Average Error: 30.8 → 0.3
Time: 34.9s
Precision: 64
\[\frac{1 - \cos x}{x \cdot x}\]
↓
\[\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{e^{\log \left(1 + \cos x\right)}}\]
double f(double x) {
        double r1996743 = 1.0;
        double r1996744 = x;
        double r1996745 = cos(r1996744);
        double r1996746 = r1996743 - r1996745;
        double r1996747 = r1996744 * r1996744;
        double r1996748 = r1996746 / r1996747;
        return r1996748;
}
↓
double f(double x) {
        double r1996749 = x;
        double r1996750 = sin(r1996749);
        double r1996751 = r1996750 / r1996749;
        double r1996752 = r1996751 * r1996751;
        double r1996753 = 1.0;
        double r1996754 = cos(r1996749);
        double r1996755 = r1996753 + r1996754;
        double r1996756 = log(r1996755);
        double r1996757 = exp(r1996756);
        double r1996758 = r1996752 / r1996757;
        return r1996758;
}


\frac{1 - \cos x}{x \cdot x}
↓
\frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{e^{\log \left(1 + \cos x\right)}}

Error

Bits error versus x

Derivation

  1. Initial program 30.8

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

  3. Applied flip--30.9

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

  4. Applied associate-/l/30.9

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

  5. Simplified15.2

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

  6. Taylor expanded around -inf 15.2

    \[\leadsto \color{blue}{\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \left(\cos x + 1\right)}}\]

  7. Simplified0.3

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

  9. Applied add-exp-log0.3

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

  10. Final simplification0.3

    \[\leadsto \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{e^{\log \left(1 + \cos x\right)}}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1 (cos x)) (* x x)))