OpenVMS VAX RTL Mathematics (MTH$) Manual

Updated: 12 December 1998

OpenVMS VAX RTL Mathematics (MTH$) Manual

Contents

Index

MTH$HATAND2

Given sine and cosine, the Arc Tangent in Degrees (H-Floating Value) with Two Arguments routine returns the angle (in degrees) whose tangent is given by the quotient of sine and cosine (sine/cosine).

Format

MTH$HATAND2 h-degrees ,sine ,cosine

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 (sine/cosine). The h-degrees argument is the address of an H-floating number that is this angle. MTH$HATAND2 writes the address of the angle into h-degrees.
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$HATAND2, 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$HATAND2, cosine specifies an H-floating number.

Description

The angle in degrees whose tangent is Y/X is computed below. The value of f is defined in the description of MTH$zCOSH.

Value of Input Arguments Angle Returned

X = 0 or Y/X > 2 ^(f+1) 90* (sign Y)

X > 0 and Y/X <= 2 ^(f+1) zATAND(Y/X)

X < 0 and Y/X <= 2 ^(f+1) 180 * (sign Y) + zATAND(Y/X)

Value of Input Arguments	Angle Returned
X = 0 or Y/X > 2 ^(f+1)	90* (sign Y)
X > 0 and Y/X <= 2 ^(f+1)	zATAND(Y/X)
X < 0 and Y/X <= 2 ^(f+1)	180 * (sign Y) + zATAND(Y/X)

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HATAND2 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 Digital.

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.

MTH$HATANH

Given the hyperbolic tangent of an angle, the Hyperbolic Arc Tangent (H-Floating Value) routine returns the hyperbolic arc tangent (as an H-floating value) of that angle.

Format

MTH$HATANH h-atanh ,hyperbolic-tangent

RETURNS

None.

Arguments

h-atanh

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Hyperbolic arc tangent of the hyperbolic tangent specified by hyperbolic-tangent. The h-atanh argument is the address of an H-floating number that is this hyperbolic arc tangent. MTH$HATANH writes the address of the hyperbolic arc tangent into h-atanh.
hyperbolic-tangent

OpenVMS usage: floating_point

type: H_floating

access: read only

mechanism: by reference

Hyperbolic tangent of an angle. The hyperbolic-tangent argument is the address of a floating-point number that is this hyperbolic tangent. For MTH$HATANH, hyperbolic-tangent specifies an H-floating number.

Description

The hyperbolic arc tangent function is computed as follows:

Value of x Value Returned

|X| < 1 zATANH(X) = zLOG((X+1)/(X-1))/2

|X| => 1 An invalid argument is signaled

Value of x	Value Returned
\|X\| < 1	zATANH(X) = zLOG((X+1)/(X-1))/2
\|X\| => 1	An invalid argument is signaled

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HATANH 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 Digital.

MTH$_INVARGMAT Invalid argument: |X| => 1. LIB$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$HCOS

The Cosine of Angle Expressed in Radians (H-Floating Value) routine returns the cosine of a given angle (in radians) as an H-floating value.

Format

MTH$HCOS h-cosine ,angle-in-radians

Corresponding JSB Entry Point

MTH$HCOS_R5

RETURNS

None.

Arguments

h-cosine

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Cosine of the angle specified by angle-in-radians. The h-cosine argument is the address of an H-floating number that is this cosine. MTH$HCOS writes the address of the cosine into h-cosine.
angle-in-radians

OpenVMS usage: floating_point

type: H_floating

access: read only

mechanism: by reference

Angle (in radians). The angle-in-radians argument is the address of a floating-point number. For MTH$HCOS, angle-in-radians specifies an H-floating number.

Description

See MTH$xSINCOS for the algorithm used to compute the cosine.

Condition Value Signaled

SS$_ROPRAND Reserved operand. The MTH$HCOS 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 Digital.

MTH$HCOSD

The Cosine of Angle Expressed in Degrees (H-Floating Value) routine returns the cosine of a given angle (in degrees) as an H-floating value.

Format

MTH$HCOSD h-cosine ,angle-in-degrees

Corresponding JSB Entry Point

MTH$HCOSD_R5

RETURNS

None.

Arguments

h-cosine

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Cosine of the angle specified by angle-in-degrees. The h-cosine argument is the address of an H-floating number that is this cosine. MTH$HCOSD writes this cosine into h-cosine.
angle-in-degrees

OpenVMS usage: floating_point

type: H_floating

access: read only

mechanism: by reference

Angle (in degrees). The angle-in-degrees argument is the address of a floating-point number. For MTH$HCOSD, angle-in-degrees specifies an H-floating number.

Description

See the MTH$SINCOSD routine for the algorithm used to compute the cosine.

Condition Value Signaled

SS$_ROPRAND Reserved operand. The MTH$HCOSD 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 Digital.

MTH$HCOSH

The Hyperbolic Cosine (H-Floating Value) routine returns the hyperbolic cosine of the input value as an H-floating value.

Format

MTH$HCOSH h-cosh ,floating-point-input-value

RETURNS

None.

Arguments

h-cosh

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Hyperbolic cosine of the input value specified by floating-point-input-value. The h-cosh argument is the address of an H-floating number that is this hyperbolic cosine. MTH$HCOSH writes the address of the hyperbolic cosine into h-cosh.
floating-point-input-value

OpenVMS usage: floating_point

type: H_floating

access: read only

mechanism: by reference

The input value. The floating-point-input-value argument is the address of this input value. For MTH$HCOSH, floating-point-input-value specifies an H-floating number.

Description

Computation of the hyperbolic cosine depends on the magnitude of the input argument. The range of the function is partitioned using four data-type-dependent constants: a(z), b(z), and c(z). The subscript z indicates the data type. The constants depend on the number of exponent bits (e) and the number of fraction bits (f) associated with the data type (z).
The values of e and f are as follows:
e = 15 f = 113

The values of the constants in terms of e and f are:

Variable Value

a(z) 2 ^-f/2

b(z) (f+1)/2*ln(2)

c(z) 2 ^e-1*ln(2)

Based on the above definitions, zCOSH(X) is computed as follows:

Value of X Value Returned

|X| < a(z) 1

a(z) <= |X| < .25 Computed using a power series expansion in |X| ²

.25 <= |X| < b(z) (zEXP(|X|) + 1/zEXP(|X|))/2

b(z) <= |X| < c(z) zEXP(|X|)/2

c(z) <= |X| Overflow occurs

Variable	Value
a(z)	2 ^-f/2
b(z)	(f+1)/2*ln(2)
c(z)	2 ^e-1*ln(2)

Value of X	Value Returned
\|X\| < a(z)	1
a(z) <= \|X\| < .25	Computed using a power series expansion in \|X\| ²
.25 <= \|X\| < b(z)	(zEXP(\|X\|) + 1/zEXP(\|X\|))/2
b(z) <= \|X\| < c(z)	zEXP(\|X\|)/2
c(z) <= \|X\|	Overflow occurs

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HCOSH 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 Digital.

MTH$_FLOOVEMAT Floating-point overflow in Math Library: the absolute value of floating-point-input-value is greater than about yyy; LIB$SIGNAL copies the reserved operand to the signal mechanism vector. The result is the reserved operand -0.0 unless a condition handler changes the signal mechanism vector. The value of yyy is 11356.523.

MTH$HEXP

The Exponential (H-Floating Value) routine returns the exponential of the input value as an H-floating value.

Format

MTH$HEXP h-exp ,floating-point-input-value

Corresponding JSB Entry Point

MTH$HEXP_R6

RETURNS

None.

Arguments

h-exp

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Exponential of the input value specified by floating-point-input-value. The h-exp argument is the address of an H-floating number that is this exponential. MTH$HEXP writes the address of the exponential into h-exp.
floating-point-input-value

OpenVMS usage: floating_point

type: H_floating

access: read only

mechanism: by reference

The input value. The floating-point-input-value argument is the address of a floating-point number. For MTH$HEXP, floating-point-input-value specifies an H-floating number.

Description

The exponential of x is computed as:

Value of x Value Returned

x > c(z) Overflow occurs

x <= -c(z) 0

|x| < 2 ^-(f+1) 1

Otherwise 2 ^Y * 2 ^U * 2 ^W

where: Y = INTEGER(x*ln2(E)) V = FRAC(x*ln2(E)) * 16 U = INTEGER(V)/16 W = FRAC(V)/16 2^W = polynomial approximation of degree 14 for z = H.
See also MTH$HCOS for definitions of f and c(z).

Value of x	Value Returned
x > c(z)	Overflow occurs
x <= -c(z)	0
\|x\| < 2 ^-(f+1)	1
Otherwise	2 ^Y * 2 ^U * 2 ^W

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HEXP 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 Digital.

MTH$_FLOOVEMAT Floating-point overflow in Math Library: floating-point-input-value is greater than yyy; LIB$SIGNAL copies the reserved operand to the signal mechanism vector. The result is the reserved operand -0.0 unless a condition handler changes the signal mechanism vector. The value of yyy is approximately 11355.830 for MTH$HEXP.

MTH$_FLOUNDMAT Floating-point underflow in Math Library: floating-point-input-value is less than or equal to yyy and the caller (CALL or JSB) has set hardware floating-point underflow enable. The result is set to 0.0. If the caller has not enabled floating-point underflow (the default), a result of 0.0 is returned but no error is signaled. The value of yyy is approximately --11356.523 for MTH$HEXP.

MTH$HLOG

The Natural Logarithm (H-Floating Value) routine returns the natural (base e) logarithm of the input argument as an H-floating value.

Format

MTH$HLOG h-natlog ,floating-point-input-value

Corresponding JSB Entry Point

MTH$HLOG_R8

RETURNS

None.

Arguments

h-natlog

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Natural logarithm of floating-point-input-value. The h-natlog argument is the address of an H-floating number that is this natural logarithm. MTH$HLOG writes the address of this natural logarithm into h-natlog.
floating-point-input-value

OpenVMS usage: floating_point

type: H_floating

access: read only

mechanism: by reference

The input value. The floating-point-input-value argument is the address of a floating-point number that is this value. For MTH$HLOG, floating-point-input-value specifies an H-floating number.

Description

Computation of the natural logarithm routine is based on the following:

ln(X*Y) = ln(X) + ln(Y)
ln(1+X) = X - X²/2 + X³/3 - X⁴/4 ...
for |X| < 1
ln(X) = ln(A) + 2* (V + V³/3 + V⁵/5 + V⁷/7 ...)
where V = (X-A)/(X+A), A > 0,
and p(y) = 2 * (1 + y/3 + y²/5 ...)
For x = 2ⁿ*f, where n is an integer and f is in the interval of 0.5 to 1, define the following quantities:
If n => 1, then N = n-1 and F = 2f

If n <= 0, then N = n and F = f

From (1) it follows that:
ln(X) = N*ln(2) + ln(F)

Based on the previous relationships, zLOG is computed as follows:

If |F-1| < 2^-5,
zLOG(X) = N*zLOG(2) + W + W*p(W),
where W = F-1.
Otherwise,
zLOG(X) = N*zLOG(2) + zLOG(A) + V*p(V²),
where V = (F-A)/(F+A) and A and zLOG(A)
are obtained by table lookup.

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HLOG 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 Digital.

MTH$_LOGZERNEG Logarithm of zero or negative value. Argument floating-point-input-value is less than or equal to 0.0. LIB$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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Index

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OpenVMS usage:	floating_point
type:	H_floating
access:	write only
mechanism:	by reference

SS$_ROPRAND	Reserved operand. The MTH$HATAND2 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 Digital.
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.

SS$_ROPRAND	Reserved operand. The MTH$HATANH 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 Digital.
MTH$_INVARGMAT	Invalid argument: \|X\| => 1. LIB$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.

SS$_ROPRAND	Reserved operand. The MTH$HCOSH 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 Digital.
MTH$_FLOOVEMAT	Floating-point overflow in Math Library: the absolute value of floating-point-input-value is greater than about yyy; LIB$SIGNAL copies the reserved operand to the signal mechanism vector. The result is the reserved operand -0.0 unless a condition handler changes the signal mechanism vector. The value of yyy is 11356.523.

SS$_ROPRAND	Reserved operand. The MTH$HEXP 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 Digital.
MTH$_FLOOVEMAT	Floating-point overflow in Math Library: floating-point-input-value is greater than yyy; LIB$SIGNAL copies the reserved operand to the signal mechanism vector. The result is the reserved operand -0.0 unless a condition handler changes the signal mechanism vector. The value of yyy is approximately 11355.830 for MTH$HEXP.
MTH$_FLOUNDMAT	Floating-point underflow in Math Library: floating-point-input-value is less than or equal to yyy and the caller (CALL or JSB) has set hardware floating-point underflow enable. The result is set to 0.0. If the caller has not enabled floating-point underflow (the default), a result of 0.0 is returned but no error is signaled. The value of yyy is approximately --11356.523 for MTH$HEXP.

SS$_ROPRAND	Reserved operand. The MTH$HLOG 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 Digital.
MTH$_LOGZERNEG	Logarithm of zero or negative value. Argument floating-point-input-value is less than or equal to 0.0. LIB$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.