Integer square root (rounded to nearest integer) operation

sqrt

I(

x

) = round(^p

x

) if

x

²

I

and

x

^0

=

invalid

if

x

²

I

and

x <

0

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E) 5.1.6 Divisibility and even/odd test operations

divides

I :

I

^

I

^!

Boolean

divides

I(

x;y

) =

true

if

x;y

²

I

and

x

^j

y

=

false

if

x;y

²

I

and not

x

^j

y

NOTES

1 dividesI(0;0) =^false, since 0 does not divide anything, not even 0.

2 dividesI cannot be implemented as, e.g., eqI(0;rem^f_I(y;x)), since the remainder functions are^undenedfor a zero second argument.

even

I :

I

^!

Boolean

even

I(

x

) =

true

if

x

²

I

and 2^j

x

=

false

if

x

²

I

and not 2^j

x odd

I :

I

^!

Boolean

odd

I(

x

) =

true

if

x

²

I

and not 2^j

x

=

false

if

x

²

I

and 2^j

x 5.1.7 Additional integer division and remainder operations

quot

I :

I

^

I

^!

I

^[f

integer over ow ; invalid

^g

quot

I(

x;y

) =

result

I(^d

x=y

^e) if

x;y

²

I

and

y

⁶= 0

=

invalid

if

x

²

I

and

y

= 0

pad

I :

I

^

I

^!

I

^[f

invalid

^g

pad

I(

x;y

) = (^d

x=y

^e

y

)^,

x

if

x;y

²

I

and

y

⁶= 0

=

invalid

if

x

²

I

and

y

= 0

remc

I :

I

^

I

^!

I

^[f

integer over ow ; invalid

^g

remc

I(

x;y

) =

result

I(

x

^,(^d

x=y

^e

y

))if

x;y

²

I

and

y

⁶= 0

=

invalid

if

x

²

I

and

y

= 0

divr

I :

I

^

I

^!

I

^[f

integer over ow ; invalid

^g

divr

I(

x;y

) =

result

I(round(

x=y

)) if

x;y

²

I

and

y

⁶= 0

=

invalid

if

x

²

I

and

y

= 0

remr

I :

I

^

I

^!

I

^[f

integer over ow ; invalid

^g

remr

I(

x;y

) =

result

I(

x

^,(round(

x=y

)^

y

))

if

x;y

²

I

and

y

⁶= 0

=

invalid

if

x

²

I

and

y

= 0

NOTE { remcI and remrI can over ow only for unsigned integer datatypes (minI = 0).

ISO/IEC CD 10967-2.3:1998(E) Third Committee Draft 5.1.8 Greatest common divisor and least common multiple operations

gcd

I :

I

^

I

^!

I

^[f

integer over ow ; invalid

^g

gcd

I(

x;y

) =

result

I(max^f

v

^{2Z j}

v

^j

x

and

v

^j

y

^g)

if

x;y

²

I

and (

x

⁶= 0 or

y

⁶= 0)

=

invalid

if

x

= 0 and

y

= 0 and +¹ is not available NOTES

1 Returning 0 for gcd_I(0;0), as is sometimes suggested, would be incorrect, since the greatest common divisor for 0 and 0 is innity.

2 gcd_I will over ow only if bounded_I=^true, minint_I =^,maxint_I^,1, and both arguments to gcd_I are minint_I. The greatest common divisor is then^,minint_I, which is then not in I.

lcm

I :

I

^

I

^!

I

^[f

integer over ow

^g

lcm

I(

x;y

) =

result

I(min^f

v

^{2Z j}

x

^j

v

and

y

^j

v

and

v >

0^g)

if

x;y

²

I

and

x

⁶= 0 and

y

⁶= 0

= 0 if

x;y

²

I

and (

x

= 0 or

y

= 0)

NOTE 3 { lcm_I(x;y) over ows for many arguments: e.g., if x and y are relative primes, then the least common multiple is^jx^y^j, which may be greater than maxint_I.

gcd seq

I : [

I

]^!

I

^[f

integer over ow ; invalid

^g

gcd seq

I([

x

¹

;:::;x

n])

=

result

I(max^f

v

^{2Z j}

v

^j

x

i for all

i

^2f1

;:::;n

^gg)

if ^f

x

¹

;:::;x

n^g

I

and ^f0^g6=^f

x

¹

;:::;x

n^g

=

invalid

if ^f0^g=^f

x

¹

;:::;x

n^gand +¹ is not available

lcm seq

I : [

I

]^!

I

^[f

integer over ow

^g

lcm seq

I([

x

¹

;:::;x

n])

=

result

I(min^f

v

^{2Z j}

x

i^j

v

for all

i

^2f1

;:::;n

^gand

v >

0^g) if ^f

x

¹

;:::;x

n^g

I

and 0^62f

x

¹

;:::;x

n^g

= 0 if ^f

x

¹

;:::;x

n^g

I

and 0^2f

x

¹

;:::;x

n^g

5.1.9 Support operations for extended integer range

These operations can be used to implement extended range integer datatypes, and unbounded integer datatypes.

add wrap

I :

I

^

I

^!

I

add wrap

I(

x;y

) =

wrap

I(

x

+

y

) if

x;y

²

I add ov

I :

I

^

I

^!f,1

;

0

;

1^g

add ov

I(

x;y

) = ((

x

+

y

)^,

add wrap

I(

x;y

))

=

(

maxint

I^,

minint

I+ 1) if

x;y

²

I

and

I

⁶=^Z

= 0 if

x;y

²

I

and

I

=^Z

sub wrap

I :

I

^

I

^!

I

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E) sub wrap

I(

x;y

) =

wrap

I(

x

^,

y

) if

x;y

²

I

sub ov

I :

I

^

I

^!f,1

;

0

;

1^g

sub ov

I(

x;y

) = ((

x

^,

y

)^,

sub wrap

I(

x;y

))

=

(

maxint

I^,

minint

I + 1) if

x;y

²

I

and

I

⁶=^Z

= 0 if

x;y

²

I

and

I

=^Z

mul wrap

I :

I

^

I

^!

I

mul wrap

I(

x;y

) =

wrap

I(

x

^

y

) if

x;y

²

I mul ov

I :

I

^

I

^!

I

mul ov

I(

x;y

) = ((

x

^

y

)^,

mul wrap

I(

x;y

))

=

(

maxint

I ^,

minint

I + 1) if

x;y

²

I

and

I

⁶=^Z

= 0 if

x;y

²

I

and

I

=^Z

NOTE { The add ovI and sub ovI will only return^,1 (for negative over ow), 0 (no over ow), and 1 (for positive over ow).

5.2 Additional basic oating point operations

Clause 5.2 of ISO/IEC 10967-1 species oating point datatypes and a number of operations on values of a oating point datatype. In this clause some additional operations on values of a oating point datatype are specied.

NOTE { Further operations on values of a oating point datatype, for elementary oating point numerical functions, are specied in clause 5.3.

F

is a oating point type conforming to ISO/IEC 10967-1. Floating point datatypes con-forming to ISO/IEC 10967-1 usually do contain^,

0

, innity, and

NaN

values. Therefore, in this clause there are specications for such values as arguments.

5.2.1 The rounding and oating point result helper functions

Floating point rounding helper functions:

down

F :^R!

F

^

is a rounding function. It rounds towards negative innity.

NOTE 1 { F^is dened in ISO/IEC 10967-1. It is the unbounded extension of F.

up

F :^R!

F

^

is a rounding function. It rounds towards positive innity.

nearest

F :^R!

F

^

is a rounding function, that is partially implementation dened. It rounds to nearest. The handling of ties is implementation dened, but must be sign symmetric. If iec 559F =

true

, the semantics of

nearest

F is completely determined: ties are rounded to even last digit by

nearest

F.

result

F is a helper function that is partially implementation dened. The specication from ISO/IEC 10967-1 is repeated here, but here details regarding continuation values upon over ow and under ow are given.

NOTE 2 { These details are intended to be in accordance with IEC 559 wheniec 559F =

true.

ISO/IEC CD 10967-2.3:1998(E) Third Committee Draft

and under ow is only recorded in indicator

=

under ow

(

x

) if iec 559F =

true

and

x

⁶= 0

5.2.2 Floating point maximum and minimum operations

What the maximum and minimum operations should return on one quiet

NaN

(

qNaN

) input depends on the context. Sometimes

qNaN

is the appropriate result, sometimes the non-

NaN

argument is the appropriate result. Therefore, two variants (each) of the oating point maxi-mum and minimaxi-mum operations are specied here, and the programmer can decide which one is appropriate to use at each particular place of usage, if both are included in the ISO/IEC 10967-2 binding.

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E)

= +¹ if

y

= +¹and

x

²

F

^[f+¹

;

^,

0

^g

=

x

if

y

=^,

0

and

x

²

F

and

x

^0

=^,

0

if

y

=^,

0

and

x

²

F

and

x <

0

=

x

if

y

=^,1and

x

²

F

^[f,1

;

^,

0

^g

=

qNaN

if

x

is a quiet NaN and

y

is not a signalling NaN

=

qNaN

if

y

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN

min

F :

F

^

F

^!

F

min

F(

x;y

) = min^f

x;y

^g if

x;y

²

F

=

y

if

x

= +¹ and

y

²

F

^[f,1

;

^,

0

^g

=^,

0

if

x

=^,

0

and

y

²

F

and

y

^0

=

y

if

x

=^,

0

and ((

y

²

F

and

y <

0) or

y

=^,

0

)

=^,1 if

x

=^,1and

y

²

F

^[f+¹

;

^,

0

^g

=

x

if

y

= +¹and

x

²

F

^[f+¹

;

^,

0

^g

=^,

0

if

y

=^,

0

and

x

²

F

and

x

^0

=

x

if

y

=^,

0

and

x

²

F

and

x <

0

=^,1 if

y

=^,1and

x

²

F

^[f,1

;

^,

0

^g

=

qNaN

if

x

is a quiet NaN and

y

is not a signalling NaN

=

qNaN

if

y

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN

mmax

F :

F

^

F

^!

F

mmax

F(

x;y

) =

max

F(

x;y

) if

x;y

²

F

^[f+¹

;

^,

0 ;

^,1g

=

x

if

x

²

F

^[f+¹

;

^,

0 ;

^,1gand

y

is a quiet NaN

=

y

if

y

²

F

^[f+¹

;

^,

0 ;

^,1g and

x

is a quiet NaN

=

qNaN

if

x

is a quiet NaN and

y

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN

mmin

F :

F

^

F

^!

F

mmin

F(

x;y

) =

min

F(

x;y

) if

x;y

²

F

^[f+¹

;

^,

0 ;

^,1g

=

x

if

x

²

F

^[f+¹

;

^,

0 ;

^,1gand

y

is a quiet NaN

=

y

if

y

²

F

^[f+¹

;

^,

0 ;

^,1g and

x

is a quiet NaN

=

qNaN

if

x

is a quiet NaN and

y

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN NOTE { If one of the arguments to mmaxF or mminF is a quiet NaN, that argument is ignored.

max seq

F : [

F

]^!

F

^[f,1

; invalid

^g

max seq

F([

x

¹

;:::;x

n])

=^,1 if

n

= 0 and ^,1is available

=

invalid

(

qNaN

) if

n

= 0 and ^,1is not available

=

x

¹ if

n

= 1 and

x

¹ is not a NaN

=

qNaN

if

n

= 1 and

x

¹ is a quiet NaN

=

invalid

(

qNaN

) if

n

= 1 and

x

¹ is a signalling NaN

=

max

F(

max seq

F([

x ;:::;x

n ])

;x

n)

ISO/IEC CD 10967-2.3:1998(E) Third Committee Draft

if

n

^2

min seq

F : [

F

]^!

F

^[f+¹

; invalid

^g

min seq

F([

x

¹

;:::;x

n])

= +¹ if

n

= 0 and +¹ is available

=

invalid

(

qNaN

) if

n

= 0 and +¹ is not available

=

x

¹ if

n

= 1 and

x

¹ is not a NaN

=

qNaN

if

n

= 1 and

x

¹ is a quiet NaN

=

invalid

(

qNaN

) if

n

= 1 and

x

¹ is a signalling NaN

=

min

F(

min seq

F([

x

¹

;:::;x

n^,1])

;x

n) if

n

^2

mmax seq

F : [

F

]^!

F

^[f,1

; invalid

^g

mmax seq

F([

x

¹

;:::;x

n])

=^,1 if

n

= 0 and ^,1is available

=

invalid

(

qNaN

) if

n

= 0 and ^,1is not available

=

x

¹ if

n

= 1 and

x

¹ is not a signalling NaN

=

invalid

(

qNaN

) if

n

= 1 and

x

¹ is a signalling NaN

=

mmax

F(

mmax seq

F([

x

¹

;:::;x

n^,1])

;x

n) if

n

^2

mmin seq

F : [

F

]^!

F

^[f+¹

; invalid

^g

mmin seq

F([

x

¹

;:::;x

n])

= +¹ if

n

= 0 and +¹ is available

=

invalid

(

qNaN

) if

n

= 0 and +¹ is not available

=

x

¹ if

n

= 1 and

x

¹ is not a signalling NaN

=

invalid

(

qNaN

) if

n

= 1 and

x

¹ is a signalling NaN

=

mmin

F(

mmin seq

F([

x

¹

;:::;x

n^,1])

;x

n) if

n

^2

5.2.3 Floating point positive dierence (monus, diminish) operation dim

F :

F

^

F

^!

F

^[f

oating over ow ; under ow

^g

dim

F(

x;y

) =

result

F(max^f0

;x

^,

y

)^g

;rnd

F) if

x;y

²

F

=

dim

F(0

;y

) if

x

=^,

0

and

y

²

F

^[f,1

;

^,

0 ;

+^1g

=

dim

F(

x;

0) if

y

=^,

0

and

x

²

F

^[f,1

;

+^1g

= +¹ if

x

= +¹ and

y

²

F

^{[f,1 g}

=

invalid

(

qNaN

) if

x

= +¹ and

y

= +¹

= 0 if

x

=^,1 and

y

²

F

^[f+^{1 g}

=

invalid

(

qNaN

) if

x

=^,1 and

y

=^,1

= 0 if

y

= +¹ and

x

²

F

= +¹ if

y

=^,1 and

x

²

F

=

qNaN

if

x

is a quiet NaN and

y

is not a signalling NaN

=

qNaN

if

y

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN NOTE { dimF cannot be implemented by maxF(0;subF(x;y)), since this latter expression has other over ow properties.

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E) 5.2.4 Round, oor, and ceiling operations

rounding

F :

F

^!

F

^[f,

0

^g

rounding

F(

x

) = round(

x

) if

x

²

F

and (

x

^0 or round(

x

)⁶= 0)

=

neg

F(0) if

x

²

F

and

x <

0 and round(

x

) = 0

=^,

0

if

x

=^,

0

= +¹ if

x

= +¹

=^,1 if

x

=^,1

=

qNaN

if

x

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN oor_F :

F

^!

F

oor_F(

x

) =^b

x

^c if

x

²

F

=^,

0

if

x

=^,

0

= +¹ if

x

= +¹

=^,1 if

x

=^,1

=

qNaN

if

x

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN

ceiling

F :

F

^!

F

^[f,

0

^g

ceiling

F(

x

) =^d

x

^e if

x

²

F

and (

x

^0 or ^d

x

^e6= 0)

=

neg

F(0) if

x

²

F

and

x <

0 and ^d

x

^e= 0

=^,

0

if

x

=^,

0

= +¹ if

x

= +¹

=^,1 if

x

=^,1

=

qNaN

if

x

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN NOTES

1 The result in the second case for roundingF and ceilingF is 0, if ^,0 is not in the type corresponding to F, otherwise it is^,0.

2 oor_F(x) = negF(ceilingF(negF(x))), ceilingF(x) = negF( oor_F(negF(x))), and roundingF(x) = negF(roundingF(negF(x))).

Negative zeroes, if available, are handed in such a way as to maintain these identites.

3 Truncate to integer is specied in ISO/IEC 10967-1:1994, by the name intpartF.

rounding rest

F :

F

^!

F rounding rest

F(

x

)

=

x

^,round(

x

) if

x

²

F

= 0 if

x

=^,

0

=

invalid

(

qNaN

) if

x

= +¹

=

invalid

(

qNaN

) if

x

=^,1

=

qNaN

if

x

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN oor rest_F :

F

^!

F

ISO/IEC CD 10967-2.3:1998(E) Third Committee Draft

oor rest_F(

x

) =

x

^,b

x

^c if

x

²

F

= 0 if

x

=^,

0

=

invalid

(

qNaN

) if

x

= +¹

=

invalid

(

qNaN

) if

x

=^,1

=

qNaN

if

x

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN

ceiling rest

F :

F

^!

F

ceiling rest

F(

x

)

=

x

^,d

x

^e if

x

²

F

= 0 if

x

=^,

0

=

invalid

(

qNaN

) if

x

= +¹

=

invalid

(

qNaN

) if

x

=^,1

=

qNaN

if

x

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN

NOTE 4 { The rest after truncation is specied in ISO/IEC 10967-1:1994, by the name fractpart_F.

5.2.5 Operation for remainder after division and round to integer (IEEE remainder) irem

F :

F

^

F

^!

F

^[f,

0 ; under ow ; invalid

^g

irem

F(

x;y

) =

result

F(

x

^,(round(

x=y

)^

y

)

;nearest

F)

if

x;y

²

F

and

y

⁶= 0 and

(

x

^0 or

x

^,(round(

x=y

)^

y

)⁶= 0)

=^,

0

if

x;y

²

F

and

y

⁶= 0 and

x <

0 and

x

^,(round(

x=y

)^

y

) = 0

=^,

0

if

x

=^,

0

and

y

²

F

^[f,1

;

+^1g and

y

⁶= 0

=

x

if

x

²

F

and

y

^2f,1

;

+^1g

=

invalid

(

qNaN

) if

x

²

F

^[f,1

;

^,

0 ;

+^1g and

y

=^,

0

=

invalid

(

qNaN

) if

x

²

F

^[f,

0

^g and

y

= 0

=

invalid

(

qNaN

) if

x

^2f,1

;

+^1g and

y

²

F

^[f,1

;

+^1g

=

qNaN

if

x

is a quiet NaN and

y

is not a signalling NaN

=

qNaN

if

y

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN

5.2.6 Square root and reciprocal square root operations sqrt

F :

F

^!

F

^[f

invalid

^g

sqrt

F(

x

) =

nearest

F(^p

x

) if

x

²

F

and

x

^0

=^,

0

if

x

=^,

0

=

invalid

(

qNaN

) if (

x

²

F

and

x <

0) or

x

=^,1

= +¹ if

x

= +¹

=

qNaN

if

x

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN

rec sqrt

F :

F

^!

F

^[f

invalid ; pole

^g

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E) rec sqrt

F(

x

) =

rnd

F(1

=

^p

x

) if

x

²

F

and

x >

0

=

pole

(+¹) if

x

²

F

and

x

= 0

=

pole

(+¹) if

x

=^,

0

= 0 if

x

= +¹

=

invalid

(

qNaN

) if (

x

²

F

and

x <

0) or

x

=^,1

=

qNaN

if

x

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN

5.2.7 Support operations for extended oating point precision add lo

F :

F

^

F

^!

F

^[f

oating over ow ; under ow

^g

add lo

F(

x;y

) =

result

F((

x

+

y

)^,

rnd

F(

x

+

y

)

;rnd

F)

if

x;y;add

F(

x;y

)²

F

=

under ow

(0)? if

add

F(

x;y

) =

under ow

(

u

)

= 0? if

add

F(

x;y

) =

oating over ow

(+¹)

= 0? if

add

F(

x;y

) =

oating over ow

(^,1)

=

add lo

F(0

;y

) if

x

=^,

0

and

y

²

F

^[f,1

;

^,

0 ;

+^1g

=

add lo

F(

x;

0) if

y

=^,

0

and

x

²

F

^[f,1

;

+^1g

=

invalid

(

qNaN

)? if

x

^2f,1

;

+^1g and

y

²

F

^[f,1

;

+^1g

=

invalid

(

qNaN

)? if

y

^2f,1

;

+^1g and

x

²

F

=

qNaN

if

x

is a quiet NaN and

y

is not a signalling NaN

=

qNaN

if

y

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN

sub lo

F :

F

^

F

^!

F

^[f

oating over ow ; under ow

^g

sub lo

F(

x;y

) =

result

F((

x

^,

y

)^,

rnd

F(

x

^,

y

)

;rnd

F)

if

x;y;sub

F(

x;y

)²

F

=

under ow

(0)? if

sub

F(

x;y

) =

under ow

(

u

)

=

oating over ow

(^,1)?0?

if

sub

F(

x;y

) =

oating over ow

(+¹)

=

oating over ow

(+¹)?0?

if

sub

F(

x;y

) =

oating over ow

(^,1)

=

sub lo

F(0

;y

) if

x

=^,

0

and

y

²

F

^[f,1

;

^,

0 ;

+^1g

=

sub lo

F(

x;

0) if

y

=^,

0

and

x

²

F

^[f,1

;

+^1g

=

invalid

(

qNaN

)? if

x

^2f,1

;

+^1g and

y

²

F

^[f,1

;

+^1g

=

invalid

(

qNaN

)? if

y

^2f,1

;

+^1g and

x

²

F

=

qNaN

if

x

is a quiet NaN and

y

is not a signalling NaN

=

qNaN

if

y

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN NOTES

1 If rnd styleF = nearest, then, in the absence of notications, add loF and sub loF returns exact results.

2 sub loF(x;y) = add loF(x;negF(y)).

mul lo

F :

F

^

F

^!

F

^[f

oating over ow ; under ow

^g

mul lo

F(

x;y

) =

result

F((

x

^

y

)^,

rnd

F(

x

^

y

)

;rnd

F)

if

x;y;mul

F(

x;y

)²

F

=

under ow

(0)? if

mul

F(

x;y

) =

under ow

(

u

)

ISO/IEC CD 10967-2.3:1998(E) Third Committee Draft

= 0 if

x;y

²

F

and

mul

F(

x;y

) =^,

0

=

oating over ow

(^,1)?0?

if

mul

F(

x;y

) =

oating over ow

(+¹)

=

oating over ow

(+¹)?0?

if

mul

F(

x;y

) =

oating over ow

(^,1)

=

mul lo

F(0

;y

) if

x

=^,

0

and

y

²

F

^[f,1

;

^,

0 ;

+^1g

=

mul lo

F(

x;

0) if

y

=^,

0

and

x

²

F

^[f,1

;

+^1g

=

invalid

(

qNaN

)? if

x

^2f,1

;

+^1g and

y

²

F

^[f,1

;

+^1g

=

invalid

(

qNaN

)? if

y

^2f,1

;

+^1g and

x

²

F

=

qNaN

if

x

is a quiet NaN and

y

is not a signalling NaN

=

qNaN

if

y

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN NOTE 3 { In the absence of notications, mul loF returns an exact result.

div rest

F :

F

^

F

^!

F

^[f

oating over ow ; under ow ; invalid

^g

div rest

F(

x;y

)=

result

F(

x

^,(

y

^

rnd

F(

x=y

))

;rnd

F)

if

x;y;div

F(

x;y

)²

F

=

result

F(

x

^,(

y

^

u

)

;rnd

F)

if

div

F(

x;y

) =

under ow

(

u

) and

z

²

F

=

x

if

x;y

²

F

and

(

div

F(

x;y

) =^,

0

or

div

F(

x;y

) =

under ow

(^,

0

))

=

invalid

(

qNaN

) if

x

²

F

and

y

= 0

=

oating over ow

(^,1)?0?

if

div

F(

x;y

) =

oating over ow

(+¹)

=

oating over ow

(+¹)?0?

if

div

F(

x;y

) =

oating over ow

(^,1)

=

div rest

F(0

;y

) if

x

=^,

0

and

y

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

y

=^,

0

and

x

²

F

^[f,1

;

+^1g

=

invalid

(

qNaN

)? if

x

^2f,1

;

+^1g and

y

²

F

^[f,1

;

+^1g

=

invalid

(

qNaN

)? if

y

^2f,1

;

+^1g and

x

²

F

=

qNaN

if

x

is a quiet NaN and

y

is not a signalling NaN

=

qNaN

if

y

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN

sqrt rest

F :

F

^!

F

^[f

under ow ; invalid

^g

sqrt rest

F(

x

) =

result

F(

x

^,(

sqrt

F(

x

)^

sqrt

F(

x

))

;rnd

F) if

x

²

F

and

x

^0

=^,

0

if

x

=^,

0

=

invalid

(

qNaN

) if (

x

²

F

and

x <

0) or

x

=^,1

=

invalid

(

qNaN

)?0? if

x

= +¹

=

qNaN

if

x

is a quiet NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN NOTE 4 { sqrt restF(x) is exact when there is no^{under ow}. add3F :

F

^

F

^

F

^!

F

^[f

oating over ow ; under ow

^g

add3F(

x;y;z

) =

result

F((

x

+

y

) +

z;rnd

F)

if

x;y;z

²

F

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E)

not

y

nor

z

is a signalling NaN

=

qNaN

if

y

is a quiet NaN and

not

x

nor

z

is a signalling NaN

=

qNaN

if

z

is a quiet NaN and

not

x

nor

y

is a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

not

y

nor

z

is a signalling NaN

=

qNaN

if

y

is a quiet NaN and

not

x

nor

z

is a signalling NaN

=

qNaN

if

z

is a quiet NaN and

not

x

nor

y

is a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

ISO/IEC CD 10967-2.3:1998(E) Third Committee Draft

not

y

nor

z

is a signalling NaN

=

qNaN

if

y

is a quiet NaN and

not

x

nor

z

is a signalling NaN

=

qNaN

if

z

is a quiet NaN and

not

x

nor

y

is a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

not

y

nor

z

is a signalling NaN

=

qNaN

if

y

is a quiet NaN and

not

x

nor

z

is a signalling NaN

=

qNaN

if

z

is a quiet NaN and

not

x

nor

y

is a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN or

z

is a signalling NaN

For the following operation

F

⁰is a oating point type conforming to ISO/IEC 10967-1.

NOTE 7 { It is expected that p_F⁰ > p_F, i.e. F⁰has higher precision than F, but that is not required.

mul

F^!F⁰ :

F

^

F

^!

F

^0[f,

0 ; oating over ow ; under ow

^g

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E)

NOTE 8 { Converting a signalling^NaNresults in a notication of^invalid. See clause 5.4.

5.2.8 Exact summation operation

An exact summation operation is useful for computing high accuracy sums, even if only the rst element of the resulting list is ultimately kept.

In order to be able to specify the exact sum operation, which sums a sequence of oating point numbers returning a sequence of oating point numbers of decreasing magnitude, by

p

F, a number of helper functions are needed.

=

sNaN

if

x

is a signalling NaN or

y

is a signalling NaN The extended real summation helper function:

ISO/IEC CD 10967-2.3:1998(E) Third Committee Draft

=

rnd

(

x

) :

seq result

F(

x

^,

rnd

(

x

)

;rnd

)

if

rnd

(

x

)⁶= 0 and

rnd

(

x

)²

F

and (

denorm

F =

true

or ^j

x

^jfminN_F)

= [

rnd

(

x

^,fminN_F)

;

fminN_F]

if ^,fminN_F

< x

and

x <

0 and

denorm

F =

false

= [

rnd

(

x

+ fminN_F)

;

^,fminN_F]

if 0

< x

and

x <

fminN_F and

denorm

F =

false

The exact summation operation:

sum

F : [

F

]^![

F

]^[f

oating over ow

^g

sum

F([

x

¹

;:::;x

n])

=

seq result

F(

sum

([

x

¹

;:::;x

n])

;nearest

F)

if

sum

([

x

¹

;:::;x

n])^2R and

n

^1

= [

sum

([

x

¹

;:::;x

n])] if

sum

([

x

¹

;:::;x

n])^2f,1

;

^,

0 ;

+^1g and

n

^1

= [^,

0

] if

n

= 0 and ^,

0

is available

= [0] if

n

= 0 and ^,

0

is not available

= [

qNaN

] if

sum

([

x

¹

;:::;x

n]) is a quiet NaN

=

invalid

([

qNaN

]) if

sum

([

x

¹

;:::;x

n]) is a signalling NaN NOTE { sum_F(sum_F(a)) = sum_F(a), and sum_F(sum_F(a)++sum_F(b)) = sum_F(a++b) if there is no notication (where ++ is sequence concatenation). Thus sum_F([]) = sum_F([^,0]).

5.3 Elementary transcendental oating point operations 5.3.1 Specication format

5.3.1.1 Maximum error requirements

The specications for each of the transcendental operations use an approximation helper function.

The approximation helper functions are ideally identical to the true mathematical functions.

However, that would imply that the maximum error for the corresponding operation was merely 0.5 ulp. This part of ISO/IEC 10967 does not require that the maximum error is only 0.5 ulp, but may be a bit bigger. To express this, the approximation helper functions need not be identical to the mathematical elementary transcendental functions, but are allowed to be approximate.

The approximation helper functions for the individual operations in this subclause have maxi-mum error parameters that describe the maximaxi-mum relative error of the helper function composed with

nearest

F, for normalised results. The maximum error parameter also describe the maximum absoluteerror for subnormal continuation values if

denorm

F =

true

. The relevant maximum er-ror parameters shall be available to programs.

That for a helper function

h

F, approximating

f

, the maximum error is

max error op

F means that for all arguments

x;:::

²

F

^

:::

the following inequality is true:

j

f

(

x;:::

)^,

nearest

F(

h

F(

x;:::

))^j

max error op

F ^

r

^{eF(f(x;:::)),pF}

NOTES

1 Partially conforming implementations may have greater values for maximum error param-eters than stipulated below. See annex B.

2 For most positive (and not too small) return values t, the true result is thus claimed to be in the interval [t^,(max error opF ^ulpF(t));t + (max error opF ^ulpF(t))]. But if the return value is exactly r_nF for some n^2Z, then the true result is claimed to be in the interval [t^,(max error opF ^ulpF(t)=rF);t + (max error opF^ulpF(t))], Similarly for negative return values.

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E)

The results of the approximating helper functions in this clause must be exact for certain arguments as detailed below, and may be exact for all arguments. If the approximating helper function is exact for all arguments, then the corresponding maximum error parameter should be 0.5, the minimum value.

5.3.1.2 The trans result helper function

The

trans result

F helper function is similar to the

result

F helper function extended with spec-ications for the continuation value on over ow, and it also returns^,

0

for negative under ows that round (or are ushed) to zero, if possible. (Those extentions are implied in ISO/IEC 10967-1 for IEC 559 conforming implementations.) But

trans result

F is simplied compared to

result

F

concerning

under ow

:

trans result

F always under ows for nonzero arguments that have an absolute value less than fminN_F, whereas

result

F does not always under ow then.

In addition, the rounding is xed to

nearest

F, rather than being parameterised. This is user visible only in the cases where the operation's approximation helper function is (required to be) exact, but where that value is not representable in

F

, e.g.

e

or



.

trans result

F :^R!

F

^[f

under ow ; oating over ow

^g

The approximation helper functions are required to be zero exactly at the points where the approximated mathematical function is exactly zero. At points where the approximation helper functions are not zero, they are required to have the same sign as the approximated mathematical function at that point.

For the radian trigonometric helper functions, this sign requirement is imposed only for argu-ments,

x

, such that ^j

x

^j

big angle r

F (see clause 5.3.6).

NOTE { For the operations, the continuation value after an ^{under ow} may be zero (or negative zero) as given by trans result_F, even though the approximation helper function is not zero at that point. Such zero results are required to be accompanied by an^{under ow}

ISO/IEC CD 10967-2.3:1998(E) Third Committee Draft

notication. When appropriate, zero may also be returned for IEC 559 innities arguments.

See the individual specications.

5.3.1.4 Monotonicity requirements

When the maximum error is tight, i.e. 0.5 ulp, that implies that the approximation helper func-tions must be monotonous on the same intervals as the corresponding exact function is strictly monotonous. When the maximum error is greater than 0.5 ulp, and the rounding is not directed, a numerical function is not automatically monotonous where the corresponding exact function is.

The approximation helper functions in this clause are required to be monotonous on the same intervals as the mathematical functions they are approximating are monotonous. There is no general requirement that the approximation helper functions are strictly monotonous on the same intervals as the corresponding exact function is strictly monotonous, however, since such a requirement cannot be made due to that all oating point types are discrete, not continuous.

For the radian trigonometric helper functions, this monotonicity requirement is imposed only for arguments,

x

, such that ^j

x

^j

big angle r

F (see clause 5.3.6).

The unit argument trigonometric and unit argument inverse trigonometric approximating helper functions are excepted from the monotonicity requirement for the angular unit argument.

5.3.2 Hypotenuse operation

Maximum error parameter for the

hypot

F operation:

max error hypot

F ²

F

The

max error hypot

F parameter is required to be in the interval [0

:

5

;

1].

The

hypot

^_F approximation helper function:

hypot

^_F :

F

^

F

^!R

hypot

^_F(

x;y

)returns a close approximation to^p

x

²+

y

²in^R, with maximum error

max error hypot

F. Further requirements on the

hypot

^_F approximation helper function:

hypot

^_F(

x;y

) =

hypot

^_F(

y;x

)

hypot

^_F(^,

x;y

) =

hypot

^_F(

x;y

)

hypot

^_F(

x;y

)^max^fj

x

^j

;

^j

y

^jg

hypot

^_F(

x;y

)^j

x

^j+^j

y

^j

hypot

^_F(

x;y

)^1 if ^p

x

²+

y

^21

hypot

^_F(

x;y

)^1 if ^p

x

²+

y

^21 The

hypot

F operation:

hypot

F :

F

^

F

^!

F

^[f

under ow ; oating over ow

^g

hypot

F(

x;y

) =

trans result

F(

hypot

^_F(

x;y

))

if

x;y

²

F

=

hypot

F(0

;y

) if

x

=^,

0

and

y

²

F

^[f,1

;

^,

0 ;

+^1g

=

hypot

F(

x;

0) if

y

=^,

0

and

x

²

F

^[f,1

;

+^1g

= +¹ if

x

^2f,1

;

+^1g and

y

²

F

^[f,1

;

+^1g

= +¹ if

y

^2f,1

;

+^1g and

x

²

F

=

qNaN

if

x

is a quiet NaN and

y

is not a signalling NaN

=

qNaN

if

y

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

y

is a signalling NaN

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E) 5.3.3 Operations for exponentiations and logarithms

There are two maximum error parameters for approximate exponentiations and logarithms:

max error exp

F ²

F max error power

F ²

F

The

max error exp

F parameter is required to be in the interval [0

:

5

;

1

:

5^

rnd error

F].

The

max error power

F parameter is required to be in the interval [

max error exp

F

;

2^

The

max error power

F parameter is required to be in the interval [

max error exp

F

;

2^

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