Merged kennykb-numerics-branch back to the head; TIPs 132 and 232

1 files changed, 1361 insertions, 0 deletions

diff --git a/generic/tclStrToD.c b/generic/tclStrToD.c
new file mode 100755
index 0000000..2d622a7
--- /dev/null
+++ b/generic/tclStrToD.c
@@ -0,0 +1,1361 @@
+/*
+ *----------------------------------------------------------------------
+ *
+ * tclStrToD.c --
+ *
+ *	This file contains a TclStrToD procedure that handles conversion
+ *	of string to double, with correct rounding even where extended
+ *	precision is needed to achieve that.  It also contains a
+ *	TclDoubleDigits procedure that handles conversion of double
+ *	to string (at least the significand), and several utility functions
+ *	for interconverting 'double' and the integer types.
+ *
+ * Copyright (c) 2005 by Kevin B. Kenny.  All rights reserved.
+ *
+ * See the file "license.terms" for information on usage and redistribution
+ * of this file, and for a DISCLAIMER OF ALL WARRANTIES.
+ *
+ * RCS: @(#) $Id: tclStrToD.c,v 1.2 2005/05/10 18:34:49 kennykb Exp $
+ *
+ *----------------------------------------------------------------------
+ */
+
+#include <tclInt.h>
+#include <stdio.h>
+#include <stdlib.h>
+#include <float.h>
+#include <limits.h>
+#include <math.h>
+#include <ctype.h>
+#include <tommath.h>
+
+/*
+ * The stuff below is a bit of a hack so that this file can be used in
+ * environments that include no UNIX, i.e. no errno: just arrange to use
+ * the errno from tclExecute.c here.
+ */
+
+#ifdef TCL_GENERIC_ONLY
+#define NO_ERRNO_H
+#endif
+
+#ifdef NO_ERRNO_H
+extern int errno;			/* Use errno from tclExecute.c. */
+#define ERANGE 34
+#endif
+
+#if ( FLT_RADIX == 2 ) && ( DBL_MANT_DIG == 53 ) && ( DBL_MAX_EXP == 1024 )
+#define IEEE_FLOATING_POINT
+#endif
+
+/*
+ * gcc on x86 needs access to rounding controls.  It is tempting to
+ * include fpu_control.h, but that file exists only on Linux; it is
+ * missing on Cygwin and MinGW.
+ */
+
+#if defined(__GNUC__) && defined(__i386)
+typedef unsigned int fpu_control_t __attribute__ ((__mode__ (__HI__)));
+#define _FPU_GETCW(cw) __asm__ ("fnstcw %0" : "=m" (*&cw))
+#define _FPU_SETCW(cw) __asm__ ("fldcw %0" : : "m" (*&cw))
+#endif
+
+/* The powers of ten that can be represented exactly as IEEE754 doubles. */
+
+#define MAXPOW 22
+static double pow10 [MAXPOW+1];
+
+static int mmaxpow;		/* Largest power of ten that can be
+				 * represented exactly in a 'double'. */
+
+/* Inexact higher powers of ten */
+
+static CONST double pow_10_2_n [] = {
+    1.0,
+    100.0,
+    10000.0,
+    1.0e+8,
+    1.0e+16,
+    1.0e+32,
+    1.0e+64,
+    1.0e+128,
+    1.0e+256
+};
+
+/* Logarithm of the floating point radix. */
+
+static int log2FLT_RADIX;
+
+/* Number of bits in a double's significand */
+
+static int mantBits;
+
+/* Table of powers of 5**(2**n), up to 5**256 */
+
+static mp_int pow5[9];
+
+/* The smallest representable double */
+
+static double tiny;
+
+/* The maximum number of digits to the left of the decimal point of a
+ * double. */
+
+static int maxDigits;
+
+/* The maximum number of digits to the right of the decimal point in a
+ * double. */
+
+static int minDigits;
+
+/* Number of mp_digit's needed to hold the significand of a double */
+
+static int mantDIGIT;
+
+/* Static functions defined in this file */
+
+static double RefineResult _ANSI_ARGS_((double approx, CONST char* start,
+					int nDigits, long exponent));
+static double ParseNaN _ANSI_ARGS_(( int signum, CONST char** end ));
+static double SafeLdExp _ANSI_ARGS_(( double fraction, int exponent ));
+
+/*
+ *----------------------------------------------------------------------
+ *
+ * TclStrToD --
+ *
+ *	Scans a double from a string.
+ *
+ * Results:
+ *	Returns the scanned number. In the case of underflow, returns
+ *	an appropriately signed zero; in the case of overflow, returns
+ *	an appropriately signed HUGE_VAL.
+ *
+ * Side effects:
+ *	Stores a pointer to the end of the scanned number in '*endPtr',
+ *	if endPtr is not NULL. If '*endPtr' is equal to 's' on return from
+ *	this function, it indicates that the input string could not be
+ *	recognized as a number.
+ *	In the case of underflow or overflow, 'errno' is set to ERANGE.
+ *
+ *------------------------------------------------------------------------
+ */
+
+double
+TclStrToD( CONST char* s, 
+				/* String to scan */
+	   CONST char ** endPtr )
+				/* Pointer to the end of the scanned number */
+{
+
+    CONST char* p = s;
+    CONST char* startOfSignificand = NULL;
+				/* Start of the significand in the
+				 * string */
+    int signum = 0;		/* Sign of the significand */
+    double exactSignificand = 0.0;
+				/* Significand, represented exactly
+				 * as a floating-point number */
+    int seenDigit = 0;		/* Flag == 1 if a digit has been seen */
+    int nSigDigs = 0;		/* Number of significant digits presented */
+    int nDigitsAfterDp = 0;	/* Number of digits after the decimal point */
+    int nTrailZero = 0;		/* Number of trailing zeros in the 
+				 * significand */
+    long exponent = 0;		/* Exponent */
+    int seenDp = 0;		/* Flag == 1 if decimal point has been seen */
+
+    char c;			/* One character extracted from the input */
+
+    /* 
+     * v must be 'volatile double' on gc-ix86 to force correct rounding
+     * to IEEE double and not Intel double-extended.
+     */
+
+    volatile double v;		/* Scanned value */
+    int machexp;		/* Exponent of the machine rep of the
+				 * scanned value */
+    int expt2;			/* Exponent for computing first
+				 * approximation to the true value */
+    int i, j;
+
+    /*
+     * With gcc on x86, the floating point rounding mode is double-extended.
+     * This causes the result of double-precision calculations to be rounded
+     * twice: once to the precision of double-extended and then again to the
+     * precision of double.  Double-rounding introduces gratuitous errors of
+     * 1 ulp, so we need to change rounding mode to 53-bits.
+     */
+
+#if defined(__GNUC__) && defined(__i386)
+    fpu_control_t roundTo53Bits = 0x027f;
+    fpu_control_t oldRoundingMode;
+    _FPU_GETCW( oldRoundingMode );
+    _FPU_SETCW( roundTo53Bits );
+#endif
+
+    /* Discard leading whitespace */
+
+    while ( isspace( *p ) ) {
+	++p;
+    }
+
+    /* Determine the sign of the significand */
+
+    switch( *p ) {
+	case '-':
+	    signum = 1;
+	    /* FALLTHROUGH */
+	case '+':
+	    ++p;
+    }
+
+    /* Discard leading zeroes */
+
+    while ( *p == '0' ) {
+	seenDigit = 1;
+	++p;
+    }
+
+    /* 
+     * Scan digits from the significand. Simultaneously, keep track
+     * of the number of digits after the decimal point. Maintain
+     * a pointer to the start of the significand. Keep "exactSignificand"
+     * equal to the conversion of the DBL_DIG most significant digits.
+     */
+
+    for ( ; ; ) {
+	c = *p;
+	if ( c == '.' && !seenDp ) {
+	    seenDp = 1;
+	    ++p;
+	} else if ( isdigit(c) ) {
+	    if ( c == '0' ) {
+		if ( startOfSignificand != NULL ) {
+		    ++nTrailZero;
+		}
+	    } else {
+		if ( startOfSignificand == NULL ) {
+		    startOfSignificand = p;
+		} else if ( nTrailZero ) {
+		    if ( nTrailZero + nSigDigs < DBL_DIG ) {
+			exactSignificand *= pow10[ nTrailZero ];
+		    } else if ( nSigDigs < DBL_DIG ) {
+			exactSignificand *= pow10[ DBL_DIG - nSigDigs ];
+		    }
+		    nSigDigs += nTrailZero;
+		}
+		if ( nSigDigs < DBL_DIG ) {
+		    exactSignificand = 10. * exactSignificand + (c - '0');
+		}
+		++nSigDigs;
+		nTrailZero = 0;
+	    }
+	    if ( seenDp ) {
+		++nDigitsAfterDp;
+	    }
+	    seenDigit = 1;
+	    ++p;
+	} else {
+	    break;
+	}
+    }
+
+    /*
+     * At this point, we've scanned the significand, and p points
+     * to the character beyond it.  "startOfSignificand" is the first
+     * non-zero character in the significand. "nSigDigs" is the number
+     * of significant digits of the significand, not including any
+     * trailing zeroes. "exactSignificand" is a floating point number
+     * that represents, without loss of precision, the first
+     * min(DBL_DIG,n) digits of the significand.  "nDigitsAfterDp"
+     * is the number of digits after the decimal point, again excluding
+     * trailing zeroes.
+     *
+     * Now scan 'E' format
+     */
+
+    exponent = 0;
+    if ( seenDigit && ( *p == 'e' || *p == 'E' ) ) {
+	CONST char* stringSave = p;
+	++p;
+	c = *p;
+	if ( isdigit( c ) || c == '+' || c == '-' ) {
+	    errno = 0;
+	    exponent = strtol( p, (char**)&p, 10 );
+	    if ( errno == ERANGE ) {
+		if ( exponent > 0 ) {
+		    v = HUGE_VAL;
+		} else {
+		    v = 0.0;
+		}
+		*endPtr = p;
+		goto returnValue;
+	    }
+	}
+	if ( p == stringSave + 1 ) {
+	    p = stringSave;
+	    exponent = 0;
+	}
+    }
+    exponent = exponent + nTrailZero - nDigitsAfterDp;
+
+    /*
+     * If we come here with no significant digits, we might still be
+     * looking at Inf or NaN.  Go parse them.
+     */
+
+    if ( !seenDigit ) {
+
+	/* Test for Inf */
+
+	if ( c == 'I' || c == 'i' ) {
+
+	    if ( ( p[1] == 'N' || p[1] == 'n' )
+		 && ( p[2] == 'F' || p[2] == 'f' ) ) {
+		p += 3;
+		if ( ( p[0] == 'I' || p[0] == 'i' )
+		     && ( p[1] == 'N' || p[1] == 'n' )
+		     && ( p[2] == 'I' || p[2] == 'i' )
+		     && ( p[3] == 'T' || p[3] == 't' )
+		     && ( p[4] == 'Y' || p[1] == 'y' ) ) {
+		    p += 5;
+		}
+		errno = ERANGE;
+		v = HUGE_VAL;
+		if ( endPtr != NULL ) {
+		    *endPtr = p;
+		}
+		goto returnValue;
+	    }
+
+
+#ifdef IEEE_FLOATING_POINT
+
+	    /* IEEE floating point supports NaN */
+
+	} else if ( (c == 'N' || c == 'n' )
+		    && ( sizeof(Tcl_WideUInt) == sizeof( double ) ) ) {
+	    
+	    if ( ( p[1] == 'A' || p[1] == 'a' )
+		 && ( p[2] == 'N' || p[2] == 'n' ) ) {
+		p += 3;
+		
+	        if ( endPtr != NULL ) {
+		    *endPtr = p;
+		}
+		
+		/* Restore FPU mode word */
+
+#if defined(__GNUC__) && defined(__i386)
+		_FPU_SETCW( oldRoundingMode );
+#endif
+		return ParseNaN( signum, endPtr );
+
+	    }
+#endif
+
+	}
+
+	goto error;
+    }
+
+    /*
+     * We've successfully scanned; update the end-of-element pointer.
+     */
+
+    if ( endPtr != NULL ) {
+	*endPtr = p;
+    }
+
+    /* Test for zero. */
+
+    if ( nSigDigs == 0 ) {
+	v = 0.0;
+	goto returnValue;
+    }
+
+    /*
+     * The easy cases are where we have an exact significand and
+     * the exponent is small enough that we can compute the value
+     * with only one roundoff.  In addition to the cases where we
+     * can multiply or divide an exact-integer significand by an
+     * exact-integer power of 10, there is also David Gay's case
+     * where we can scale the significand by a power of 10 (still
+     * keeping it exact) and then multiply by an exact power of 10.
+     * The last case enables combinations like 83e25 that would
+     * otherwise require high precision arithmetic.
+     */
+
+    if ( nSigDigs <= DBL_DIG ) {
+	if ( exponent >= 0 ) {
+	    if ( exponent <= mmaxpow ) {
+		v = exactSignificand * pow10[ exponent ];
+		goto returnValue;
+	    } else {
+		int diff = DBL_DIG - nSigDigs;
+		if ( exponent - diff <= mmaxpow ) {
+		    volatile double factor = exactSignificand * pow10[ diff ];
+		    v = factor * pow10[ exponent - diff ];
+		    goto returnValue;
+		}
+	    }
+	} else {
+	    if ( exponent >= -mmaxpow ) {
+		v = exactSignificand / pow10[ -exponent ];
+		goto returnValue;
+	    }
+	}
+    }
+
+    /* 
+     * We don't have one of the easy cases, so we can't compute the
+     * scanned number exactly, and have to do it in multiple precision.
+     * Begin by testing for obvious overflows and underflows.
+     */
+
+    if ( nSigDigs + exponent - 1 > maxDigits ) {
+	v = HUGE_VAL;
+	errno = ERANGE;
+	goto returnValue;
+    }
+    if ( nSigDigs + exponent - 1 < minDigits ) {
+	errno = ERANGE;
+	v = 0.;
+	goto returnValue;
+    }
+
+    /*
+     * Nothing exceeds the boundaries of the tables, at least.
+     * Compute an approximate value for the number, with
+     * no possibility of overflow because we manage the exponent
+     * separately.
+     */
+
+    if ( nSigDigs > DBL_DIG ) {
+	expt2 = exponent + nSigDigs - DBL_DIG;
+    } else {
+	expt2 = exponent;
+    }
+    v = frexp( exactSignificand, &machexp );
+    if ( expt2 > 0 ) {
+	v = frexp( v * pow10[ expt2 & 0xf ], &j );
+	machexp += j;
+	for ( i = 4; i < 9; ++i ) {
+	    if ( expt2 & ( 1 << i ) ) {
+		v = frexp( v * pow_10_2_n[ i ], &j );
+		machexp += j;
+	    }
+	}
+    } else {
+	v = frexp( v / pow10[ (-expt2) & 0xf ], &j );
+	machexp += j;
+	for ( i = 4; i < 9; ++i ) {
+	    if ( (-expt2) & ( 1 << i ) ) {
+		v = frexp( v / pow_10_2_n[ i ], &j );
+		machexp += j;
+	    }
+	}
+    }
+
+    /*
+     * A first approximation is that the result will be v * 2 ** machexp.
+     * v is greater than or equal to 0.5 and less than 1.
+     * If machexp > DBL_MAX_EXP * log2(FLT_RADIX), there is an overflow.
+     * Constrain the result to the smallest representible number to avoid
+     * premature underflow.
+     */
+
+    if ( machexp > DBL_MAX_EXP * log2FLT_RADIX ) {
+	v = HUGE_VAL;
+	errno = ERANGE;
+	goto returnValue;
+    }
+
+    v = SafeLdExp( v, machexp );
+    if ( v < tiny ) {
+	v = tiny;
+    }
+
+    /* We have a first approximation in v. Now we need to refine it. */
+
+    v = RefineResult( v, startOfSignificand, nSigDigs, exponent );
+
+    /* In a very few cases, a second iteration is needed. e.g., 457e-102 */
+    
+    v = RefineResult( v, startOfSignificand, nSigDigs, exponent );
+
+    /* Handle underflow */
+
+  returnValue:
+    if ( nSigDigs != 0 && v == 0.0 ) {
+	errno = ERANGE;
+    }
+
+    /* Return a number with correct sign */
+
+    if ( signum ) {
+	v = -v;
+    }
+
+    /* Restore FPU mode word */
+    
+#if defined(__GNUC__) && defined(__i386)
+    _FPU_SETCW( oldRoundingMode );
+#endif
+
+    return v;	
+    
+    /* Come here on an invalid input */
+
+  error:
+    if ( endPtr != NULL ) {
+	*endPtr = s;
+    }
+
+    /* Restore FPU mode word */
+    
+#if defined(__GNUC__) && defined(__i386)
+    _FPU_SETCW( oldRoundingMode );
+#endif
+    return 0.0;
+
+}
+
+/*
+ *----------------------------------------------------------------------
+ *
+ * RefineResult --
+ *
+ *	Given a poor approximation to a floating point number, returns
+ *	a better one (The better approximation is correct to within
+ *	1 ulp, and is entirely correct if the poor approximation is
+ *	correct to 1 ulp.)
+ *
+ * Results:
+ *	Returns the improved result.
+ *
+ *----------------------------------------------------------------------
+ */
+
+static double
+RefineResult( double approxResult, 
+				/* Approximate result of conversion */
+	      CONST char* sigStart,
+				/* Pointer to start of significand in
+				 * input string. */
+	      int nSigDigs,	/* Number of significant digits */
+	      long exponent )	/* Power of ten to multiply by significand */
+{
+
+    int M2, M5;			/*  Powers of 2 and of 5 needed to put
+				 * the decimal and binary numbers over
+				 * a common denominator. */
+    double significand;		/* Sigificand of the binary number */
+    int binExponent;		/* Exponent of the binary number */
+
+    int msb;			/* Most significant bit position of an
+				 * intermediate result */
+    int nDigits;		/* Number of mp_digit's in an intermediate
+				 * result */
+    mp_int twoMv;		/* Approx binary value expressed as an
+				 * exact integer scaled by the multiplier 2M */
+    mp_int twoMd;		/* Exact decimal value expressed as an
+				 * exact integer scaled by the multiplier 2M */
+    int scale;			/* Scale factor for M */
+    int multiplier;		/* Power of two to scale M */
+    double num, den;		/* Numerator and denominator of the
+				 * correction term */
+    double quot;		/* Correction term */
+    double minincr;		/* Lower bound on the absolute value
+				 * of the correction term. */
+    int i;
+    CONST char* p;
+
+    /*
+     * The first approximation is always low.  If we find that
+     * it's HUGE_VAL, we're done.
+     */
+
+    if ( approxResult == HUGE_VAL ) {
+	return approxResult;
+    }
+
+    /*
+     * Find a common denominator for the decimal and binary fractions.
+     * The common denominator will be 2**M2 + 5**M5.
+     */
+
+    significand = frexp( approxResult, &binExponent );
+    i = mantBits - binExponent;
+    if ( i < 0 ) {
+	M2 = 0;
+    } else {
+	M2 = i;
+    }
+    if ( exponent > 0 ) {
+	M5 = 0;
+    } else {
+	M5 = -exponent;
+	if ( (M5-1) > M2 ) {
+	    M2 = M5-1;
+	}
+    }
+
+    /* 
+     * The floating point number is significand*2**binExponent.
+     * The 2**-1 bit of the significand (the most significant) 
+     * corresponds to the 2**(binExponent+M2 + 1) bit of 2*M2*v.
+     * Allocate enough digits to hold that quantity, then
+     * convert the significand to a large integer, scaled
+     * appropriately. Then multiply by the appropriate power of 5.
+     */
+
+    msb = binExponent + M2;  /* 1008 */
+    nDigits = msb / DIGIT_BIT + 1;
+    mp_init_size( &twoMv, nDigits );
+    i = ( msb % DIGIT_BIT + 1 ); 
+    twoMv.used = nDigits;
+    significand *= SafeLdExp( 1.0, i );
+    while ( -- nDigits >= 0 ) {
+	twoMv.dp[nDigits] = (mp_digit) significand;
+	significand -= (mp_digit) significand;
+	significand = SafeLdExp( significand, DIGIT_BIT );
+    }
+    for ( i = 0; i <= 8; ++i ) {
+	if ( M5 & ( 1 << i ) ) {
+	    mp_mul( &twoMv, pow5+i, &twoMv );
+	}
+    }
+    
+    /* 
+     * Collect the decimal significand as a high precision integer.
+     * The least significant bit corresponds to bit M2+exponent+1
+     * so it will need to be shifted left by that many bits after
+     * being multiplied by 5**(M5+exponent).
+     */
+
+    mp_init( &twoMd ); mp_zero( &twoMd );
+    i = nSigDigs;
+    for ( p = sigStart ; ; ++p ) {
+	char c = *p;
+	if ( isdigit( c ) ) {
+	    mp_mul_d( &twoMd, (unsigned) 10, &twoMd );
+	    mp_add_d( &twoMd, (unsigned) (c - '0'), &twoMd );
+	    --i;
+	    if ( i == 0 ) break;
+	}
+    }
+    for ( i = 0; i <= 8; ++i ) {
+	if ( (M5+exponent) & ( 1 << i ) ) {
+	    mp_mul( &twoMd, pow5+i, &twoMd );
+	}
+    }
+    mp_mul_2d( &twoMd, M2+exponent+1, &twoMd );
+    mp_sub( &twoMd, &twoMv, &twoMd );
+
+    /*
+     * The result, 2Mv-2Md, needs to be divided by 2M to yield a correction
+     * term. Because 2M may well overflow a double, we need to scale the
+     * denominator by a factor of 2**binExponent-mantBits
+     */
+
+    scale = binExponent - mantBits - 1;
+
+    mp_set( &twoMv, 1 );
+    for ( i = 0; i <= 8; ++i ) {
+	if ( M5 & ( 1 << i ) ) {
+	    mp_mul( &twoMv, pow5+i, &twoMv );
+	}
+    }
+    multiplier = M2 + scale + 1;
+    if ( multiplier > 0 ) {
+	mp_mul_2d( &twoMv, multiplier, &twoMv );
+    } else if ( multiplier < 0 ) {
+	mp_div_2d( &twoMv, -multiplier, &twoMv, NULL );
+    }
+
+    /*
+     * If the result is less than unity, the error is less than 1/2 unit
+     * in the last place, so there's no correction to make.
+     */
+
+    if ( mp_cmp_mag( &twoMd, &twoMv ) == MP_LT ) {
+	return approxResult;
+    }
+
+    /* 
+     * Convert the numerator and denominator of the corrector term
+     * accurately to floating point numbers.
+     */
+
+    num = TclBignumToDouble( &twoMd );
+    den = TclBignumToDouble( &twoMv );
+
+    quot = SafeLdExp( num/den, scale );
+    minincr = SafeLdExp( 1.0, binExponent - mantBits );
+
+    if ( quot < 0. && quot > -minincr ) {
+	quot = -minincr;
+    } else if ( quot > 0. && quot < minincr ) {
+	quot = minincr;
+    }
+
+    mp_clear( &twoMd );
+    mp_clear( &twoMv );
+
+    
+    return approxResult + quot;
+}
+
+/*
+ *----------------------------------------------------------------------
+ *
+ * ParseNaN --
+ *
+ *	Parses a "not a number" from an input string, and returns the
+ *	double precision NaN corresponding to it.
+ *
+ * Side effects:
+ *	Advances endPtr to follow any (hex) in the input string.
+ *
+ *	If the NaN is followed by a left paren, a string of spaes
+ *	and hexadecimal digits, and a right paren, endPtr is advanced
+ *	to follow it.
+ *
+ * The string of hexadecimal digits is OR'ed into the resulting
+ * NaN, and the signum is set as well.  Note that a signalling NaN
+ * is never returned.
+ *
+ *----------------------------------------------------------------------
+ */
+
+double
+ParseNaN( int signum,		/* Flag == 1 if minus sign has been
+				 * seen in front of NaN */
+	  CONST char** endPtr )
+				/* Pointer-to-pointer to char following "NaN" 
+				 * in the input string */
+{
+    CONST char* p = *endPtr;
+    char c;
+    union {
+	Tcl_WideUInt iv;
+	double dv;
+    } theNaN;
+
+    /* Scan off a hex number in parentheses.  Embedded blanks are ok. */
+
+    theNaN.iv = 0;
+    if ( *p == '(' ) {
+	++p;
+	for ( ; ; ) {
+	    c = *p++;
+	    if ( isspace(c) ) {
+		continue;
+	    } else if ( c == ')' ) {
+		*endPtr = p;
+		break;
+	    } else if ( isdigit(c) ) {
+		c -= '0';
+	    } else if ( c >= 'A' && c <= 'F' ) {
+		c = c - 'A' + 10;
+	    } else if ( c >= 'a' && c <= 'f' ) {
+		c = c - 'a' + 10;
+	    } else {
+		theNaN.iv = ( ((Tcl_WideUInt) 0x7ff8) << 48 )
+		    | ( ((Tcl_WideUInt) signum) << 63 );
+		return theNaN.dv;
+	    }
+	    theNaN.iv = (theNaN.iv << 4) | c;
+	}
+    }
+
+    /* 
+     * Mask the hex number down to the least significant 51 bits.
+     */
+
+    theNaN.iv &= ( ((Tcl_WideUInt) 1) << 51 ) - 1;
+    if ( signum ) {
+	theNaN.iv |= ((Tcl_WideUInt) 0xfff8) << 48;
+    } else {
+	theNaN.iv |= ((Tcl_WideUInt) 0x7ff8) << 48;
+    }
+
+    *endPtr = p;
+    return theNaN.dv;
+}
+
+/*
+ *----------------------------------------------------------------------
+ *
+ * TclDoubleDigits --
+ *
+ *	Converts a double to a string of digits.
+ *
+ * Results:
+ *	Returns the position of the character in the string
+ *	after which the decimal	point should appear.  Since
+ *	the string contains only significant digits, the
+ *	position may be less than zero or greater than the
+ *	length of the string.
+ *
+ * Side effects:
+ *	Stores the digits in the given buffer and sets 'signum'
+ *	according to the sign of the number.
+ *
+ *----------------------------------------------------------------------
+ */
+
+int
+TclDoubleDigits( char * strPtr,	/* Buffer in which to store the result,
+				 * must have at least 18 chars */
+		 double v,	/* Number to convert. Must be
+				 * finite, and not NaN */
+		 int *signum )	/* Output: 1 if the number is negative.
+				 * Should handle -0 correctly on the
+				 * IEEE architecture. */
+{
+
+    double f;			/* Significand of v */
+
+    int e;			/* Power of FLT_RADIX that satisfies
+				 * v = f * FLT_RADIX**e */
+
+
+    int low_ok;
+    int high_ok;
+
+    mp_int r;			/* Scaled significand */
+    mp_int s;			/* Divisor such that v = r / s */
+    mp_int mplus;		/* Scaled epsilon: (r + 2* mplus) ==
+				 * v(+) where v(+) is the floating point
+				 * successor of v. */
+    mp_int mminus;		/* Scaled epsilon: (r - 2*mminus ) ==
+				 * v(-) where v(-) is the floating point
+				 * predecessor of v. */
+    mp_int temp;
+
+    int rfac2 = 0;		/* Powers of 2 and 5 by which large */
+    int rfac5 = 0;		/* integers should be scaled        */
+    int sfac2 = 0;
+    int sfac5 = 0;
+    int mplusfac2 = 0;
+    int mminusfac2 = 0;
+    
+    double a;
+    char c;
+    int i, k, n;
+
+    /* 
+     * Take the absolute value of the number, and report the number's
+     * sign.  Take special steps to preserve signed zeroes in IEEE floating
+     * point. (We can't use fpclassify, because that's a C9x feature and
+     * we still have to build on C89 compilers.)
+     */
+
+#ifndef IEEE_FLOATING_POINT
+    if ( v >= 0.0 ) {
+	*signum = 0;
+    } else {
+	*signum = 1;
+	v = -v;
+    }
+#else
+    union {
+	Tcl_WideUInt iv;
+	double dv;
+    } bitwhack;
+    bitwhack.dv = v;
+    if ( bitwhack.iv & ( (Tcl_WideUInt) 1 << 63 ) ) {
+	*signum = 1;
+	bitwhack.iv &= ~( (Tcl_WideUInt) 1 << 63 );
+	v = bitwhack.dv;
+    } else {
+	*signum = 0;
+    }
+#endif
+
+    /* Handle zero specially */
+
+    if ( v == 0.0 ) {
+	*strPtr++ = '0';
+	*strPtr++ = '\0';
+	return 1;
+    }
+
+    /* 
+     * Develop f and e such that v = f * FLT_RADIX**e, with
+     * 1.0/FLT_RADIX <= f < 1.
+     */
+
+    f = frexp( v, &e );
+    n = e % log2FLT_RADIX;
+    if ( n > 0 ) {
+	n -= log2FLT_RADIX;
+	e += 1;
+    }
+    f *= ldexp( 1.0, n );
+    e = ( e - n ) / log2FLT_RADIX;
+    if ( f == 1.0 ) {
+	f = 1.0 / FLT_RADIX;
+	e += 1;
+    }
+
+    /*
+     * If the original number was denormalized, adjust e and f to be
+     * denormal as well.
+     */
+
+    if ( e < DBL_MIN_EXP ) {
+	n = mantBits + ( e - DBL_MIN_EXP ) * log2FLT_RADIX;
+	f = ldexp( f, ( e - DBL_MIN_EXP ) * log2FLT_RADIX );
+	e = DBL_MIN_EXP;
+	n = ( n + DIGIT_BIT - 1 ) / DIGIT_BIT;
+    } else {
+	n = mantDIGIT;
+    }
+
+    /*
+     * Now extract the base-2**DIGIT_BIT digits of f into a multi-precision
+     * integer r.  Preserve the invariant v = r * 2**rfac2 * FLT_RADIX**e
+     * by adjusting e.
+     */
+    
+    a = f;
+    n = mantDIGIT;
+    mp_init_size( &r, n );
+    r.used = n;
+    r.sign = MP_ZPOS;
+    i = ( mantBits % DIGIT_BIT );
+    if ( i == 0 ) {
+	i = DIGIT_BIT;
+    }
+    while ( n > 0 ) {
+	a *= ldexp( 1.0, i );
+	i = DIGIT_BIT;
+	r.dp[--n] = (mp_digit) a;
+	a -= (mp_digit) a;
+    }
+    e -= DBL_MANT_DIG;
+
+    low_ok = high_ok = ( mp_iseven( &r ) );
+
+    /* 
+     * We are going to want to develop integers r, s, mplus, and mminus
+     * such that v = r / s, v(+)-v / 2 = mplus / s; v-v(-) / 2 = mminus / s
+     * and then scale either s or r, mplus, mminus by an appropriate
+     * power of ten.
+     *
+     * We actually do this by keeping track of the powers of 2 and 5
+     * by which f is multiplied to yield v and by which 1 is multiplied
+     * to yield s, mplus, and mminus.
+     */
+
+    if ( e >= 0 ) {
+
+	int bits = e * log2FLT_RADIX;
+
+	if ( f != 1.0 / FLT_RADIX ) {
+
+	    /* Normal case, m+ and m- are both FLT_RADIX**e */
+
+	    rfac2 += bits + 1;
+	    sfac2 = 1;
+	    mplusfac2 = bits;
+	    mminusfac2 = bits;
+
+	} else {
+
+	    /* 
+	     * If f is equal to the smallest significand, then we need another
+	     * factor of FLT_RADIX in s to cope with stepping to
+	     * the next smaller exponent when going to e's predecessor.
+	     */
+
+	    rfac2 += bits + log2FLT_RADIX - 1;
+	    sfac2 = 1 + log2FLT_RADIX;
+	    mplusfac2 = bits + log2FLT_RADIX;
+	    mminusfac2 = bits;
+
+	}
+
+    } else {
+
+	/* v has digits after the binary point */
+
+	if ( e <= DBL_MIN_EXP - DBL_MANT_DIG
+	     || f != 1.0 / FLT_RADIX ) {
+
+	    /* 
+	     * Either f isn't the smallest significand or e is
+	     * the smallest exponent.  mplus and mminus will both be 1.
+	     */
+
+	    rfac2 += 1;
+	    sfac2 = 1 - e * log2FLT_RADIX;
+	    mplusfac2 = 0;
+	    mminusfac2 = 0;
+
+	} else {
+
+	    /* 
+	     * f is the smallest significand, but e is not the smallest
+	     * exponent.  We need to scale by FLT_RADIX again to cope
+	     * with the fact that v's predecessor has a smaller exponent.
+	     */
+
+	    rfac2 += 1 + log2FLT_RADIX;
+	    sfac2 = 1 + log2FLT_RADIX * ( 1 - e );
+	    mplusfac2 = FLT_RADIX;
+	    mminusfac2 = 0;
+
+	}
+    }
+
+    /* 
+     * Estimate the highest power of ten that will be
+     * needed to hold the result.
+     */
+
+    k = (int) ceil( log( v ) / log( 10. ) );
+    if ( k >= 0 ) {
+	sfac2 += k;
+	sfac5 = k;
+    } else {
+	rfac2 -= k;
+	mplusfac2 -= k;
+	mminusfac2 -= k;
+	rfac5 = -k;
+    }
+
+    /*
+     * Scale r, s, mplus, mminus by the appropriate powers of 2 and 5.
+     */
+
+    mp_init_set( &mplus, 1 );
+    for ( i = 0; i <= 8; ++i ) {
+	if ( rfac5 & ( 1 << i ) ) {
+	    mp_mul( &mplus, pow5+i, &mplus );
+	}
+    }
+    mp_mul( &r, &mplus, &r );
+    mp_mul_2d( &r, rfac2, &r );
+    mp_init_copy( &mminus, &mplus );
+    mp_mul_2d( &mplus, mplusfac2, &mplus );
+    mp_mul_2d( &mminus, mminusfac2, &mminus );
+    mp_init_set( &s, 1 );
+    for ( i = 0; i <= 8; ++i ) {
+	if ( sfac5 & ( 1 << i ) ) {
+	    mp_mul( &s, pow5+i, &s );
+	}
+    }
+    mp_mul_2d( &s, sfac2, &s );
+
+    /*
+     * It is possible for k to be off by one because we used an
+     * inexact logarithm.
+     */
+
+    mp_init( &temp );
+    mp_add( &r, &mplus, &temp );
+    i = mp_cmp_mag( &temp, &s );
+    if ( i > 0 || ( high_ok && i == 0 ) ) {
+	mp_mul_d( &s, 10, &s );
+	++k;
+    } else {
+	mp_mul_d( &temp, 10, &temp );
+	i = mp_cmp_mag( &temp, &s );
+	if ( i < 0 || ( high_ok && i == 0 ) ) {
+	    mp_mul_d( &r, 10, &r );
+	    mp_mul_d( &mplus, 10, &mplus );
+	    mp_mul_d( &mminus, 10, &mminus );
+	    --k;
+	}
+    }
+
+    /*
+     * At this point, k contains the power of ten by which we're
+     * scaling the result.  r/s is at least 1/10 and strictly less
+     * than ten, and v = r/s * 10**k.  mplus and mminus give the
+     * rounding limits.
+     */
+
+    for ( ; ; ) {
+	int tc1, tc2;
+	mp_mul_d( &r, 10, &r );
+	mp_div( &r, &s, &temp, &r ); /* temp = 10r / s; r = 10r mod s */
+	i = temp.dp[0];
+	mp_mul_d( &mplus, 10, &mplus );
+	mp_mul_d( &mminus, 10, &mminus );
+	tc1 = mp_cmp_mag( &r, &mminus );
+	if ( low_ok ) {
+	    tc1 = ( tc1 <= 0 );
+	} else {
+	    tc1 = ( tc1 < 0 );
+	}
+	mp_add( &r, &mplus, &temp );
+	tc2 = mp_cmp_mag( &temp, &s );
+	if ( high_ok ) {
+	    tc2 = ( tc2 >= 0 );
+	} else {
+	    tc2= ( tc2 > 0 );
+	}
+	if ( ! tc1 ) {
+	    if ( !tc2 ) {
+		*strPtr++ = '0' + i;
+	    } else {
+		c = (char) (i + '1');
+		break;
+	    }
+	} else {
+	    if ( !tc2 ) {
+		c = (char) (i + '0');
+	    } else {
+		mp_mul_2d( &r, 1, &r );
+		n = mp_cmp_mag( &r, &s );
+		if ( n < 0 ) {
+		    c = (char) (i + '0');
+		} else {
+		    c = (char) (i + '1');
+		}
+	    }
+	    break;
+	}
+    };
+    *strPtr++ = c;
+    *strPtr++ = '\0';
+
+    /* Free memory */
+
+    mp_clear_multi( &r, &s, &mplus, &mminus, &temp, NULL );
+    return k;
+	    
+}
+
+/*
+ *----------------------------------------------------------------------
+ *
+ * TclInitDoubleConversion --
+ *
+ *	Initializes constants that are needed for conversions to and
+ *      from 'double'
+ *
+ * Results:
+ *	None.
+ *
+ * Side effects:
+ *	The log base 2 of the floating point radix, the number of
+ *	bits in a double mantissa, and a table of the powers of five
+ *	and ten are computed and stored.
+ *
+ *----------------------------------------------------------------------
+ */
+
+void
+TclInitDoubleConversion( void )
+{
+    int i;
+    int x;
+    double d;
+    if ( frexp( (double) FLT_RADIX, &log2FLT_RADIX ) != 0.5 ) {
+	Tcl_Panic( "This code doesn't work on a decimal machine!" );
+    }
+    --log2FLT_RADIX;
+    mantBits = DBL_MANT_DIG * log2FLT_RADIX;
+    d = 1.0;
+    x = (int) (DBL_MANT_DIG * log((double) FLT_RADIX) / log( 5.0 ));
+    if ( x < MAXPOW ) {
+	mmaxpow = x;
+    } else {
+	mmaxpow = MAXPOW;
+    }
+    for ( i = 0; i <= mmaxpow; ++i ) {
+	pow10[i] = d;
+	d *= 10.0;
+    }
+    for ( i = 0; i < 9; ++i ) {
+	mp_init( pow5 + i );
+    }
+    mp_set( pow5, 5 );
+    for ( i = 0; i < 8; ++i ) {
+	mp_sqr( pow5+i, pow5+i+1 );
+    }
+    tiny = SafeLdExp( 1.0, DBL_MIN_EXP * log2FLT_RADIX - mantBits );
+    maxDigits = (int) ((DBL_MAX_EXP * log((double) FLT_RADIX)
+			+ 0.5 * log(10.))
+		       / log( 10. ));
+    minDigits = (int) floor ( ( DBL_MIN_EXP - DBL_MANT_DIG )
+			      * log( (double) FLT_RADIX ) / log( 10. ) );
+    mantDIGIT = ( mantBits + DIGIT_BIT - 1 ) / DIGIT_BIT;
+}
+
+/*
+ *----------------------------------------------------------------------
+ *
+ * TclFinalizeDoubleConversion --
+ *
+ *	Cleans up this file on exit.
+ *
+ * Results:
+ *	None
+ *
+ * Side effects:
+ *	Memory allocated by TclInitDoubleConversion is freed.
+ *
+ *----------------------------------------------------------------------
+ */
+
+void
+TclFinalizeDoubleConversion()
+{
+    int i;
+    for ( i = 0; i < 9; ++i ) {
+	mp_clear( pow5 + i );
+    }
+}
+
+/*
+ *----------------------------------------------------------------------
+ *
+ * TclBignumToDouble --
+ *
+ *	Convert an arbitrary-precision integer to a native floating 
+ *	point number.
+ *
+ * Results:
+ *	Returns the converted number.  Sets errno to ERANGE if the
+ *	number is too large to convert.
+ *
+ *----------------------------------------------------------------------
+ */
+
+double
+TclBignumToDouble( mp_int* a )
+				/* Integer to convert */
+{
+    mp_int b;
+    int bits;
+    int shift;
+    int i;
+    double r;
+
+    /* Determine how many bits we need, and extract that many from 
+     * the input. Round to nearest unit in the last place. */
+
+    bits = mp_count_bits( a );
+    if ( bits > DBL_MAX_EXP * log2FLT_RADIX ) {
+	errno = ERANGE;
+	return HUGE_VAL;
+    }
+    shift = mantBits + 1 - bits;
+    mp_init( &b );
+    if ( shift > 0 ) {
+	mp_mul_2d( a, shift, &b );
+    } else if ( shift < 0 ) {
+	mp_div_2d( a, -shift, &b, NULL );
+    } else {
+	mp_copy( a, &b );
+    }
+    mp_add_d( &b, 1, &b );
+    mp_div_2d( &b, 1, &b, NULL );
+
+    /* Accumulate the result, one mp_digit at a time */
+
+    r = 0.0;
+    for ( i = b.used-1; i >= 0; --i ) {
+	r = ldexp( r, DIGIT_BIT ) + b.dp[i];
+    }
+    mp_clear( &b );
+
+    /* Scale the result to the correct number of bits. */
+
+    r = ldexp( r, bits - mantBits );
+
+    /* Return the result with the appropriate sign. */
+
+    if ( a->sign == MP_ZPOS ) {
+	return r;
+    } else {
+	return -r;
+    }
+}		
+
+/*
+ *----------------------------------------------------------------------
+ *
+ * SafeLdExp --
+ *
+ *	Do an 'ldexp' operation, but handle denormals gracefully.
+ *
+ * Results:
+ *	Returns the appropriately scaled value.
+ *
+ * On some platforms, 'ldexp' fails when presented with a number
+ * too small to represent as a normalized double.  This routine
+ * does 'ldexp' in two steps for those numbers, to return correctly
+ * denormalized values.
+ *
+ *----------------------------------------------------------------------
+ */
+
+static double
+SafeLdExp( double fract, int expt )
+{
+    int minexpt = DBL_MIN_EXP * log2FLT_RADIX;
+    volatile double a, b, retval;
+    if ( expt < minexpt ) {
+	a = ldexp( fract, expt - mantBits - minexpt );
+	b = ldexp( 1.0, mantBits + minexpt );
+	retval = a * b;
+    } else {
+	retval = ldexp( fract, expt );
+    }
+    return retval;
+}
+
+/*
+ *----------------------------------------------------------------------
+ *
+ * TclFormatNaN --
+ *
+ *	Makes the string representation of a "Not a Number"
+ *
+ * Results:
+ *	None.
+ *
+ * Side effects:
+ *	Stores the string representation in the supplied buffer,
+ *	which must be at least TCL_DOUBLE_SPACE characters.
+ *
+ *----------------------------------------------------------------------
+ */
+
+void
+TclFormatNaN( double value,	/* The Not-a-Number to format */
+	      char* buffer )	/* String representation */
+{
+#ifndef IEEE_FLOATING_POINT
+    strcpy( buffer, "NaN" );
+    return;
+#else
+
+    union {
+	double dv;
+	Tcl_WideUInt iv;
+    } bitwhack;
+
+    bitwhack.dv = value;
+    if ( bitwhack.iv & ((Tcl_WideUInt) 1 << 63 ) ) {
+	bitwhack.iv &= ~ ((Tcl_WideUInt) 1 << 63 );
+	*buffer++ = '-';
+    }
+    *buffer++ = 'N'; *buffer++ = 'a'; *buffer++ = 'N';
+    bitwhack.iv &= (((Tcl_WideUInt) 1) << 51) - 1;
+    if ( bitwhack.iv != 0 ) {
+	sprintf( buffer, "(%" TCL_LL_MODIFIER "x)", bitwhack.iv );
+    } else {
+	*buffer = '\0';
+    }
+
+#endif
+}