Average Error: 31.5 → 0.1
Time: 17.1s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x}{x}\right)\right) \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}\]
\frac{1 - \cos x}{x \cdot x}
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x}{x}\right)\right) \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}
double f(double x) {
        double r562145 = 1.0;
        double r562146 = x;
        double r562147 = cos(r562146);
        double r562148 = r562145 - r562147;
        double r562149 = r562146 * r562146;
        double r562150 = r562148 / r562149;
        return r562150;
}
↓
double f(double x) {
        double r562151 = x;
        double r562152 = sin(r562151);
        double r562153 = r562152 / r562151;
        double r562154 = log1p(r562153);
        double r562155 = expm1(r562154);
        double r562156 = 2.0;
        double r562157 = r562151 / r562156;
        double r562158 = tan(r562157);
        double r562159 = r562158 / r562151;
        double r562160 = r562155 * r562159;
        return r562160;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  3. Applied flip--31.6

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

  4. Simplified16.1

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

  6. Applied *-un-lft-identity16.1

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

  7. Applied *-un-lft-identity16.1

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

  8. Applied distribute-lft-out16.1

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

  9. Applied times-frac16.1

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

  10. Applied times-frac0.3

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

  11. Simplified0.3

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

  12. Simplified0.1

    \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x}}\]
  13. Using strategy rm

  14. Applied expm1-log1p-u0.1

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x}{x}\right)\right)} \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}\]

  15. Final simplification0.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x}{x}\right)\right) \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))