Dedication MET Professor Donald Knuth (Stanford) MET AF ONT with EX and Professor William Kahan (Berkeley) Extending T EX and Floating-Point Arithmetic AF Nelson H. F. Beebe ONT Department of Mathematics T University of Utah Salt Lake City, UT 84112-0090 MET AF ONT MET AF ONT USA MET AF ONT restricts input numbers to 12 integer bits: T EX Users Group Conference 2007 talk... – p. 1/30 T EX Users Group Conference 2007 talk... – p. 2/30 Arithmetic in T EX and Arithmetic in MET MET Binary integer arithmetic with ≥ 32 bits (T EX \count registers) % mf expr EX and EX and gimme an expr: 4095 >> 4095 Fixed-point arithmetic with sign bit, overflow bit, ≥ 14 gimme an expr: 4096 integer bits, and 16 fractional bits (T EX \dimen , ! Enormous number has been reduced. \muskip , and \skip registers) >> 4095.99998 MET AF ONT to work AF AF Overflow detected on division and multiplication but not gimme an expr: infinity >> 4095.99998 on addition (flaw (NHFB), feature (DEK)) gimme an expr: epsilon >> 0.00002 gimme an expr: 1/epsilon Gyrations sometimes needed in ! Arithmetic overflow. with fixed-point numbers ONT ONT >> 32767.99998 gimme an expr: 1/3 >> 0.33333 T Uh, oh. A little while ago one of the quantities T that I was computing got too large, so I’m afraid gimme an expr: 3*(1/3) >> 0.99998 gimme an expr: 1.2 - 2.3 >> -1.1 your answers will be somewhat askew. You’ll probably have to adopt different tactics next gimme an expr: 1.2 - 2.4 >> -1.2 gimme an expr: 1.3 - 2.4 >> -1.09999 time. But I shall try to carry on anyway. T EX Users Group Conference 2007 talk... – p. 3/30 T EX Users Group Conference 2007 talk... – p. 4/30
Historical remarks [cont] Historical remarks MET MET EX and EX and It is difficult today to appreciate that probably the biggest problem facing Floating Point Arithmetic . . . The subject programmers in the early 1950s was scaling numbers so as to achieve is not at all as trivial as most people think, acceptable precision from a fixed-point machine. and it involves a surprising amount of AF AF interesting information. Martin Campbell-Kelly Programming the Mark I: Donald E. Knuth The Art of Computer Programming: Early Programming Activity ONT ONT at the University of Manchester Seminumerical Algorithms , (1998) T Annals of the History of Computing T 2 (2) 130–168 (1980) T EX Users Group Conference 2007 talk... – p. 5/30 T EX Users Group Conference 2007 talk... – p. 6/30 Why no floating-point arithmetic? Historical remarks [cont] MET MET System dependence in precision , range , rounding , Computer hardware designers can make their underflow , overflow machines much more pleasant to use, EX and EX and for example by providing Base varies: 2 , 3 (Setun), 4 (Illiac II), 8 (Burroughs), floating-point arithmetic 10 , 16 (IBM S/360), 256 (Illiac III), 10000 (Maple) which satisfies simple mathematical laws. Bizarre behavior when T EX was developed: The facilities presently available on most AF AF x × y � = y × x (early Crays) machines make the job of rigorous error analysis hopelessly difficult , but properly x � = 1 . 0 × x (Pr1me) designed operations would encourage x + x � = 2 × x (Pr1me) numerical analysts to provide better ONT ONT x � = y but 1 . 0 / ( x − y ) gets zero-divide error subroutines which have certified accuracy. T T wrap between underflow and overflow (PDP-10) Donald E. Knuth job termination on overflow or zero-divide (most) Computer Programming as an Art ACM Turing Award Lecture (1973) No standardization: almost every vendor had unique floating-point system T EX Users Group Conference 2007 talk... – p. 7/30 T EX Users Group Conference 2007 talk... – p. 8/30
Why no floating-point . . . [cont]? Why no floating-point . . . [cont]? MET MET Language dependence: Input/output problem requires base conversion, and is hard (e.g., conversion from 128-bit binary format can Algol, Pascal, and SAIL ( real ) EX and EX and require more than 11 500 decimal digits) Fortran ( REAL , DOUBLE PRECISION , and sometimes DEK wrote A simple program whose proof isn’t REAL*16 ) (1990) about T EX’s conversions between fixed-point C/C++ ( double , float added in 1989, long double binary and decimal in 1999) AF AF Most languages do not guarantee exact base Java and C# (only float and double , but arithmetic system is badly botched: see Kahan and conversion Darcy’s How Java’s Floating-Point Hurts T EX guarantees identical line-breaking and ONT ONT MET AF ONT has no floating-point at all, and generates Everyone Everywhere ) page-breaking across all platforms (floating-point T T Compiler dependence: multiple precisions mapped to arithmetic used only for interword glue calculations) just one BSD compilers still provide no 80-bit format after 27 identical fonts on all systems years in hardware T EX Users Group Conference 2007 talk... – p. 9/30 T EX Users Group Conference 2007 talk... – p. 10/30 IEEE 754 binary standard (1985) IEEE 754 binary standard [cont] MET MET Preliminary version first implemented in Intel 8087 chip Nonstop computing model: sticky flags record (1980) exceptions EX and EX and Three formats defined: 32-bit, 64-bit, and 80-bit. Four rounding modes: 128-bit format available on some Alpha, IA-64, and to nearest with ties to even (default) SPARC systems. to + ∞ AF AF Nonzero normal numbers are rational : to −∞ x = ( − 1) s f × 2 p , where f ∈ [1 , 2) to zero (historical chopping) Signed zero ±∞ generated from large/small and finite/0 ONT ONT Largest stored exponent represents Infinity when NaN generated from 0/0, ∞ − ∞ , ∞ / ∞ , and any f = 0 , quiet and signaling NaN (Not-a-Number) when T T operation with a NaN operand f � = 0 NaN returned from functions when result is undefined in real arithmetic (e.g., √− 1 ) Smallest stored exponent allows f to have leading zeros with gradual underflow to subnormal values T EX Users Group Conference 2007 talk... – p. 11/30 T EX Users Group Conference 2007 talk... – p. 12/30
IEEE 754R Precision and range Remark on floating-point arithmetic MET MET Contrary to popular misconception, even in some books Binary EX and EX and and compilers, floating-point arithmetic is not fuzzy . 32-bit 24b ( ≈ 7D) 1e-45 1e-38 3e+38 64-bit 53b ( ≈ 15D) 4e-324 2e-308 1e+308 Results are exact if they are representable 80-bit 64b ( ≈ 19D) 3e-4951 3e-4932 1e+4932 Multiplication by power of base is always exact, in 128-bit 113b ( ≈ 34D) 6e-4966 3e-4932 1e+4932 absence of underflow and overflow AF AF 256-bit 234b ( ≈ 70D) 2e-315723 5e-315653 3e+315652 Subtraction of numbers of like signs and exponents is Decimal exact 32-bit 7D 1e-101 1e-95 1e+96 ONT ONT 64-bit 16D 1e-398 1e-383 1e+384 T T 128-bit 34D 1e-6176 1e-6143 1e+6144 256-bit 70D 1e-1572932 1e-1572863 1e+15782864 T EX Users Group Conference 2007 talk... – p. 13/30 T EX Users Group Conference 2007 talk... – p. 14/30 Binary versus decimal Binary versus decimal [cont] MET MET IEEE 854 Standard for Radix-Independent humans less uncomfortable with decimal arithmetic Floating-Point Arithmetic (1987, 1994) EX and EX and sales tax: 5% of 0 . 70 = 0 . 0349999 . . . in all binary older Cobol standards require 18D fixed-point precisions, instead of exact decimal 0 . 035 . Thus, Cobol 2002 requires 32D fixed-point and floating-point significant cumulative rounding errors in businesses with many small transactions (food, telephone, . . . ) Proposals to add decimal arithmetic to C and C++ AF AF (2005, 2006) financial computations need fixed-point decimal arithmetic 25 years of Rexx and NetRexx scripting languages give valuable experience in arbitrary-precision decimal hand calculators use decimal arithmetic ONT ONT arithmetic additional decimal rounding rules (8 instead of 4) T T excellent IBM decNumber library provides open source decimal arithmetic eliminates most base-conversion decimal floating-point arithmetic with a billion ( 10 9 ) problems digits of precision and exponent magnitudes up to 999 999 999 T EX Users Group Conference 2007 talk... – p. 15/30 T EX Users Group Conference 2007 talk... – p. 16/30
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