The UK votes YES to accept this document as a Final CD.

In addition to the comments below on the document the UK would also like to stress the following points:

1. We understand that, in some programming communities, integer values stored in a floating point datatype are treated differently from other floating point values.

However, we strongly support a clear separation of values which are always exact, i.e. the integer datatype, from values which are always approximate, i.e. the floating point datatype, within this Standard. In our view it is not possible for an approximate value to be deemed to be exact.

This does not preclude a specific programming language from allowing such a conversion.

2. Regarding the distinguishing of infinities it is our view that it is important to distinguish between user-created ones, e.g. poles created by divide by zero, and others, such as accumulated values becoming too large, if at all possible. We would like to see LIA-1 amended to allow for this at its next revision.

3. Clause 5.5 page 10

We are not happy about an implementation being REQUIRED to provide a lower level of accuracy at a higher performance. We believe that

a) this should only be optional and

b) that the user should be provided with sufficient information in such a case as to be able to make an informed decision as to whether the speed gain is worth the loss of accuracy.

- --------------------------------------------------------------------------

The following comments on the document relate to minor technical or other issues.

Note: Maths text is \TeX\ formatted.

1. Clause 5.6

Replace heading "Extensions" by "IEC 559 Special values". Does this change need to be made anywhere else?

2. General

The disposition of comments (N2391, p11, clause 3 The major issues, point 4) said that the IEC special values would be in bold type. But '-0' and the others are not clearly in bold. It is essential that these values are not confused with normal mathematics.

This should be checked with the special values possibly specified differently, i.e. in a different font to the Bold one currently being used.

3. Clause 9.3 Page 16

Why is there a difference in the second case of the exceptions on page 16 from the corresponding integral power functions on page 17? Shouldn't it be

$$\hbox{\it POWER}_{FF}(0,0)=\hbox{\bf bad{\_}argument}$$

rather than

$$\hbox{\it POWER}_{FF}(0,0)=\hbox{\bf pole}(\infty)?$$

I.e. for the first TWO exceptions, the respective if-clauses should be

1. if $x<0$ or $x=y=0$;

2. if $y<0$.

If on the other hand the original formulation is the intended one, then there should be additional rationale explaining the difference.

4. Clause 10.3 Page 19

There is a typographic error on the first line (filename "plia2.tex" should not be there!)

5. Clause 10.3 Page 19

Line 2 of the axioms appears to be in contradiction with line 2 of the special values clause, unless a clause is added to the latter to make it read

$$\hbox{\it LOG}_{FF}(1,w)=\hbox{\bf bad{\_}argument}

\quad\hbox{if $w=0$ or $w=\pm\infty$}$$

The if-clause in line 1 could be simplified to

if $w\le 1$.

6. Clause 11.14 Page 26

Line 2 of definition, "pi" should be Greek symbol.

7. Clause 11.14 Page 27

Last two cases of exceptions, letters 'x' and 'u' should be italic.

8. Clause 11.17 Page 28

Typo on 4th exception -- too much white space!

9. Clause 17.8 Page 51

Insert line break between "w" and "MIN2 F"

10. Clause 20.1 Page 57

It seems that the if-clause in the second line of exceptions simplifies to if $x^2+y^2<fmin_N^2$ unless it matters precisely which of the quantities ($\max{x^2,y^2}$, $\min{x^2,y^2}$ or $x^2+y^2$) are considered. In that case a (clearer) comment to this effect is in order. (SEE NEXT ITEM.)

11. Clause 20.1 Page 57

First line of exceptions seems to contradict rationale in A.20.1, (line 2, page 79), which states that HYPOT must never produce an underflow. (SEE PREV. ITEM.)

12. Clause 10.3 Page 19

This is more in the nature of a philosophical question than a comment about the draft. Why would it be a bad idea to implement logarithms with fractional bases via

$$\hbox{\it LOG}(1/b,x)=-\hbox{\it LOG}(b,x) \quad\hbox{if $b\in (1,\infty)$}(??)$$

and thereby extend {\it LOG} to bases $b\in (0,1)$?

- -------------------end of UK comments ---------------------------

------------------------------------------------------------------------

DIN - NI - 22.11

c/o Prof. Dr. Wolfgang V. Walter

Technische Universitaet Dresden

Institut f. Wissenschaftliches Rechnen

Fachrichtung Mathematik

Mommsenstr. 13

D - 01062 Dresden

Germany

Tel: +49 - 351 / 463-3996 or -4266 (secr.)

FAX: +49 - 351 / 463-7114

e-mail: wwalter@math.tu-dresden.de

=======================================================================

Germany believes that CD 10967-2, also known as LIA-2, is NOT ready

for registration as a final CD and should not be approved.

=======================================================================

We agree with most of Fred Tydeman's comments which he sent DIN (and also X3J11) on March 10, 1997, except for 31) and 33). Mr. Tydeman lists 101 minor and major technical problems and typos which are indicative of the failings of the current draft.

In essence, we believe that the current draft of LIA-2 should be extended, corrected, or made more explicit in the following areas:

- --------------------end of first set of comments -----------------

Comments by Mr Tydeman (who has been in Germany for 19 months), submitted to DIN NI-22 and to ANSI X3J11:

- -----

We have reviewed ISO/IEC 10967-2.2:1996(E), Second edition, December, 1996. Information technology -- Language independent arithmetic -- Part 2: Elementary numerical functions. That document will be called LIA-2 in the rest of this note.

The above document is the second committee draft submitted to SC22 to become a Final Committee Draft (FCD). The closing date for the ballot is June 19, 1997 (document SC22 N2397).

We have also reviewed four other documents that are related:

1) ISO/IEC 10967-1:1994(E) Information technology -- Language independent arithmetic -- Part 1: Integer and floating point arithmetic. That document will be called LIA-1 in the rest of this note.

2) CEI/IEC 559: 1989 Second edition, 1989-01, Binary floating-point arithmetic for microprocessor systems (also ANSI/IEEE Std 754-1985 IEEE Standard for Binary Floating-Point Arithmetic). That document will be called IEEE-754 in the rest of this note.

3) ISO/IEC 10967-2:1995(E), First edition, November, 1995. Information technology -- Language independent arithmetic -- Part 2: Elementary numerical functions.

4) SC22/WG11 N434 Disposition of Comments on first edition LIA-2.

================================================================

In summary, we believe that the LIA-2 document should NOT be approved.

There are still too many errors in it.

================================================================

The following numbered comments are in page number order.

The comments have been classified into four categories:

[major] Cause for "No" vote unless resolved in satisfactory way.

[minor] Not cause for "No" vote, but should be addressed.

[typo] Typographical or obvious very minor error.

[info] For your information.

1) [minor] Page ix. Please add a sentence saying if notes are normative or for information only.

2) [minor] Page x, last paragraph, last sentence. How does a program select the library best suited to its application? What parameters are used to make this selection? Our impression is there is one math library that conforms to LIA-2.

3) [minor] Page 1, section 1.1, 2nd paragraph, change "exceptional input value" to "IEC 559 special value" or "IEC 559 continuation value".

4) [minor] Page 2, section 2.1 and following. Change "operations" to "numerical functions". This document is about numerical functions, not arithmetic operations.

5) [minor] Page 2, section 2.2, 1st paragraph, what does ", and no others" mean? Is an input or output operation (which most languages have) considered part of "others"?

6) [minor] Page 7, item 22. Change "digits in the fraction" to "radix-r digits in the fraction f in 0.f*r**e". We are used to scientific notation where there is one digit before the decimal point and several fraction digits after the decimal point.

7) [minor] Page 8, item 34. Change "normalized value x" to "normalized value x (0.f*r**p)".

8) [minor] Page 8, section 5, 3rd paragraph. Change "error limit, and extensions" to "error limit, and IEC 559 special values".

15) [minor] Page 11, section 6, 6th paragraph. Change "signaling NaN as an input, then a bad_argument notification shall result" to "signaling NaN as an input, then a bad_argument notification shall result with a quiet NaN as the continuation value".

16) [minor] Page 11, section 6, 6th paragraph. Add "Functions with one quiet NaN input return that same quiet NaN. Functions with multiple quiet NaN inputs return one of the input quiet NaNs."

17) [minor] Page 15, section 9.2, in the Note. Remove the word "only".

19) [minor] Page 17, section 9.6, title. Change "known base" to "integer base".

20) [minor] Page 17, section 9.6.1 and 9.6.2, Axioms. The domains are not consistent. One mentions fmax, the other fmin. It seems that the domains should be the same domain as in the Definition section.

21) [minor] Page 17, section 9.6.1, section Exceptions. Change "x <=" to "x <". On a binary machine, log2(fminN) is an exact integer, so underflow will not happen for exp2(log2(fminN)).

22) [minor] Page 18, section 9.6.2, section Exceptions. Change "x <=" to "x <". On a decimal machine, log10(fminN) is an exact integer, so underflow will not happen for exp10(log10(fminN)).

25) [minor] Page 19, section 10.2, in the Note. Remove the word "only".

26) [typo] Page 19, section 10.3, before Signature. What is "plia2.tex"?

27) [type] Page 19, section 10.3, IEC 559 Special Values. What is the ">/" symbol in "if w >/ 1"? Should it be w not element of F or w <= 1?

28) [minor] Page 19, section 10.3, IEC 559 Special Values. Remove, or explain better, the log(1,w) = bad_argument case. log of 1 for any valid base is 0.

30) [minor] Page 19, section 10.4, title. Change "known base" to "integer base".

31) [minor] Page 19, section 10.4.1, many sections. Replace lower case "log_2" with "log2" where the 2 is a subscript.

33) [minor] Page 20, section 10.4.2, many sections. Replace lower case "log_10" with "log10" where the 10 is a subscript.

35) [minor] Pages 21 to 28, sections 11.1 to 11.18, Exceptions. Many of the definitions of underflow are in terms of fmin which should be fminN.

36) [minor] Page 21, section 11.1, Exceptions. For the underflow case, change the "1 <= |x|" to "0 <= |x|".

37) [minor] Page 21, section 11.2, Axioms. Add SIN(u/2,u) = 0.0

38) [minor] Page 21, section 11.2, Exceptions. For the underflow case, change the "u/8 <= |x|" to "0 <= |x|".

39) [typo] Page 22, section 11.5, Axioms. The exponent "(-1-p/2)" needs to be fixed so that it is all an exponent.

40) [typo] Page 28, section 11.17, Exceptions. "if ( 2|u," is missing some text.

41) [typo] Page 29, section 12.1, Signature. Change "bad_argument underflow" to "bad_argument, underflow",

42) [minor] Page 29, section 12.1, Axioms. Change "|x| <= fmax" to "|x| <= 1".

43) [minor] Page 30, section IEC 559 Special Values. Here and in many following functions, there is a function N() that is not defined (that we can find). We assume it is nearest rounding.

48) [typo] Page 42, section 15.1, title. Change "Convert to float" to "Convert to integer".

49) [minor] Page 43, section 15.3, Definition. Change "<= x <=" to "< x <". Since you allow round to nearest to round up or down, maxint + 1/2 could round to maxint + 1 which is overflow.

50) [minor] Page 45, section 15.9, Exceptions. Change it so that floor(0.0) is 0.0, not underflow. For example, if 0 < |x| < fminNG.

51) [minor] Page 45, section 15.10, Exceptions. Change it so that ceiling(0.0) is 0.0, not underflow.

52) [minor] Page 45, section 15.10, IEC 559 Special Values. Why are -0, infinity, and NaN allowed to be in one floating type (F), but not another floating type (G)? Why is that case allowed for here, but not in DPROD (section 19.1)?

53) [typo] Page 45, section 15.10 and higher, make the "if"s line up vertically.

54) [minor] Page 45, section 15.11, Exceptions. Change it so that nearest(0.0) is 0.0, not underflow.

55) [minor] Page 46, section 15.12, Exceptions. Change it so that truncate(0.0) is 0.0, not underflow.

56) [minor] Page 48, sections 16.2.1 to 16.2.4, IEC Special Values. Add "f(SNaN) causes no notification". Need some way to print SNaNs.

57) [minor] Page 49, section 16.3, 2nd paragraph. Since compile time literal conversion and runtime input conversion need not be identical, there should be an error bound on the difference between them.

58) [typo] Page 51, sections 17.6 and 17.8, IEC Special Values. Add some spaces before F(w,v) = F(v,w) and after the +INF.

59) [minor] Page 51, section 18, first paragraph. Change "2p-3" to "2p" or explain why the -3.

60) [minor] Page 52, section 18.2.1, IEC 559 Special Values. Change the bad_argument cases to be 0.

61) [minor] Page 53, section 18.2.2, IEC 559 Special Values. Change the bad_argument cases to be 0.

62) [minor] Page 53, section 18.2.3, IEC 559 Special Values. Change the bad_argument cases to be 0.

63) [minor] Page 53, section 18.2.4, Error limit. What is "xhi(x,y)"?

64) [minor] Page 54, section 18.2.5, IEC 559 Special Values. Change the bad_argument cases to be 0.

65) [minor] Page 56, section 19.2, IEC 559 Special Values. Change the bad_argument cases to be 0.

66) [minor] Page 57, section 20.1. The text description says that underflow does not occur, yet underflow shows up in the exceptions section.

67) [typo] Page 57, section 20.1, IEC 559 Special Values. Change "(w.v)" to "(w,v)".

68) [minor] Page 57, section 20.3, IEC 559 Special Values. Remove the DIM(NaN,w) case.

69) [minor] Page 58, new section. Please add an ulp(x) function that returns the value of an ulp of the argument x. ulp(small) can underflow if denormals are not representable. ulp(0.0) is bad_argument. ulp(infinity) is bad_argument. ulp(-x) == ulp(x).

70) [minor] Page 59, section A, Acknowledgements. Remove the X3 from X3T2. This may need to be elsewhere in the document.

71) [minor] Add WG14/J11 to the list of standards committees that provided advice.

72) [minor] Page 59, section A, Project milestones. Change "August

1996" to "December 1996".

73) [typo] Page 60, section A.2, last line. Change "LIA=2" to "LIA-2".

74) [typo] Page 61, section A.5.2. Change "log" to "LOG".

76) [minor] Page 63, section A.5.5. Remove "It should be mentioned that an implementation can usually provide numerical operations with higher accuracy than that associated with the underlying basic arithmetic operations." How can "sin()" be more accurate than "+"?

77) [minor] Page 63, section A.5.6, 2nd paragraph. The statement "LIA-2 requires conformance to IEC 559 by implementations which provide these special values" needs to be changed. Just because an implementation supports -0, +/- infinity, and NaN, does not mean that it is capable of conforming to IEC 559. There are systems that do not support denormal numbers, but do have those special values. There are systems that provide Signaling NaNs, but no Quiet NaNs, but do the rest.

78) [typo] Page 63, section A.5.6, 3rd paragraph. The reference to [37] is wrong because reference [37] appears to be two documents combined into one item.

79) [minor] Page 64, section A.5.6.4, 1st paragraph. Remove "(unjustified)".

82) [minor] Page 66, section A.9. Add a reference to a new entry in the bibliography to which X/Open Portability Guide. Is it XPG3, XPG4, or something else? They have changed the meaning of the math library with the various releases.

84) [minor] Page 74, section A.15.1. Change "exceeds maxint" to "exceeds maxint+1".

85) [minor] Page 74, section A.15.2. Change "minint exceeds" to "minint-1 exceeds".

86) [typo] Page 74, section A.15.3. The second paragraph should be

section A.15.4 TRUNCATE.

87) [typo] Pages 74 and 75. Increase the number of each section after A.15.3 by 1.

88) [minor] Page 74, old sections A.15.4, A.15.5, A.15.6, A.15.7. fmax need not always be an integer. The IBM 1620 had a two decimal digit exponent and variable length significand that could be longer than decimal 99 digits (as we recall).

89) [minor] The statement that "the argument must lie between two integral values" is false. Consider the exact values 0.0 and 1.0; neither is between.

90) [typo] Page 75, old section A.15.8. Change "th%" to "the" and fix the fmaxG where the G overlaps the x.

91) [minor] Page 75, old sections A.15.8, A.15.9, A.15.10, A.15.11. Need to add words to exclude zero from raising underflow in the last sentence.

92) [minor] Page 75, section A.16. Change "All language" to "Most language". ALGOL-60 has no I/O. Change the first "I/O" to "Input/Output (I/O)".

93) [minor] Page 76, section A.18, 1st paragraph. Remove "and/or range" or explain how using one gets more exponent range by using doubled representation. Change "highest" to "most precise".

94) [typo] Page 77, section A.18.2.3. Change "MULHI" to "MULLO".

95) [typo] Page 77, section A.18.2.4. Change "DIVHI" to "DIVLO".

96) [minor] Page 79, section A.20.1. Remove "It must never produce an underflow." Remove "This claim is not always valid."

97) [minor] Page 99, section C, item [1]. Add "second edition" to the IEC 559 reference. Change "854-1987" to "754-1985".

98) [minor] Page 99, section C, item [14]. We believe that the 1994 standard has been published, so remove "(to be published)".

99) [minor] Page 100, section C. Add item [23.5] ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic.

100) [info] Page 100, section C, item [28]. There is a corrected version of that Handbook of Mathematical Functions that has been published by Dover.

101) [minor] Page 101, section C, item [37]. This appears to be two references in one and needs to be split into two. The Sorenson/Tang reference appears to be correct in the first draft of LIA-2. In addition, the Floating-Point C Extensions (FPCE) reference is out of date. It should be: Numerical C Technical Report, X3/TR-17:1995.

- ---------------------end of Germany comments ----------------------

The US national body votes to 'Disapprove' the ballot for Final Commitee Draft (FCD) for ISO/IEC 10967-2.2:1996(E), Second edition, December, 1996. Information technology -- Language independent arithmetic -- Part 2: Elementary numerical functions. That document will be called LIA-2 in the rest of this document.

The following are the US comments in support of the vote to Disapprove. The comments have been classified into four categories and are presented by category:

- - M [major] These are major technical errors or ambiguities that are the cause of the US vote to 'Disapprove';

- - m [minor] These are technical errors or ambiguities that the US would like to see resolved, but did not cause the US to vote 'No';

- - E [editorial] These are typographical or obviously very minor errors;

- - I [info] These are noted for informational purposes only.

If the M category comments can be resolved, the US reserves the right to change its 'No' vote to 'Yes'.

M [MAJOR] COMMENTS

[M] Page 9, section 5.2 Definition. The definitions given in sections 8 and higher are not consistent. Some exclude all domains where exceptions happen, others exclude some of the domain where exceptions happen. Contrast 11.1 sin() and 11.7 tan(). You may need to add a paragraph to this section explaining the intent.

[M] Page 9, section 5.2, item b). Change the "OP" in the exponent to lower case so that the error is calculated in terms of the correct mathematical result, instead of the computed result. The program can still estimate the error based upon the calculated value, but the error limits for conformance must be defined in terms of the true mathematical value. It matters the most when the true value is just below a power of the radix and the calculated value is just above that same power of the radix. Using the calculated value to compute the error bound results in an error "radix" times as large as using the true value.

[M] Page 10, section 5.3. It is not clear if axioms and the functions are dependent upon rounding or independent of the rounding. For example, if the user changes the rounding to plus infinity, calls a function, then changes the rounding to minus infinity and calls the same function with the same argument (so as to get the bounds of the result), must the function honor the rounding? May the function honor the rounding? Or, must the function be independent of the rounding? If rounding matters, than many of the axioms in sections 8 and higher are false, in particular, those that state that f(x) is x if x is small.

[M] Page 10, section 5.5, 2nd paragraph. What is the meaning of "each such operation provided by an implementation must also be provided at lower accuracy and higher performance."? This is new with the second edition. It also is a contradiction to the Introduction on page x where it is a recommendation. In reading the response to the Japan's comments to the first edition "It is intended that an implementation provide a library of reasonably high accuracy, and also provide one of lower accuracy, with error bounds three or four times greater than those for high accuracy.", it appears that there is supposed to be an accurate and slow library and a sloppy and fast library. That intent needs to be made clearer. For example, is the lower accuracy library still required to meet the accuracy requirement of LIA-2? How does the program select the accurate or sloppy version? Is it done at compile time, link time or at runtime?

[M] Page 11, section 6, 4th paragraph. First choice is to remove the entire paragraph, second choice is to change it so that functions may optionally signal underflow for denormal arguments. It appears that you tried to apply the idea that f(x) is x (not underflow) for denormal x to the rest of the document, but you did not do a consistent job. For example, you say sin(x) for a denormal is not underflow, yet sinh(x) and arcsin(x) for a denormal are underflow.

[M] Page 16, section 9.3, Exceptions. Rewrite the case of x < 0 into cases that cover integers and non-integers.POWER( -2.0, 2.0 ) and POWER( -3.0, 3.0 ) are well defined. The US disagrees with the assumption in A.9.3 that integer floating-point values are approximate.

[M] Page 18, section 10.1, IEC 559 Special Values. Remove the bad_argument exception for ln(+INF). Make it consistent with sqrt().

[M] Page 19, section 10.2, IEC 559 Special Values. Remove the bad_argument exception for ln1p(+INF). Make it consistent with sqrt().

[M] Page 19, section 10.3, IEC 559 Special Values. Remove the bad_argument from log(+inf,w) = bad_argument (+inf) case. Make it consistent with sqrt().

[M] Page 20, section 10.4.1, IEC 559 Special Values. Remove the bad_argument exception for log_2(+INF). Make it consistent with sqrt().

[M] Page 20, section 10.4.2, IEC 559 Special Values. Remove the bad_argument exception for log_10(+INF). Make it consistent with sqrt().

[M] Page 32, section 12.8, IEC 559 Special Values. Remove the bad_argument exception for arctan(+INF,w) and replace it with pi/2. Make it consistent with sqrt().

[M] Page 32, section 12.8, IEC 559 Special Values. Remove the bad_argument exception for arctan(-INF,w) and replace it with -pi/2. Make it consistent with sqrt().

[M] Page 34, section 12.14, IEC 559 Special Values. Remove the bad_argument exception for arccot(+INF,w) and replace it with +0. Make it consistent with sqrt().

[M] Page 34, section 12.14, IEC 559 Special Values. Remove the bad_argument exception for arccot(-INF,w) and replace it with -0. Make it consistent with sqrt().

[M] Page 62, section A.5.2. Change "OP(x,...)" to "op(x,...)" in the exponent of r for the error bound so that it computes the error in terms of the mathematical function and not the computer function.

[M] Page 65, section A.6. The idea that some underflows are better than other underflows, that is, some should be acknowledged but others ignored, should either be removed or made optional. Also, there is a factual error: Underflow can happen for a normal as well as a denormal input argument. Consider, sin(x), where x is the smallest normalized number. Since the series for sin() is x - x**3/3! ..., the mathematical result is a little less than the input, so in round to zero or round to minus infinity, the computed result is the largest denormal and hence underflow.

[M] Page 65, section A.8.1, 2nd paragraph. The idea that some infinities are different from other infinities, that is, +inf from addition is different from +inf from multiplication is different from +inf from divide by zero, needs to be removed. It sounds like you want the math functions to check the exception flags and treat an input of +inf and overflow different from an input of +inf and divide-by-zero.

[M] Page 66, section A.9.3, first paragraph. The idea that an integer, just because it is in floating-point format, is not an integer, needs to be removed. Once that is done, then there are special cases that need to be added. For example, POWER( -3.0, 3.0 ) should be -27.0, not bad_argument.

Here is the IEC 559 specification adopted by WG14/X3J11 into C9X, the revision of C currently in progress:

pow(x, +/- 0) returns 1 for any x.

pow(x, +INFINITY) returns +INFINITY for |x| > 1. pow(x, +INFINITY) returns +0 for |x| < 1. pow(x, -INFINITY) returns +0 for |x| > 1.

pow(x,- -INFINITY) returns +INFINITY for |x| < 1. pow(+INFINITY, y) returns +INFINITY for y > 0.

pow(+INFINITY, y) returns +0 for y < 0.

pow(-INFINITY, y) returns -INFINITY for y an odd integer > 0.

pow(-INFINITY, y) returns +INFINITY for y > 0 and not an odd integer.

pow(-INFINITY, y) returns -0 for y an odd integer < 0. pow(-INFINITY, y)

returns +0 for y < 0 and not an odd integer. pow(x, y) returns one of its NaN arguments if y is a NaN, or if x is a NaN and y is nonzero.

pow(+/- 1, +/- INFINITY) returns a NaN and raises the invalid exception.

pow(x, y) returns a NaN and raises the invalid exception for finite x < 0 and finite nonintegral y.

pow(+/- 0, y) returns +/- INFINITY and raises the divide-by-zero exception for y an odd integer < 0.

pow(+/- 0, y) returns +INFINITY and raises the divide-by-zero exception for y < 0 and not an odd integer.

pow(+/- 0, y) returns +/- 0 for y an odd integer > 0.

pow(+/- 0, y) returns +0 for y > 0 and not an odd integer.

m [MINOR] COMMENTS

[m] Page ix. Please add a sentence saying if notes are normative or for information only.

[m] Page x, last paragraph, last sentence. How does a program select the library best suited to its application? What parameters are used to make this selection? Our impression is there is one math library that conforms to LIA-2.

[m] Page 1, section 1.1, 2nd paragraph, change "exceptional input value" to "IEC 559 special value" or "IEC 559 continuation value".

[m] Page 2, section 2.1 and following. Change "operations" to "numerical functions". This document is about numerical functions, not arithmetic operations.

[m] Page 2, section 2.2, 1st paragraph, what does ", and no others" mean? Is an input or output operation (which most languages have) considered part of "others"?

[m] Page 7, item 22. Change "digits in the fraction" to "radix-r digits in the fraction f in 0.f*r**e". We are used to scientific notation where there is one digit before the decimal point and several fraction digits after the decimal point.

[m] Page 8, item 34. Change "normalized value x" to "normalized value x (0.f*r**p)".

[m] Page 8, section 5, 3rd paragraph. Change "error limit, and extensions" to "error limit, and IEC 559 special values".

[m] Page 11, section 5.6 title. Change "Extensions" to "IEC 559 Special Values".

[m] Page 11, section 6, 6th paragraph. Change "signaling NaN as an input, then a bad_argument notification shall result" to "signaling NaN as an input, then a bad_argument notification shall result with a quiet NaN as the continuation value".

[m] Page 11, section 6, 6th paragraph. Add "Functions with one quiet NaN input return that same quiet NaN. Functions with multiple quiet NaN inputs return one of the input quiet NaNs."

[m] Page 15, section 9.2, in the Note. Remove the word "only".

[m] Page 17, section 9.6, title. Change "known base" to "integer base".

[m] Page 17, section 9.6.1 and 9.6.2, Axioms. The domains are not consistent. One mentions fmax, the other fmin. It seems that the domains should be the same domain as in the Definition section.

[m] Page 17, section 9.6.1, section Exceptions. Change "x <=" to "x <". On a binary machine, log2(fminN) is an exact integer, so underflow will not happen for exp2(log2(fminN)).

[m] Page 18, section 9.6.2, section Exceptions. Change "x <=" to "x <". On a decimal machine, log10(fminN) is an exact integer, so underflow will not happen for exp10(log10(fminN)).

[m] Page 19, section 10.2, in the Note. Remove the word "only".

[m] Page 19, section 10.3, IEC 559 Special Values. Remove, or explain better, the log(1,w) = bad_argument case. log of 1 for any valid base is 0.

[m] Page 19, section 10.4, title. Change "known base" to "integer base".

[m] Page 19, section 10.4.1, many sections. Replace lower case "log_2" with "log2" where the 2 is a subscript.

[m] Page 20, section 10.4.2, many sections. Replace lower case "log_10" with "log10" where the 10 is a subscript.

[m] Pages 21 to 28, sections 11.1 to 11.18, Exceptions. Many of the definitions of underflow are in terms of fmin which should be fminN.

[m] Page 21, section 11.1, Exceptions. For the underflow case, change the "1 <= |x|" to "0 <= |x|".

[m] Page 21, section 11.2, Axioms. Add SIN(u/2,u) = 0.0

[m] Page 21, section 11.2, Exceptions. For the underflow case, change the "u/8 <= |x|" to "0 <= |x|".

[m] Page 29, section 12.1, Axioms. Change "|x| <= fmax" to "|x| <= 1".

[m] Page 30, section IEC 559 Special Values. Here and in many following functions, there is a function N() that is not defined (that we can find). We assume it is nearest rounding.

[m] Page 43, section 15.3, Definition. Change "<= x <=" to "< x <". Since you allow round to nearest to round up or down, maxint + 1/2 could round to maxint + 1 which is overflow.

[m] Page 45, section 15.9, Exceptions. Change it so that floor(0.0) is 0.0, not underflow. For example, if 0 < |x| < fminNG.

[m] Page 45, section 15.10, Exceptions. Change it so that ceiling(0.0) is 0.0, not underflow.

[m] Page 45, section 15.10, IEC 559 Special Values. Why are -0, infinity, and NaN allowed to be in one floating type (F), but not another floating type (G)? Why is that case allowed for here, but not in DPROD (section 19.1)?

[m] Page 45, section 15.11, Exceptions. Change it so that nearest(0.0) is 0.0, not underflow.

[m] Page 46, section 15.12, Exceptions. Change it so that truncate(0.0) is 0.0, not underflow.

[m] Page 48, sections 16.2.1 to 16.2.4, IEC Special Values. Add "f(SNaN) causes no notification". Need some way to print SNaNs.

[m] Page 49, section 16.3, 2nd paragraph. Since compile time literal conversion and runtime input conversion need not be identical, there should be an error bound on the difference between them.

[m] Page 51, section 18, first paragraph. Change "2p-3" to "2p" or explain why the -3.

[m] Page 52, section 18.2.1, IEC 559 Special Values. Change the bad_argument cases to be 0.

[m] Page 53, section 18.2.2, IEC 559 Special Values. Change the bad_argument cases to be 0.

[m] Page 53, section 18.2.3, IEC 559 Special Values. Change the bad_argument cases to be 0.

[m] Page 53, section 18.2.4, Error limit. What is "xhi(x,y)"? Explain or correct if in error.

[m] Page 54, section 18.2.5, IEC 559 Special Values. Change the bad_argument cases to be 0.

[m] Page 56, section 19.2, IEC 559 Special Values. Change the bad_argument cases to be 0.

[m] Page 57, section 20.1. The text description says that underflow does not occur, yet underflow shows up in the exceptions section.

[m] Page 57, section 20.3, IEC 559 Special Values. Remove the DIM(NaN,w) case.

[m] Page 58, new section. Please add an ulp(x) function that returns the value of an ulp of the argument x. ulp(small) can underflow if denormals are not representable. ulp(0.0) is bad_argument. ulp(infinity) is bad_argument. ulp(-x) == ulp(x).

[m] Page 59, section A, Acknowledgements. Remove the X3 from X3T2. This may need to be elsewhere in the document.

[m] Page 59, Add WG14/J11 to the list of standards committees that provided advice.

[m] Page 59, section A, Project milestones. Change "August 1996" to "December 1996".

[m] Page 63, section A.5.5. Remove "It should be mentioned that an implementation can usually provide numerical operations with higher accuracy than that associated with the underlying basic arithmetic operations." How can "sin()" be more accurate than "+"?

[m] Page 63, section A.5.6, 2nd paragraph. The statement "LIA-2 requires conformance to IEC 559 by implementations which provide these special values" needs to be changed. Just because an implementation supports -0, +/- infinity, and NaN, does not mean that it is capable of conforming to IEC 559. There are systems that do not support denormal numbers, but do have those special values. There are systems that provide Signaling NaNs, but no Quiet NaNs, but do the rest.

[m] Page 64, section A.5.6.4, 1st paragraph. Remove "(unjustified)".

[m] Page 66, section A.9. Add a reference to a new entry in the bibliography to which X/Open Portability Guide. Is it XPG3, XPG4, or something else? They have changed the meaning of the math library with the various releases.

[m] Page 74, section A.15.1. Change "exceeds maxint" to "exceeds maxint+1".

[m] Page 74, section A.15.2. Change "minint exceeds" to "minint-1 exceeds".

[m] Page 74, old sections A.15.4, A.15.5, A.15.6, A.15.7. fmax need not always be an integer. The IBM 1620 had a two decimal digit exponent and variable length significand that could be longer than decimal 99 digits (as we recall).

[m] The statement that "the argument must lie between two integral values" is false. Consider the exact values 0.0 and 1.0; neither is between.

[m] Page 75, old sections A.15.8, A.15.9, A.15.10, A.15.11. Need to add words to exclude zero from raising underflow in the last sentence.

[m] Page 75, section A.16. Change "All language" to "Most language". ALGOL-60 has no I/O. Change the first "I/O" to "Input/Output (I/O)".

[m] Page 76, section A.18, 1st paragraph. Remove "and/or range" or explain how using one gets more exponent range by using doubled representation. Change "highest" to "most precise".

[m] Page 79, section A.20.1. Remove "It must never produce an underflow." Remove "This claim is not always valid."

[m] Page 99, section C, item [1]. Add "second edition" to the IEC 559 reference. Change "854-1987" to "754-1985".

[m] Page 99, section C, item [14]. We believe that the 1994 standard has been published, so remove "(to be published)".

[m] Page 100, section C. Add item [23.5] ANSI/IEEE Std 754-1985, IEEE Standard for Binary Floating-Point Arithmetic.

[m] Page 101, section C, item [37]. This appears to be two references in one and needs to be split into two. The Sorenson/Tang reference appears to be correct in the first draft of LIA-2. In addition, the Floating-Point C Extensions (FPCE) reference is out of date. It should be: Numerical C Technical Report, X3/TR-17:1995.

E [EDITORIAL] COMMENTS

[E] Page 19, section 10.3, before Signature. What is "plia2.tex"?

[E] Page 19, section 10.3, IEC 559 Special Values. What is the ">/" symbol in "if w >/ 1"? Should it be w not element of F or w <= 1?

[E] Page 22, section 11.5, Axioms. The exponent "(-1-p/2)" needs to be fixed so that it is all an exponent.

[E] Page 28, section 11.17, Exceptions. "if ( 2|u," is missing some text.

[E] Page 29, section 12.1, Signature. Change "bad_argument underflow" to "bad_argument, underflow".

[E] Page 42, section 15.1, title. Change "Convert to float" to "Convert to integer".

[E] Page 45, section 15.10 and higher, make the "if"s line up vertically.

[E] Page 51, sections 17.6 and 17.8, IEC Special Values. Add some spaces before F(w,v) = F(v,w) and after the +INF.

[E] Page 57, section 20.1, IEC 559 Special Values. Change "(w.v)" to "(w,v)".

[E] Page 60, section A.2, last line. Change "LIA=2" to "LIA-2".

[E] Page 61, section A.5.2. Change "log" to "LOG".

[E] Page 63, section A.5.6, 3rd paragraph. The reference to [37] is wrong because reference [37] appears to be two documents combined into one item.

[E] Page 74, section A.15.3. The second paragraph should be section A.15.4 TRUNCATE.

[E] Pages 74 and 75. Increase the number of each section after A.15.3 by 1.

[E] Page 75, old section A.15.8. Change "th%" to "the" and fix the fmaxG where the G overlaps the x.

[E] Page 77, section A.18.2.3. Change "MULHI" to "MULLO".

[E] Page 77, section A.18.2.4. Change "DIVHI" to "DIVLO".

I [INFORMATIONAL] COMMENTS

[I] Page 100, section C, item [28]. There is a corrected version of that Handbook of Mathematical Functions that has been published by Dover.