IEEE-754-1985-R1990.pdf

ANSlllEEE Std 754-1985 754-85 B IEEE Standard for Binary Floating-point Arithmetic Published by The Institute of Electrical and Electroniis Engineers, Inc 345 East 47th Street, New York, NY 10017, USA August 12,1985 SHI0116 Copyright The Institute of Electrical and Electronics Engineers, Inc. Provided by IHS under license with IEEELicensee=IHS Employees/1111111001, User=O'Connor, Maurice Not for Resale, 04/28/2007 23:54:33 MDTNo reproduction or networking permitted without license from IHS -,-,- ANSMEEE Std 754-1985 An American Nationul Standud IEEE Standard for Binary Floating-Point Arithmetic Sponsor Standards Committee of the IEEE Computer Society Approved March 21, 1985 IEEE Standards Board Approved July 26, 1986 American National Standards Institute Copyright 1986 by The Inetitute of Electrical and Electronics Engineers, Inc 345 East 47th Street, New York, NY 10017, USA No pH of this publication m y be mpnniuced in any form, in an electronic retrimof IEEE which have expressed an interest in participating in the development of the standard. Use of an IEEE Standard is wholly voluntary. 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Comments on standards and requests for interpretations should be ad- dressed to: Secretary, IEEE Standards Board 345 East 47th Street New York, NY 10017 USA Copyright The Institute of Electrical and Electronics Engineers, Inc. Provided by IHS under license with IEEELicensee=IHS Employees/1111111001, User=O'Connor, Maurice Not for Resale, 04/28/2007 23:54:33 MDTNo reproduction or networking permitted without license from IHS -,-,- B Foreword (This Foreword is not a part of ANSI/IEEE S t d 764-1986, IEEE Standard for Binary Floating-Point Arithmetic.) 'his standard is a product of the Floating-Point Working Group -of the“icroprocessor Standards Subcommittee of the Standards Committee of the IEEE Computer Society. This work was sponsored by the Technical Committee on Microprocessors and Minicomputers. Draft 8.0 of this standard was pub- lished to solicit public comments.' Implementation techniques can be found in An Implementation Guide to a Proposed Standard for Floating-point Arithmetic by Jerome T. Coonen: which was based on a stiU earlier draft of the proposal. This standard defines a family of commercially feasible ways for new systems to perform binary floating-point arithmetic. The issues of retrofitting were not considered. Among the desiderata that guided the formulation of this standard were (1) Facilitate movement of existing programs from diverse computers to those that adhere to this standard. (2) Enhance the capabilities and safety available to programmers who, though not expert in numerical methods, may well be attempting to produce numerically sophisticated programs. However, we recog- nize that utility and safety are sometimes antagonists. (3) Encourage experts to develop and distribute robust and efficient numerical programs that are portable, by way of minor editing and recompilation, onto any computer that conforms to this standard and possesses adequate capacity. When restricted to a declared subset of the standard, these programs should produce identical results on all conforming systems. (4) Provide direct support for (a) Execution-time diagnosis of anomalies (b) Smoother handling of exceptions (c) Interval arithmetic at a reasonable cost (a) Standard elementary functions such as exp and cos (b) Very high precision (multiword) arithmetic (c) Coupling of numerical and symbolic algebraic computation (5) Provide for development of (6) Enable rather than preclude further refinements and extensions. computer Magazine vol 14, no 3, March 1981. Womputer Magazine vol 13, no 1, January 1980. Copyright The Institute of Electrical and Electronics Engineers, Inc. Provided by IHS under license with IEEELicensee=IHS Employees/1111111001, User=O'Connor, Maurice Not for Resale, 04/28/2007 23:54:33 MDTNo reproduction or networking permitted without license from IHS -,-,- Members of the Floating-Point Working Group of the Microprocessor Standards Subcommittee and those who participated by correspondence were as follows: David Stevenson, Ckairman Andrew Allison William Ames Mike Arya Janis Baron Steve Baume1 Dileep Bhandarkar Joel Boney EH. Bristol Werner Buchholz Jim Bunch Ed Burdick Gary R. Burke Paul Clemente W.J. Cody Jerome T. Coonen Jim Crapuchettes Itzhak Davidesko Wayne Davison RH. Delp James Demmel DOM Denman A l v i n Despain Augustin A. Dubrulle Tom Eggen Philip J. Faillace Richard Fateman David Feign Don Feinberg Stuart Feldman Eugene Fisher Paul F. Flanagan Gordon Force Lloyd Fosdick Robert Fraley Howard Fullmer Daniel D. Gajski David M. Gay C.W. Gear Martin Graham David Gustavson Guy K. Haas Kenton Hanson Chuck Hastings David Hou if the two nearest representable values are equally near, the one with its least significant bit zero shall be deliv- ered. However, an infinitely precise result with magnitude at least 2Emax(2-2-P) shall round to 03 with no change in sign; here E, and p are de- termined by the destination format (see Section 3) unless overridden by a rounding precision mode (4.3). 4.2 Directed Roundings. An implementation shall also provide three user-selectable directed rounding modes: round toward fa, round toward -03, and round toward O. When rounding toward +a the result shall be the format's value (possibly +a) closest to and no less than the infinitely precise result. When rounding toward -P the result shall be the for- mat's value (possibly -03) closest to and no greater than the infinitely precise result. When rounding toward O the result shall be the for- mat's value closest to and no greater in magni- tude than the infinitely precise result. 4.3 Rounding Precision. Normally, a result is rounded to the precision of its destination. How- ever, some systems deliver results only to double or extended destinations. On such a system the user, which may be a high-level language com- piler, shall be able to specify that B result be rounded instead to single precision, though it may be stored in the double or extended format with its wider exponent range? Similarly, a sys- tem that delivers results only to double extended destinations shall permit the user to specify rounding to. single or double .precision. Note that to meet the specifications in 4.1, the result cannot suffer more than one rounding error. 4Control of rounding precision is intended to allow systems whose destinations are always double or extended to mimic, in the absence of over/underflow, the precisions of systems with single and double destinations. An implementation should not provide operations that combine double or ex- tended operands to produce a single result, nor operations that combine double extended operands to produce a double result, with only one rounding. Copyright The Institute of Electrical and Electronics Engineers, Inc. Provided by IHS under license with IEEELicensee=IHS Employees/1111111001, User=O'Connor, Maurice Not for Resale, 04/28/2007 23:54:33 MDTNo reproduction or networking permitted without license from IHS -,-,- 754-85 m4805702 OOl1795L b E BINARY FLOATING-POINT ARITHMETIC 5. Operations A l l conforming implementations of this stand- ard shall provide operations to add, subtract, multiply, divide, extract the square root, find the remainder, round to integer in floating-point for- mat, convert between different floating-point for- mats, convert between floating-point and integer formats, convert binary -decimal, and com- pare. Whether copying without change of format is considered an operation is an implementation option. Except for binary - decimal conversion, each of the operations shall be performed as if it first produced an intermediate result correct to W e precision and with unbounded range, and then coerced t h i s intermediate result to fit in the destination's format (see Sections 4 and 7). Sec- tion 6 augments the following specifications to cover +O, +a, and NaN; Section 7 enumerates exceptions caused by exceptional operands and exceptional results. D 5 . 1 Arithmetic. An implementation shall provide the add, subtract, multiply, divide, and remainder operations for any two operands of the same for- mat, for each supported format; it should also provide the operations for operands of differing formats, The destination format (regardless of the rounding precision control of 4.3) shall be at least as wide as the wider operand's format. A l l results shall be rounded as specified in Section 4. When y # O, the remainder T = x REM y is defined regardless of the rounding mode by the mathematical relation T = x - y X n, where n is the integer nearest the exact value x/ whenever In -x/yl= 4, then n is even Thus, the re- mainder is always exact. If T = O, its sign shall be that of x. Precision control (4.3) shall not apply to the remainder operation. 5.2 Square Root. The square root operation shall be provided in all supported formats. The result is defined and has a positive sign for all operands 2 O, except that G shall be -0. The destination format shall be at least as wide as the operand's. The result shall be rounded as specified in Section 4 . ANSI/IEEE Std 764-1985 shall be rounded as specified in Section 4. Con- version to a wider precision is exact. 5.4 Conversion Between Floating-point and Integer Formats. It shall be possible to convert between all supported floating-point formats and all supported integer formats. Conversion to in- teger shall be effected by rounding as specified in Section 4. Conversions between floating-point integers and integer formats shall be exact unless an exception arises as specified in 7.1. 5.5 Round Floating-point Number to Integer Value. It shall be possible to round a floating- point number to an integral valued floating-point number in the same format. The rounding shall be as specified in Section 4, with the understand- ing that when rounding to nearest, if the differ- ence between the unrounded operand and the rounded result is exactly one half, the rounded result is even. 5.6 Binary - Decimal Conversion. Conversion between decimal strings in at least one format and binary floating-point numbers in all sup- ported basic formats shall be provided for num- bers throughout the ranges specified in Table 2. The integers M and N in Tables 2 and 3 are such that the decimal strings have values +M X lO+N. On input, trailing zeros shall be appended to or stripped from M (up to the limits specified in Table 2) so as to minimize N. When the destina- tion is a decimal string, its least significant digit should be located by format specifications for purposes of rounding. When the integer M lies outside the range specified in Tables 2 and 3, that is, when M 2 lo9 for single or lW7 for double, the implementor may, at his option, alter all significant digits after the ninth for single and seventeenth for double to other decimal digits, typically O. Conversions shall be correctly rounded as specified in Section 4 for operands lying within Table 2 Decimal Conversion Ranges 5.3 Floating-point Format Conversions. It Format shall be possible to convert floating-point num- bers between all supported formats. If the con- Sinae version is to a narrower precision, the result Decimal to Binary Binary to Decimal MaxM MaxN MaxM MaxN 109-1 99 109-1 63 1017-1 999 1017-1 340 D 11 Copyright The Institute of Electrical and Electronics Engineers, Inc. Provided by IHS under license with IEEELicensee=IHS Employees/1111111001, User=O'Connor, Maurice Not for Resale, 04/28/2007 23:54:33 MDTNo reproduction or networking permitted without license from IHS -,-,- ANSI/IEEE Std 764-1985 the ranges specified in Table 3. Otherwise, for rounhg to nearest, the error in the converted result shall not exceed by more than 0.47 units in the destination's least significant digit the error that is incurred by the rounding specifications of Section 4, provided that exponent overhnder- flow does not occur. In the directed rounding modes the error shall have the correct sign and shall not exceed 1.47 units in the last place. Conversions shall be monotonic, that is, in- creasing the value of a binary floating-point num- ber shall not decrease its value when converted to a decimal string; and increasing the value of a decimal string shall not decrease its value when converted to a binary floating-point number. When rounding to nearest, conversion from bi- nary to decimal and back to binary shall be the identity as long as the decimal string is carried to the maximum precision specified in Table 2, namely, 9 digits for single and 17 digits for dou- ble? If decimal to binary conversion overhder- flows, the response is as specified in Section 7. Overhnderflow and NaNs and infinities encoun- tered during binary to decimal conversion should be indicated to the user by appropriate strings. NaNs encoded in decimal strings are not spec- ified in this standard, To avoid inconsistencies, the procedures used for binary - decimal conversion should give the same results regardle