cos2 (problem 3.4.1)

Average Error: 30.7 → 0.1

Time: 11.6s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r602195 = 1.0;
        double r602196 = x;
        double r602197 = cos(r602196);
        double r602198 = r602195 - r602197;
        double r602199 = r602196 * r602196;
        double r602200 = r602198 / r602199;
        return r602200;
}

↓

double f(double x) {
        double r602201 = x;
        double r602202 = sin(r602201);
        double r602203 = r602202 / r602201;
        double r602204 = 2.0;
        double r602205 = r602201 / r602204;
        double r602206 = tan(r602205);
        double r602207 = r602201 / r602206;
        double r602208 = r602203 / r602207;
        return r602208;
}

Error

Bits error versus x

Derivation

1. Initial program 30.7

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

3. Applied flip-- 30.8

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

4. Simplified 15.5

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

6. Applied *-un-lft-identity 15.5

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

7. Applied times-frac 15.5

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

8. Simplified 15.5

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

9. Simplified 15.3

   \[\leadsto \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{x \cdot x}\]
10. Using strategy rm

11. Applied associate-/l* 15.5

    \[\leadsto \color{blue}{\frac{\sin x}{\frac{x \cdot x}{\tan \left(\frac{x}{2}\right)}}}\]
12. Using strategy rm

13. Applied *-un-lft-identity 15.5

    \[\leadsto \frac{\sin x}{\frac{x \cdot x}{\color{blue}{1 \cdot \tan \left(\frac{x}{2}\right)}}}\]

14. Applied times-frac 0.4

    \[\leadsto \frac{\sin x}{\color{blue}{\frac{x}{1} \cdot \frac{x}{\tan \left(\frac{x}{2}\right)}}}\]

15. Applied associate-/r* 0.1

    \[\leadsto \color{blue}{\frac{\frac{\sin x}{\frac{x}{1}}}{\frac{x}{\tan \left(\frac{x}{2}\right)}}}\]

16. Simplified 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))