Compute Math Function Using One Input Value (CMF1)

Op Code (Hex)	Extender	Operand 1	Operand 2	Operand 3	Operand [4-5]
CMF1 100B		Receiver	Controls	Source

CMF1B 1C0B	Branch options	Receiver	Controls	Source	Branch targets

CMF1I 180B	Indicator options	Receiver	Controls	Source	Indicator targets

Operand 1: Numeric variable scalar.

Operand 2: Character(2) scalar.

Operand 3: Numeric scalar.

Operand 4-5:

Branch Form-Branch point, instruction pointer, relative instruction number, or absolute instruction number.
Indicator Form-Numeric variable scalar or character variable scalar.

Description

The mathematical function, indicated by the controls operand, is performed on the source operand value and the result is placed in the receiver operand.

The calculation is always done in floating-point.

The result of the operation is copied into the receiver operand.

The controls operand must be a character scalar that specifies which mathematical function is to be performed. It must be at least 2 bytes in length and has the following format:

Offset
Dec	Hex	Field Name	Data Type and Length
0	0	Controls operand	Char(2)
		Hex 0001 = Sine Hex 0002 = Arc sine Hex 0003 = Cosine Hex 0004 = Arc cosine Hex 0005 = Tangent Hex 0006 = Arc tangent Hex 0007 = Cotangent Hex 0010 = Exponential function Hex 0011 = Logarithm based e (natural logarithm) Hex 0012 = Sine hyperbolic Hex 0013 = Cosine hyperbolic Hex 0014 = Tangent hyperbolic Hex 0015 = Arc tangent hyperbolic Hex 0020 = Square root All other values are reserved
2	2	--- End ---

The controls operand mathematical functions are as follows:

Hex 0001-Sine
The sine of the numeric value of the source operand, whose value is considered to be in radians, is computed and placed in the receiver operand.
The result is in the range:
```
-1 <= SIN(x) <= 1
```
Hex 0002-Arc sine
The arc sine of the numeric value of the source operand is computed and the result (in radians) is placed in the receiver operand.
The result is in the range:
```
-pi/2 <= ASIN(x) <= +pi/2
```
Hex 0003-Cosine
The cosine of the numeric value of the source operand, whose value is considered to be in radians, is computed and placed in the receiver operand.
The result is in the range:
```
-1 <= COS(x) <= 1
```
Hex 0004-Arc cosine
The arc cosine of the numeric value of the source operand is computed and the result (in radians) is placed in the receiver operand.
The result is in the range:
```
0 <= ACOS(x) <= pi
```
Hex 0005-Tangent
The tangent of the source operand, whose value is considered to be in radians, is computed and the result is placed in the receiver operand.
The result is in the range:
```
-infinity <= TAN(x) <= +infinity
```
Hex 0006-Arc tangent
The arc tangent of the source operand is computed and the result (in radians) is placed in the receiver operand.
The result is in the range:
```
-pi/2 <= ATAN(x) <= pi/2
```
Hex 0007-Cotangent
The cotangent of the source operand, whose value is considered to be in radians, is computed and the result is placed in the receiver operand.
The result is in the range:
```
-infinity <= COT(x) <= +infinity
```
Hex 0010-Exponential function
The computation e power (source operand) is performed and the result is placed in the receiver operand.
The result is in the range:
```
0 <= EXP(x) <= +infinity
```
Hex 0011-Logarithm based e (natural logarithm)
The natural logarithm of the source operand is computed and the result is placed in the receiver operand.
The result is in the range:
```
-infinity <= LN(x) <= +infinity
```
Hex 0012-Sine hyperbolic
The sine hyperbolic of the numeric value of the source operand is computed and the result (in radians) is placed in the receiver operand.
The result is in the range:
```
-infinity <= SINH(x) <= +infinity
```
Hex 0013-Cosine hyperbolic
The cosine hyperbolic of the numeric value of the source operand is computed and the result (in radians) is placed in the receiver operand.
The result is in the range:
```
+1 <= COSH(x) <= +infinity
```
Hex 0014-Tangent hyperbolic
The tangent hyperbolic of the numeric value of the source operand is computed and the result (in radians) is placed in the receiver operand.
The result is in the range:
```
-1 <= TANH(x) <= +1
```
Hex 0015-Arc tangent hyperbolic
The inverse of the tangent hyperbolic of the numeric value of the source operand is computed and the result (in radians) is placed in the receiver operand.
The result is in the range:
```
-infinity <= ATANH(x) <= +infinity
```
Hex 0020-Square root
The square root of the numeric value of the source operand is computed and placed in the receiver operand.
The result is in the range:
```
0 <= SQRT(x) <= +infinity
```

The following chart shows some special cases for certain arguments (X) of the different mathematical functions which take one argument value.

X	Masked	UnMasked					Maximum	Minimum
Function	NaN	NaN	+Infinity	-Infinity	+0	-0	Value	Value	Other
Sine	g	A(e)	A(f)	A(f)	+0	-0	A(1,f)	A(1,f)	B(3)
Arc sine	g	A(e)	A(f)	A(f)	+0	-0	A(6,f)	A(6,f)	-
Cosine	g	A(e)	A(f)	A(f)	+1	+1	A(1,f)	A(1,f)	B(3)
Arc cosine	g	A(e)	A(f)	A(f)	+pi/2	+pi/2	A(6,f)	A(6,f)	-
Tangent	g	A(e)	A(f)	A(f)	+0	-0	A(1,f)	A(1,f)	B(3)
Arc Tangent	g	A(e)	+pi/2	-pi/2	+0	-0	-	-	-
Cotangent	g	A(e)	A(f)	A(f)	+inf	-inf	A(1,f)	A(1,f)	B(3)
E Exponent	g	A(e)	+inf	+0	+1	+1	C(4,a)	D(5,b)	-
Logarithm	g	A(e)	+inf	A(f)	-inf	-inf	-	-	A(2,f)
Sine hyperbolic	g	A(e)	+inf	-inf	+0	-0	-	-	-
Cosine hyperbolic	g	A(e)	+inf	+inf	+1	+1	-	-	-
Tangent hyperbolic	g	A(e)	+1	-1	+0	-0	-	-	-
Arc tangent hyperbolic	g	A(e)	A(f)	A(f)	+0	-0	A(6,f)	A(6,f)	-
Square root	g	A(e)	+inf	A(f)	+0	-0	-	-	A(2,f)
Special cases for single argument math functions

Capital letters in the chart indicate the exceptions, small letters indicate the returned results, and Arabic numerals indicate the limits of the arguments (X) as defined in the following lists:

A =	Floating-point invalid operand (hex 0C09) exception (no result stored if unmasked; if masked, occurrence bit is set)
B =	Floating-point inexact result (hex 0C0D) exception (result is stored whether or not exception is masked)
C =	Floating-point overflow (hex 0C06) exception (no result is stored if unmasked; if masked, occurrence bit is set)
D =	Floating-point underflow (hex 0C07) exception (no result is stored if unmasked; occurrence bit is always set)
a =	Result follows the rules that depend on round mode
b =	Result is +0 or a denormalized value
c =	Result is +infinity
d =	Result is -infinity
e =	Result is the masked form of the input NaN
f =	Result is the system default masked NaN
g =	Result is the input NaN
inf =	Result is infinity
1 =	\| pi * 2**50 \|=Hex 432921FB54442D18
2 =	Argument is in the range: -inf < x < -0
3 =	\| pi * 2**26 \|=Hex 41A921FB54442D18
4 =	1n(2**1023) Hex 40862E42FEFA39EF
5 =	1n(2**-1021.4555)=Hex C086200000000000
6 =	Argument is in the range: -1 <= x <= +1

The following chart provides accuracy data for the mathematical functions that can be invoked by this instruction.

Function Name	Sample Selection			Accuracy data
	Sample Selection			Relative Error(e)		Absolute Error(E)
	A	Range of x	D	MAX(e)	SD(e)	MAX(E)	SD(E)
Arc cosine	9	0<=x<=3.14	U			8.26* 10**-14	2.1110*-15
Arc sine	10	-1.57<=x<=1.57	U	1.0210*-13	2.6610*-15
Arc tangent	1	-pi/2<x<pi/2	1			3.3310*-16	9.5710*-17
Arc tangent hyperbolic	14	-3<=x<=3	U			1.0610*-14	1.7910*-15
Cosine		(See Sine below)
Cosine hyperbolic		(See Sine Hyperbolic below)
Cotangent	11	-10<=x<=100 .000001<=x<=.001 4000<=x<=4000000	U U U	4.8310-16 4.3610*-16 5.7210**-16	1.4810-16 1.4910*-16 1.4610**-16
Exponential	2	-100<=x<=300	U	5.7010*-14	1.1310*-14
Natural Logarithm	3	0.5<=x<=1.5	U			2.7710*-16	8.0110*-17
Natural Logarithm	4	-100<=x<=700	E	2.1710*-16	7.3710*-17
Sine cosine	5	-10<=x<=100 .000001<=x<=.001 4000<=x<=4000000	U U U			2.2210-16 2.2210*-16 2.2210**-16	1.3110-16 1.5610*-16 1.2810**-16
Sine cosine	6	-10<=x<=100 .000001<=x<=.001 4000<=x<=4000000	U U U			3.3310-16 4.3310*-19 3.3310**-16	8.3910-17 1.2810*-19 8.1710**-17
Sine/cosine hyperbolic	12	-100<=x<=300	U	6.3110*-16	1.9710*-16
Square root	7	-100<=x<=700	E	4.1310*-16	1.2710*-16
Tangent	8	-10<=x<=100 .000001<=x<=.001 4000<=x<=4000000	U U U	4.5910-16 4.4210*-16 4.7710**-16	1.5410-16 1.4410*-16 1.4310**-16	3.2510*-19	8.0610*-20
Tangent hyperbolic	13	-100<=x<=300	U	8.3510*-16	3.8710*-17	2.2210*-16	3.1710*-17
Note: Algorithm Notes: f(x) = x, and g(x) = ATAN(TAN(x)). f(x) = ex, and g(x) = e(1n(ex)). f(x) = 1n(x), and g(x) = 1n(e(1n(x))). f(x) = x, and g(x) = 1n(ex). Sum of squares algorithm. f(x) = 1, and g(x) = SIN(x))2 + (COS(x))*2. Double angle algorithm. f(x) - SIN(2x), and g(x) = 2(SIN(x)COS(x)). f(x) = e(x, and g(x) = (SQR(ex))2. f(x) = TAN(x), and g(x) = SIN(x) / COS(x). f(x) = x, and g(x) = ACOS(COS(x)). f(x) = x, and g(x) = ASIN(SIN(x)). f(x) = COT(x), and g(x) = COS(x) / SIN(x). f(x) = SINH(2x), and g(x) = 2(SINH(x)*COSH(x)). f(x) = TANH(x), and g(x) = SINH(x) / COSH(x). f(x) = x, and g(x) = ATANH(TANH(x)). Distribution Note: The sample input arguments were tangents of numbers, x, uniformly distributed between -pi/2 and +pi/2.
Accuracy data for CMF1 mathematical functions

The vertical columns in the accuracy data chart have the following meanings:

Function Name: This column identifies the principal mathematical functions evaluated with entries arranged in alphabetical order by function name.
Sample Selection: This column identifies the selection of samples taken for a particular math function through the following subcolumns:
- A: identifies the algorithm used against the argument, x, to gather the accuracy samples. The numbers in this column refer to notes describing the functions, f(x) and g(x), which were calculated to test for the anticipated relation where f(x) should equal g(x). An accuracy sample then, is an evaluation of the degree to which this relation held true. The algorithm used to sample the arctangent function, for example, defines g(x) to first calculate the tangent of x to provide an appropriate distribution of input arguments for the arctangent function. Since f(x) is defined simply as the value of x, the relation to be evaluated is then x=ARCTAN(TAN(x)). This type of algorithm, where a function and its inverse are used in tandem, is the usual type employed to provide the appropriate comparison values for the evaluation.
- "Range of x": gives the range of x used to obtain the accuracy samples. The test values for x are uniformly distributed over this range. It should be noted that x is not always the direct input argument to the function being tested; it is sometimes desirable to distribute the input arguments in a nonuniform fashion to provide a more complete test of the function (see column D below). For each function, accuracy data is given for one or more segments within the valid range of x. In each case, the numbers given are the most meaningful to the function and range under consideration.
- D: identifies the distribution of arguments input to the particular function being sampled. The letter E indicates an exponential distribution. The letter U indicates a uniform distribution. A number refers to a note providing detailed information regarding the distribution.

Accuracy Data: The maximum relative error and standard deviation of the relative error are generally useful and revealing statistics; however, they are useless for the range of a function where its value becomes zero. This is because the slightest error in the argument can cause an unpredictable fluctuation in the magnitude of the answer. When a small argument error would have this effect, the maximum absolute error and standard deviation of the absolute error are given for the range.

Relative Error (e): The maximum relative error and standard deviation (root mean square) of the relative error are defined:

MAX(e) =

MAX( ABS((f(x) - g(x) ) / f(x)))

where: MAX selects the largest of its arguments and ABS takes the absolute value of its argument.

SD(e) =

SQR( (1/N) SUMSQ((f(x) - g(x) ) / f(x)))

where: SQR takes the square root of its argument and SUMSQ takes the summation of the squares of its arguments over all of the test cases.

Absolute Error (E): The maximum absolute error produced during the testing and the standard deviation (root mean square) of the absolute error are:

MAX(E) =
MAX( ABS( f(x) - g(x) ) )
where: the operators are those defined above.
SD(E) =
SQR( (1/N) SUMSQ( f(x) - g(x) ) )
where: the operators are those defined above.

Limitations (Subject to Change)

The following are limits that apply to the functions performed by this instruction.

The source and receiver operands must both be specified as floating-point with the same length (4 bytes for short format or 8 bytes for long format).

Resultant Conditions

Positive-The algebraic value of the receiver operand is positive.
Negative-The algebraic value of the receiver operand is negative.
Zero-The algebraic value of the receiver operand is zero.
Unordered-The value assigned to the floating-point result is NaN.

Authorization Required

None

Lock Enforcement

None

Exceptions

06 Addressing

0601 Space Addressing Violation
0602 Boundary Alignment
0603 Range

08 Argument/Parameter

0801 Parameter Reference Violation

0C Computation

0C06 Floating-Point Overflow
0C07 Floating-Point Underflow
0C09 Floating-Point Invalid Operand
0C0D Floating-Point Inexact Result

10 Damage Encountered

1004 System Object Damage State
1044 Partial System Object Damage

1C Machine-Dependent

1C03 Machine Storage Limit Exceeded

20 Machine Support

2002 Machine Check
2003 Function Check

22 Object Access

2201 Object Not Found
2202 Object Destroyed
2203 Object Suspended
2208 Object Compressed
220B Object Not Available

24 Pointer Specification

2401 Pointer Does Not Exist
2402 Pointer Type Invalid

2C Program Execution

2C04 Branch Target Invalid

2E Resource Control Limit

2E01 User Profile Storage Limit Exceeded
2E02 Security Audit Journal Failure

32 Scalar Specification

3201 Scalar Type Invalid
3203 Scalar Value Invalid

36 Space Management

3601 Space Extension/Truncation

44 Protection Violation

4401 Object Domain or Hardware Storage Protection Violation
4402 Literal Values Cannot Be Changed