Argument angular-unit tangentus operation

The

tanu

^_F approximation helper function:

tanu

^_F :

F

^R!R

tanu

^_F(

u;x

) returns a close approximation to tan(

x

^2^

=u

) in ^Rif

u

⁶= 0, with maximum error

max error tanu

F(

u

).

Further requirements on the

tanu

^_F approximation helper function:

tanu

^_F(

u;n

^

u

+

x

) =

tanu

^_F(

u;x

) if

n

^2Z and

u

²

F

and

u

⁶= 0

tanu

^_F(

u;u=

8) = 1 if

u

²

F

and

u

⁶= 0

tanu

^_F(

u;

3^

u=

8) =^,1 if

u

²

F

and

u

⁶= 0

tanu

^_F(

u;

^,

x

) =^,

tanu

^_F(

u;x

) if

u

²

F

and

u

⁶= 0

tanu

^_F(^,

u;x

) =^,

tanu

^_F(

u;x

) if

u

²

F

and

u

⁶= 0 NOTE 1 { tanu^_F(u;x)^x^2^=u if^jx^2^=u^j<fminN_F. The

tanu

F operation:

tanu

F :

F

^

F

^!

F

^[f,

0 ; pole ; oating over ow ; under ow ; invalid ; angle too big

^g

tanu

F(

u;x

) =

trans result

F(

tanu

^_F(

u;x

))

if

cycle

F(

u;x

)²

F

and

cycle

F(

u;x

)^62f,

u=

2

;

^,

u=

4

;u=

4^g

=^,

0

if

cycle

F(

u;x

)²

F

and

cycle

F(

u;x

) =^,

u=

2

=^,

0

if

cycle

F(

u;x

) =^,

0

=

pole

(+¹) if

cycle

F(

u;x

) =

u=

4

=

pole

(^,1) if

cycle

F(

u;x

) =^,

u=

4

=

angle too big

(

qNaN

) if

u

²

G

F and

x

²

F

and ^j

x=u

^j

> big angle u

F

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

x

²

F

^[f,

0

^g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1g and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

²

F

and

x

^2f,1

;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN NOTE 2 { The^{p ole}notication can arise for tanuF(u;x) only when u=4 is in F.

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E) 5.3.7.6 Argument angular-unit cotangentus operation

The

cotu

^_F approximation helper function:

cotu

^_F :

F

^R!R

cotu

^_F(

u;x

) returns a close approximation to cot(

x

^2^

=u

) in ^R if

u

⁶= 0, with maximum error

max error tanu

F(

u

).

Further requirements on the

cotu

^_F approximation helper function:

cotu

^_F(

u;n

^

u

+

x

) =

cotu

^_F(

u;x

) if

n

^2Z and

u

²

F

and

u

⁶= 0

cotu

^_F(

u;u=

8) = 1 if

u

²

F

and

u

⁶= 0

cotu

^_F(

u;

3^

u=

8) =^,1 if

u

²

F

and

u

⁶= 0

cotu

^_F(

u;

^,

x

) =^,

cotu

^_F(

u;x

) if

u

²

F

and

u

⁶= 0

cotu

^_F(^,

u;x

) =^,

cotu

^_F(

u;x

) if

u

²

F

and

u

⁶= 0 The

cotu

F operation:

cotu

F :

F

^

F

^!

F

^[f,

0 ; pole ; oating over ow ; under ow ; invalid ; angle too big

^g

cotu

F(

u;x

) =

trans result

F(

cotu

^_F(

u;x

))

if

cycle

F(

u;x

)²

F

and

cycle

F(

u;x

)^62f,

u=

2

;

^,

u=

4

;

0

;u=

2^g

=^,

0

if

cycle

F(

u;x

)²

F

and

cycle

F(

u;x

) =^,

u=

4

=

pole

(+¹) if

cycle

F(

u;x

) = 0

=

pole

(^,1) if

cycle

F(

u;x

) =^,

0

=

pole

(+¹) if

cycle

F(

u;x

) =

u=

2

=

pole

(^,1) if

cycle

F(

u;x

) =^,

u=

2

=

angle too big

(

qNaN

) if

u

²

G

F and

x

²

F

and ^j

x=u

^j

> big angle u

F

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

x

²

F

^[f,

0

^g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1gand

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

²

F

and

x

^2f,1

;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN

5.3.7.7 Argument angular-unit secantus operation

The

secu

^_F approximation helper function:

secu

^_F :

F

^R!R

secu

^_F(

u;x

) returns a close approximation to sec(

x

^2^

=u

) in ^R if

u

⁶= 0, with maximum error

max error tanu

F(

u

).

Further requirements on the

secu

^_F approximation helper function:

secu

^_F(

u;n

^

u

+

x

) =

secu

^_F(

u;x

) if

n

^2Z and

u

²

F

and

u

⁶= 0

secu

^_F(

u;

0) = 1 if

u

²

F

and

u

⁶= 0

secu

^_F(

u;u=

6) = 2 if

u

²

F

and

u

⁶= 0

secu

^_F(

u;u=

3) =^,2 if

u

²

F

and

u

⁶= 0

secu

^_F(

u;u=

2) =^,1 if

u

²

F

and

u

⁶= 0

secu

^_F(

u;

^,

x

) =

secu

^_F(

u;x

) if

u

²

F

and

u

⁶= 0

secu

^_F(^,

u;x

) =

secu

^_F(

u;x

) if

u

²

F

and

u

⁶= 0

secu

^_F(

u;x

) = 1 if^j

x

^2^

=u

^j

<

0

:

5^p

epsilon

F

The

secu

F operation:

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

F :

F

^

F

^!

F

^[f

pole ; oating over ow ; invalid ; angle too big

^g

secu

F(

u;x

) =

trans result

F(

secu

^_F(

u;x

))

if

cycle

F(

u;x

)²

F

and

cycle

F(

u;x

)^62f,

u=

4

;u=

4^g

= 1 if

cycle

F(

u;x

) =^,

0

=

pole

(+¹) if

cycle

F(

u;x

) =

u=

4

=

pole

(+¹) if

cycle

F(

u;x

) =^,

u=

4

=

angle too big

(

qNaN

) if

u

²

G

F and

x

²

F

and ^j

x=u

^j

> big angle u

F

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

x

²

F

^[f,

0

^g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1g and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

²

F

and

x

^2f,1

;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN

5.3.7.8 Argument angular-unit cosecantus operation

The

cscu

^_F approximation helper function:

cscu

^_F :

F

^R!R

cscu

^_F(

u;x

) returns a close approximation to csc(

x

^2^

=u

) in ^R if

u

⁶= 0, with maximum error

max error tanu

F(

u

).

Further requirements on the

cscu

^_F approximation helper function:

cscu

^_F(

u;n

^

u

+

x

) =

cscu

^_F(

u;x

) if

n

^2Z and

u

²

F

and

u

²0

cscu

^_F(

u;u=

12) = 2 if

u

²

F

and

u

⁶= 0

cscu

^_F(

u;u=

4) = 1 if

u

²

F

and

u

⁶= 0

cscu

^_F(

u;

5^

u=

12) = 2 if

u

²

F

and

u

⁶= 0

cscu

^_F(

u;

^,

x

) =^,

cscu

^_F(

u;x

) if

u

²

F

and

u

⁶= 0

cscu

^_F(^,

u;x

) =^,

cscu

^_F(

u;x

) if

u

²

F

and

u

⁶= 0 The

cscu

F operation:

cscu

F :

F

^

F

^!

F

^[f

pole ; oating over ow ; invalid ; angle too big

^g

cscu

F(

u;x

) =

trans result

F(

cscu

^_F(

u;x

))

if

cycle

F(

u;x

)²

F

and

cycle

F(

u;x

)^62f,

u=

2

;

0

;u=

2^g

=

pole

(+¹) if

cycle

F(

u;x

) = 0

=

pole

(^,1) if

cycle

F(

u;x

) =^,

0

=

pole

(+¹) if

cycle

F(

u;x

) =

u=

2

=

pole

(^,1) if

cycle

F(

u;x

) =^,

u=

2

=

angle too big

(

qNaN

) if

u

²

G

F and

x

²

F

and ^j

x=u

^j

> big angle u

F

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

x

²

F

^[f,

0

^g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1g and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

²

F

and

x

^2f,1

;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E) 5.3.7.9 Argument angular-unit arcus sinus operation

The

arcsinu

^_F approximation helper function:

arcsinu

^_F :

F

^

F

^!R

arcsinu

^_F(

u;x

) returns a close approximation to arcsin(

x

)^

u=

(2^



) in ^R, with maximum error

max error sinu

F(

u

).

Further requirements on the

arcsinu

^_F approximation helper function:

arcsinu

^_F(

u;

1

=

2) =

u=

12

arcsinu

^_F(

u;

1) =

u=

4

arcsinu

^_F(

u;

^,

x

) =^,

arcsinu

^_F(

u;x

)

arcsinu

^_F(^,

u;x

) =^,

arcsinu

^_F(

u;x

)

NOTE { arcsinu^_F(u;x)^u=(2^) if^jx^j<fminN_F. The

arcsinu

F operation:

arcsinu

F :

F

^

F

^!

F

^[f,

0 ; under ow ; invalid

^g

arcsinu

F(

u;x

)

=

trans result

F(

arcsinu

^_F(

u;x

))

if

u

²

G

F and

x

²

F

and ^j

x

^j1 and

x

⁶= 0

= 0 if

u

²

G

F and

u >

0 and

x

= 0

=^,

0

if

u

²

G

F and

u >

0 and

x

=^,

0

=^,

0

if

u

²

G

F and

u <

0 and

x

= 0

= 0 if

u

²

G

F and

u <

0 and

x

=^,

0

=

invalid

(

qNaN

) if

u

²

G

F and

x

²

F

and ^j

x

^j

>

1

=

invalid

(

qNaN

) if

u

²

G

F and

x

^2f,1

;

+^1g

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1gand

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN

5.3.7.10 Argument angular-unit arcus cosinus operation

The

arccosu

^_F approximation helper function:

arccosu

^_F :

F

^

F

^!R

arccosu

^_F(

u;x

) returns a close approximation to arccos(

x

)^

u=

(2^



) in ^R, with maximum error

max error sinu

F(

u

).

Further requirements on the

arccosu

^_F approximation helper function:

arccosu

^_F(

u;

1

=

2) =

u=

6

arccosu

^_F(

u;

0) =

u=

4

arccosu

^_F(

u;

^,1

=

2) =

u=

3

arccosu

^_F(

u;

^,1) =

u=

2

arccosu

^_F(^,

u;x

) =^,

arccosu

^_F(

u;x

) The

arccosu

F operation:

arccosu

F :

F

^

F

^!

F

^[f

under ow ; invalid

^g

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

F(

u;x

)

=

trans result

F(

arccosu

^_F(

u;x

))

if

u

²

G

F and

x

²

F

and ^j

x

^j1

=

trans result

F(

u=

4) if

u

²

G

F and

x

=^,

0

=

invalid

(

qNaN

) if

u

²

G

F and

x

²

F

and ^j

x

^j

>

1

=

invalid

(

qNaN

) if

u

²

G

F and

x

^2f,1

;

+^1g

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1g and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN

5.3.7.11 Argument angular-unit arcus operation

The

arcu

^_F approximation helper function:

arcu

^_F :

F

^

F

^

F

^!R

arcu

^_F(

u;x;y

) returns a close approximation to

arc

(

x;y

)^

u=

(2^



) in^R, with maximum error

max error tanu

F(

u

).

Further requirements on the

arcu

^_F approximation helper function:

arcu

^_F(

u;x;x

) =

u=

8 if

x >

0

arcu

^_F(

u;

0

;y

) =

u=

4 if

y >

0

arcu

^_F(

u;x;

^,

x

) = 3^

u=

8 if

x <

0

arcu

^_F(

u;x;

0) =

u=

2 if

x <

0

arcu

^_F(

u;x;

^,

y

) =^,

arcu

^_F(

u;x;y

) if

y

⁶= 0 or

x >

0

arcu

^_F(^,

u;x;y

) =^,

arcu

^_F(

u;x;y

)

The

arcu

F operation:

arcu

F :

F

^

F

^

F

^!

F

^[f,

0 ; under ow ; invalid

^g

arcu

F(

u;x;y

) =

trans result

F(

arcu

^_F(

u;x;y

))

if

u

²

G

F and

x;y

²

F

and (

x <

0 or

y

⁶= 0)

=

mul

F(

u;

0) if

u

²

G

F and

x

²

F

and

x >

0 and

y

= 0

=

invalid

(0) if

u

²

G

F and

x

= 0 and

y

= 0

=

arcu

F(

u;

0

;y

) if

u

²

G

F and

x

=^,

0

and

y

²

F

^[f,1

;

^,

0 ;

+^1g

=

neg

F(

arcu

F(

u;x;

0)) if

u

²

G

F and

y

=^,

0

and

x

²

F

^[f,1

;

+^1g

=

mul

F(0

;u

) if

u

²

G

F and

x

= +¹ and

y

²

F

and

y

^0

=

mul

F(

neg

F(0)

;u

) if

u

²

G

F and

x

= +¹ and

y

²

F

and

y <

0

=

nearest

F(

u=

8)?inval? if

u

²

G

F and

x

= +¹ and

y

= +¹

=

nearest

F(

u=

4) if

u

²

G

F and

x

²

F

and

y

= +¹

=

nearest

F(3^

u=

8)?inval?if

u

²

G

F and

x

=^,1 and

y

= +¹

=

nearest

F(

u=

2) if

u

²

G

F and

x

=^,1 and

y

²

F

and

y

^0

=

nearest

F(^,

u=

2) if

u

²

G

F and

x

=^,1 and

y

²

F

and

y <

0

=

nearest

F(^,3^

u=

8)?inval?if

u

²

G

F and

x

=^,1 and

y

=^,1

=

nearest

F(^,

u=

4) if

u

²

G

F and

x

²

F

and

y

=^,1

=

nearest

F(^,

u=

8)?inval? if

u

²

G

F and

x

= +¹ and

y

=^,1

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

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

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1gand

x;y

²

F

^[f,1

;

^,

0 ;

+^1g

=

qNaN

if

u

is a quiet NaN and not

x

nor

y

is a signalling NaN

=

qNaN

if

x

is a quiet NaN and not

u

nor

y

is a signalling NaN

=

qNaN

if

y

is a quiet NaN and not

u

nor

x

is a signalling NaN

=

invalid

(

qNaN

) if

u

is a signalling NaN or

x

is a signalling NaN or

y

is a signalling NaN

5.3.7.12 Argument angular-unit arcus tangentus operation

The

arctanu

^_F approximation helper function:

arctanu

^_F :

F

^

F

^!R

arctanu

^_F(

u;x

) returns a close approximation to arctan(

x

)^

u=

(2^



) in ^R, with maximum error

max error tanu

F(

u

).

Further requirements on the

arctanu

^_F approximation helper function:

arctanu

^_F(

u;

1) =

u=

8

arctanu

^_F(

u;x

) =

u=

4 if

arctanu

^_F(

u;x

)⁶= arctan(

x

)^

u=

(2^



) and

x >

3^

r

F

=epsilon

F

arctanu

^_F(

u;

^,

x

) =^,

arctanu

^_F(

u;x

)

arctanu

^_F(^,

u;x

) =^,

arctanu

^_F(

u;x

)

NOTE 1 { arctanu^_F(u;x)^u=(2^) if^jx^j<fminN_F The

arctanu

F operation:

arctanu

F :

F

^

F

^!

F

^[f,

0 ; invalid ; under ow

^g

arctanu

F(

u;x

)

=

trans result

F(

arctanu

^_F(

u;x

))

if

u

²

G

F and

x

²

F

and

x

⁶= 0

= 0 if

u

²

G

F and

u >

0 and

x

= 0

=^,

0

if

u

²

G

F and

u >

0 and

x

=^,

0

=^,

0

if

u

²

G

F and

u <

0 and

x

= 0

= 0 if

u

²

G

F and

u <

0 and

x

=^,

0

=

trans result

F(^,

u=

4) if

u

²

G

F and

x

=^,1

=

trans result

F(

u=

4) if

u

²

G

F and

x

= +¹

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1gand

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN NOTE 2 { arctanuF(u;x)^arcuF(u;1;x).

5.3.7.13 Argument angular-unit arcus cotangentus operation

The

arccotu

^_F and

arcctgu

^_F approximation helper functions:

arccotu

^_F :

F

^

F

^!R

arcctgu

^_F :

F

^

F

^!R

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

^_F(

u;x

) returns a close approximation to arccot(

x

)^

u=

(2^



) in ^R, with maximum error

max error tanu

F(

u

).

arcctgu

^_F(

u;x

) returns a close approximation to arcctg(

x

)^

u=

(2^



) in ^R, with maximum error

max error tanu

F(

u

).

There are two reasonable ways of selecting the principle value for the inverse of the cot oper.

It is best to leave it to the user/programmer to decide which one is the most appropriate in a particular application. LIA-2 species both of them. Selecting just one is premature at the LIA level.

Further requirements on the

arccotu

^_F and

arccotu

^_F approximation helper functions:

arccotu

^_F(

u;

1) =

u=

8

arccotu

^_F(

u;

0) =

u=

4

arccotu

^_F(

u;

^,1) = 3^

u=

8

arccotu

^_F(

u;x

)^

u=

2 if

u >

0

arccotu

^_F(

u;x

)^

u=

2 if

u <

0

arccotu

^_F(

u;x

) =

u=

2 if

arccotu

^_F(

u;x

)⁶= arccot(

x

)^

u=

(2^



) and

x <

^,3^

r

F

=epsilon

F

arccotu

^_F(^,

u;x

) =^,

arccotu

^_F(

u;x

)

arcctgu

^_F(

u;x

) =

arccotu

^_F(

u;x

) if

x

^0

arcctgu

^_F(

u;

^,

x

) =^,

arcctgu

^_F(

u;x

)

The

arccotu

F operation:

arccotu

F :

F

^

F

^!

F

^[f

invalid ; under ow

^g

arccotu

F(

u;x

) =

trans result

F(

arccotu

^_F(

u;x

))

if

u

²

G

F and

x

²

F

=

trans result

F(

u=

4) if

u

²

G

F and

x

=^,

0

=

trans result

F(

u=

2) if

u

²

G

F and

x

=^,1

= 0 if

u

²

G

F and

u >

0 and

x

= +¹

=^,

0

if

u

²

G

F and

u <

0 and

x

= +¹

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1g and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN NOTE { arccotuF(u;x)^arcuF(u;x;1).

The

arcctgu

F operation:

arcctgu

F :

F

^

F

^!

F

^[f

invalid ; under ow

^g

arcctgu

F(

u;x

) =

trans result

F(

arcctgu

^_F(

u;x

))

if

u

²

G

F and

x

²

F

=

trans result

F(^,

u=

4) if

u

²

G

F and

x

=^,

0

=^,

0

if

u

²

G

F and

u >

0 and

x

=^,1

= 0 if

u

²

G

F and

u >

0 and

x

= +¹

= 0 if

u

²

G

F and

u <

0 and

x

=^,1

=^,

0

if

u

²

G

F and

u <

0 and

x

= +¹

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

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

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1gand

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN

5.3.7.14 Argument angular-unit arcus secantus operation

The

arcsecu

^_F approximation helper function:

arcsecu

^_F :

F

^

F

^!R

arcsecu

^_F(

u;x

) returns a close approximation to arcsec(

x

)^

u=

(2^



) in ^R, with maximum error

max error tanu

F(

u

).

Further requirements on the

arcsecu

^_F approximation helper function:

arcsecu

^_F(

u;

2) =

u=

6

arcsecu

^_F(

u;

^,2) =

u=

3

arcsecu

^_F(

u;

^,1) =

u=

2

arcsecu

^_F(

u;x

)^

u=

4 if

x >

0 and

u >

0

arcsecu

^_F(

u;x

)^

u=

4 if

x <

0 and

u >

0

arcsecu

^_F(

u;x

) =

u=

4 if

arcsecu

^_F(

u;x

)⁶= arcsec(

x

)^

u=

(2^



) and

j

x

^j

>

3^

r

F

=epsilon

F

arcsecu

^_F(^,

u;x

) =^,

arcsecu

^_F(

u;x

) The

arcsecu

F operation:

arcsecu

F :

F

^

F

^!

F

^[f

under ow ; invalid

^g

arcsecu

F(

u;x

) =

trans result

F(

arcsecu

^_F(

u;x

))

if

u

²

G

F and

x

²

F

and (

x

^,1 or

x

^1)

=

invalid

(

qNaN

) if

u

²

G

F and

x

²

F

and ^,1

< x <

1

=

invalid

(

qNaN

) if

u

²

G

F and

x

=^,

0

=

trans result

F(

u=

4) if

u

²

G

F and

x

=^,1

=

trans result

F(

u=

4) if

u

²

G

F and

x

= +¹

=

invalid

(

qNaN

) if

u

²

F

and ^j

u

^j

< min angular unit

F and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1gand

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN

5.3.7.15 Argument angular-unit arcus cosecantus operation

The

arccscu

^_F approximation helper function:

arccscu

^_F :

F

^

F

^!R

arccscu

^_F(

u;x

) returns a close approximation to arccsc(

x

)^

u=

(2^



) in ^R, with maximum error

max error tanu

F(

u

).

Further requirements on the

arccscu

^_F approximation helper function:

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

^_F(

u;

2) =

u=

12

arccscu

^_F(

u;

1) =

u=

4

arccscu

^_F(

u;

^,

x

) =^,

arccscu

^_F(

u;x

)

arccscu

^_F(^,

u;x

) =^,

arccscu

^_F(

u;x

) The

arccscu

F operation:

arccscu

F :

F

^

F

^!

F

^[f

under ow ; invalid

^g

arccscu

F(

u;x

) =

trans result

F(

arccscu

^_F(

u;x

))

if

u

²

G

F and

x

²

F

and (

x

^1 or

x

^,1)

=

invalid

(

qNaN

) if

u

²

G

F and

x

²

F

and ^,1

< x <

1

=

invalid

(

qNaN

) if

u

²

G

F and

x

=^,

0

=^,

0

if

u

²

G

F and

u >

0 and

x

=^,1

= 0 if

u

²

G

F and

u >

0 and

x

= +¹

= 0 if

u

²

G

F and

u <

0 and

x

=^,1

=^,

0

if

u

²

G

F and

u <

0 and

x

= +¹

=

invalid

(

qNaN

) if

u

²

F

and

u < min angular unit

F and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

invalid

(

qNaN

) if

u

^2f,1

;

^,

0 ;

+^1g and

x

²

F

^[f,1

;

^,

0 ;

+^1g

=

qNaN

if

x

is a quiet NaN and

u

is not a signalling NaN

=

qNaN

if

u

is a quiet NaN and

x

is not a signalling NaN

=

invalid

(

qNaN

) if

x

is a signalling NaN or

u

is a signalling NaN

5.3.8 Operations for degree trigonometrics and inverse degree trigonometrics deg

F :

F

^!

F

^[f,

0 ; angle too big

^g

deg

F(

x

) =

unit

F(360

;x

)

sind

F :

F

^!

F

^[f,

0 ; under ow ; angle too big

^g

sind

F(

x

) =

sinu

F(360

;x

)

cosd

F :

F

^!

F

^[f

under ow ; angle too big

^g

cosd

F(

x

) =

cosu

F(360

;x

)

cossind

F :

F

^!

F

^(

F

^[f,

0

^g)^[f

under ow ; angle too big

^g

cossind

F(

x

) =

cossinu

F(360

;x

)

tand

F :

F

^!

F

^[f,

0 ; pole ; oating over ow ; under ow ; angle too big

^g

tand

F(

x

) =

tanu

F(360

;x

)

cotd

F :

F

^!

F

^[f

pole ; oating over ow ; under ow ; angle too big

^g

cotd

F(

x

) =

cotu

F(360

;x

)

secd

F :

F

^!

F

^[f

pole ; oating over ow ; angle too big

^g

secd

F(

x

) =

secu

F(360

;x

)

cscd

F :

F

^!

F

^[f

pole ; oating over ow ; angle too big

^g

cscd

F(

x

) =

cscu

F(360

;x

)

arcsind

F :

F

^!

F

^[f

under ow ; invalid

^g

arcsind

F(

x

) =

arcsinu

F(360

;x

)

arccosd

F :

F

^!

F

^[f

invalid

^g

arccosd

F(

x

) =

arccosu

F(360

;x

)

arcd

F :

F

^

F

^!

F

^[f

under ow ; invalid

^g

arcd

F(

x;y

) =

arcu

F(360

;x;y

)

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

F :

F

^!

F

^[f

under ow

^g

arctand

F(

x

) =

arctanu

F(360

;x

)

arccotd

F :

F

^!

F

^[f

under ow

^g

arccotd

F(

x

) =

arccotu

F(360

;x

)

arcctgd

F :

F

^!

F

^[f

under ow

^g

arcctgd

F(

x

) =

arcctgu

F(360

;x

)

arcsecd

F :

F

^!

F

^[f

invalid

^g

arcsecd

F(

x

) =

arcsecu

F(360

;x

)

arccscd

F :

F

^!

F

^[f

under ow ; invalid

^g

arccscd

F(

x

) =

arccscu

F(360

;x

)

5.3.9 Operations for angular-unit conversions

5.3.9.1 Converting radian angle to argument angular-unit angle

Dene the mathematical function:

rad to cycle

:^RR!R

rad to cycle

(

x;v

)

= arccos(cos(

x

))^

v=

(2^



)

if sin(

x

)^0 and

v

⁶= 0

=^,arccos(cos(

x

))^

v=

(2^



)

if sin(

x

)

<

0 and

v

⁶= 0 The

rad to cycle

^_F approximation helper function:

rad to cycle

^_F :^R

F

^!R

rad to cycle

^_F(

x;v

) returns a close approximation to

rad to cycle

(

x;v

) in ^R, with maximum error

max error rad

F, if ^j

x

^j

big angle r

F.

Further requirements on the

rad to cycle

^_F approximation helper function:

rad to cycle

^_F(

n

^2^



+

=

6

;v

) =

v=

12 if

n

^2Z and ^j

n

^2^



+

=

6^j

big angle r

F

rad to cycle

^_F(

n

^2^



+

=

4

;v

) =

v=

8 if

n

^2Z and ^j

n

^2^



+

=

4^j

big angle r

F

rad to cycle

^_F(

n

^2^



+

=

3

;v

) =

v=

6 if

n

^2Z and ^j

n

^2^



+

=

3^j

big angle r

F

rad to cycle

^_F(

n

^2^



+

=

2

;v

) =

v=

4 if

n

^2Z and ^j

n

^2^



+

=

2^j

big angle r

F

rad to cycle

^_F(

n

^2^



+ 2^

=

3

;v

) =

v=

3

if

n

^2Z and ^j

n

^2^



+ 2^

=

3^j

big angle r

F

rad to cycle

^_F(

n

^2^



+ 3^

=

4

;v

) = 3^

v=

8

if

n

^2Z and ^j

n

^2^



+ 3^

=

4^j

big angle r

F

rad to cycle

^_F(

n

^2^



+ 5^

=

6

;v

) = 5^

v=

12

if

n

^2Z and ^j

n

^2^



+ 5^

=

6^j

big angle r

F

rad to cycle

^_F(

n

^2^



+

;v

) =

v=

2 if

n

^2Z and ^j

n

^2^



+



^j

big angle r

F

rad to cycle

^_F(^,

x;v

) =^,

rad to cycle

^_F(

x;v

)

if

rad to cycle

(

x;v

)⁶=

v=

2

rad to cycle

^_F(

x;

^,

v

) =^,

rad to cycle

^_F(

x;v

)

if

rad to cycle

(

x;v

)⁶=

v=

2 The

rad to cycle

F operation:

rad to cycle

F :

F

^

F

^!

F

^[f

under ow ; angle too big ; invalid

^g

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

5.3.9.2 Converting argument angular-unit angle to radian angle

Dene the mathematical function: The

cycle to rad

^_F approximation helper function:

cycle to rad

^_F :

F

^R!R

cycle to rad

^_F(

u;x

) returns a close approximation to

cycle to rad

(

u;x

) in ^R, if

u

⁶= 0, with maximum error

max error rad

F.

Further requirements on the

cycle to rad

^_F approximation helper function:

cycle to rad

^_F(

u;n

^

u

+

x

) =

cycle to rad

^_F(

u;x

)

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E) 5.3.9.3 Converting argument unit angle to (another) argument

angular-unit angle

The

cycle to cycle

^_F approximation helper function:

cycle to cycle

^_F :

F

^R

F

^!R

cycle to cycle

^_F(

u;x;v

) returns a close approximation to

cycle to cycle

(

u;x;v

) in ^R if

u

⁶= 0 and^j

x=u

^j

big angle u

F, with maximum error

max error rad

F.

Further requirements on the

cycle to cycle

^_F approximation helper function:

cycle to cycle

^_F(

u;n

^

u

+

x;v

) =

cycle to cycle

^_F(

u;x;v

)

ISO/IEC CD 10967-2.3:1998(E) Third Committee Draft 5.3.9.4 Degree angle conversions to and from other angular units

rad to deg

F :

F

^!

F

^[f

under ow ; angle too big

^g

rad to deg

F(

x

)=

rad to cycle

F(

x;

360)

deg to rad

F :

F

^!

F

^[f,

0 ; under ow ; angle too big

^g

deg to rad

F(

x

) =

cycle to rad

F(360

;x

)

cycle to deg

F :

F

^

F

^!

F

^[f,

0 ; under ow ; angle too big ; invalid

^g

cycle to deg

F(

u;x

) =

cycle to cycle

F(

u;x;

360)

deg to cycle

F :

F

^

F

^!

F

^[f,

0 ; under ow ; angle too big ; invalid

^g

deg to cycle

F(

x;v

) =

cycle to cycle

F(360

;x;v

)

5.4 Conversion operations

Fixed point string formats and oating point string formats should have formats for ^,

0

, +¹,

,1, quiet and signalling NaNs. Integer string formats may have formats for such values.

NOTES

1 In ordinary string formats for numerals, the string \Hello world!" is an example of a signalling NaN.

2 This part of ISO/IEC 10967does not specify any string formats, not even for the special values^,0, +¹,^,1, and quiet NaN, but possibilities include the strings used in the text of this part of ISO/IEC 10967, as well as strings like \+innity" or \positiva oandligheten", etc, and the strings used may depend on preference settings. String formats for numerical values, and if and how they may depend on preference settings, is also an issue for bindings or programming language specications. It is not an issue for this part of ISO/IEC 10967.

5.4.1 Integer to integer conversions

I

and

I

⁰ represent dierent integer data types (even if the sets

I

and

I

⁰ are the same). At least one of

I

and

I

⁰conform to LIA-1.

NOTE 1 { If both are I and I⁰ are conforming to ISO/IEC 10967-1, then this conversion is covered by ISO/IEC 10967-1. This operation generalises the cvtI^!I⁰ of ISO/IEC 10967-1:1994, with respect to that one of the integer types in the conversion need not be conforming to ISO/IEC 10967-1.

convert

I^!I⁰ :

I

^!

I

^0[f

integer over ow

^g

convert

I^!I⁰(

x

)=

result

I⁰(

x

) if

x

²

I

=^,

0

if

x

=^,

0

and ^,

0

is available in the target type

=^,1 if

x

=^,1 and^,1 is available in the target type

= +¹ if

x

= +¹ and +¹ is available in the target type

= 0 if

x

=^,

0

and ^,

0

is not available in the target type

=

integer over ow

if

x

=^,1 and^,1 is not available in the target type

=

integer over ow

if

x

= +¹ and +¹ is not available in the target type

=

qNaN

if

x

is a quiet NaN and quiet NaN is available in the target type

=

invalid

(

qNaN

) if

x

is a signalling NaN [

sNaN

without notication?]

NOTE 2 { This covers, among other things, \input" and \output" of integer type values, including the possibility of over ow (for either input or output).

Third Committee Draft ISO/IEC CD 10967-2.3:1998(E) 5.4.2 Floating point to integer conversions

I

is an ISO/IEC 10967-1 conforming integer type.

F

is an ISO/IEC 10967-1 conforming oating point type.

NOTE { The operations in this clause are more specic than the oating point to integer conversion in ISO/IEC 10967-1:1994 which allows any rounding.

rounding

F^!I :

F

^!

I

^[f

integer over ow

^g

rounding

F^!I(

x

)

=

result

I(round(

x

)) if

x

²

F

=^,

0

if

x

=^,

0

and ^,

0

is available in the target type

=^,1 if

x

=^,1and ^,1is available in the target type

= +¹ if

x

= +¹ and +¹ is available in the target type

= 0 if

x

=^,

0

and ^,

0

is not available in the target type

=

integer over ow

if

x

=^,1and ^,1is not available in the target type

=

integer over ow

if

x

= +¹ and +¹ is not available in the target type

=

qNaN

if

x

is a quiet NaN and quiet NaN is available in the target type

=

invalid

(

qNaN

) if

x

is a signalling NaN [

sNaN

without notication?]

oor_F^!_I :

F

^!

I

^[f

integer over ow

^g

oor_F^!_I(

x

) =

result

I(^b

x

^c) if

x

²

F

=^,

0

if

x

=^,

0

and ^,

0

is available in the target type

=^,1 if

x

=^,1and ^,1is available in the target type

= +¹ if

x

= +¹ and +¹ is available in the target type

= 0 if

x

=^,

0

and ^,

0

is not available in the target type

=

integer over ow

if

x

=^,1and ^,1is not available in the target type

=

integer over ow

if

x

= +¹ and +¹ is not available in the target type

=

qNaN

if

x

is a quiet NaN and quiet NaN is available in the target type

=

invalid

(

qNaN

) if

x

is a signalling NaN [

sNaN

without notication?]

ceiling

F^!I :

F

^!

I

^[f

integer over ow

^g

ceiling

F^!I(

x

) =

result

I(^d

x

^e) if

x

²

F

=^,

0

if

x

=^,

0

and ^,

0

is available in the target type

=^,1 if

x

=^,1and ^,1is available in the target type

= +¹ if

x

= +¹ and +¹ is available in the target type

= 0 if

x

=^,

0

and ^,

0

is not available in the target type

=

integer over ow

if

x

=^,1and ^,1is not available in the target type

=

integer over ow

if

x

= +¹ and +¹ is not available in the target type

=

qNaN

if

x

is a quiet NaN and quiet NaN is available in the target type

=

invalid

(

qNaN

) if

x

is a signalling NaN [

sNaN

without notication?]

5.4.3 Integer to oating point conversions convert

^nearest_I^!_F :

I

^!

F

^[f

oating over ow

^g

convert

^nearest_I^!_F (

x

)=

result

F(

x;nearest

F) if

x

²

I

=^,

0

if

x

=^,

0

and ^,

0

is available in the target type

=^,1 if

x

=^,1and ^,1is available in the target type

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

= +¹ if

x

= +¹ and +¹ is available in the target type

= 0 if

x

=^,

0

and ^,

0

is not available in the target type

=

oating over ow

if

x

=^,1 and^,1 is not available in the target type

=

oating over ow

if

x

= +¹ and +¹ is not available in the target type

=

qNaN

if

x

is a quiet NaN and quiet NaN is available in the target type

=

invalid

(

qNaN

) if

x

is a signalling NaN [

sNaN

without notication?]

The following two operations are to support interval arithmetic.

convert

^down_I^!_F :

I

^!

F

^[f

oating over ow

^g

=

oating over ow

if

x

=^,1 and^,1 is not available in the target type

=

oating over ow

if

x

= +¹ and +¹ is not available in the target type

=

qNaN

if

x

is a quiet NaN and quiet NaN is available in the target type

=

invalid

(

qNaN

) if

x

is a signalling NaN [

sNaN

without notication?]

convert

^up_I^!_F :

I

^!

F

^[f

oating over ow

^g

=

oating over ow

if

x

=^,1 and^,1 is not available in the target type

=

oating over ow

if

x

= +¹ and +¹ is not available in the target type

=

qNaN

if

x

is a quiet NaN and quiet NaN is available in the target type

=

invalid

(

qNaN

) if x is a signalling NaN [

sNaN

without notication?]

NOTE { Integer to nearest oating point conversions are covered by ISO/IEC 10967-1. I.e.

cvtI^!F = convert^nearest_I^!_F , when both I and F conform to LIA-1.

5.4.4 Floating point to oating point conversions

F

and

F

⁰ represent dierent oating point data types (even if the sets

F

and

F

⁰ are the same).

At least one of

F

and

F

⁰ conform to LIA-1.

=

oating over ow

if

x

=^,1 and^,1 is not available in the target type

=

oating over ow

if

x

= +¹ and +¹ is not available in the target type

=

qNaN

if

x

is a quiet NaN and quiet NaN is

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

available in the target type

=

invalid

(

qNaN

) if x is a signalling NaN [

sNaN

without notication?]

The following two operations are to support interval arithmetic.

convert

^down_F^!_F⁰ :

F

^!

F

^0[f

oating over ow ; under ow

^g

convert

^down_F^!_F⁰(

x

)

=

result

F⁰(

x;down

F⁰) if

x

²

F

=^,

0

if

x

=^,

0

and ^,

0

is available in the target type

=^,1 if

x

=^,1and ^,1is available in the target type

= +¹ if

x

= +¹ and +¹ is available in the target type

= 0 if

x

=^,

0

and ^,

0

is not available in the target type

=

oating over ow

if

x

=^,1and ^,1is not available in the target type

=

oating over ow

if

x

= +¹ and +¹ is not available in the target type

=

qNaN

if

x

is a quiet NaN and quiet NaN is available in the target type

=

invalid

(

qNaN

) if

x

is a signalling NaN [

sNaN

without notication?]

convert

^up_F^!_F⁰ :

F

^!

F

^0[f

oating over ow ; under ow

^g

convert

^up_F^!_F⁰(

x

)

=

result

F⁰(

x;up

F⁰) if

x

²

F

=^,

0

if

x

=^,

0

and ^,

0

is available in the target type

=^,1 if

x

=^,1and ^,1is available in the target type

= +¹ if

x

= +¹ and +¹ is available in the target type

= 0 if

x

=^,

0

and ^,

0

is not available in the target type

=

oating over ow

if

x

=^,1and ^,1is not available in the target type

=

oating over ow

if

x

= +¹ and +¹ is not available in the target type

=

qNaN

if

x

is a quiet NaN and quiet NaN is available in the target type

=

invalid

(

qNaN

) if

x

is a signalling NaN [

sNaN

without notication?]

NOTES

1 Floating point to nearest oating point conversions are covered by ISO/IEC 10967-1 when

1 Floating point to nearest oating point conversions are covered by ISO/IEC 10967-1 when

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