Average Error: 31.7 → 0.1
Time: 16.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\sin x}{x} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{x \cdot \cos \left(\frac{1}{2} \cdot x\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\sin x}{x} \cdot \frac{\sin \left(\frac{1}{2} \cdot x\right)}{x \cdot \cos \left(\frac{1}{2} \cdot x\right)}
double f(double x) {
        double r827426 = 1.0;
        double r827427 = x;
        double r827428 = cos(r827427);
        double r827429 = r827426 - r827428;
        double r827430 = r827427 * r827427;
        double r827431 = r827429 / r827430;
        return r827431;
}
↓
double f(double x) {
        double r827432 = x;
        double r827433 = sin(r827432);
        double r827434 = r827433 / r827432;
        double r827435 = 0.5;
        double r827436 = r827435 * r827432;
        double r827437 = sin(r827436);
        double r827438 = cos(r827436);
        double r827439 = r827432 * r827438;
        double r827440 = r827437 / r827439;
        double r827441 = r827434 * r827440;
        return r827441;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  3. Applied flip--31.8

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

  4. Simplified15.5

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

  6. Applied *-un-lft-identity15.5

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

  7. Applied times-frac15.5

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

  8. Applied times-frac0.3

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

  9. Simplified0.3

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

  10. Simplified0.1

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

  11. Taylor expanded around inf 0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))