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float d = b*b - 4*a*c; // discriminant float sqrtd = sqrt(d); float x1 = (-b + sqrtd)/(2*a); float x2 = (-b - sqrtd)/(2*a); cout << "The solutions are:" << endl; cout << "\tx1 = " << x1 << endl; cout << "\tx2 = " << x2 << endl; cout << "Check:" << endl; cout << "\ta*x1*x1 + b*x1 + c = " << a*x1*x1 + b*x1 + c << endl; cout << "\ta*x2*x2 + b*x2 + c = " << a*x2*x2 + b*x2 + c << endl; } The quadratic formula requires computing the square root b 2 4ac . This is done on the line float sqrtd = sqrt(d); which calls the square root function sqrt() defined in the header file <cmath>. The last two lines of the program check the solutions by substituting them back into the original quadratic equation. If the resulting expression on the left evaluates to 0 then the solutions are correct. This run solves the equation 2x2 + 1x 3 = 0 correctly: Enter the coefficients of a quadratic equation: a: 2 b: 1 c: -3 The equation is: 2*x*x + 1*x + -3 = 0 The solutions are: x1 = 1 x2 = -1.5 Check: a*x1*x1 + b*x1 + c = 0 a*x2*x2 + b*x2 + c = 0
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But this run attempts to solve the equation x2 + 10000000000x + 1 = 0 and fails: Enter the coefficients of a quadratic equation: a: 1 b: 1e10 c: 1 The equation is: 1*x*x + 1e10*x + 1 = 0 The solutions are: x1 = 0 x2 = -1e10 Check: a*x1*x1 + b*x1 + c = 1 a*x2*x2 + b*x2 + c = 1 The first solution, x1 = 0, is obviously incorrect: the resulting quadratic expression ax12 + bx1 + c evaluates to 1 instread of 0. The second solution, x2 = 1e10 = 10,000,000,000 is even worse. The correct solutions are x1 = 0.00000 00000 99999 99999 99999 99519 and x2 = 9,999,999,999.99999 99999.
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Numeric overflow and round-off errors are examples of run-time errors, which are errors that occur while the program is running. Such errors are more serious than compile-time errors such as neglecting to declare a variable or forgetting a semicolon because they are usually harder to detect and locate. Compile-time errors are caught by the compiler, which usually gives a pretty good report on where they are. But run-time errors are detected only when the user notices that the results are incorrect. Even if the program crashes, it still may be difficult to find where the problem is in the program.
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EXAMPLE 2.16 Other Kinds of Run-Time Errors
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Here are two more runs of the quadratic formula program in Example 2.15: Enter the coefficients of a quadratic equation: a: 1 b: 2 c: 3 The equation is: 1*x*x + 2*x + 3 = 0 The solutions are: x1 = nan x2 = nan Check: a*x1*x1 + b*x1 + c = nan a*x2*x2 + b*x2 + c = nan The quadratic equation 1x2 + 2x + 3 = 0 has no real solution because the discriminant b2 4ac is negative. When the program runs, the square root function sqrt(d) fails because d < 0. It returns the symbolic constant nan which stands for not a number. Then every subsequent numeric operation that uses this constant results in the same value. That s why the check values come out as nan at the end of the run. This run attempts to solve the equation 0x2 + 2x + 5 = 0. That equation has the solution x = 2.5. But the quadratic formula fails because a = 0: Enter the coefficients of a quadratic equation: a: 0 b: 2 c: 5 The equation is: 0*x*x + 2*x + 5 = 0 The solutions are: x1 = nan x2 = -inf Check: a*x1*x1 + b*x1 + c = nan a*x2*x2 + b*x2 + c = nan Notice that x1 comes out as nan, but x2 comes out as -inf. The symbol inf stands for infinity. That s what you get when you divide a nonzero number by zero. The quadratic formula computes x2 as 4 b b 2 4ac ( 2 ) ( 2 ) 4( 0 )( 5 ) 2 2 x 2 = ------------------------------------- = --------------------------------------------------------- = -------------- = ----0 0 2a 2 (0 ) which becomes -inf. But it computes x1 as b + b 2 4ac (2 ) + ( 2) 4( 0 )( 5 ) 2+2 0 x 1 = ------------------------------------- = -------------------------------------------------------- = --------------- = -2a 2( 0 ) 0 0 which becomes nan.
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The three symbols inf, -inf, and nan are numeric constants. The usual numeric operators can be applied to them, although the results are usually useless. For example, you can multiply nan by any number, but the result will still be nan. 2.13 THE E-FORMAT FOR FLOATING-POINT VALUES When input or output, floating-point values may be specified in either of two formats: fixedpoint and scientific. The output in Example 2.16 illustrates both: 333.333 has fixed-point format, and -5.68434e 14 has scientific format.
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