cos2 (problem 3.4.1)

Average Error: 31.3 → 0.1

Time: 1.4m

Precision: 64

Math

\[\frac{1 - \cos x}{x \cdot x}\]

↓

\[\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}\]

C

double f(double x) {
        double r3369353 = 1.0;
        double r3369354 = x;
        double r3369355 = cos(r3369354);
        double r3369356 = r3369353 - r3369355;
        double r3369357 = r3369354 * r3369354;
        double r3369358 = r3369356 / r3369357;
        return r3369358;
}

↓

double f(double x) {
        double r3369359 = x;
        double r3369360 = 2.0;
        double r3369361 = r3369359 / r3369360;
        double r3369362 = tan(r3369361);
        double r3369363 = r3369362 / r3369359;
        double r3369364 = sin(r3369359);
        double r3369365 = r3369364 / r3369359;
        double r3369366 = r3369363 * r3369365;
        return r3369366;
}

Error

Bits error versus x

Derivation

1. Initial program 31.3

   \[\frac{1 - \cos x}{x \cdot x}\]
2. Using strategy rm

3. Applied flip-- 31.5

   \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x}\]

4. Simplified 15.8

   \[\leadsto \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{x \cdot x}\]
5. Using strategy rm

6. Applied *-un-lft-identity 15.8

   \[\leadsto \frac{\frac{\sin x \cdot \sin x}{1 + \color{blue}{1 \cdot \cos x}}}{x \cdot x}\]

7. Applied *-un-lft-identity 15.8

   \[\leadsto \frac{\frac{\sin x \cdot \sin x}{\color{blue}{1 \cdot 1} + 1 \cdot \cos x}}{x \cdot x}\]

8. Applied distribute-lft-out 15.8

   \[\leadsto \frac{\frac{\sin x \cdot \sin x}{\color{blue}{1 \cdot \left(1 + \cos x\right)}}}{x \cdot x}\]

9. Applied times-frac 15.8

   \[\leadsto \frac{\color{blue}{\frac{\sin x}{1} \cdot \frac{\sin x}{1 + \cos x}}}{x \cdot x}\]

10. Applied times-frac 0.3

    \[\leadsto \color{blue}{\frac{\frac{\sin x}{1}}{x} \cdot \frac{\frac{\sin x}{1 + \cos x}}{x}}\]

11. Simplified 0.3

    \[\leadsto \color{blue}{\frac{\sin x}{x}} \cdot \frac{\frac{\sin x}{1 + \cos x}}{x}\]

12. Simplified 0.1

    \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x}}\]

13. Final simplification 0.1

    \[\leadsto \frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))