OpenVMS VAX RTL Mathematics (MTH$) Manual

Updated: 12 December 1998

OpenVMS VAX RTL Mathematics (MTH$) Manual

Contents

Index

MTH$HLOG2

The Base 2 Logarithm (H-Floating Value) routine returns the base 2 logarithm of the input value specified by floating-point-input-value as an H-floating value.

Format

MTH$HLOG2 h-log2 ,floating-point-input-value

RETURNS

None.

Arguments

h-log2

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Base 2 logarithm of floating-point-input-value. The h-log2 argument is the address of an H-floating number that is this base 2 logarithm. MTH$HLOG2 writes the address of this logarithm into h-log2.
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 input value. For MTH$HLOG2, floating-point-input-value specifies an H-floating number.

Description

The base 2 logarithm function is computed as follows:
zLOG2(X) = zLOG2(E) * zLOG(X)

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HLOG2 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.

MTH$HLOG10

The Common Logarithm (H-Floating Value) routine returns the common (base 10) logarithm of the input argument as an H-floating value.

Format

MTH$HLOG10 h-log10 ,floating-point-input-value

Corresponding JSB Entry Point

MTH$HLOG10_R8

RETURNS

None.

Arguments

h-log10

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Common logarithm of the input value specified by floating-point-input-value. The h-log10 argument is the address of an H-floating number that is this common logarithm. MTH$HLOG10 writes the address of the common logarithm into h-log10.
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$HLOG10, floating-point-input-value specifies an H-floating number.

Description

The common logarithm function is computed as follows:
zLOG10(X) = zLOG10(E) * zLOG(X)

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HLOG10 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.

MTH$HSIN

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

Format

MTH$HSIN h-sine ,angle-in-radians

Corresponding JSB Entry Point

MTH$HSIN_R5

RETURNS

None.

Arguments

h-sine

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

The sine of the angle specified by angle-in-radians. The h-sine argument is the address of an H-floating number that is this sine. MTH$HSIN writes the address of the sine into h-sine.
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 that is this angle. For MTH$HSIN, angle-in-radians specifies an H-floating number.

Description

See MTH$xSINCOS for the algorithm used to compute this sine.

Condition Value Signaled

SS$_ROPRAND Reserved operand. The MTH$HSIN 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$HSIND

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

Format

MTH$HSIND h-sine ,angle-in-degrees

Corresponding JSB Entry Point

MTH$HSIND_R5

RETURNS

None.

Arguments

h-sine

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Sine of the angle specified by angle-in-degrees. MTH$HSIND writes into h-sine the address of an H-floating number that is this sine.
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 an H-floating number that is this angle.

Description

See MTH$xSINCOSD for the algorithm used to compute the sine.

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HSIND 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$_FLOUNDMAT Floating-point underflow in Math Library. The absolute value of the input angle is less than 180/Pi sign*2 ^-m (where m = 16,384 for H-floating).

MTH$HSINH

The Hyperbolic Sine (H-Floating Value) routine returns the hyperbolic sine of the input value specified by floating-point-input-value as an H-floating value.

Format

MTH$HSINH h-sinh ,floating-point-input-value

RETURNS

None.

Arguments

h-sinh

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Hyperbolic sine of the input value specified by floating-point-input-value. The h-sinh argument is the address of an H-floating number that is this hyperbolic sine. MTH$HSINH writes the address of the hyperbolic sine into h-sinh.
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$HSINH, floating-point-input-value specifies an H-floating number.

Description

Computation of the hyperbolic sine function depends on the magnitude of the input argument. The range of the function is partitioned using three 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, zSINH(X) is computed as follows:

Value of X Value Returned

|X| < a(z) X

a(z) <= |X| < 1.0 zSINH(X) is computed using a power series expansion in |X| ²

1.0 <= |X| < b(z) (zEXP(X) - zEXP(-X))/2

b(z) <= |X| < c(z) SIGN(X)*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)	X
a(z) <= \|X\| < 1.0	zSINH(X) is computed using a power series expansion in \|X\| ²
1.0 <= \|X\| < b(z)	(zEXP(X) - zEXP(-X))/2
b(z) <= \|X\| < c(z)	SIGN(X)*zEXP(\|X\|)/2
c(z) <= \|X\|	Overflow occurs

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HSINH 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$_FLOOVEMAT Floating-point overflow in Math Library: the absolute value of floating-point-input-value is greater than yyy. LIB$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. The value of yyy is approximately 11356.523.

MTH$HSQRT

The Square Root (H-Floating Value) routine returns the square root of the input value floating-point-input-value as an H-floating value.

Format

MTH$HSQRT h-sqrt ,floating-point-input-value

Corresponding JSB Entry Point

MTH$HSQRT_R8

RETURNS

None.

Arguments

h-sqrt

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Square root of the input value specified by floating-point-input-value. The h-sqrt argument is the address of an H-floating number that is this square root. MTH$HSQRT writes the address of the square root into h-sqrt.
floating-point-input-value

OpenVMS usage: floating_point

type: H_floating

access: read only

mechanism: by reference

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

Description

The square root of X is computed as follows:

If X < 0 , an error is signaled.

Let X = 2 ^K * F

where:

K is the exponential part of the floating-point data

F is the fractional part of the floating-point data

If K is even:
X = 2 ^(2*P) * F,
zSQRT(X) = 2 ^P * zSQRT(F),
1/2 <= F < 1 , where P = K/2

If K is odd:
X = 2 ^(2*P+1) * F = 2 ^(2*P+2) * (F/2) ,
zSQRT(X) = 2 ^(P+1) * zSQRT(F/2),
1/4 <= F/2 < 1/2 , where p = (K-1)/2

Let F' = A*F + B, when K is even:

A = 0.95F6198 (hex)

B = 0.6BA5918 (hex)

Let F' = A* (F/2) + B, when K is odd:

A = 0.D413CCC (hex)

B = 0.4C1E248 (hex)

Let K' = P, when K is even

Let K' = P+1, when K is odd

Let Y[0] = 2^K' * F' be a straight line approximation within the given interval using coefficients A and B, which minimize the absolute error at the midpoint and endpoint.
Starting with Y[0], n Newton-Raphson iterations are performed:

Y[n+1] = 1/2 * (Y[n] + X/Y[n])

where n = 5 for H-floating.

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HSQRT 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$_SQUROONEG Square root of negative number. Argument floating-point-input-value is less than 0.0. LIB$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.

MTH$HTAN

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

Format

MTH$HTAN h-tan ,angle-in-radians

Corresponding JSB Entry Point

MTH$HTAN_R5

RETURNS

None.

Arguments

h-tan

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Tangent of the angle specified by angle-in-radians. The h-tan argument is the address of an H-floating number that is this tangent. MTH$HTAN writes the address of the tangent into h-tan.
angle-in-radians

OpenVMS usage: floating_point

type: H_floating

access: read only

mechanism: by reference

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

Description

When the input argument is expressed in radians, the tangent function is computed as follows:

If |X| < 2^(-f/2) , then zTAN(X) = X (see the section on MTH$zCOSH for the definition of f)
Otherwise, call MTH$zSINCOS to obtain zSIN(X) and zCOS(X); then

If zCOS(X) = 0 , signal overflow
Otherwise, zTAN(X) = zSIN(X)/zCOS(X)

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HTAN 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.

MTH$HTAND

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

Format

MTH$HTAND h-tan ,angle-in-degrees

Corresponding JSB Entry Point

MTH$HTAND_R5

RETURNS

None.

Arguments

h-tan

OpenVMS usage: floating_point

type: H_floating

access: write only

mechanism: by reference

Tangent of the angle specified by angle-in-degrees. The h-tan argument is the address of an H-floating number that is this tangent. MTH$HTAND writes the address of the tangent into h-tan.
angle-in-degrees

OpenVMS usage: floating_point

type: H_floating

access: read only

mechanism: by reference

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

Description

When the input argument is expressed in degrees, the tangent function is computed as follows:

If |X| < (180/Pi sign)*2^(-2/(e-1)) and underflow signaling is enabled, underflow is signaled (see the section on MTH$zCOSH for the definition of e).
Otherwise, if |X| < (180/Pi sign)*2^(-f/2) , then zTAND(X) = (Pi sign/180)*X . See the description of MTH$zCOSH for the definition of f.
Otherwise, call MTH$zSINCOSD to obtain zSIND(X) and zCOSD(X).

Then, if zCOSD(X) = 0 , signal overflow
Else, zTAND(X) = zSIND(X)/zCOSD(X)

Condition Values Signaled

SS$_ROPRAND Reserved operand. The MTH$HTAND 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.

Contents

Index

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

SS$_ROPRAND	Reserved operand. The MTH$HLOG2 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.

SS$_ROPRAND	Reserved operand. The MTH$HLOG10 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.

SS$_ROPRAND	Reserved operand. The MTH$HSIND 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$_FLOUNDMAT	Floating-point underflow in Math Library. The absolute value of the input angle is less than 180/Pi sign*2 ^-m (where m = 16,384 for H-floating).

SS$_ROPRAND	Reserved operand. The MTH$HSINH 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$_FLOOVEMAT	Floating-point overflow in Math Library: the absolute value of floating-point-input-value is greater than yyy. LIB$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. The value of yyy is approximately 11356.523.

	If X < 0 , an error is signaled.
	Let X = 2 ^K * F
	where:
	K is the exponential part of the floating-point data
	F is the fractional part of the floating-point data
	If K is even: X = 2 ^(2P) F, zSQRT(X) = 2 ^P * zSQRT(F), 1/2 <= F < 1 , where P = K/2
	If K is odd: X = 2 ^(2P+1) F = 2 ^(2P+2) (F/2) , zSQRT(X) = 2 ^(P+1) * zSQRT(F/2), 1/4 <= F/2 < 1/2 , where p = (K-1)/2
	Let F' = A*F + B, when K is even:
	A = 0.95F6198 (hex)
	B = 0.6BA5918 (hex)
	Let F' = A* (F/2) + B, when K is odd:
	A = 0.D413CCC (hex)
	B = 0.4C1E248 (hex)
	Let K' = P, when K is even
	Let K' = P+1, when K is odd

SS$_ROPRAND	Reserved operand. The MTH$HSQRT 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$_SQUROONEG	Square root of negative number. Argument floating-point-input-value is less than 0.0. LIB$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.

SS$_ROPRAND	Reserved operand. The MTH$HTAN 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.

SS$_ROPRAND	Reserved operand. The MTH$HTAND 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.