Section 3.4

Group 4

The legal instructions of group 4 are concerned with the division process. Function-numbers 46 and 47 are unassigned and if encountered, will cause entry to the Monitor Routine, the program being suspended.

The functions numbered 40 to 45 each use two operands. One of these is the signed 48-bit number y. The other is either the signed 48-bit number x or the standard double-length number x:. In all cases x or x: is the dividend (numerator) and y is the divisor (denominator), i.e. the instructions divide x or x: by y.

In all cases if y = 0 the Monitor Program will be entered and the program suspended.

The results occupy either one word or two, depending on the actual function used. Single word results are written into Z in 3-address form and into X in 2-address form. Results which occupy two words are written into Z and Z+1 in 3-address form and into X and X+1 in 2-address form. Nothing should be assumed about the quotient and remainder of a division instruction which overflows.

Function Number 40 [0100 000]

(3) z_I' + (z*_I'/y_I) = x_I/y_I; 0 ≤ z*_I'/y_I < 1

(2) As 3-address with x_I' and x*_I' in place of z_I' and z*_I' respectively.

Unrounded division of integers. Divide x_I by y_I, writing the unrounded integral quotient into Z and the remainder into Z+1. The remainder is always of the same sign as y and numerically less than y. The quotient and remainder, in 3-address form, satisfy

z_I'.y_I + z*_I' = x_I

OVR is set if x is -2⁴⁷ and y is -1.

The results obtained for selected values of x_I and y_I are tabulated below.

x_I	y_I	z_I'	z*_I'
+37	+5	+7	+2
-37	+5	-8	+3
+37	-5	-8	-3
-37	-5	+7	-2
+1	+5	0	+1
+1	-5	-1	-4

Note also that:-

i) z_I' = quotient, z*_F' = remainder when x_F is divided y_Fii) z_F' = unrounded quotient when x_F is divided by y_I.

Although the remainder is such that it can in turn be divided by y to give the non-negative l.s. half of a double-length quotient it is not necessary to do so, since the 42-instruction gives a result in this form.

Function Number 41 [0100 001]

(3) z_I' = (x_I/y_I)_r or (x_F/y_F)_r; -½ ≤ x/y - z_I' < ½

(2) As 3-address but with x_I' in place of z_I'.

Rounded division of integers. Divide x by y, to produce a rounded integral quotient, writing this into Z or X.

The rounding is such that if the rounded quotient is subtracted from the true quotient of x_I/y_I the difference is numerically less than ½ or equal to -½.**(See below)

** Editors note: This should be “and greater than or equal to -½”

In fact the rounded quotient is the nearest integer to x/y and is the algebraically greater if x/y is an odd multiple of ½.

The results of some selected cases are tabulated below.

x_I y_I z_I'

+13 +2 +7

-13 +2 -6

+13 -2 -6

-13 -2 +7

+37 +5 +7

-37 +5 -7

OVR is set if x = -2⁴⁷ and y = -1.

Function Number 42 [0010 010]

(3) z:_M' = (x_I/y_I)_r or (x_F/y_F)_r; -½ e ≤ (x/y) - z:_M' < ½ e

(2) As 3-address but with x:_M' in place of z:_M'.

Division with rounded double-length quotient. Divide x by y to give a mixed-number quotient, writing the signed integral part into Z or X and the non-negative fractional part into Z+1 or X+1. OVR is set if

x_I = -2⁴⁷ and y_I = -1

(or the fractional interpretations of those words).

The results given in some selected cases are tabulated below.

x_I	y_I	z_I'	z*_F'
+37	+8	+4	+0.625
-37	+8	-5	+0.375
+37	-8	-5	+0.375
-37	-8	+4	+0.625
+1	+8	0	+0.125
+1	-8	-1	+0.875

The quotient is rounded to the nearest multiple of 2^-47 and to the algebraically greater if the true value of x/y is an odd multiple of 2^-48.

Note also that:-
i) z:_F' = rounded d.l. quotient when x_F is divided by y_I;
ii) z:_I' = rounded d.l. quotient when x_I is divided by y_F.

Function Number 43 [0100 011]

(3) z_F' = (x_F/y_F)_r or (x_I/y_I)_r ; -½ e ≤ x/y - z_F' < ½ e , |x| < |y| or x = -y

(2) As 3-address but with x_F' in place of z_F'.

Rounded division of x by y producing a fractional quotient, which is written into Z, or X. The operands must be such that x is numerically less than y or is equal to -y. OVR is set if neither of these conditions is satisfied. If x and y have the same scaling factor (e.g. if both are fractions or if both are integers) than z' is a fraction.

The result is the multiple of 2^-47 nearest to the true value of x/y and is the algebraically larger if x/y is an odd multiple of 2^-48.

Note also that z_I’ = (x_I/y_F)_r

This instruction can be used, following a 44-instruction, to produce a rounded standard double-length quotient when x: is divided by y.

Function Number 44 [0100 100]

(3) z_I' + (z*_I'/y_I) = x:_I/y_I ; 0 ≤ z*_I'/y_I < 1

(2) As 3-address but with x_I' and x*_I' in place of z_I' and z*_I'.

Unrounded division with double-length dividend (numerator).

The standard d.l. number x: is divided by the s.l. number y producing an integral quotient and a remainder. The quotient is written into Z or X and the remainder into Z+1 or X+1. OVR is set if z' exceeds single-length capacity. If x: is not in standard form the OMP is entered; the OMP does a 'partial justify' and obeys the instruction.

For comments and examples relating to the values of quotients and remainders please see under function-number 40; in essence functions 40 and 44 differ only in using single-and double-length dividends respectively.

If the remainder from a 44-instruction is divided by the original divisor (using a 43-instruction) the result is necessarily a non-negative fraction. Thus if a is a standard d.l. number in A100 and A101 and if b is an s.l. number in A200 then after obeying the instructions

44 A100 A200 A7
43 A8 A200

A7 and A8 will together contain the standard d.l. mid-point (rounded) value of a/b, with the signed integral part in A7 and the non-negative fractional part in A8.

Function number 45 [0100 101]

(3) z_F' = (x:_F/y_F)_r or (x:_M/y_I)_r ; -½ e ≤ (x:_F/y_F) - z_F' < ½ e
(2) As 3-address but with x_F' in place of z_F'

Rounded division with double-length dividend. Divide the standard d.l. number x: by the s.l. number y to produce a rounded fractional quotient, writing this quotient into Z or X. The operands must be such that |x:_F| < |y_F| or x:_F = -y_F. OVR is set if neither of these conditions is satisfied. If x: is not in standard form, the O&MP is entered; the OMP does a 'partial justify' and obeys the instruction.

The result is that multiple of 2^-47 which is nearest to the true value of x:_F/y_F being the algebraically larger if the true quotient is an odd multiple of 2^-48.

Note also that

z_I' = (x:_M/y_F)_r or (x:_I/y_I)_r ; -½ ≤ (x:_M/y_F) - z_I' < ½

i.e. the quotient may be regarded as the rounded integral quotient when x:_M is divided by y_F or when x:_I is divided by y_I.

Function number 46 [0100 110]

Function number 47 [0100 111]

Illegal instructions