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Could you please help me to understand the mathematics behind MIPS (million instructions per second) rating formula?

The formula for MIPS is:

$$ \text{MIPS} = \frac{ \text{Instruction count}}{\text{Execution time} \ \times \ 10^6}$$

For example, there are 12 instructions and they are executed in 4 seconds. So the CPU's performance is $\frac{12 \ \text{instructions} }{4 \ \text{sec}} = 3 \ \text{instr}/\text{sec}$. MIPS would be $3\times10^{-6}$.

Question: How do we go from $3$ instructions per second to millions instructions per second? What is the logic/mathematics behind dividing by $10^6 = 1 \text{ million}$?

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Perhaps the unit conversion confuses you?

$$ \frac{i\,\text{instr}}{t\,\text{sec}} = \frac{i\,\text{instr}}{t\,\text{sec}} \times 1 = \frac{i\,\text{instr}}{t\,\text{sec}} \times \frac{\text{millions}}{10^6} = \frac{i}{t \times 10^6} \times \frac{\text{millions}\,\text{instr}}{\text{sec}} $$

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