Average Error: 30.9 → 0.1
Time: 15.9s
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 r699635 = 1.0;
        double r699636 = x;
        double r699637 = cos(r699636);
        double r699638 = r699635 - r699637;
        double r699639 = r699636 * r699636;
        double r699640 = r699638 / r699639;
        return r699640;
}
↓
double f(double x) {
        double r699641 = x;
        double r699642 = 2.0;
        double r699643 = r699641 / r699642;
        double r699644 = tan(r699643);
        double r699645 = r699644 / r699641;
        double r699646 = sin(r699641);
        double r699647 = r699645 * r699646;
        double r699648 = r699647 / r699641;
        return r699648;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 30.9

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

  3. Applied flip--31.0

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

  4. Simplified15.5

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

  6. Applied *-un-lft-identity15.5

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

  7. Applied times-frac15.5

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

  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 2019162 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))