Average Error: 30.2 → 0.1
Time: 15.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}
double f(double x) {
        double r1012223 = 1.0;
        double r1012224 = x;
        double r1012225 = cos(r1012224);
        double r1012226 = r1012223 - r1012225;
        double r1012227 = r1012224 * r1012224;
        double r1012228 = r1012226 / r1012227;
        return r1012228;
}
↓
double f(double x) {
        double r1012229 = x;
        double r1012230 = 2.0;
        double r1012231 = r1012229 / r1012230;
        double r1012232 = tan(r1012231);
        double r1012233 = r1012232 / r1012229;
        double r1012234 = sin(r1012229);
        double r1012235 = r1012234 / r1012229;
        double r1012236 = r1012233 * r1012235;
        return r1012236;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.2

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

  3. Applied flip--30.3

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

  4. Simplified14.9

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

  6. Applied *-un-lft-identity14.9

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

  7. Applied times-frac14.9

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

  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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))