Fixed a bug on example code at 5.6. Getting Accurate Results

[enochjung]

@lshqqytiger benchmarked the example codes on ThunderX2 CN9980 processor, but the performance did not meet our expectations.
Interestingly, Case 2 performed faster than Case 1.

• Case 1

  int checksum = 0;
  
  const clock_t start = clock();
  
  for (int a = 0; a < 1000; a++)
      for (int b = 0; b < 1000; b++)
          checksum ^= gcd(a, b);
  
  const clock_t end = clock();

• Case 2

  int a[1000], b[1000];
  
  for (int i = 0; i < 1000; i++)
      a[i] = rand() % 1000, b[i] = rand() % 1000;
  
  int checksum = 0;
  
  const clock_t start = clock();
  
  for (int t = 0; t < 1000; t++)
      for (int i = 0; i < 1000; i++)
          checksum += gcd(a[i], b[i]);
  
  const clock_t end = clock();

Case 1 took 0.104947 seconds, while Case 2 took 0.075065 seconds.

The faster speed of Case 2 was due to a high branch prediction rate. In Case 2, there were only 1,000 combinations of gcd arguments, enabling the branch predictor to memorize the entire sequence of comparison and perform branch prediction successfully.

To ensure the example code executes 1 million combinations of gcd arguments as intended, I modified the code as follows:

• Case 3

  int a[1000], b[1000];
  
  for (int i = 0; i < 1000; i++)
      a[i] = rand() % 1000, b[i] = rand() % 1000;
  
  int checksum = 0;
  
  const clock_t start = clock();
  
  for (int t = 0; t < 1000; t++)
      for (int i = 0; i < 1000; i++)
          checksum += gcd(a[t], b[i]);  // modified from a[i] to a[t]
  
  const clock_t end = clock();

Case 3 took 0.122731 seconds, performing the expected result.