Average Error: 31.3 → 0.1
Time: 42.4s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right) \cdot x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right) \cdot x}
double f(double x) {
        double r1106347 = 1.0;
        double r1106348 = x;
        double r1106349 = cos(r1106348);
        double r1106350 = r1106347 - r1106349;
        double r1106351 = r1106348 * r1106348;
        double r1106352 = r1106350 / r1106351;
        return r1106352;
}
↓
double f(double x) {
        double r1106353 = x;
        double r1106354 = sin(r1106353);
        double r1106355 = r1106354 / r1106353;
        double r1106356 = 2.0;
        double r1106357 = r1106353 / r1106356;
        double r1106358 = sin(r1106357);
        double r1106359 = r1106355 * r1106358;
        double r1106360 = cos(r1106357);
        double r1106361 = r1106360 * r1106353;
        double r1106362 = r1106359 / r1106361;
        return r1106362;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

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

  3. Applied flip--31.4

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

  4. Simplified15.1

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

  6. Applied *-un-lft-identity15.1

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

  7. Applied times-frac15.1

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

  8. Simplified0.2

    \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)\right)}\]
  9. Using strategy rm

  10. Applied tan-quot0.2

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

  11. Applied associate-*r/0.2

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

  12. Applied frac-times0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))