Average Error: 31.4 → 0.2
Time: 16.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\frac{\sin x}{x}}{x} \cdot \sin \left(\frac{1}{2} \cdot x\right)}{\cos \left(\frac{1}{2} \cdot x\right)}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\frac{\sin x}{x}}{x} \cdot \sin \left(\frac{1}{2} \cdot x\right)}{\cos \left(\frac{1}{2} \cdot x\right)}
double f(double x) {
        double r723239 = 1.0;
        double r723240 = x;
        double r723241 = cos(r723240);
        double r723242 = r723239 - r723241;
        double r723243 = r723240 * r723240;
        double r723244 = r723242 / r723243;
        return r723244;
}
↓
double f(double x) {
        double r723245 = x;
        double r723246 = sin(r723245);
        double r723247 = r723246 / r723245;
        double r723248 = r723247 / r723245;
        double r723249 = 0.5;
        double r723250 = r723249 * r723245;
        double r723251 = sin(r723250);
        double r723252 = r723248 * r723251;
        double r723253 = cos(r723250);
        double r723254 = r723252 / r723253;
        return r723254;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

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

  3. Applied flip--31.5

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

  4. Simplified15.7

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

  6. Applied *-un-lft-identity15.7

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

  7. Applied times-frac15.7

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

  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 15.5

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))