1This directory contains source for a library of binary -> decimal 2and decimal -> binary conversion routines, for single-, double-, 3and extended-precision IEEE binary floating-point arithmetic, and 4other IEEE-like binary floating-point, including "double double", 5as in 6 7 T. J. Dekker, "A Floating-Point Technique for Extending the 8 Available Precision", Numer. Math. 18 (1971), pp. 224-242 9 10and 11 12 "Inside Macintosh: PowerPC Numerics", Addison-Wesley, 1994 13 14The conversion routines use double-precision floating-point arithmetic 15and, where necessary, high precision integer arithmetic. The routines 16are generalizations of the strtod and dtoa routines described in 17 18 David M. Gay, "Correctly Rounded Binary-Decimal and 19 Decimal-Binary Conversions", Numerical Analysis Manuscript 20 No. 90-10, Bell Labs, Murray Hill, 1990; 21 http://cm.bell-labs.com/cm/cs/what/ampl/REFS/rounding.ps.gz 22 23(based in part on papers by Clinger and Steele & White: see the 24references in the above paper). 25 26The present conversion routines should be able to use any of IEEE binary, 27VAX, or IBM-mainframe double-precision arithmetic internally, but I (dmg) 28have so far only had a chance to test them with IEEE double precision 29arithmetic. 30 31The core conversion routines are strtodg for decimal -> binary conversions 32and gdtoa for binary -> decimal conversions. These routines operate 33on arrays of unsigned 32-bit integers of type ULong, a signed 32-bit 34exponent of type Long, and arithmetic characteristics described in 35struct FPI; FPI, Long, and ULong are defined in gdtoa.h. File arith.h 36is supposed to provide #defines that cause gdtoa.h to define its 37types correctly. File arithchk.c is source for a program that 38generates a suitable arith.h on all systems where I've been able to 39test it. 40 41The core conversion routines are meant to be called by helper routines 42that know details of the particular binary arithmetic of interest and 43convert. The present directory provides helper routines for 5 variants 44of IEEE binary floating-point arithmetic, each indicated by one or 45two letters: 46 47 f IEEE single precision 48 d IEEE double precision 49 x IEEE extended precision, as on Intel 80x87 50 and software emulations of Motorola 68xxx chips 51 that do not pad the way the 68xxx does, but 52 only store 80 bits 53 xL IEEE extended precision, as on Motorola 68xxx chips 54 Q quad precision, as on Sun Sparc chips 55 dd double double, pairs of IEEE double numbers 56 whose sum is the desired value 57 58For decimal -> binary conversions, there are three families of 59helper routines: one for round-nearest (or the current rounding 60mode on IEEE-arithmetic systems that provide the C99 fegetround() 61function, if compiled with -DHonor_FLT_ROUNDS): 62 63 strtof 64 strtod 65 strtodd 66 strtopd 67 strtopf 68 strtopx 69 strtopxL 70 strtopQ 71 72one with rounding direction specified: 73 74 strtorf 75 strtord 76 strtordd 77 strtorx 78 strtorxL 79 strtorQ 80 81and one for computing an interval (at most one bit wide) that contains 82the decimal number: 83 84 strtoIf 85 strtoId 86 strtoIdd 87 strtoIx 88 strtoIxL 89 strtoIQ 90 91The latter call strtoIg, which makes one call on strtodg and adjusts 92the result to provide the desired interval. On systems where native 93arithmetic can easily make one-ulp adjustments on values in the 94desired floating-point format, it might be more efficient to use the 95native arithmetic. Routine strtodI is a variant of strtoId that 96illustrates one way to do this for IEEE binary double-precision 97arithmetic -- but whether this is more efficient remains to be seen. 98 99Functions strtod and strtof have "natural" return types, float and 100double -- strtod is specified by the C standard, and strtof appears 101in the stdlib.h of some systems, such as (at least some) Linux systems. 102The other functions write their results to their final argument(s): 103to the final two argument for the strtoI... (interval) functions, 104and to the final argument for the others (strtop... and strtor...). 105Where possible, these arguments have "natural" return types (double* 106or float*), to permit at least some type checking. In reality, they 107are viewed as arrays of ULong (or, for the "x" functions, UShort) 108values. On systems where long double is the appropriate type, one can 109pass long double* final argument(s) to these routines. The int value 110that these routines return is the return value from the call they make 111on strtodg; see the enum of possible return values in gdtoa.h. 112 113Source files g_ddfmt.c, misc.c, smisc.c, strtod.c, strtodg.c, and ulp.c 114should use true IEEE double arithmetic (not, e.g., double extended), 115at least for storing (and viewing the bits of) the variables declared 116"double" within them. 117 118One detail indicated in struct FPI is whether the target binary 119arithmetic departs from the IEEE standard by flushing denormalized 120numbers to 0. On systems that do this, the helper routines for 121conversion to double-double format (when compiled with 122Sudden_Underflow #defined) penalize the bottom of the exponent 123range so that they return a nonzero result only when the least 124significant bit of the less significant member of the pair of 125double values returned can be expressed as a normalized double 126value. An alternative would be to drop to 53-bit precision near 127the bottom of the exponent range. To get correct rounding, this 128would (in general) require two calls on strtodg (one specifying 129126-bit arithmetic, then, if necessary, one specifying 53-bit 130arithmetic). 131 132By default, the core routine strtodg and strtod set errno to ERANGE 133if the result overflows to +Infinity or underflows to 0. Compile 134these routines with NO_ERRNO #defined to inhibit errno assignments. 135 136Routine strtod is based on netlib's "dtoa.c from fp", and 137(f = strtod(s,se)) is more efficient for some conversions than, say, 138strtord(s,se,1,&f). Parts of strtod require true IEEE double 139arithmetic with the default rounding mode (round-to-nearest) and, on 140systems with IEEE extended-precision registers, double-precision 141(53-bit) rounding precision. If the machine uses (the equivalent of) 142Intel 80x87 arithmetic, the call 143 _control87(PC_53, MCW_PC); 144does this with many compilers. Whether this or another call is 145appropriate depends on the compiler; for this to work, it may be 146necessary to #include "float.h" or another system-dependent header 147file. 148 149Source file strtodnrp.c gives a strtod that does not require 53-bit 150rounding precision on systems (such as Intel IA32 systems) that may 151suffer double rounding due to use of extended-precision registers. 152For some conversions this variant of strtod is less efficient than the 153one in strtod.c when the latter is run with 53-bit rounding precision. 154 155When float or double are involved, the values that the strto* routines 156return for NaNs are determined by gd_qnan.h, which the makefile 157generates by running the program whose source is qnan.c. For other 158types, default NaN values are specified in g__fmt.c and may need 159adjusting. Note that the rules for distinguishing signaling from 160quiet NaNs are system-dependent. For cross-compilation, you need to 161determine arith.h and gd_qnan.h suitably, e.g., using the arithmetic 162of the target machine. 163 164C99's hexadecimal floating-point constants are recognized by the 165strto* routines (but this feature has not yet been heavily tested). 166Compiling with NO_HEX_FP #defined disables this feature. 167 168When compiled with -DINFNAN_CHECK, the strto* routines recognize C99's 169NaN and Infinity syntax. Moreover, unless No_Hex_NaN is #defined, the 170strto* routines also recognize C99's NaN(...) syntax: they accept 171(case insensitively) strings of the form NaN(x), where x is a string 172of hexadecimal digits and spaces; if there is only one string of 173hexadecimal digits, it is taken for the fraction bits of the resulting 174NaN; if there are two or more strings of hexadecimal digits, each 175string is assigned to the next available sequence of 32-bit words of 176fractions bits (starting with the most significant), right-aligned in 177each sequence. Strings of hexadecimal digits may be preceded by "0x" 178or "0X". 179 180For binary -> decimal conversions, I've provided a family of helper 181routines: 182 183 g_ffmt 184 g_dfmt 185 g_ddfmt 186 g_xfmt 187 g_xLfmt 188 g_Qfmt 189 g_ffmt_p 190 g_dfmt_p 191 g_ddfmt_p 192 g_xfmt_p 193 g_xLfmt_p 194 g_Qfmt_p 195 196which do a "%g" style conversion either to a specified number of decimal 197places (if their ndig argument is positive), or to the shortest 198decimal string that rounds to the given binary floating-point value 199(if ndig <= 0). They write into a buffer supplied as an argument 200and return either a pointer to the end of the string (a null character) 201in the buffer, if the buffer was long enough, or 0. Other forms of 202conversion are easily done with the help of gdtoa(), such as %e or %f 203style and conversions with direction of rounding specified (so that, if 204desired, the decimal value is either >= or <= the binary value). 205On IEEE-arithmetic systems that provide the C99 fegetround() function, 206if compiled with -DHonor_FLT_ROUNDS, these routines honor the current 207rounding mode. For pedants, the ...fmt_p() routines are similar to the 208...fmt() routines, but have an additional final int argument, nik, 209that for conversions of Infinity or NaN, determines whether upper, 210lower, or mixed case is used, whether (...) is added to NaN values, 211and whether the sign of a NaN is reported or suppressed: 212 213 nik = ic + 6*(nb + 3*ns), 214 215where ic with 0 <= ic < 6 controls the rendering of Infinity and NaN: 216 217 0 ==> Infinity or NaN 218 1 ==> infinity or nan 219 2 ==> INFINITY or NAN 220 3 ==> Inf or NaN 221 4 ==> inf or nan 222 5 ==> INF or NAN 223 224nb with 0 <= nb < 3 determines whether NaN values are rendered 225as NaN(...): 226 227 0 ==> no 228 1 ==> yes 229 2 ==> no for default NaN values; yes otherwise 230 231ns = 0 or 1 determines whether the sign of NaN values reported: 232 233 0 ==> distinguish NaN and -NaN 234 1 ==> report both as NaN 235 236For an example of more general conversions based on dtoa(), see 237netlib's "printf.c from ampl/solvers". 238 239For double-double -> decimal, g_ddfmt() assumes IEEE-like arithmetic 240of precision max(126, #bits(input)) bits, where #bits(input) is the 241number of mantissa bits needed to represent the sum of the two double 242values in the input. 243 244The makefile creates a library, gdtoa.a. To use the helper 245routines, a program only needs to include gdtoa.h. All the 246source files for gdtoa.a include a more extensive gdtoaimp.h; 247among other things, gdtoaimp.h has #defines that make "internal" 248names end in _D2A. To make a "system" library, one could modify 249these #defines to make the names start with __. 250 251Various comments about possible #defines appear in gdtoaimp.h, 252but for most purposes, arith.h should set suitable #defines. 253 254Systems with preemptive scheduling of multiple threads require some 255manual intervention. On such systems, it's necessary to compile 256dmisc.c, dtoa.c gdota.c, and misc.c with MULTIPLE_THREADS #defined, 257and to provide (or suitably #define) two locks, acquired by 258ACQUIRE_DTOA_LOCK(n) and freed by FREE_DTOA_LOCK(n) for n = 0 or 1. 259(The second lock, accessed in pow5mult, ensures lazy evaluation of 260only one copy of high powers of 5; omitting this lock would introduce 261a small probability of wasting memory, but would otherwise be harmless.) 262Routines that call dtoa or gdtoa directly must also invoke freedtoa(s) 263to free the value s returned by dtoa or gdtoa. It's OK to do so whether 264or not MULTIPLE_THREADS is #defined, and the helper g_*fmt routines 265listed above all do this indirectly (in gfmt_D2A(), which they all call). 266 267By default, there is a private pool of memory of length 2000 bytes 268for intermediate quantities, and MALLOC (see gdtoaimp.h) is called only 269if the private pool does not suffice. 2000 is large enough that MALLOC 270is called only under very unusual circumstances (decimal -> binary 271conversion of very long strings) for conversions to and from double 272precision. For systems with preemptively scheduled multiple threads 273or for conversions to extended or quad, it may be appropriate to 274#define PRIVATE_MEM nnnn, where nnnn is a suitable value > 2000. 275For extended and quad precisions, -DPRIVATE_MEM=20000 is probably 276plenty even for many digits at the ends of the exponent range. 277Use of the private pool avoids some overhead. 278 279Directory test provides some test routines. See its README. 280I've also tested this stuff (except double double conversions) 281with Vern Paxson's testbase program: see 282 283 V. Paxson and W. Kahan, "A Program for Testing IEEE Binary-Decimal 284 Conversion", manuscript, May 1991, 285 ftp://ftp.ee.lbl.gov/testbase-report.ps.Z . 286 287(The same ftp directory has source for testbase.) 288 289Some system-dependent additions to CFLAGS in the makefile: 290 291 HU-UX: -Aa -Ae 292 OSF (DEC Unix): -ieee_with_no_inexact 293 SunOS 4.1x: -DKR_headers -DBad_float_h 294 295If you want to put this stuff into a shared library and your 296operating system requires export lists for shared libraries, 297the following would be an appropriate export list: 298 299 dtoa 300 freedtoa 301 g_Qfmt 302 g_ddfmt 303 g_dfmt 304 g_ffmt 305 g_xLfmt 306 g_xfmt 307 gdtoa 308 strtoIQ 309 strtoId 310 strtoIdd 311 strtoIf 312 strtoIx 313 strtoIxL 314 strtod 315 strtodI 316 strtodg 317 strtof 318 strtopQ 319 strtopd 320 strtopdd 321 strtopf 322 strtopx 323 strtopxL 324 strtorQ 325 strtord 326 strtordd 327 strtorf 328 strtorx 329 strtorxL 330 331When time permits, I (dmg) hope to write in more detail about the 332present conversion routines; for now, this README file must suffice. 333Meanwhile, if you wish to write helper functions for other kinds of 334IEEE-like arithmetic, some explanation of struct FPI and the bits 335array may be helpful. Both gdtoa and strtodg operate on a bits array 336described by FPI *fpi. The bits array is of type ULong, a 32-bit 337unsigned integer type. Floating-point numbers have fpi->nbits bits, 338with the least significant 32 bits in bits[0], the next 32 bits in 339bits[1], etc. These numbers are regarded as integers multiplied by 3402^e (i.e., 2 to the power of the exponent e), where e is the second 341argument (be) to gdtoa and is stored in *exp by strtodg. The minimum 342and maximum exponent values fpi->emin and fpi->emax for normalized 343floating-point numbers reflect this arrangement. For example, the 344P754 standard for binary IEEE arithmetic specifies doubles as having 34553 bits, with normalized values of the form 1.xxxxx... times 2^(b-1023), 346with 52 bits (the x's) and the biased exponent b represented explicitly; 347b is an unsigned integer in the range 1 <= b <= 2046 for normalized 348finite doubles, b = 0 for denormals, and b = 2047 for Infinities and NaNs. 349To turn an IEEE double into the representation used by strtodg and gdtoa, 350we multiply 1.xxxx... by 2^52 (to make it an integer) and reduce the 351exponent e = (b-1023) by 52: 352 353 fpi->emin = 1 - 1023 - 52 354 fpi->emax = 1046 - 1023 - 52 355 356In various wrappers for IEEE double, we actually write -53 + 1 rather 357than -52, to emphasize that there are 53 bits including one implicit bit. 358Field fpi->rounding indicates the desired rounding direction, with 359possible values 360 FPI_Round_zero = toward 0, 361 FPI_Round_near = unbiased rounding -- the IEEE default, 362 FPI_Round_up = toward +Infinity, and 363 FPI_Round_down = toward -Infinity 364given in gdtoa.h. 365 366Field fpi->sudden_underflow indicates whether strtodg should return 367denormals or flush them to zero. Normal floating-point numbers have 368bit fpi->nbits in the bits array on. Denormals have it off, with 369exponent = fpi->emin. Strtodg provides distinct return values for normals 370and denormals; see gdtoa.h. 371 372Compiling g__fmt.c, strtod.c, and strtodg.c with -DUSE_LOCALE causes 373the decimal-point character to be taken from the current locale; otherwise 374it is '.'. 375 376Source files dtoa.c and strtod.c in this directory are derived from 377netlib's "dtoa.c from fp" and are meant to function equivalently. 378When compiled with Honor_FLT_ROUNDS #defined (on systems that provide 379FLT_ROUNDS and fegetround() as specified in the C99 standard), they 380honor the current rounding mode. Because FLT_ROUNDS is buggy on some 381(Linux) systems -- not reflecting calls on fesetround(), as the C99 382standard says it should -- when Honor_FLT_ROUNDS is #defined, the 383current rounding mode is obtained from fegetround() rather than from 384FLT_ROUNDS, unless Trust_FLT_ROUNDS is also #defined. 385 386Compile with -DUSE_LOCALE to use the current locale; otherwise 387decimal points are assumed to be '.'. With -DUSE_LOCALE, unless 388you also compile with -DNO_LOCALE_CACHE, the details about the 389current "decimal point" character string are cached and assumed not 390to change during the program's execution. 391 392On machines with a 64-bit long double and perhaps a 113-bit "quad" 393type, you can invoke "make Printf" to add Printf (and variants, such 394as Fprintf) to gdtoa.a. These are analogs, declared in stdio1.h, of 395printf and fprintf, etc. in which %La, %Le, %Lf, and %Lg are for long 396double and (if appropriate) %Lqa, %Lqe, %Lqf, and %Lqg are for quad 397precision printing. 398 399Please send comments to David M. Gay (dmg at acm dot org, with " at " 400changed at "@" and " dot " changed to "."). 401