Average Error: 30.9 → 0.1
Time: 37.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{1}{\frac{x}{\tan \left(\frac{x}{2}\right)}} \cdot \frac{\sin x}{x}\]
double f(double x) {
        double r2654105 = 1.0;
        double r2654106 = x;
        double r2654107 = cos(r2654106);
        double r2654108 = r2654105 - r2654107;
        double r2654109 = r2654106 * r2654106;
        double r2654110 = r2654108 / r2654109;
        return r2654110;
}
↓
double f(double x) {
        double r2654111 = 1.0;
        double r2654112 = x;
        double r2654113 = 2.0;
        double r2654114 = r2654112 / r2654113;
        double r2654115 = tan(r2654114);
        double r2654116 = r2654112 / r2654115;
        double r2654117 = r2654111 / r2654116;
        double r2654118 = sin(r2654112);
        double r2654119 = r2654118 / r2654112;
        double r2654120 = r2654117 * r2654119;
        return r2654120;
}


\frac{1 - \cos x}{x \cdot x}
↓
\frac{1}{\frac{x}{\tan \left(\frac{x}{2}\right)}} \cdot \frac{\sin x}{x}

Error

Bits error versus x

Derivation

  1. Initial program 30.9

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

  3. Applied flip--31.0

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

  4. Applied associate-/l/31.0

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

  5. Simplified14.7

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

  6. Taylor expanded around inf 14.7

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

  7. Simplified14.9

    \[\leadsto \frac{\sin x \cdot \sin x}{\color{blue}{\left(x + \cos x \cdot x\right) \cdot x}}\]
  8. Using strategy rm

  9. Applied times-frac0.3

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

  10. Simplified0.1

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

  12. Applied *-un-lft-identity0.1

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

  13. Applied associate-/l*0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019101 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))