ISO,IEC 10967-3 standard.Language independent arithmetic

Working draft

ISO/IEC WD 10967-3.1:2001(E)

csc(z) = 1= sin(z)

csc(x) = ~ csch(~ x) = ~ csch( ~ x) csc(~ y) = ~ csch( y) = ~ csch(y)

csc(x +~ y) = ~ csch( y +~ x) = ~ csch(y ~ x)

The csci(F ) operation:

csci(F ) : i(F ) ! i(F ) [ funder ow; over ow; in nitaryg csci(F )(^ y) = negi(F )(^ cschF (y))

The cscc(F ) approximation helper function:

cscc(F ) : C ! C

cscc(F )(z) returns a close approximation to csc(z) in C with maximum error max error tanc(F ).

Further requirements on the cscc(F ) approximation helper function are:

cscc(F )(conj(z)) = conj(cscc(F )(z)) cscc(F )( z) = cscc(F )(z)

The relationship to the cscF and cschF for real-valued operations shall be:

if z 2 C if z 2 C

approximation helper functions in an associated library

csc	(x) = csc				(x)				if x	2 R
c(F )				F
csc	(~		y) =	~			csch	(y)	if y	2 R
c(F )							F

The cscc(F ) operation:

cscc(F ) : c(F ) ! c(F ) [ funder ow; over ow; in nitary; absolute precision under owg

cscc(F )(x +^ y)
= result	(csc	(x +~		y); nearestF )
c(F )	c(F )
			if x +^ y 2 c(F ) and jxj 6 big			angle		rF and
				(x 6= 0 or y 6= 0)
= in nitary((+1) +^ (+1))
			if x = 0 and y = 0
= conj c(F )(cscc(F )(x +^ 0))
			if y = 0
= negF (cscc(F )(0 +^ negF (y)))
			if x = 0 and y 6= 0
= mulF ( 0; sinF (x)) +^ mulF (divF ( 1; y); cosF (x))
= radc(F )(x +^ y)			if y 2 f1; +1g
			otherwise

5.3.2.8 Radian arc sine

NOTE 1 { arcsin( z) = arcsin(z) arcsin(z) = =2 arccos(z)

arcsin(conj(z)) = conj(arcsin(z)) if Im(z) 6= 0 or jRe(z)j 6 1

Arcsin(x +~ y) = ~ Arcsinh( y +~ x) = ~ Arcsinh(y ~ x)

The arcsinF !c(F ) operation: arcsinF !c(F ) : F ! c(F [ f 0g)

5.3.2 Operations for radian trigonometric elementary functions

33

ISO/IEC WD 10967-3.1:2001(E)

Working draft

arcsinF !c(F )(x)= downF ( =2) +^ negF (arccoshF (x))

if (x 2 F and x > 1) or x = +1

= arcsinF (x) +^ mulF ( 0; x)

if (x 2 F and jxj < 1) or x = 0

= upF ( =2) +^ arccoshF (x)

if (x 2 F and x 6 1) or x = 1

= no

resultF !c(F )(x)

otherwise

The arcsini(F ) operation:

arcsini(F ) : i(F ) ! i(F )

arcsini(F )(^ y)

= ^ arcsinhF (y)

The arcsinc(F ) approximation helper function:

arcsin

:

C ! C

c(F )

arcsin

(z) returns a close approximation to arcsin(z) in

C

with maximum error max

error

sin

.

c(F )

c(F )

Further requirements on the arcsinc(F ) approximation helper function are:

arcsin

(conj(z)) = conj(arcsin

(z))

if z

2 C

and (Re(z)

6

1 or Im(z) = 0)

c(F )

c(F )

6

arcsin

(

z) = arcsin

(z)

if z

2 C

and (Re(z)

6

1 or Im(z) = 0)

c(F )

c(F )

6

6

Re(arcsin

(x)) =

=2

if x

2 R

and x

1

c(F )

Re(arcsin

(x)) = =2

if x

2 R

and x

>

1

c(F )

>

>

Im(arcsin

(z))

0

if z

2 C

and Im(z)

0

c(F )

The relationship to the arcsinF , arcsinhF , and arccoshF

approximation helper functions in

an associated library for real-valued operations shall be:

arcsin

(x) = arcsin

(x)

if x

2 R

and

j

x

j 6

1

c(F )

(~

y) = ~

F

(y)

if y

arcsin

arcsinh

2 R

c(F )

F

6

Im(arcsin

(x)) = arccosh

( x)

if x

2 R

and x

1

c(F )

F

Im(arcsin

(x)) = arccosh

(x)

if x

2 R

and x

>

1

c(F )

F

arcsin

(x) = =2 +~

arccosh (x)

if x

2 R

and x

>

1

c(F )

F

The arcsin#

range limitation helper function:

c(F )

arcsin#

(z) = max

f

upF (

=2); min

f

Re(arcsin

(z)); downF

( =2)

gg

+~

Im(arcsin

(z))

c(F )

c(F )

c(F )

The arcsinc(F ) operation:

arcsinc(F ) : c(F ) ! c(F ) [ funder owg

arcsinc(F )(x +^ y)

(arcsin#

= result

(x +~

y); nearestF )

c(F )

c(F )

if x +^ y 2 c(F )

= negc(F )(arcsinc(F )(0 +^ negF (y))

if x = 0

= conj c(F )(arcsinc(F )(x +^ 0))

= arcF (y; x) +^ (+1)

if y = 0 and x 6= 0

if x 2 f1; +1g and

((y 2 F and y > 0) or y = +1)

= arcF (negF (y); x) +^ (1)

if x 2 f1; +1g and

34	Speci cations for imaginary and complex datatypes and operations

Working draft			ISO/IEC WD 10967-3.1:2001(E)
			((y 2 F and y < 0) or y = 1)
	= no		resultc(F )(x +^ y)
			otherwise
NOTE 2 {	The inverse of sin is multi-valued, the real part may have any integer multiple of

2 added to it, and the result is also in the solution set. The arcsin function (returning the principal value for the inverse) branch cuts at fx j x 2 R and jxj > 1g. Thus arcsinc(F )(x + ^ 0) 6= arcsinc(F )(x +^ ( 0)) when jxj > 1.

5.3.2.9Radian arc cosine

NOTE 1 { arccos( z) = ~ arccos(z)

arccos(conj(z)) = conj(arccos(z)) if Im(z) 6= 0 or jRe(z)j 6 1 arccos(z) = =2 arcsin(z)

Arccos(x +~ y) = ~ Arccosh(x +~ y)

The arccosF !c(F ) operation:

arccosF !c(F ) : F ! c(F [ f 0g)

arccosF !c(F )(x) = 0 +^ arccoshF (x)

if (x 2 F and x > 1) or x = +1

= arccosF (x) +^ mulF (0; x)

if (x 2 F and jxj < 1) or x = 0

= downF ( ) +^ negF (arccoshF (negF (x)))

if (x 2 F and x 6 1) or x = 1

= no

resultF !c(F )(x)

otherwise

The arccosi(F )!c(F ) operation:

arccosi(F )!c(F ) : i(F ) ! c(F )

arccosi(F )!c(F )(^ y)

= nearestF ( =2) +^ negF (arcsinhF (y))

The arccosc(F ) approximation helper function:

arccos

:

C ! C

c(F )

arccos

(z) returns a close approximation to arccos(z) in

C

with maximum error max

error

sin

.

c(F )

c(F )

Further requirements on the arccosc(F ) approximation helper function are:

arccos

(conj(z)) = conj(arccos

(z))

if z

2 C

and (

Re(z)

j 6

1 or Im(z) = 0)

c(F )

c(F )

j

6

6

Re(arccos

(x)) =

if x

2 R

and x

1

c(F )

Re(arccos

(x)) = 0

if x

2 R

and x

>

1

c(F )

j 6

Im(arccos

(

z)) =

Im(arccos

(z)) if z

2 C

and (

Re(z)

1 or Im(z) = 0)

c(F )

>

c(F )

6

j

6

Im(arccos

(z))

0

if im(z)

0

c(F )

The relationship to the arccosF

and arccoshF

approximation helper functions in an associated

library for real-valued operations shall be:

arccos

(x) = arccos

(x)

if x

2 R

and

x

1

c(F )

F

j

j 6

Im(arccos

(x)) = arccosh (x)

if x

2 R

and x

6

1

c(F )

F

Im(arccos

(x)) =

arccosh (x)

if x

2 R

and x

>

1

c(F )

F

arccos

(x) =

~

arccosh

(x)

if x

2 R

and x

>

1

c(F )

F

arccos

(~

x0) = =2

~

arcsinh

(x0)

if x0

2 R

c(F )

F

5.3.2 Operations for radian trigonometric elementary functions

35

ISO/IEC WD 10967-3.1:2001(E)

Working draft

arccos

(x) =

~

arccosh

(

x)

if x

2 R

and x

6

1

c(F )

F

The arccos#

range limitation helper function:

c(F )

arccos#

(x +~

y) = min

f

Re(arccos

(x +~

y)); downF ( )

g

+~

Im(arccos

(x +~

y))

c(F )

c(F )

c(F )

The arccosc(F ) operation:

arccosc(F ) : c(F ) ! c(F [ f 0g)

arccosc(F )(x +^ y)

(arccos#

= result

(x +~

y); nearestF )

c(F )

c(F )

y

c(F ) and (x0

= 0 or

x

> 1)

if x +^

2

j

j

= arccosF (x) +^

( 0)

if x0

x

1

6

= 0 and

if x = 0

j j 6

= arccosc(F )(0 +^ y)

= conj c(F )(arccosc(F )(x +^ 0))

if y = 0 and x 6= 0

=arcF (x; y) +^ (1) if x 2 f1; +1g and ((y 2 F and y > 0)orx'=+1)

=arcF (x; negF (y)) +^ (+1)

if x 2 f1; +1g and ((y 2 F and y < 0)orx'=1)

= no resultc(F )(x +^ y)

otherwise

NOTE 2 { The inverse of cos is multi-valued, the real part may have any integer multiple of 2 added to it, and the result is also in the solution set. The arccos function (returning the principal value for the inverse) branch cuts at fx j x 2 R and jxj > 1g. Thus arccosc(F )(x + ^ 0) 6= arccosc(F )(x +^ ( 0)) when jxj > 1.

5.3.2.10Radian arc tangent

NOTE 1 { arctan( z) = arctan(z)

arctan(conj(z)) = conj(arctan(z)) if Re(z) 6= 0 or jIm(z)j 6 1 arctan(z) = =2 arccot(z)

Arctan(x +~ y) = ~ Arctanh( y +~ x) = ~ Arctanh(y ~ x)

The arctani(F ) operation:

arctani(F ) : i(F ) ! i(F ) [ fin nitary; invalidg arctani(F )(^ y) = ^ arctanhF (y)

The arctani(F )!c(F ) operation:

arctani(F )!c(F ) : i(F ) ! c(F [ f 0g) [ fin nitaryg

arctani(F )!c(F )(^ y)

= downF ( =2) +^ negF (arccothF (negF (y)))

if (y 2 F and y < 1) or x = 1

= mulF ( 0; y) +^ arctanhF (y)

if (y 2 F and jyj 6 1) or y = 0

= upF ( =2) +^ arccothF (y)

if (y 2 F and y > 1) or x = +1 = no resultF !c(F )(^ y) otherwise

The arctanc(F ) approximation helper function:

36	Speci cations for imaginary and complex datatypes and operations

c(F )

= conj

= negc(F )

Working draft

ISO/IEC WD 10967-3.1:2001(E)

arctan

:

C ! C

c(F )

arctan

(z) returns a close approximation to arctan(z) in

C

with maximum error max

error

tan

c(F )

.

c(F )

Further requirements on the arctanc(F ) approximation helper function are:

arctan

(conj(z)) = conj(arctan

(z))

if z

2 C

c(F )

c(F )

arctan

(

z) =

arctan

(z)

if z

2 C

and (Re(z) < 1 or Im(z) = 0)

c(F )

c(F )

6

Re(arctan

(~

y)) =

=2

if y

2 R

and y > 1

c(F )

>

>

Re(arctan

(z))

0

if Re(z)

0)

c(F )

The relationship to the arctanF , arctanhF , and arccothF

approximation helper functions in

an associated library for real-valued operations shall be:

arctan

(x) = arctan

(x)

if x

2 R

c(F )

(~

y) = ~

F

(y)

if y

and

< 1

arctan

arctanh

2 R

j

y

j

c(F )

F

Im(arctan

(~

y)) = ~

arccoth (y)

if y

2 R

and

j

y

j

> 1

c(F )

F

The arctan#

range limitation helper function:

c(F )

arctan#

(z) = max

f

upF (

=2); min Re(arctan

(z)); downF ( =2)

gg

+~

Im(arctan

(z))

c(F )

f

c(F )

c(F )

The arctanc(F ) operation:

arctanc(F ) : c(F ) ! c(F ) [ funder ow; in nitaryg

arctanc(F )(x +^ y)

(arctan#

= result

(x +~

y); nearestF )

c(F )

c(F )

if x +^ y 2 c(F ) and (x 6= 0 or jyj =6 1) = in nitary(nearestF ( =4) +^ (mulF (y; +1)))

if x = 0 and jyj = 1

(arctanc(F )(0 +^ negF (y)) if x = 0

(arctanc(F )(x +^ 0))

if y = 0 and x 6= 0

= signbF (x) downF ( =2) +^ divF c(F )(1; x +^ negF (y)))

if (x 2 f 1; +1g and y 2 F [ f 1; +1g) or

(x 2 F [ f 1; +1g and y 2 f 1; +1g)

= no resultc(F )(x +^ y)

otherwise

NOTE 2 { The inverse of tan is multi-valued, the real part may have any integer multiple of 2 (even any integer multiple of ) added to it, and the result is also in the solution set. The arctan function (returning the principal value for the inverse) branch cuts at f^ y j y 2 F and jyj > 1g. Thus arctanc(F )(0 +^ y) 6= arctanc(F )(( 0) +^ y) when jyj > 1.

5.3.2.11Radian arc cotangent

NOTE 1 { arccot( z) = arccot(z)

arccot(conj(z)) = conj(arccot(z)) if Re(z) 6= 0 or jIm(z)j > 1 arccot(z) = =2 arctan(z)

Arccot(x +~ y) = ~ Arccoth( y +~ x) arccot(z) = arctan(1=z)

The arccoti(F ) operation:

5.3.2 Operations for radian trigonometric elementary functions

37

ISO/IEC WD 10967-3.1:2001(E)

Working draft

arccoti(F ) : i(F ) ! i(F ) [ funder ow; in nitary; invalidg

arccoti(F )(^ y)

= ^ arccothF (y)

The arccoti(F )!c(F ) operation:

arccoti(F )!c(F ) : i(F ) ! c(F ) [ funder ow; in nitaryg

arccoti(F )!c(F )(^ y)

= upF ( =2) +^ arctanhF (y)

if (y 2 F and 1 < y and y < 0) or y = 0

= mulF ( 0; y) +^ arccothF (y)

if y 2 F and jyj > 1

= divF ( 1; y) +^ arccothF (y)

if y 2 f1; +1g

= downF ( =2) +^ arctanhF (y)

if y 2 F and 0 6 y and y < 1

= no

resultF !c(F )(^ y)

otherwise

The arccotc(F ) approximation helper function:

arccot

:

C ! C

c(F )

arccot

(z) returns a close approximation to arccot(z) in

C

with maximum error max

error

tan

c(F )

.

c(F )

Further requirements on the arccotc(F ) approximation helper function are:

arccot

(conj(z)) = conj(arccot

(z))

if z

2 C

c(F )

c(F )

arccot

(

z) =

arccot

(z)

if z

2 C

and (Im(z) = 0 or

j

Re(z)

j

< 1)

c(F )

c(F )

6

Re(arccot

(~

y)) = =2

if y

2 R

and 0 < y < 1

c(F )

>

>

Re(arccot

(z))

0

if Re(z)

0

c(F )

The relationship to the arccotF , arccothF

and arctanhF

approximation helper functions in an

associated library for real-valued operations shall be:

arccot

(x) = arccot

(x)

if x

2 R

c(F )

(~

y) = ~

F

( y)

if y

arccot

arccoth

2 R

c(F )

F

The arccot#

range limitation helper function:

c(F )

arccot#

(z) = max

f

upF ( =2); min

f

Re(arccot

(z)); downF ( =2)

gg

+~

Im(arccot

(z))

c(F )

c(F )

c(F )

The arccotc(F ) operation:

arccotc(F ) : c(F ) ! c(F [ f 0g) [ funder ow; in nitaryg

arccotc(F )(x +^ y)

(arccot#

= result

(x +~

y); nearestF )

c(F )

c(F )

if x +^ y 2 c(F ) and (jyj =6 1 or x 6= 0) and y 6= 0

=arccotF (x) +^ ( 0) if x is not a NaN and y = 0

=negc(F )(arccotc(F )(0 +^ negF (y))

if x = 0

= conj c(F )(arccotc(F )(x +^ 0))

if y = 0 and x 6= 0

= mulF (signbF (x); 0) +^ mulF (signbF (y); 0)

if (x 2 f1; +1g and y 2 F [ f1; +1g) or (x 2 F [ f1; +1g and y 2 f1; +1g)

38	Speci cations for imaginary and complex datatypes and operations

= no resultF !c(F )(x)

Working draft			ISO/IEC WD 10967-3.1:2001(E)
	= in nitary(downF ( =4); x) +^ (mulF (y; 1))
			if x = 0 and jyj = 1
	= no		resultc(F )(x +^ y)
			otherwise

NOTE 2 { The inverse of cot is multi-valued, the real part may have any integer multiple of 2 (even any integer multiple of ) added to it, and the result is also in the solution set. The arccot function (returning the principal value for the inverse) branch cuts at f^ y j y 2 R and jyj < 1g. Thus arccotc(F )(0 +^ y) 6= arccotc(F )(( 0) +^ y) when jyj < 1 or y = 0.

5.3.2.12Radian arc secant

NOTE 1 { arcsec( z) = arcsec(z)

arcsec(conj(z)) = conj(arcsec(z)) if Im(z) 6= 0 or jRe(z)j > 1 arcsec(z) = =2 arccsc(z)

Arcsec(x +~ y) = ~ Arcsech(x +~ y) arcsec(z) = arccos(1=z)

The arcsecF !c(F ) operation:

arcsecF !c(F ) : F ! c(F [ f 0g) [ funder ow; in nitaryg

arcsecF !c(F )(x) = downF ( ) +^ negF (arcsechF (x))?

if (x 2 F and 0 6 x < 1)

=0 +^ arcsechF (x) if (x 2 F and 1 < x < 0)

=arcsecF (x) +^ 0 if (x 2 F and x > 1) or x = +1

=arcsecF (x) +^ ( 0) if (x 2 F and x 6 1) or x = 1

=in nitary(nearestF ( =4) +^ (+1))

if x = 0 and y = 0 otherwise

The arcseci(F )!c(F ) operation:

arcseci(F )!c(F ) : i(F ) ! c(F ) [ funder ow; in nitaryg

arcseci(F )!c(F )(^ y)

= nearestF ( =2) +^ arccschF (y)

The arcsecc(F ) approximation helper function:

arcsec

:

C ! C

c(F )

arcsec

(z) returns a close approximation to arcsec(z) in

C

with maximum error max

error

tan

c(F )

.

c(F )

Further requirements on the arcsecc(F ) approximation helper function are:

arcsec

(conj(z)) = conj(arcsec

(z))

if z

2 C

and (Im(z) = 0 or

j

Re(z)

j >

1

c(F )

>

c(F )

>

6

Im(arcsec

(z))

0

if Im(z)

0

c(F )

The relationship to the arcsecF and arcsechF

approximation helper functions in an associated

library for real-valued operations shall be:

arcsec

(x) = arcsec

(x)

if x

2 R

and

x

1

c(F )

F

j j >

The arcsec#

range limitation helper function:

c(F )

arcsec#

(z)

= min

f

Re(arcsec

(z)); downF ( =2)

g

+~

Im(arcsec

(z))

if Re(z)

>

1

c(F )

c(F )

c(F )

6

= min

f

Re(arcsec

(z)); downF ( )

g

+~

Im(arcsec

(z))

if Re(z)

1

c(F )

c(F )

= arcsecc(F )(z)

otherwise

5.3.2 Operations for radian trigonometric elementary functions

39

= arccosc(F )

= arccosc(F )

ISO/IEC WD 10967-3.1:2001(E) Working draft

The arcsecc(F ) operation:

arcsecc(F ) : c(F ) ! c(F ) [ funder ow; in nitaryg

arcsecc(F )(x +^ y)

= resultc(F )(arcsec#c(F )(x +~ y); nearestF )

if x +^ y 2 c(F ) and (x 6= 0 or y 6= 0)

= arcsecc(F )(0 +^ negF (y))

if x = 0

= conj c(F )(arcsecc(F )(x +^ 0))

if y = 0 and x 6= 0 and x 6= 0

(divF;c(F )(1; x +^ y))

if x 2 f 1; +1g and y 2 F [ f 1; +1g

(divF;c(F )(1; x +^ y))

if y 2 f 1; +1g and x 2 F [ f 1; +1g

= in nitary(nearest( =4) +^ (+1))

if y = 0 and x = 0 = in nitary(nearest(3 =4) +^ (+1))

if y = 0 and x = 0

= no resultc(F )(x +^ y)

otherwise

NOTE 2 { The inverse of sec is multi-valued, the real part may have any integer multiple of 2 added to it, and the result is also in the solution set. The arcsec function (returning the principal value for the inverse) branch cuts at fx j x 2 R and 1 < x < 1g. Thus arcsecc(F )(x +^ 0) 6= arcsecc(F )(x +^ ( 0)) when 1 < x < 1 or x = 0.

5.3.2.13Radian arc cosecant

NOTE 1 { arccsc( z) = arccsc(z)

arccsc(conj(z)) = conj(arccsc(z)) if Im(z) 6= 0 or jRe(z)j > 1 arccsc(z) = =2 arcsec(z)

Arcsec(x +~ y) = ~ Arcsech( y +~ x) arccsc(z) = arcsin(1=z)

The arccscF !c(F ) operation:

arccscF !c(F ) : F ! c(F [ f 0g) [ funder ow; in nitaryg

arccscF !c(F )(x) = arccscF (x) +^ ( 0) if (x 2 F and x > 1) or x = +1 = downF ( =2) +^ negF (arcsechF (x))??

if (x 2 F and 0 6 x 6 1) = upF ( =2) +^ arcsechF (x)??

if (x 2 F and 1 6 x < 0) or x = 0

=arccscF (x) +^ 0 if (x 2 F and x < 1) or x = 1

=in nitary(?upF ( =4) +^ (+1))

if x = 0 and y = 0 = no resultF !c(F )(x) otherwise

The arccsci(F ) operation:

arccsci(F ) : i(F ) ! i(F ) [ funder ow; in nitaryg arccsci(F )(^ y) = ^ arccschF (y)

40	Speci cations for imaginary and complex datatypes and operations

= arcsinc(F )(divF;c(F )

= arcsinc(F )(divF;c(F )

Working draft

ISO/IEC WD 10967-3.1:2001(E)

The arccscc(F ) approximation helper function:

arccsc

:

C ! C

c(F )

arccsc

(z) returns a close approximation to arccsc(z) in

C

with maximum error max

error

tan

c(F )

.

c(F )

Further requirements on the arccscc(F ) approximation helper function are:

arccsc

(conj(z)) = conj(arccsc

(z))

if z

2 C

and (Im(z) = 0 or

j

Re(z)

j

>

0)

c(F )

c(F )

6

arccsc

(

z) = arccsc

(z)

if z

2 C

and (Im(z) = 0 or

j

Re(z)

j

>

0)

c(F )

c(F )

6

Im(arccsc

(z))

6

0

if Im(z)

>

0

c(F )

The relationship to the arccscF and arccschF

approximation helper functions in an associated

library for real-valued operations shall be:

arccsc

(x) = arccsc (x)

if x

2 R

and x

j >

1

c(F )

(~

y) = ~

F

(

y)

if y

j

arccsc

arccsch

2 R

c(F )

F

The arccsc#

range limitation helper function:

c(F )

arccsc#

(z) = max

f

upF (

=2); min

f

Re(arccsc

(z)); downF ( =2)

gg

+~

Im(arccsc

(z))

c(F )

c(F )

c(F )

The arccscc(F ) operation:

arccscc(F ) : c(F ) ! c(F [ f 0g) [ funder ow; in nitaryg

arccscc(F )(x +^ y)

(arccsc#

= result

(x +~

y); nearestF )

c(F )

c(F )

= arccscF (x) +^ ( 0)

if x +^ y 2 c(F ) and (y 6= 0 or 0 < jxj < 1)

if y = 0 and jxj > 1

= negc(F )(arccscc(F )(0 +^ negF (y))

if x = 0

= conj c(F )(arccscc(F )(x +^ 0))

if y = 0 and x 6= 0 (1; x +^ y))

if x 2 f1; +1g and y 2 F [ f1; +1g

(1; x +^ y))

if y 2 f1; +1g and x 2 F [ f1; +1g

= in nitary(?upF ( =4) +^ (1))

if x = 0 and y = 0

= no resultc(F )(x +^ y)

otherwise

NOTE 2 { The inverse of csc is multi-valued, the real part may have any integer multiple of 2 added to it, and the result is also in the solution set. The arccsc function (returning the principal value for the inverse) branch cuts at fx j x 2 R and 1 < x < 1g. Thus arccscc(F )(x +^ 0) 6= arccscc(F )(x +^ ( 0)) when 1 < x < 1 or x = 0.

5.3.3Operations for hyperbolic elementary functions

Note that the correspondences speci ed below to other ISO/IEC 10967 operations are exact, not approximate.

5.3.3 Operations for hyperbolic elementary functions

41

ISO/IEC WD 10967-3.1:2001(E)

Working draft

5.3.3.1Hyperbolic normalisation

radhF : F ! F		if x 2 F [ f1; 0; +1; qNaNg
radhF (x)	= x
	= invalid(qNaN)	if x is a signalling NaN
radhi(F ) : i(F ) ! i(F ) [ funder ow; absolute				precision		under owg
radhi(F )(^ y)	= ^ radF (y)

radhc(F ) : c(F ) ! c(F ) [ funder ow; absolute precision under owg radhc(F )(x +^ y)

=itimesc(F )(radc(F )(y +^ negF (x)))

5.3.3.2Hyperbolic sine

NOTE { sinh( z) = sinh(z) sinh(conj(z)) = conj(sinh(z))

sinh(z +~ k 2 ) = sinh(z) if k 2 Z sinh(z) = cosh(~ =2 z)

sinh(x) = ~ sin(~ x) = ~ sin( ~ x) sinh(~ y) = ~ sin( y) = ~ sin(y)

sinh(x +~ y) = ~ sin( y +~ x) = ~ sin(y ~ x)

sinh(x +~ y) = sinh(x) cos(y) +~ cosh(x) sin(y)

The sinhi(F ) operation:

sinhi(F ) : i(F ) ! i(F ) [ funder ow; absolute precision under owg sinhi(F )(^ y) = ^ (sinF (y))

The sinhc(F ) operation:

sinhc(F ) : c(F ) ! c(F ) [ funder ow; over ow; absolute precision under owg sinhc(F )(x +^ y)

=itimesc(F )(sinc(F )(y +^ negF (x)))

5.3.3.3Hyperbolic cosine

NOTE { cosh( z) = cosh(z) cosh(conj(z)) = conj(cosh(z))

cosh(z +~ k 2 ) = cosh(z) if k 2 Z cosh(z) = sinh(~ =2 z)

cosh(x) = cos(~ x) = cos( ~ x) cosh(~ y) = cos(y)

cosh(x +~ y) = cos( y +~ x) = cos(y ~ x)

cosh(x +~ y) = cosh(x) cos(y) +~ sinh(x) sin(y)

42	Speci cations for imaginary and complex datatypes and operations