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OpenVMS VAX RTL Mathematics (MTH$) Manual
MTH$HACOS
Given the cosine of an angle, the Arc Cosine of Angle Expressed in
Radians (H-Floating Value) routine returns that angle (in radians) in
H-floating-point precision.
Format
MTH$HACOS h-radians ,cosine
Corresponding JSB Entry Point
MTH$HACOS_R8
RETURNS
None.
Arguments
h-radians
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
write only |
| mechanism: |
by reference |
Angle (in radians) whose cosine is specified by
cosine. The h-radians argument is the
address of an H-floating number that is this angle. MTH$HACOS writes
the address of the angle into h-radians.
cosine
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
read only |
| mechanism: |
by reference |
The cosine of the angle whose value (in radians) is to be returned. The
cosine argument is the address of a floating-point
number that is this cosine. The absolute value of
cosine must be less than or equal to 1. For MTH$HACOS,
cosine specifies an H-floating number.
Description
The angle in radians whose cosine is X is computed as:
| Value of Cosine |
Value Returned |
|
0
|
Pi sign/2
|
|
1
|
0
|
|
-1
|
Pi sign
|
|
0
< X
< 1
|
zATAN(zSQRT(1-X
2)/X) , where zATAN and zSQRT are the Math Library arc
tangent and square root routines, respectively, of the appropriate data
type
|
|
-1
< X
< 0
|
zATAN(zSQRT(1-X
2)/X) + Pi sign
|
|
1
< |X|
|
The error MTH$_INVARGMAT is signaled
|
Condition Values Signaled
|
SS$_ROPRAND
|
Reserved operand. The MTH$HACOS routine encountered a floating-point
reserved operand due to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit of 1 and a biased
exponent of 0. Floating-point reserved operands are reserved for future
use by Compaq.
|
|
MTH$_INVARGMAT
|
Invalid argument. The absolute value of
cosine is greater than 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism argument vector
CHF$L_MCH_SAVR0/R1. The result is the floating-point reserved operand
unless you have written a condition handler to change
CHF$L_MCH_SAVR0/R1.
|
MTH$HACOSD
Given the cosine of an angle, the Arc Cosine of Angle Expressed in
Degrees (H-Floating Value) routine returns that angle (in degrees) as
an H-floating value.
Format
MTH$HACOSD h-degrees ,cosine
Corresponding JSB Entry Point
MTH$HACOSD_R8
RETURNS
None.
Arguments
h-degrees
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
write only |
| mechanism: |
by reference |
Angle (in degrees) whose cosine is specified by
cosine. The h-degrees argument is the
address of an H-floating number that is this angle. MTH$HACOSD writes
the address of the angle into h-degrees.
cosine
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
read only |
| mechanism: |
by reference |
Cosine of the angle whose value (in degrees) is to be returned. The
cosine argument is the address of a floating-point
number that is this cosine. The absolute value of
cosine must be less than or equal to 1. For
MTH$HACOSD, cosine specifies an H-floating number.
Description
The angle in degrees whose cosine is X is computed as:
| Value of Cosine |
Angle Returned |
|
0
|
90
|
|
1
|
0
|
|
-1
|
180
|
|
0
< X
< 1
|
zATAND(zSQRT(1-X
2)/X) , where zATAND and zSQRT are the Math Library arc
tangent and square root routines, respectively, of the appropriate data
type
|
|
-1
< X
< 0
|
zATAND(zSQRT(1-X
2)/X) + 180
|
|
1
< |X|
|
The error MTH$_INVARGMAT is signaled
|
Condition Values Signaled
|
SS$_ROPRAND
|
Reserved operand. The MTH$HACOSD routine encountered a floating-point
reserved operand due to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit of 1 and a biased
exponent of 0. Floating-point reserved operands are reserved for future
use by Compaq.
|
|
MTH$_INVARGMAT
|
Invalid argument. The absolute value of
cosine is greater than 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism argument vector
CHF$L_MCH_SAVR0/R1. The result is the floating-point reserved operand
unless you have written a condition handler to change
CHF$L_MCH_SAVR0/R1.
|
MTH$HASIN
Given the sine of an angle, the Arc Sine in Radians (H-Floating Value)
routine returns that angle (in radians) as an H-floating value.
Format
MTH$HASIN h-radians ,sine
Corresponding JSB Entry Point
MTH$HASIN_R8
RETURNS
None.
Arguments
h-radians
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
write only |
| mechanism: |
by reference |
Angle (in radians) whose sine is specified by sine.
The h-radians argument is the address of an H-floating
number that is this angle. MTH$HASIN writes the address of the angle
into h-radians.
sine
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
read only |
| mechanism: |
by reference |
The sine of the angle whose value (in radians) is to be returned. The
sine argument is the address of a floating-point
number that is this sine. The absolute value of sine
must be less than or equal to 1. For MTH$HASIN, sine
specifies an H-floating number.
Description
The angle in radians whose sine is X is computed as:
| Value of Sine |
Angle Returned |
|
0
|
0
|
|
1
|
Pi sign/2
|
|
-1
|
-Pi sign/2
|
|
0
< |X|
< 1
|
zATAN(X/zSQRT(1-X
2)) , where zATAN and zSQRT are the Math Library arc tangent
and square root routines, respectively, of the appropriate data type
|
|
1
< |X|
|
The error MTH$_INVARGMAT is signaled
|
Condition Values Signaled
|
SS$_ROPRAND
|
Reserved operand. The MTH$HASIN routine encountered a floating-point
reserved operand due to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit of 1 and a biased
exponent of 0. Floating-point reserved operands are reserved for future
use by Compaq.
|
|
MTH$_INVARGMAT
|
Invalid argument. The absolute value of
sine is greater than 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism argument vector
CHF$L_MCH_SAVR0/R1. The result is the floating-point reserved operand
unless you have written a condition handler to change
CHF$L_MCH_SAVR0/R1.
|
MTH$HASIND
Given the sine of an angle, the Arc Sine in Degrees (H-Floating Value)
routine returns that angle (in degrees) as an H-floating value.
Format
MTH$HASIND h-degrees ,sine
Corresponding JSB Entry Point
MTH$HASIND_R8
RETURNS
None.
Arguments
h-degrees
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
write only |
| mechanism: |
by reference |
Angle (in degrees) whose sine is specified by sine.
The h-degrees argument is the address of an H-floating
number that is this angle. MTH$HASIND writes the address of the angle
into h-degrees.
sine
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
read only |
| mechanism: |
by reference |
Sine of the angle whose value (in degrees) is to be returned. The
sine argument is the address of a floating-point
number that is this sine. The absolute value of sine
must be less than or equal to 1. For MTH$HASIND, sine
specifies an H-floating number.
Description
The angle in degrees whose sine is X is computed as:
| Value of Sine |
Value Returned |
|
0
|
0
|
|
1
|
90
|
|
-1
|
-90
|
|
0
< |X|
< 1
|
zATAND(X/zSQRT(1-X
2)) , where zATAND and zSQRT are the Math Library arc
tangent and square root routines, respectively, of the appropriate data
type
|
|
1
< |X|
|
The error MTH$_INVARGMAT is signaled
|
Condition Values Signaled
|
SS$_ROPRAND
|
Reserved operand. The MTH$HASIND routine encountered a floating-point
reserved operand due to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit of 1 and a biased
exponent of 0. Floating-point reserved operands are reserved for future
use by Compaq.
|
|
MTH$_INVARGMAT
|
Invalid argument. The absolute value of
sine is greater than 1. LIB$SIGNAL copies the
floating-point reserved operand to the mechanism argument vector
CHF$L_MCH_SAVR0/R1. The result is the floating-point reserved operand
unless you have written a condition handler to change
CHF$L_MCH_SAVR0/R1.
|
MTH$HATAN
Given the tangent of an angle, the Arc Tangent in Radians (H-Floating
Value) routine returns that angle (in radians) as an H-floating value.
Format
MTH$HATAN h-radians ,tangent
Corresponding JSB Entry Point
MTH$HATAN_R8
RETURNS
None.
Arguments
h-radians
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
write only |
| mechanism: |
by reference |
Angle (in radians) whose tangent is specified by
tangent. The h-radians argument is
the address of an H-floating number that is this angle. MTH$HATAN
writes the address of the angle into h-radians.
tangent
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
read only |
| mechanism: |
by reference |
The tangent of the angle whose value (in radians) is to be returned.
The tangent argument is the address of a
floating-point number that is this tangent. For MTH$HATAN,
tangent specifies an H-floating number.
Description
In radians, the computation of the arc tangent function is based on the
following identities:
|
|
arctan(X) = X - X
3/3 + X
5/5 - X
7/7 + ...
|
|
|
arctan(X) = X + X*Q(X
2),
where Q(Y) = - Y/3 + Y
2/5 - Y
3/7 + ...
|
|
|
arctan(X) = X*P(X
2),
where P(Y) = 1 - Y/3 + Y
2/5 - Y
3/7 + ...
|
|
|
arctan(X) = Pi/2 - arctan(1/X)
|
|
|
arctan(X) = arctan(A) + arctan((X-A)/(1+A*X))
for any real A
|
The angle in radians whose tangent is X is computed as:
| Value of X |
Angle Returned |
|
0
<= X
<= 3/32
|
X + X * Q(X
2)
|
|
3/32
< X
<= 11
|
ATAN(A) + V* (P(V
2)) , where A and ATAN(A) are chosen by table lookup and V =
(X - A)/(1 + A*X)
|
|
11
< X
|
Pi sign/2 - W* (P(W
2)) where W = 1/X
|
|
X
< 0
|
-zATAN(|X|)
|
Condition Value Signaled
|
SS$_ROPRAND
|
Reserved operand. The MTH$HATAN routine encountered a floating-point
reserved operand due to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit of 1 and a biased
exponent of 0. Floating-point reserved operands are reserved for future
use by Compaq.
|
MTH$HATAND
Given the tangent of an angle, the Arc Tangent in Degrees (H-Floating
Value) routine returns that angle (in degrees) as an H-floating value.
Format
MTH$HATAND h-degrees ,tangent
Corresponding JSB Entry Point
MTH$HATAND_R8
RETURNS
None.
Arguments
h-degrees
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
write only |
| mechanism: |
by reference |
Angle (in degrees) whose tangent is specified by
tangent. The h-degrees argument is
the address of an H-floating number that is this angle. MTH$HATAND
writes the address of the angle into h-degrees.
tangent
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
read only |
| mechanism: |
by reference |
The tangent of the angle whose value (in degrees) is to be returned.
The tangent argument is the address of a
floating-point number that is this tangent. For MTH$HATAND,
tangent specifies an H-floating number.
Description
The computation of the arc tangent function is based on the following
identities:
|
|
arctan(X) = 180/Pi sign* (X - X
3/3 + X
5/5 - X
7/7 + ...)
|
|
|
arctan(X) = 64*X + X*Q(X
2),
where Q(Y) = 180/Pi sign*[(1- 64*Pi sign/180) - Y/3 +
Y
2/5 - Y
3/7 + Y
4/9 ...]
|
|
|
arctan(X) = X*P(X
2),
where P(Y) = 180/Pi sign*[1 - Y/3 + Y
2/5 - Y
3/7 +
Y
4/9 ...]
|
|
|
arctan(X) = 90 - arctan>(1/X)
|
|
|
arctan(X) = arctan(A) + arctan((X - A)/(1 + A*X))
|
The angle in degrees whose tangent is X is computed as:
| Tangent |
Angle Returned |
|
X
<= 3/32
|
64*X + X*Q(X
2)
|
|
3/32
< X
<= 11
|
ATAND(A) + V*P(V
2) , where A and ATAND(A) are chosen by table lookup and V =
(X - A)/(1 + A*X)
|
|
11
< X
|
90 - W * (P(W
2)) , where W = 1/X
|
|
X
< 0
|
-zATAND(|X|)
|
Condition Value Signaled
|
SS$_ROPRAND
|
Reserved operand. The MTH$HATAND routine encountered a floating-point
reserved operand due to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit of 1 and a biased
exponent of 0. Floating-point reserved operands are reserved for future
use by Compaq.
|
MTH$HATAN2
Given sine and cosine, the Arc
Tangent in Radians (H-Floating Value) with Two Arguments routine
returns the angle (in radians) as an H-floating value whose tangent is
given by the quotient of sine and
cosine (sine/cosine).
Format
MTH$HATAN2 h-radians ,sine ,cosine
RETURNS
None.
Arguments
h-radians
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
write only |
| mechanism: |
by reference |
Angle (in radians) whose tangent is specified by
(sine/cosine). The
h-radians argument is the address of an H-floating
number that is this angle. MTH$HATAN2 writes the address of the angle
into h-radians.
sine
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
read only |
| mechanism: |
by reference |
Dividend. The sine argument is the address of a
floating-point number that is this dividend. For MTH$HATAN2,
sine specifies an H-floating number.
cosine
| OpenVMS usage: |
floating_point |
| type: |
H_floating |
| access: |
read only |
| mechanism: |
by reference |
Divisor. The cosine argument is the address of a
floating-point number that is this divisor. For MTH$HATAN2,
cosine specifies an H-floating number.
Description
The angle in radians whose tangent is Y/X is computed
as follows, where f is defined in the description of MTH$zCOSH:
| Value of Input Arguments |
Angle Returned |
|
X = 0 or Y/X > 2
(f+1)
|
Pi sign/2* (sign Y)
|
|
X > 0 and Y/X
<= 2
(f+1)
|
zATAN(Y/X)
|
|
X
< 0 and Y/X
<= 2
(f+1)
|
Pi sign* (sign Y) + zATAN(Y/X)
|
Condition Values Signaled
|
SS$_ROPRAND
|
Reserved operand. The MTH$HATAN2 routine encountered a floating-point
reserved operand due to incorrect user input. A floating-point reserved
operand is a floating-point datum with a sign bit of 1 and a biased
exponent of 0. Floating-point reserved operands are reserved for future
use by Compaq.
|
|
MTH$_INVARGMAT
|
Invalid argument. Both
cosine and
sine are zero. LIB$SIGNAL copies the floating-point
reserved operand to the mechanism argument vector CHF$L_MCH_SAVR0/R1.
The result is the floating-point reserved operand unless you have
written a condition handler to change CHF$L_MCH_SAVR0/R1.
|
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