Average Error: 30.5 → 0.2
Time: 1.0m
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{x}}{\frac{\cos \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{x}}}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin \left(x \cdot \frac{1}{2}\right)}{x}}{\frac{\cos \left(x \cdot \frac{1}{2}\right)}{\frac{\sin x}{x}}}
double f(double x) {
        double r5445420 = 1.0;
        double r5445421 = x;
        double r5445422 = cos(r5445421);
        double r5445423 = r5445420 - r5445422;
        double r5445424 = r5445421 * r5445421;
        double r5445425 = r5445423 / r5445424;
        return r5445425;
}
↓
double f(double x) {
        double r5445426 = x;
        double r5445427 = 0.5;
        double r5445428 = r5445426 * r5445427;
        double r5445429 = sin(r5445428);
        double r5445430 = r5445429 / r5445426;
        double r5445431 = cos(r5445428);
        double r5445432 = sin(r5445426);
        double r5445433 = r5445432 / r5445426;
        double r5445434 = r5445431 / r5445433;
        double r5445435 = r5445430 / r5445434;
        return r5445435;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.5

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

  3. Applied flip--30.6

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

  4. Applied associate-/l/30.6

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

  5. Simplified14.7

    \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}\]
  6. Using strategy rm

  7. Applied clear-num14.7

    \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}{\sin x \cdot \sin x}}}\]
  8. Using strategy rm

  9. Applied times-frac15.1

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

  10. Applied add-cube-cbrt15.1

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{x \cdot x}{\sin x} \cdot \frac{1 + \cos x}{\sin x}}\]

  11. Applied times-frac15.1

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{x \cdot x}{\sin x}} \cdot \frac{\sqrt[3]{1}}{\frac{1 + \cos x}{\sin x}}}\]

  12. Simplified15.1

    \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x}} \cdot \frac{\sqrt[3]{1}}{\frac{1 + \cos x}{\sin x}}\]

  13. Simplified15.0

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

  14. Taylor expanded around inf 14.8

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

  15. Simplified0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2019120 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))