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EXAMPLE 2.12 Integer Overflow
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This program repeatedly multiplies n by 1000 until it overflows. int main() { // prints n until it overflows: int n=1000; cout << "n = " << n << endl; n *= 1000; // multiplies n by 1000 cout << "n = " << n << endl; n *= 1000; // multiplies n by 1000 cout << "n = " << n << endl; n *= 1000; // multiplies n by 1000 cout << "n = " << n << endl; } n = 1000 n = 1000000 n = 1000000000 n = -727379968 This shows that the computer that ran this program cannot multiply 1,000,000,000 by 1000 correctly.
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This program is similar to the one in Example 2.12. It repeatedly squares x until it overflows. int main() { // prints x until it overflows: float x=1000.0; cout << "x = " << x << endl; x *= x; // multiplies n by itself; i.e., it squares x cout << "x = " << x << endl; x *= x; // multiplies n by itself; i.e., it squares x cout << "x = " << x << endl; x *= x; // multiplies n by itself; i.e., it squares x cout << "x = " << x << endl; x *= x; // multiplies n by itself; i.e., it squares x cout << "x = " << x << endl; } x = 1000 x = 1e+06 x = 1e+12 x = 1e+24 x = inf This shows that, starting with x = 1000, this computer cannot square x correctly more than three times. The last output is the special symbol inf which stands for infinity.
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Note the difference between integer overflow and floating-point overflow. The last output in Example 2.12 is the negative integer 727,379,968 instead of the correct value of 1,000,000,000,000 = 1012. The last output in Example 2.13 is the infinity symbol inf instead of the correct value of 1048. Integer overflow wraps around to negative integers. Floating-point overflow sinks into the abstract notion of infinity.
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2.12 ROUND-OFF ERROR Round-off error is another kind of error that often occurs when computers do arithmetic on rational numbers. For example, the number 1/3 might be stored as 0.333333, which is not exactly equal to 1/3. The difference is called round-off error. In some cases, these errors can cause serious problems. EXAMPLE 2.14 Round-off Error
This program does some simple arithmetic to illustrate roundoff error: int main() { // illustrates round-off error:: double x = 1000/3.0;cout << "x = " << x << endl; // x = 1000/3 double y = x - 333.0;cout << "y = " << y << endl; // y = 1/3 double z = 3*y - 1.0;cout << "z = " << z << endl; // z = 3(1/3) - 1 if (z == 0) cout << "z == 0.\n"; else cout << "z does not equal 0.\n"; // z != 0 } x = 333.333 y = 0.333333 z = -5.68434e-14 z does not equal 0. In exact arithmetic, the variables would have the values x = 333 1/3, y = 1/3, and z = 0. But 1/3 cannot be represented exactly as a floating-point value. The inaccuracy is reflected in the residue value for z.
Example 2.14 illustrates an inherent problem with using floating-point types within conditional tests of equality. The test (z == 0) will fail even if z is very nearly zero, which is likely to happen when z should algebraically be zero. So it is better to avoid tests for equality with floating-point types. The next example shows that round-off error can be difficult to recognize. EXAMPLE 2.15 Hidden Round-off Error
This program implements the quadratic formula to solve quadratic equations. #include <cmath> // defines the sqrt() function #include <iostream> using namespace std; int main() { // implements the quadratic formula float a, b, c; cout << "Enter the coefficients of a quadratic equation:" << endl; cout << "\ta: "; cin >> a; cout << "\tb: "; cin >> b; cout << "\tc: "; cin >> c; cout << "The equation is: " << a << "*x*x + " << b << "*x + " << c << " = 0" << endl;
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