Average Error: 31.2 → 0.1
Time: 33.1s
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 r1084989 = 1.0;
        double r1084990 = x;
        double r1084991 = cos(r1084990);
        double r1084992 = r1084989 - r1084991;
        double r1084993 = r1084990 * r1084990;
        double r1084994 = r1084992 / r1084993;
        return r1084994;
}
↓
double f(double x) {
        double r1084995 = x;
        double r1084996 = 2.0;
        double r1084997 = r1084995 / r1084996;
        double r1084998 = tan(r1084997);
        double r1084999 = r1084998 / r1084995;
        double r1085000 = sin(r1084995);
        double r1085001 = r1085000 / r1084995;
        double r1085002 = r1084999 * r1085001;
        return r1085002;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Initial program 31.2

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

  3. Applied flip--31.3

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

  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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))