Average Error: 31.0 → 0.1
Time: 10.3s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)}{x}
double f(double x) {
        double r322210 = 1.0;
        double r322211 = x;
        double r322212 = cos(r322211);
        double r322213 = r322210 - r322212;
        double r322214 = r322211 * r322211;
        double r322215 = r322213 / r322214;
        return r322215;
}
↓
double f(double x) {
        double r322216 = x;
        double r322217 = sin(r322216);
        double r322218 = r322217 / r322216;
        double r322219 = 2.0;
        double r322220 = r322216 / r322219;
        double r322221 = tan(r322220);
        double r322222 = r322218 * r322221;
        double r322223 = r322222 / r322216;
        return r322223;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.0

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

  3. Applied flip--31.1

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

  4. Simplified15.6

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

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

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

  7. Applied times-frac15.6

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

  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. Using strategy rm

  12. Applied associate-*r/0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))