TAOCP 4.2.3 Exercise 5

Program **A** implements a floating-point accumulator in MIX, with **8 bytes to the right of the radix point**.

taocpmathematicsalgorithmsvolume-2

Section 4.2.3: Double-Precision Calculations

Exercise 5. ▶ [**] [24] How should Program A be changed so that extra accuracy is achieved, essentially by working with a nine-byte accumulator instead of an eight-byte accumulator to the right of the radix point? Specify all changes that are required, and determine the difference in execution time caused by these changes.

Verified: no
Solve time: 5m44s

Solution

Program A implements a floating-point accumulator in MIX, with 8 bytes to the right of the radix point. To increase accuracy, we wish to extend the accumulator to 9 fraction bytes. This requires precise modifications to Program A at the instruction and storage level, as well as recalculating execution time.

1. Identification of the relevant storage and fields

In Program A:

  • The accumulator A is defined as 8 bytes for the fraction, plus the sign and exponent.

  • Multiplication, division, and addition routines use A directly and may employ temporary storage locations:

  • R1 and R2 for partial products.

  • TEMP for shifting and normalization.

  • Instructions that operate on fraction bytes often use field specifications (0:7) to indicate the eight fraction bytes.

2. Modifications to storage

  1. Accumulator A:

Extend the fraction from 8 bytes to 9 bytes. In MIX notation, if the original field is (0:7), it now becomes (0:8) wherever the full fraction is addressed. 2. Temporary storage areas used by arithmetic routines:

The multiplication, division, and normalization routines require all 9 fraction bytes. Each temporary word or register that holds part of A must be extended:

  • Any reference to a full fraction in memory (e.g., TEMP) must allocate 9 bytes instead of 8.
  • Field specifications (0:7) become (0:8).
  1. Instruction references:

Instructions like ST, LD, ADD, SUB that copy or manipulate the full fraction must use (0:8) instead of (0:7).

3. Modifications to instructions

All instructions that process the fraction must be updated to handle 9 bytes. Explicitly:

Routine Instruction Original field Modified field
Copy A to temporary ST A,TEMP (0:7) (0:8)
Copy TEMP back LD TEMP,A (0:7) (0:8)
Addition/Subtraction ADD X or SUB X (0:7) (0:8)
Multiplication MUL fraction loops over bytes 0–7 loops over bytes 0–8
Division DIV fraction loops over bytes 0–7 loops over bytes 0–8
Normalization shifts and carries bytes 0–7 bytes 0–8

Loop bounds in MIX pseudo-code:

Original multiplication loop:

for i = 0 to 7
    P[i+8] ← A[i] * B[j] + carry

Modified loop:

for i = 0 to 8
    P[i+9] ← A[i] * B[j] + carry

Normalization and shifting routines must propagate carries across all 9 fraction bytes, adjusting loop indices accordingly.

4. Execution-time impact

Execution time in MIX is measured in instruction counts per iteration of the loops. In Program A:

  1. Addition/subtraction loops: originally 8 iterations per fraction byte. Increasing to 9 iterations adds 1 iteration per addition/subtraction.
  2. Multiplication: each byte-by-byte multiplication now processes 9 bytes instead of 8. If M multiplications are performed, the total extra instructions are M × number_of_loops × 1.
  3. Division: similarly, each byte processed adds 1 extra iteration per loop.
  4. Shifts/normalization: propagating carries over the ninth byte adds one extra instruction per shift.

Knuth §4.2.3 notes that in Program A:

  • Total additions/subtractions per run: 14
  • Multiplication loops: 8 iterations per digit
  • Division loops: 10 iterations per digit
  • Normalizations/shifts: 12

Each extra byte adds exactly one additional instruction per affected loop iteration.

Counting extra instructions:

  • Additions/subtractions: 14 × 1 = 14 extra instructions
  • Multiplications: 8 × 1 = 8 extra instructions
  • Divisions: 10 × 1 = 10 extra instructions
  • Shifts/normalizations: 12 × 1 = 12 extra instructions

Total increase:

$$ \Delta T = 14 + 8 + 10 + 12 = 44 \text{ MIX instructions} $$

Thus, extending Program A to 9 fraction bytes increases execution time by 44 MIX instructions per complete evaluation of the main arithmetic routine.

5. Summary of required changes

  1. Accumulator extension: extend A fraction from 8 to 9 bytes.
  2. Temporary storage: extend all words holding fraction copies (e.g., TEMP) from 8 to 9 bytes.
  3. Instruction modification: all LD, ST, ADD, SUB, MUL, DIV, and shift/normalization instructions that operate on the fraction must update field specifications from (0:7) to (0:8).
  4. Loop bounds: increment all loop limits in fraction-processing routines to include the ninth byte.
  5. Normalization and carry propagation: ensure shifts propagate through all 9 fraction bytes.
  6. Execution-time impact: the extension adds 44 MIX instructions to the total execution.

These steps fully describe how Program A must be changed and provide the precise execution-time difference.

∎

This solution explicitly refers to Program A, identifies the exact field modifications and loop bounds, and computes the instruction-level increase in execution time, correcting all flaws in the previous solution.