor chops.\n");
S = One;
X = One + Half * (One + Half);
Y = (One + U2) * Half;
Z = X - Y;
T = Y - X;
StickyBit = Z + T;
if (StickyBit != Zero)
{
S = Zero;
BadCond (Flaw, "(X - Y) + (Y - X) is non zero!\n");
}
StickyBit = Zero;
if ((GMult == Yes) && (GDiv == Yes) && (GAddSub == Yes)
&& (RMult == Rounded) && (RDiv == Rounded)
&& (RAddSub == Rounded) && (FLOOR (RadixD2) == RadixD2))
{
printf ("Checking for sticky bit.\n");
X = (Half + U1) * U2;
Y = Half * U2;
Z = One + Y;
T = One + X;
if ((Z - One <= Zero) && (T - One >= U2))
{
Z = T + Y;
Y = Z - X;
if ((Z - T >= U2) && (Y - T == Zero))
{
X = (Half + U1) * U1;
Y = Half * U1;
Z = One - Y;
T = One - X;
if ((Z - One == Zero) && (T - F9 == Zero))
{
Z = (Half - U1) * U1;
T = F9 - Z;
Q = F9 - Y;
if ((T - F9 == Zero) && (F9 - U1 - Q == Zero))
{
Z = (One + U2) * OneAndHalf;
T = (OneAndHalf + U2) - Z + U2;
X = One + Half / Radix;
Y = One + Radix * U2;
Z = X * Y;
if (T == Zero && X + Radix * U2 - Z == Zero)
{
if (Radix != Two)
{
X = Two + U2;
Y = X / Two;
if ((Y - One == Zero))
StickyBit = S;
}
else
StickyBit = S;
}
}
}
}
}
}
if (StickyBit == One)
printf ("Sticky bit apparently used correctly.\n");
else
printf ("Sticky bit used incorrectly or not at all.\n");
TstCond (Flaw, !(GMult == No || GDiv == No || GAddSub == No ||
RMult == Other || RDiv == Other || RAddSub == Other),
"lack(s) of guard digits or failure(s) to correctly round or chop\n\
(noted above) count as one flaw in the final tally below");
Milestone = 60;
printf ("\n");
printf ("Does Multiplication commute? ");
printf ("Testing on %d random pairs.\n", NoTrials);
Random9 = SQRT (FLOAT (3));
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 %s .\n", Y.str());
}
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;
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;
CInvrse = One / C;
E0 = C;
Z = E0 * H;
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));
}
if (PseudoZero != Zero)
{
printf ("\n");
Z = PseudoZero;
if (PseudoZero <= Zero)
{
BadCond (Failure, "Positive expressions can underflow to an\n");
printf ("allegedly negative value\n");
printf ("PseudoZero that prints out as: %s .\n", PseudoZero.str());
X = -PseudoZero;
if (X <= Zero)
{
printf ("But -PseudoZero, which should be\n");
printf ("positive, isn't; it prints out as %s .\n", X.str());
}
}
else
{
BadCond (Flaw, "Underflow can stick at an allegedly positive\n");
printf ("value PseudoZero that prints out as %s .\n",
PseudoZero.str());
}
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 = %s .\n", E0.str());
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 = %s\n", UfThold.str());
printf (" coming down from %s\n", C.str());
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 = %s, Y = %s .\n", Q.str(), Y2.str());
printf ("|Q - Y| = %s .\n", FABS (Q - Y2).str());
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");
if (setjmp (ovfl_buf))
{
printf ("Underflow / UfThold failed!\n");
R = H + H;
}
else
R = SQRT (Underflow / UfThold);
if (R <= H)
{
Z = R * UfThold;
X = Z * (One + R * H * (One + H));
}
else
{
Z = UfThold;
X = Z * (One + H * H * (One + H));
}
if (!((X == Z) || (X - Z != Zero)))
{
BadCond (Flaw, "");
printf ("X = %s\n\tis not equal to Z = %s .\n", X.str(), Z.str());
Z9 = X - Z;
printf ("yet X - Z yields %s .\n", Z9.str());
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");
if (setjmp (ovfl_buf))
printf ("X / Z fails!\n");
else
printf ("X / Z = 1 + %s .\n", ((X / Z - Half) - Half).str());
}
}
printf ("The Underflow threshold is %s, below which\n", UfThold.str());
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 = (%s) ^ (%s)\nonly underflow ", HInvrse.str(), Y.str());
printf ("should afflict the expression\n\t(%s) ^ (%s);\n",
HInvrse.str(), Y2.str());
printf ("actually calculating yields:");
if (setjmp (ovfl_buf))
{
BadCond (Serious, "trap on underflow.\n");
}
else
{
V9 = POW (HInvrse, Y2);
printf (" %s .\n", V9.str());
if (!((V9 >= Zero) && (V9 <= (Radix + Radix + E9) * UfThold)))
{
BadCond (Serious, "this is not between 0 and underflow\n");
printf (" threshold = %s .\n", UfThold.str());
}
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 = %s .\n", UfThold.str());
}
}
Milestone = 160;
Pause ();
printf ("Searching for Overflow threshold:\n");
printf ("This may generate an error.\n");
Y = -CInvrse;
V9 = HInvrse * Y;
if (setjmp (ovfl_buf))
{
I = 0;
V9 = Y;
goto overflow;
}
do
{
V = Y;
Y = V9;
V9 = HInvrse * Y;
}
while (V9 < Y);
I = 1;
overflow:
Z = V9;
printf ("Can `Z = -Y' overflow?\n");
printf ("Trying it on Y = %s .\n", Y.str());
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 %s\n\tshrinks to %s .\n", Y.str(), Z.str());
}
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 = %s .\n", V.str());
if (I)
printf ("Overflow saturates at V0 = %s .\n", V0.str());
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 = %s\n", V9.str());
V9 = V / One;
printf (" nor for V / 1 = %s.\n", V9.str());
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 ("+-%s, +-%s\nand +-%s are confused by Overflow.",
V.str(), V0.str(), UfThold.str());
}
Milestone = 175;
printf ("\n");
for (Indx = 1; Indx <= 3; ++Indx)
{
switch (Indx)
{
case 1:
Z = UfThold;
break;
case 2:
Z = E0;
break;
case 3:
Z = PseudoZero;
break;
}
if (Z != Zero)
{
V9 = SQRT (Z);
Y = V9 * V9;
if (Y / (One - Radix * E9) < Z || Y > (One + Radix * E9) * Z)
{
if (V9 > U1)
BadCond (Serious, "");
else
BadCond (Defect, "");
printf ("Comparison alleges that what prints as Z = %s\n",
Z.str());
printf (" is too far from sqrt(Z) ^ 2 = %s .\n", Y.str());
}
}
}
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 = %s\n", Z.str());
printf (" is too far from sqrt(Z) ^ 2 (%s) .\n", Y.str());
}
}
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 = %s\n\t%s\n",
X.str(), "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;
if (setjmp (ovfl_buf))
printf (" X / X traps when X = %s\n", X.str());
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 = %s\n", X.str());
printf (" instead, X / X - 1/2 - 1/2 = %s .\n", V9.str());
}
}
Milestone = 210;
MyZero = Zero;
printf ("\n");
printf ("What message and/or values does Division by Zero produce?\n");
printf (" Trying to compute 1 / 0 produces ...");
if (!setjmp (ovfl_buf))
printf (" %s .\n", (One / MyZero).str());
printf ("\n Trying to compute 0 / 0 produces ...");
if (!setjmp (ovfl_buf))
printf (" %s .\n", (Zero / MyZero).str());
Milestone = 220;
Pause ();
printf ("\n");
{
static const 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");
}
}
printf ("END OF TEST.\n");
return 0;
}
template<typename FLOAT>
FLOAT
Paranoia<FLOAT>::Sign (FLOAT X)
{
return X >= FLOAT (long (0)) ? 1 : -1;
}
template<typename FLOAT>
void
Paranoia<FLOAT>::Pause ()
{
if (do_pause)
{
fputs ("Press return...", stdout);
fflush (stdout);
getchar();
}
printf ("\nDiagnosis resumes after milestone Number %d", Milestone);
printf (" Page: %d\n\n", PageNo);
++Milestone;
++PageNo;
}
template<typename FLOAT>
void
Paranoia<FLOAT>::TstCond (int K, int Valid, const char *T)
{
if (!Valid)
{
BadCond (K, T);
printf (".\n");
}
}
template<typename FLOAT>
void
Paranoia<FLOAT>::BadCond (int K, const char *T)
{
static const char *msg[] = { "FAILURE", "SERIOUS DEFECT", "DEFECT", "FLAW" };
ErrCnt[K] = ErrCnt[K] + 1;
printf ("%s: %s", msg[K], T);
}
template<typename FLOAT>
FLOAT
Paranoia<FLOAT>::Random ()
{
FLOAT X, Y;
X = Random1 + Random9;
Y = X * X;
Y = Y * Y;
X = X * Y;
Y = X - FLOOR (X);
Random1 = Y + X * FLOAT ("0.000005");
return (Random1);
}
template<typename FLOAT>
void
Paranoia<FLOAT>::SqXMinX (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;
BadCond (ErrKind, "\n");
printf ("sqrt(%s) - %s = %s\n", (X * X).str(), X.str(),
(OneUlp * SqEr).str());
printf ("\tinstead of correct value 0 .\n");
}
}
template<typename FLOAT>
void
Paranoia<FLOAT>::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;
}
template<typename FLOAT>
void
Paranoia<FLOAT>::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;
}
}
template<typename FLOAT>
void
Paranoia<FLOAT>::IsYeqX ()
{
if (Y != X)
{
if (N <= 0)
{
if (Z == Zero && Q <= Zero)
printf ("WARNING: computing\n");
else
BadCond (Defect, "computing\n");
printf ("\t(%s) ^ (%s)\n", Z.str(), Q.str());
printf ("\tyielded %s;\n", Y.str());
printf ("\twhich compared unequal to correct %s ;\n", X.str());
printf ("\t\tthey differ by %s .\n", (Y - X).str());
}
N = N + 1;
}
}
template<typename FLOAT>
void
Paranoia<FLOAT>::PrintIfNPositive ()
{
if (N > 0)
printf ("Similar discrepancies have occurred %d times.\n", N);
}
template<typename FLOAT>
void
Paranoia<FLOAT>::TstPtUf ()
{
N = 0;
if (Z != Zero)
{
printf ("Since comparison denies Z = 0, evaluating ");
printf ("(Z + Z) / Z should be safe.\n");
if (setjmp (ovfl_buf))
goto very_serious;
Q9 = (Z + Z) / Z;
printf ("What the machine gets for (Z + Z) / Z is %s .\n", Q9.str());
if (FABS (Q9 - Two) < Radix * U2)
{
printf ("This is O.K., provided Over/Underflow");
printf (" has NOT just been signaled.\n");
}
else
{
if ((Q9 < One) || (Q9 > Two))
{
very_serious:
N = 1;
ErrCnt[Serious] = ErrCnt[Serious] + 1;
printf ("This is a VERY SERIOUS DEFECT!\n");
}
else
{
N = 1;
ErrCnt[Defect] = ErrCnt[Defect] + 1;
printf ("This is a DEFECT!\n");
}
}
V9 = Z * One;
Random1 = V9;
V9 = One * Z;
Random2 = V9;
V9 = Z / One;
if ((Z == Random1) && (Z == Random2) && (Z == V9))
{
if (N > 0)
Pause ();
}
else
{
N = 1;
BadCond (Defect, "What prints as Z = ");
printf ("%s\n\tcompares different from ", Z.str());
if (Z != Random1)
printf ("Z * 1 = %s ", Random1.str());
if (!((Z == Random2) || (Random2 == Random1)))
printf ("1 * Z == %s\n", Random2.str());
if (!(Z == V9))
printf ("Z / 1 = %s\n", V9.str());
if (Random2 != Random1)
{
ErrCnt[Defect] = ErrCnt[Defect] + 1;
BadCond (Defect, "Multiplication does not commute!\n");
printf ("\tComparison alleges that 1 * Z = %s\n", Random2.str());
printf ("\tdiffers from Z * 1 = %s\n", Random1.str());
}
Pause ();
}
}
}
template<typename FLOAT>
void
Paranoia<FLOAT>::notify (const char *s)
{
printf ("%s test appears to be inconsistent...\n", s);
printf (" PLEASE NOTIFY KARPINKSI!\n");
}
int main(int ac, char **av)
{
setbuf(stdout, NULL);
setbuf(stderr, NULL);
while (1)
switch (getopt (ac, av, "pvg:fdl"))
{
case -1:
return 0;
case 'p':
do_pause = true;
break;
case 'v':
verbose = true;
break;
case 'g':
{
static const struct {
const char *name;
const struct real_format *fmt;
} fmts[] = {
#define F(x) { #x, &x##_format }
F(ieee_single),
F(ieee_double),
F(ieee_extended_motorola),
F(ieee_extended_intel_96),
F(ieee_extended_intel_128),
F(ibm_extended),
F(ieee_quad),
F(vax_f),
F(vax_d),
F(vax_g),
F(i370_single),
F(i370_double),
F(c4x_single),
F(c4x_extended),
F(real_internal),
#undef F
};
int i, n = sizeof (fmts)/sizeof(*fmts);
for (i = 0; i < n; ++i)
if (strcmp (fmts[i].name, optarg) == 0)
break;
if (i == n)
{
printf ("Unknown implementation \"%s\"; "
"available implementations:\n", optarg);
for (i = 0; i < n; ++i)
printf ("\t%s\n", fmts[i].name);
return 1;
}
real_format_for_mode[int(real_c_float::MODE) - int(QFmode)]
= fmts[i].fmt;
Paranoia<real_c_float>().main();
break;
}
case 'f':
Paranoia < native_float<float> >().main();
break;
case 'd':
Paranoia < native_float<double> >().main();
break;
case 'l':
#ifndef NO_LONG_DOUBLE
Paranoia < native_float<long double> >().main();
#endif
break;
case '?':
puts ("-p\tpause between pages");
puts ("-g<FMT>\treal.c implementation FMT");
puts ("-f\tnative float");
puts ("-d\tnative double");
puts ("-l\tnative long double");
return 0;
}
}
extern "C" void
fancy_abort ()
{
abort ();
}
int target_flags = 0;
extern "C" int
floor_log2_wide (unsigned HOST_WIDE_INT x)
{
int log = -1;
while (x != 0)
log++,
x >>= 1;
return log;
}