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This directory contains source for a library of binary -> decimal and decimal -> binary conversion routines, for single-, double-, and extended-precision IEEE binary floating-point arithmetic, and other IEEE-like binary floating-point, including "double double", as in T. J. Dekker, "A Floating-Point Technique for Extending the Available Precision", Numer. Math. 18 (1971), pp. 224-242 and "Inside Macintosh: PowerPC Numerics", Addison-Wesley, 1994 The conversion routines use double-precision floating-point arithmetic and, where necessary, high precision integer arithmetic. The routines are generalizations of the strtod and dtoa routines described in David M. Gay, "Correctly Rounded Binary-Decimal and Decimal-Binary Conversions", Numerical Analysis Manuscript No. 90-10, Bell Labs, Murray Hill, 1990; http://cm.bell-labs.com/cm/cs/what/ampl/REFS/rounding.ps.gz (based in part on papers by Clinger and Steele & White: see the references in the above paper). The present conversion routines should be able to use any of IEEE binary, VAX, or IBM-mainframe double-precision arithmetic internally, but I (dmg) have so far only had a chance to test them with IEEE double precision arithmetic. The core conversion routines are strtodg for decimal -> binary conversions and gdtoa for binary -> decimal conversions. These routines operate on arrays of unsigned 32-bit integers of type ULong, a signed 32-bit exponent of type Long, and arithmetic characteristics described in struct FPI; FPI, Long, and ULong are defined in gdtoa.h. File arith.h is supposed to provide #defines that cause gdtoa.h to define its types correctly. File arithchk.c is source for a program that generates a suitable arith.h on all systems where I've been able to test it. The core conversion routines are meant to be called by helper routines that know details of the particular binary arithmetic of interest and convert. The present directory provides helper routines for 5 variants of IEEE binary floating-point arithmetic, each indicated by one or two letters: f IEEE single precision d IEEE double precision x IEEE extended precision, as on Intel 80x87 and software emulations of Motorola 68xxx chips that do not pad the way the 68xxx does, but only store 80 bits xL IEEE extended precision, as on Motorola 68xxx chips Q quad precision, as on Sun Sparc chips dd double double, pairs of IEEE double numbers whose sum is the desired value For decimal -> binary conversions, there are three families of helper routines: one for round-nearest: strtof strtod strtodd strtopd strtopf strtopx strtopxL strtopQ one with rounding direction specified: strtorf strtord strtordd strtorx strtorxL strtorQ and one for computing an interval (at most one bit wide) that contains the decimal number: strtoIf strtoId strtoIdd strtoIx strtoIxL strtoIQ The latter call strtoIg, which makes one call on strtodg and adjusts the result to provide the desired interval. On systems where native arithmetic can easily make one-ulp adjustments on values in the desired floating-point format, it might be more efficient to use the native arithmetic. Routine strtodI is a variant of strtoId that illustrates one way to do this for IEEE binary double-precision arithmetic -- but whether this is more efficient remains to be seen. Functions strtod and strtof have "natural" return types, float and double -- strtod is specified by the C standard, and strtof appears in the stdlib.h of some systems, such as (at least some) Linux systems. The other functions write their results to their final argument(s): to the final two argument for the strtoI... (interval) functions, and to the final argument for the others (strtop... and strtor...). Where possible, these arguments have "natural" return types (double* or float*), to permit at least some type checking. In reality, they are viewed as arrays of ULong (or, for the "x" functions, UShort) values. On systems where long double is the appropriate type, one can pass long double* final argument(s) to these routines. The int value that these routines return is the return value from the call they make on strtodg; see the enum of possible return values in gdtoa.h. Source files g_ddfmt.c, misc.c, smisc.c, strtod.c, strtodg.c, and ulp.c should use true IEEE double arithmetic (not, e.g., double extended), at least for storing (and viewing the bits of) the variables declared "double" within them. One detail indicated in struct FPI is whether the target binary arithmetic departs from the IEEE standard by flushing denormalized numbers to 0. On systems that do this, the helper routines for conversion to double-double format (when compiled with Sudden_Underflow #defined) penalize the bottom of the exponent range so that they return a nonzero result only when the least significant bit of the less significant member of the pair of double values returned can be expressed as a normalized double value. An alternative would be to drop to 53-bit precision near the bottom of the exponent range. To get correct rounding, this would (in general) require two calls on strtodg (one specifying 126-bit arithmetic, then, if necessary, one specifying 53-bit arithmetic). By default, the core routine strtodg and strtod set errno to ERANGE if the result overflows to +Infinity or underflows to 0. Compile these routines with NO_ERRNO #defined to inhibit errno assignments. Routine strtod is based on netlib's "dtoa.c from fp", and (f = strtod(s,se)) is more efficient for some conversions than, say, strtord(s,se,1,&f). Parts of strtod require true IEEE double arithmetic with the default rounding mode (round-to-nearest) and, on systems with IEEE extended-precision registers, double-precision (53-bit) rounding precision. If the machine uses (the equivalent of) Intel 80x87 arithmetic, the call _control87(PC_53, MCW_PC); does this with many compilers. Whether this or another call is appropriate depends on the compiler; for this to work, it may be necessary to #include "float.h" or another system-dependent header file. Source file strtodnrp.c gives a strtod that does not require 53-bit rounding precision on systems (such as Intel IA32 systems) that may suffer double rounding due to use of extended-precision registers. For some conversions this variant of strtod is less efficient than the one in strtod.c when the latter is run with 53-bit rounding precision. The values that the strto* routines return for NaNs are determined by gd_qnan.h, which the makefile generates by running the program whose source is qnan.c. Note that the rules for distinguishing signaling from quiet NaNs are system-dependent. For cross-compilation, you need to determine arith.h and gd_qnan.h suitably, e.g., using the arithmetic of the target machine. C99's hexadecimal floating-point constants are recognized by the strto* routines (but this feature has not yet been heavily tested). Compiling with NO_HEX_FP #defined disables this feature. When compiled with -DINFNAN_CHECK, the strto* routines recognize C99's NaN and Infinity syntax. Moreover, unless No_Hex_NaN is #defined, the strto* routines also recognize C99's NaN(...) syntax: they accept (case insensitively) strings of the form NaN(x), where x is a string of hexadecimal digits and spaces; if there is only one string of hexadecimal digits, it is taken for the fraction bits of the resulting NaN; if there are two or more strings of hexadecimal digits, each string is assigned to the next available sequence of 32-bit words of fractions bits (starting with the most significant), right-aligned in each sequence. For binary -> decimal conversions, I've provided just one family of helper routines: g_ffmt g_dfmt g_ddfmt g_xfmt g_xLfmt g_Qfmt which do a "%g" style conversion either to a specified number of decimal places (if their ndig argument is positive), or to the shortest decimal string that rounds to the given binary floating-point value (if ndig <= 0). They write into a buffer supplied as an argument and return either a pointer to the end of the string (a null character) in the buffer, if the buffer was long enough, or 0. Other forms of conversion are easily done with the help of gdtoa(), such as %e or %f style and conversions with direction of rounding specified (so that, if desired, the decimal value is either >= or <= the binary value). For an example of more general conversions based on dtoa(), see netlib's "printf.c from ampl/solvers". For double-double -> decimal, g_ddfmt() assumes IEEE-like arithmetic of precision max(126, #bits(input)) bits, where #bits(input) is the number of mantissa bits needed to represent the sum of the two double values in the input. The makefile creates a library, gdtoa.a. To use the helper routines, a program only needs to include gdtoa.h. All the source files for gdtoa.a include a more extensive gdtoaimp.h; among other things, gdtoaimp.h has #defines that make "internal" names end in _D2A. To make a "system" library, one could modify these #defines to make the names start with __. Various comments about possible #defines appear in gdtoaimp.h, but for most purposes, arith.h should set suitable #defines. Systems with preemptive scheduling of multiple threads require some manual intervention. On such systems, it's necessary to compile dmisc.c, dtoa.c gdota.c, and misc.c with MULTIPLE_THREADS #defined, and to provide (or suitably #define) two locks, acquired by ACQUIRE_DTOA_LOCK(n) and freed by FREE_DTOA_LOCK(n) for n = 0 or 1. (The second lock, accessed in pow5mult, ensures lazy evaluation of only one copy of high powers of 5; omitting this lock would introduce a small probability of wasting memory, but would otherwise be harmless.) Routines that call dtoa or gdtoa directly must also invoke freedtoa(s) to free the value s returned by dtoa or gdtoa. It's OK to do so whether or not MULTIPLE_THREADS is #defined, and the helper g_*fmt routines listed above all do this indirectly (in gfmt_D2A(), which they all call). By default, there is a private pool of memory of length 2000 bytes for intermediate quantities, and MALLOC (see gdtoaimp.h) is called only if the private pool does not suffice. 2000 is large enough that MALLOC is called only under very unusual circumstances (decimal -> binary conversion of very long strings) for conversions to and from double precision. For systems with preemptively scheduled multiple threads or for conversions to extended or quad, it may be appropriate to #define PRIVATE_MEM nnnn, where nnnn is a suitable value > 2000. For extended and quad precisions, -DPRIVATE_MEM=20000 is probably plenty even for many digits at the ends of the exponent range. Use of the private pool avoids some overhead. Directory test provides some test routines. See its README. I've also tested this stuff (except double double conversions) with Vern Paxson's testbase program: see V. Paxson and W. Kahan, "A Program for Testing IEEE Binary-Decimal Conversion", manuscript, May 1991, ftp://ftp.ee.lbl.gov/testbase-report.ps.Z . (The same ftp directory has source for testbase.) Some system-dependent additions to CFLAGS in the makefile: HU-UX: -Aa -Ae OSF (DEC Unix): -ieee_with_no_inexact SunOS 4.1x: -DKR_headers -DBad_float_h If you want to put this stuff into a shared library and your operating system requires export lists for shared libraries, the following would be an appropriate export list: dtoa freedtoa g_Qfmt g_ddfmt g_dfmt g_ffmt g_xLfmt g_xfmt gdtoa strtoIQ strtoId strtoIdd strtoIf strtoIx strtoIxL strtod strtodI strtodg strtof strtopQ strtopd strtopdd strtopf strtopx strtopxL strtorQ strtord strtordd strtorf strtorx strtorxL When time permits, I (dmg) hope to write in more detail about the present conversion routines; for now, this README file must suffice. Meanwhile, if you wish to write helper functions for other kinds of IEEE-like arithmetic, some explanation of struct FPI and the bits array may be helpful. Both gdtoa and strtodg operate on a bits array described by FPI *fpi. The bits array is of type ULong, a 32-bit unsigned integer type. Floating-point numbers have fpi->nbits bits, with the least significant 32 bits in bits, the next 32 bits in bits, etc. These numbers are regarded as integers multiplied by 2^e (i.e., 2 to the power of the exponent e), where e is the second argument (be) to gdtoa and is stored in *exp by strtodg. The minimum and maximum exponent values fpi->emin and fpi->emax for normalized floating-point numbers reflect this arrangement. For example, the P754 standard for binary IEEE arithmetic specifies doubles as having 53 bits, with normalized values of the form 1.xxxxx... times 2^(b-1023), with 52 bits (the x's) and the biased exponent b represented explicitly; b is an unsigned integer in the range 1 <= b <= 2046 for normalized finite doubles, b = 0 for denormals, and b = 2047 for Infinities and NaNs. To turn an IEEE double into the representation used by strtodg and gdtoa, we multiply 1.xxxx... by 2^52 (to make it an integer) and reduce the exponent e = (b-1023) by 52: fpi->emin = 1 - 1023 - 52 fpi->emax = 1046 - 1023 - 52 In various wrappers for IEEE double, we actually write -53 + 1 rather than -52, to emphasize that there are 53 bits including one implicit bit. Field fpi->rounding indicates the desired rounding direction, with possible values FPI_Round_zero = toward 0, FPI_Round_near = unbiased rounding -- the IEEE default, FPI_Round_up = toward +Infinity, and FPI_Round_down = toward -Infinity given in gdtoa.h. Field fpi->sudden_underflow indicates whether strtodg should return denormals or flush them to zero. Normal floating-point numbers have bit fpi->nbits in the bits array on. Denormals have it off, with exponent = fpi->emin. Strtodg provides distinct return values for normals and denormals; see gdtoa.h. Compiling g__fmt.c, strtod.c, and strtodg.c with -DUSE_LOCALE causes the decimal-point character to be taken from the current locale; otherwise it is '.'. Please send comments to David M. Gay (dmg at acm dot org, with " at " changed at "@" and " dot " changed to ".").