Average Error: 31.3 → 0.1
Time: 36.7s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \sin x}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \sin x}{x}
double f(double x) {
        double r961750 = 1.0;
        double r961751 = x;
        double r961752 = cos(r961751);
        double r961753 = r961750 - r961752;
        double r961754 = r961751 * r961751;
        double r961755 = r961753 / r961754;
        return r961755;
}
↓
double f(double x) {
        double r961756 = x;
        double r961757 = 2.0;
        double r961758 = r961756 / r961757;
        double r961759 = tan(r961758);
        double r961760 = r961759 / r961756;
        double r961761 = sin(r961756);
        double r961762 = r961760 * r961761;
        double r961763 = r961762 / r961756;
        return r961763;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Initial program 31.3

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

  3. Applied flip--31.4

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

  4. Simplified15.4

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

  6. Applied *-un-lft-identity15.4

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

  7. Applied times-frac15.4

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

  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-*l/0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))